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Fundamentals of Soil Behavior Third Edition
James K. Mitchell Kenichi Soga
JOHN WILEY & SONS, INC.
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This book is printed on acid-free paper. ⬁ Copyright 2005 by John Wiley & Sons, Inc. All rights reserved
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Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada
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Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Mitchell, James Kenneth, 1930– Fundamentals of soil behavior / James K. Mitchell, Kenichi Soga.—3rd ed. p. cm. ISBN-13: 978-0-471-46302-7 (cloth : alk. paper) ISBN-10: 0-471-46302-7 (cloth : alk. paper) 1. Soil mechanics. I. Soga, Kenichi. II. Title. TA710.M577 2005 624.15136—dc22 2004025690 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1
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CONTENTS
CHAPTER
1
Preface
xi
INTRODUCTION
1
1.1 1.2 1.3
CHAPTER
2
SOIL FORMATION 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10
CHAPTER
3
Soil Behavior in Civil and Environmental Engineering Scope and Organization Getting Started
Introduction The Earth’s Crust Geologic Cycle and Geological Time Rock and Mineral Stability Weathering Origin of Clay Minerals and Clay Genesis Soil Profiles and Their Development Sediment Erosion, Transport, and Deposition Postdepositional Changes in Sediments Concluding Comments Questions and Problems
SOIL MINERALOGY 3.1
3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11
3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19
Importance of Soil Mineralogy in Geotechnical Engineering Atomic Structure Interatomic Bonding Secondary Bonds Crystals and Their Properties Crystal Notation Factors Controlling Crystal Structures Silicate Crystals Surfaces Gravel, Sand, and Silt Particles Soil Minerals and Materials Formed by Biogenic and Geochemical Processes Summary of Nonclay Mineral Characteristics Structural Units of the Layer Silicates Synthesis Pattern and Classification of the Clay Minerals Intersheet and Interlayer Bonding in the Clay Minerals The 1⬊1 Minerals Smectite Minerals Micalike Clay Minerals Other Clay Minerals
1 3 3
5 5 5 6 7 8 15 16 18 25 32 33
35 35 38 38 39 40 42 44 45 45 48 49 49 49 52 55 56 59 62 64 v
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CONTENTS
3.20 3.21 3.22 3.23 3.24 3.25
CHAPTER
4
Summary of Clay Mineral Characteristics Determination of Soil Composition X-ray Diffraction Analysis Other Methods for Compositional Analysis Quantitative Estimation of Soil Components Concluding Comments Questions and Problems
4.1 4.2
Introduction Approaches to the Study of Composition and Property Interrelationships 4.3 Engineering Properties of Granular Soils 4.4 Dominating Influence of the Clay Phase 4.5 Atterberg Limits 4.6 Activity 4.7 Influences of Exchangeable Cations and pH 4.8 Engineering Properties of Clay Minerals 4.9 Effects of Organic Matter 4.10 Concluding Comments Questions and Problems
CHAPTER
5
SOIL FABRIC AND ITS MEASUREMENT 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11
CHAPTER
6
65 65 70 74 79 80 81
SOIL COMPOSITION AND ENGINEERING PROPERTIES 83
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vi
Introduction Definitions of Fabrics and Fabric Elements Single-Grain Fabrics Contact Force Characterization Using Photoelasticity Multigrain Fabrics Voids and Their Distribution Sample Acquisition and Preparation for Fabric Analysis Methods for Fabric Study Pore Size Distribution Analysis Indirect Methods for Fabric Characterization Concluding Comments Questions and Problems
SOIL–WATER–CHEMICAL INTERACTIONS 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17
Introduction Nature of Ice and Water Influence of Dissolved Ions on Water Mechanisms of Soil–Water Interaction Structure and Properties of Adsorbed Water Clay–Water–Electrolyte System Ion Distributions in Clay–Water Systems Elements of Double-Layer Theory Influences of System Variables on the Double Layer Limitations of the Gouy–Chapman Diffuse Double Layer Model Energy and Force of Repulsion Long-Range Attraction Net Energy of Interaction Cation Exchange—General Considerations Theories for Ion Exchange Soil–Inorganic Chemical Interactions Clay–Organic Chemical Interactions
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83 85 85 94 95 97 97 98 104 105 106
109 109 110 112 119 121 122 123 127 135 137 140 140
143 143 144 145 146 146 153 153 154 157 159 163 164 164 165 167 167 168
CONTENTS
6.18
CHAPTER
7
Concluding Comments Questions and Problems
Introduction Principle of Effective Stress Force Distributions in a Particulate System Interparticle Forces Intergranular Pressure Water Pressures and Potentials Water Pressure Equilibrium in Soil Measurement of Pore Pressures in Soils Effective and Intergranular Pressure Assessment of Terzaghi’s Equation Water–Air Interactions in Soils Effective Stress in Unsaturated Soils Concluding Comments Questions and Problems
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8.10 8.11 8.12 8.13 8.14 8.15 8.16 8.17 8.18
9
173 173 174 174 178 180 181 183 184 185 188 190 193 193
SOIL DEPOSITS—THEIR FORMATION, STRUCTURE, GEOTECHNICAL PROPERTIES, AND STABILITY 195 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9
CHAPTER
169 169
EFFECTIVE, INTERGRANULAR, AND TOTAL STRESS 173 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13
CHAPTER
vii
Introduction Structure Development Residual Soils Surficial Residual Soils and Taxonomy Terrestrial Deposits Mixed Continental and Marine Deposits Marine Deposits Chemical and Biological Deposits Fabric, Structure, and Property Relationships: General Considerations Soil Fabric and Property Anisotropy Sand Fabric and Liquefaction Sensitivity and Its Causes Property Interrelationships in Sensitive Clays Dispersive Clays Slaking Collapsing Soils and Swelling Soils Hard Soils and Soft Rocks Concluding Comments Questions and Problems
CONDUCTION PHENOMENA 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9
Introduction Flow Laws and Interrelationships Hydraulic Conductivity Flows Through Unsaturated Soils Thermal Conductivity Electrical Conductivity Diffusion Typical Ranges of Flow Parameters Simultaneous Flows of Water, Current, and Salts Through Soil-Coupled Flows 9.10 Quantification of Coupled Flows
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195 195 200 205 206 209 209 212 213 217 223 226 235 239 243 243 245 245 247
251 251 251 252 262 265 267 272 274 274 277
CONTENTS
9.11 9.12 9.13 9.14 9.15 9.16 9.17 9.18 9.19 9.20 9.21 9.22 9.23
Simultaneous Flows of Water, Current, and Chemicals Electrokinetic Phenomena Transport Coefficients and the Importance of Coupled Flows Compatibility—Effects of Chemical Flows on Properties Electroosmosis Electroosmosis Efficiency Consolidation by Electroosmosis Electrochemical Effects Electrokinetic Remediation Self-Potentials Thermally Driven Moisture Flows Ground Freezing Concluding Comments Questions and Problems
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viii
CHAPTER
10
11
284 288 291 294 298 303 305 305 307 310 319 320
VOLUME CHANGE BEHAVIOR
325
10.1 10.2 10.3 10.4 10.5 10.6 10.7
325 325 327 330 331 335
10.8 10.9 10.10 10.11 10.12 10.13
CHAPTER
279 282
Introduction General Volume Change Behavior of Soils Preconsolidation Pressure Factors Controlling Resistance to Volume Change Physical Interactions in Volume Change Fabric, Structure, and Volume Change Osmotic Pressure and Water Adsorption Influences on Compression and Swelling Influences of Mineralogical Detail in Soil Expansion Consolidation Secondary Compression In Situ Horizontal Stress (K0) Temperature–Volume Relationships Concluding Comments Questions and Problems
339 345 348 353 355 359 365 366
STRENGTH AND DEFORMATION BEHAVIOR
369
11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 11.11 11.12 11.13 11.14 11.15 11.16 11.17 11.18 11.19 11.20 11.21
369 370 379 383 389 393 400 404 411 415 417 422 425 432 436 438 444 447 452 456 460
Introduction General Characteristics of Strength and Deformation Fabric, Structure, and Strength Friction Between Solid Surfaces Frictional Behavior of Minerals Physical Interactions Among Particles Critical State: A Useful Reference Condition Strength Parameters for Sands Strength Parameters for Clays Behavior After Peak and Strain Localization Residual State and Residual Strength Intermediate Stress Effects and Anisotropy Resistance to Cyclic Loading and Liquefaction Strength of Mixed Soils Cohesion Fracturing of Soils Deformation Characteristics Linear Elastic Stiffness Transition from Elastic to Plastic States Plastic Deformation Temperature Effects
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CONTENTS
11.22
CHAPTER
12
Concluding Comments Questions and Problems
ix 462 462
TIME EFFECTS ON STRENGTH AND DEFORMATION 465 Introduction General Characteristics Time-Dependent Deformation–Structure Interaction Soil Deformation as a Rate Process Bonding, Effective Stresses, and Strength Shearing Resistance as a Rate Process Creep and Stress Relaxation Rate Effects on Stress–Strain Relationships Modeling of Stress–Strain–Time Behavior Creep Rupture Sand Aging Effects and Their Significance Mechanical Processes of Aging Chemical Processes of Aging Concluding Comments Questions and Problems
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12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11 12.12 12.13 12.14
465 466 470 478 481 488 489 497 503 508 511 516 517 520 520
List of Symbols
523
References
531
Index
559
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PREFACE According to the National Research Council (1989, 2005), sound geoengineering is key in meeting seven critical societal needs. They are waste management and environmental protection, infrastructure development and rehabilitation, construction efficiency and innovation, security, resource discovery and recovery, mitigation of natural hazards, and the exploration and development of new frontiers. Solution of problems and satisfactory completion of projects in each of these areas cannot be accomplished without a solid understanding of the composition, structure, and behavior of soils because virtually all of humankind’s structures and facilities are built on, in, or with the Earth. Thus, the purpose of this book remains the same as for the prior two editions; namely, the development of an understanding of the factors determining and controlling the engineering properties and behavior of soils under different conditions, with an emphasis on why they are what they are. We believe that this understanding and its prudent application can be a valuable asset in meeting these societal needs. In the 12 years since publication of the second edition, environmental problems requiring geotechnical inputs have remained very important; dealing with natural hazards and disasters such as earthquakes, floods, and landslides has demanded increased attention; risk assessment and mitigation applied to existing structures and earthworks has become a major challenge; and the roles of soil stabilization, ground improvement, and soil as a construction material have expanded enormously. These developments, as well as the introduction of new computational, geophysical, and sensing methods, new emphasis on micromechanical analysis and behavior, and, perhaps regrettably, the reduced emphasis on laboratory measurement of soil properties have required looking at soil behavior in new ways. More and more it is becoming appreciated that geochemical and microbiological phenomena and processes play an essential role in many types of geotechnical problems. Some of these considerations have been incorporated into this new edition. Although the format of the book has remained much the same as in the first two editions, the contents have been reviewed and revised in detail, with deletion of some material no longer considered to be essential and introduction of substantial new material to incorporate important recent developments. We have reorganized the material among chapters to improve the flow of topics and logic of presentation. Time effects on soil strength and deformation behavior have been separated into a new Chapter 12. Additional soil property correlations have been incorporated. The addition of sets of questions and problems at the end of each chapter provide a feature not present in the first two editions. Many of these questions and problems are open ended and without single, clearly defined answers, but they are designed to stimulate broad thinking and the realization that judgment and incorporation of concepts and methods from a range of disciplines is often needed to provide satisfactory solutions to many geoengineering problems. We are indebted to innumerable students and professional colleagues whose inquiring minds and perceptive insights have helped us clarify issues and find new and better explanations for observed processes and behavior. J. Carlos Santamarina and David Smith provided helpful suggestions on the overall content and organization. Charles J. Shackelford reviewed and provided valuable suggestions for the sections of Chapter 9 on chemical osmosis and advective and diffusive chemical flows. Other important contributions to this third edition in the form xi
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PREFACE
of valuable comments, photos, resources, and proof checking were made by Hendrikus Allersma, Khalid Alshibli, John Atkinson, Bob Behringer, Malcolm Bolton, Lis Bowman, Jim Buckman, Pierre Delage, Antonio Gens, Henry Ji, Assaf Klar, Hideo Komine, Jean-Marie Konrad, Ning Liu, Yukio Nakata, Albert Ng, Masanobu Oda, Kenneth Sutherland, Colin Thornton, Yoichi Watabe, Siam Yimsiri, and Guoping Zhang. KS thanks his wife, Mikiko, for her encouragement and special support. We dedicate this book to the memory of Virginia (‘‘Bunny’’) Mitchell, whose continuing love, support, encouragement, and patience over more than 50 years, made this and the prior two editions possible. JAMES K. MITCHELL University Distinguished Professor, Emeritus Virginia Tech, Blacksburg, Virginia
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xii
KENICHI SOGA Reader in Geomechanics University of Cambridge, Cambridge, England March 2005
References
National Research Council. 1989. Geotechnology—Its Impact on Economic Growth, the Environment, and National Security. National Academy Press, Washington, DC. National Research Council. 2005. Geological and Geotechnical Engineering in the New Millennium, National Academy Press, Washington, DC.
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CHAPTER 1
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Introduction
1.1 SOIL BEHAVIOR IN CIVIL AND ENVIRONMENTAL ENGINEERING
Civil and environmental engineering includes the conception, analysis, design, construction, operation, and maintenance of a diversity of structures, facilities, and systems. All are built on, in, or with soil or rock. The properties and behavior of these materials have major influences on the success, economy, and safety of the work. Geoengineers play a vital role in these projects and are also concerned with virtually all aspects of environmental control, including water resources, water pollution control, waste disposal and containment, and the mitigation of such natural disasters as floods, earthquakes, landslides, and volcanoes. Soils and their interactions with the environment are major considerations. Furthermore, detailed understanding of the behavior of earth materials is essential for mining, for energy resources development and recovery, and for scientific studies in virtually all the geosciences. To deal properly with the earth materials associated with any problem and project requires knowledge, understanding, and appreciation of the importance of geology, materials science, materials testing, and mechanics. Geotechnical engineering is concerned with all of these. Environmental concerns—especially those related to groundwater, the safe disposal and containment of wastes, and the cleanup of contaminated sites—has spawned yet another area of specialization; namely, environmental geotechnics, wherein chemistry and biological science are important. Geochemical and microbiological phenomena impact the composition, properties, and stability of soils and rocks to degrees only recently beginning to be appreciated. Students in civil engineering are often quite surprised, and sometimes quite confused, by their first course in engineering with soils. After studying statics,
mechanics, and structural analysis and design, wherein problems are usually quite clear-cut and well defined, they are suddenly confronted with situations where this is no longer the case. A first course in soil mechanics may not, at least for the first half to two-thirds of the course, be mechanics at all. The reason for this is simple: Analyses and designs are useless if the boundary conditions and material properties are improperly defined. Acquisition of the data needed for analysis and design on, in, and with soils and rocks can be far more difficult and uncertain than when dealing with other engineering materials and aboveground construction. There are at least three important reasons for this. 1. No Clearly Defined Boundaries. An embankment resting on a soil foundation is shown in Fig. 1.1a, and a cantilever beam fixed at one end is shown in Fig. 1.1b. The free body of the cantilever beam, Fig. 1.1c, is readily analyzed for reactions, shears, moments, and deflections using standard methods of structural analysis. However, what are the boundary conditions, and what is the free body for the embankment foundation? 2. Variable and Unknown Material Properties. The properties of most construction materials (e.g., steel, plastics, concrete, aluminum, and wood) are ordinarily known within rather narrow limits and usually can be specified to meet certain needs. Although this may be the case in construction using earth and rock fills, at least part of every geotechnical problem involves interactions with in situ soil and rock. No matter how extensive (and expensive) any boring and sampling program, only a very small percentage of the subsurface material is available for observation and testing. In most cases, more than one stratum is 1
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1
INTRODUCTION
is not the case; in fact, it is for these very reasons that geotechnical engineering offers such a great challenge for imaginative and creative work. Modern theories of soil mechanics, the capabilities of modern computers and numerical analysis methods, and our improved knowledge of soil physics and chemistry make possible the solution of a great diversity of static and dynamic problems of stress deformation and stability, the transient and steady-state flow of fluids through the ground, and the long-term performance of earth systems. Nonetheless, our ability to analyze and compute often exceeds considerably our ability to understand, measure, and characterize a problem or process. Thus, understanding and the ability to conceptualize soil and rock behavior become all the more important. The objectives of this book are to provide a basis for the understanding of the engineering properties and behavior of soils and the factors controlling changes with time and to indicate why this knowledge is important and how it is used in the solution of geotechnical and geoenvironmental problems. It is easier to state what this book is not, rather than what it is. It is not a book on soil or rock mechanics; it is not a book on soil exploration or testing; it is not a book that teaches analysis or design; and it is not a book on geotechnical engineering practice. Excellent books and references dealing with each of these important areas are available. It is a book on the composition, structure, and behavior of soils as engineering materials. It is intended for students, researchers, and practicing engineers who seek a more in-depth knowledge of the nature and behavior of soils than is provided by classical and conventional treatments of soil mechanics and geotechnical engineering. Here are some examples of the types of questions that are addressed in this book:
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2
Figure 1.1 The problem of boundary conditions in geo-
technical problems: (a) embankment on soil foundation, (b) cantilever beam, and (c) free body diagram for analysis of propped cantilever beam.
present, and conditions are nonhomogeneous and anisotropic. 3. Stress and Time-Dependent Material Properties. Soils, and also some rocks, have mechanical properties that depend on both the stress history and the present stress state. This is because the volume change, stress–strain, and strength properties depend on stress transmission between particles and particle groups. These stresses are, for the most part, generated by body forces and boundary stresses and not by internal forces of cohesion, as is the case for many other materials. In addition, the properties of most soils change with time after placement, exposure, and loading. Because of these stress and time dependencies, any given geotechnical problem may involve not just one or two but an almost infinite number of different materials. Add to the above three factors the facts that soil and rock properties may be susceptible to influences from changes in temperature, pressure, water availability, and chemical and biological environment, and one might conclude that successful application of mechanics to earth materials is an almost hopeless proposition. It has been amply demonstrated, of course, that such
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• What are soils composed of? Why? • How does geological history influence soil properties?
• How are engineering properties and behavior re• • • • • • • •
lated to composition? What is clay? Why are clays plastic? What are friction and cohesion? What is effective stress? Why is it important? Why do soils creep and exhibit stress relaxation? Why do some soils swell while others do not? Why does stability failure sometimes occur at stresses less than the measured strength? Why and how are soil properties changed by disturbance?
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GETTING STARTED
• How do changes in environmental conditions
• • • • • • • • •
Developing answers to questions such as these requires application of concepts from chemistry, geology, biology, materials science, and physics. Principles from these disciplines are introduced as necessary to develop background for the phenomena under study. It is assumed that the reader has a basic knowledge of applied mechanics and soil mechanics, as well as a general familiarity with the commonly used engineering properties of soils and their determination.
1.2
nature of clay particles, the types and concentrations of chemicals in a soil can influence significantly its behavior in a variety of ways. Soil water and the clay– water–electrolyte system are then analyzed in Chapter 6. An analysis of interparticle forces and total and effective stresses, with a discussion of why they are important, is given in Chapter 7. The remaining chapters draw on the preceding developments for explanations of phenomena and soil properties of interest in geotechnical and geoenvironmental engineering. The formation of soil deposits, their resulting structures and relationships to geotechnical properties and stability are covered in Chapter 8. The next three chapters deal with those soil properties that are of primary importance to the solution of most geoengineering problems: the flows of fluids, chemicals, electricity, and heat and their consequences in Chapter 9; volume change behavior in Chapter 10; and deformation and strength and deformation behavior in Chapter 11. Finally, Chapter 12 on time effects on strength and deformation recognizes that soils are not inert, static materials, but rather how a given soil responds under different rates of loading or at some time in the future may be quite different than how it responds today.
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•
change properties? What are some practical consequences of the prolonged exposure of clay containment barriers to waste chemicals? What controls the rate of flow of water, heat, chemicals, and electricity through soils? How are the different types of flows through soil interrelated? Why is the residual strength of a soil often much less than its peak strength? How do soil properties change with time after deposition or densification and why? How do temperature changes influence the mechanical properties of soils? What is soil liquefaction, and why is it important? What causes frost heave, and how can it be prevented? What clay types are best suited for sealing waste repositories? What biological processes can occur in soils and why are they important in engineering problems?
SCOPE AND ORGANIZATION
The topics covered in this book begin with consideration of soil formation in Chapter 2 and soil mineralogy and compositional analysis of soil in Chapter 3. Relationships between soil composition and engineering properties are developed in Chapter 4. Soil composition by itself is insufficient for quantification of soil properties for specific situations, because the soil fabric, that is, the arrangements of particles, particle groups, and pores, may play an equally important role. This topic is covered in Chapter 5. Water may make up more than half the volume of a soil mass, it is attracted to soil particles, and the interactions between water and the soil surfaces influence the behavior. In addition, owing to the colloidal
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3
1.3
GETTING STARTED
Find an article about a problem, a project, or issue that involves some aspect of geotechnical soil behavior as an important component. The article can be from the popular press, from a technical journal or magazine, such as the Journal of Geotechnical and Geoenvironmental Engineering of the American Society of Civil Engineers, Ge´otechnique, The Canadian Geotechnical Journal, Soils and Foundations, ENR, or elsewhere. 1. Read the article and prepare a one-page informative abstract. (An informative abstract summarizes the important ideas and conclusions. A descriptive abstract, on the other hand, simply states the article contents.) 2. Summarize the important geotechnical issues that are found in the article and write down what you believe you should know about to understand them well enough to solve the problem, resolve the issue, advise a client, and the like. In other words, what is in the article that you believe the subject matter in this book should prepare you to deal with? Do not exceed two pages.
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CHAPTER 2
2.1
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Soil Formation
INTRODUCTION
The variety of geomaterials encountered in engineering problems is almost limitless, ranging from hard, dense, large pieces of rock, through gravel, sand, silt, and clay to organic deposits of soft, compressible peat. All these materials may exist over a wide range of densities and water contents. A number of different soil types may be present at any site, and the composition may vary over intervals as small as a few millimeters. It is not surprising, therefore, that much of the geoengineer’s effort is directed at the identification of soils and the evaluation of the appropriate properties for use in a particular analysis or design. Perhaps what is surprising is that the application of the principles of mechanics to a material as diverse as soil meets with as much success as it does. To understand and appreciate the characteristics of any soil deposit require an understanding of what the material is and how it reached its present state. This requires consideration of rock and soil weathering, the erosion and transportation of soil materials, depositional processes, and postdepositional changes in sediments. Some important aspects of these processes and their effects are presented in this chapter and in Chapter 8. Each has been the subject of numerous books and articles, and the amount of available information is enormous. Thus, it is possible only to summarize the subject and to encourage consultation of the references for more detail.
2.2
(acid) rocks predominate beneath the continents, and basaltic (basic) rocks predominate beneath the oceans. Because of these lithologic differences, the continental crust average density of 2.7 is slightly less than the oceanic crust average density of 2.8. The elemental compositions of the whole Earth and the crust are indicated in Fig. 2.1. There are more than 100 elements, but 90 percent of Earth consists of iron, oxygen, silicon, and magnesium. Less iron is found in the crust than in the core because its higher density causes it to sink. Silicon, aluminum, calcium, potassium, and sodium are more abundant in the crust than in the core because they are lighter elements. Oxygen is the only anion that has an abundance of more than 1 percent by weight; however, it is very abundant by volume. Silicon, aluminum, magnesium, and oxygen are the most commonly observed elements in soils. Within depths up to 2 km, the rocks are 75 percent secondary (sedimentary and metamorphic) and 25 percent igneous. From depths of 2 to 15 km, the rocks are about 95 percent igneous and 5 percent secondary. Soils may extend from the ground surface to depths of several hundred meters. In many cases the distinction between soil and rock is difficult, as the boundary between soft rock and hard soil is not precisely defined. Earth materials that fall in this range are sometimes difficult to deal with in engineering and construction, as it is not always clear whether they should be treated as soils or rocks. A temperature gradient of about 1C per 30 m exists between the bottom of Earth’s crust at 1200C and the surface.1 The rate of cooling as molten rock magma
THE EARTH’S CRUST
The continental crust covers 29 percent of Earth’s surface. Seismic measurements indicate that the continental crust is about 30 to 40 km thick, which is 6 to 8 times thicker than the crust beneath the ocean. Granitic
1 In some localized areas, usually within regions of recent crustal movement (e.g., fault lines, volcanic zones) the gradient may exceed 20C per 100 m. Such regions are of interest both because of their potential as geologic hazards and because of their possible value as sources of geothermal energy.
5
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6
2
100% 90% 80%
SOIL FORMATION Other Sand
1.E-11 1.E-13 1.E-15 1.E-17
Sand Clayey Sand
1.E-01 1.E-03 1.E-05 1.E-07 1.E-09
Hydraulic Conductivity (m/s)
Hydraulic conductivity (m/s)
1.E-01 1.E-03 1.E-05 1.E-07 1.E-09
1.E-11 1.E-13 1.E-15 1.E-17
Sand > Clayey Sand 1.E-19 1.E-21 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 Matric Suction (kPa)
20
40 60 Saturation (%)
80
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1.E-19 1.E-21 0
(a)
100
(b)
1.E+00
Sand
1.E-02
Clayey Sand
1.E-04 1.E-06 1.E-08 1.E-10 1.E-12
Relative Permeability kr
1.E+00 Relative Permeability kr
263
1.E-02 1.E-04 1.E-06 1.E-08 1.E-10 1.E-12
1.E-14
1.E-14
1.E-16 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 Matric Suction (kPa)
1.E-16
(c)
Sand Clayey Sand
0
20
40 60 Saturation (%)
80
100
(d)
Figure 9.11 Hydraulic conductivity of partially saturated sand and clayey sand as a function of matric suction and degree of saturation (from Stephens, 1996).
tion 7.12. Various methods to measure the hydraulic conductivity of unsaturated soils are available (Klute, 1986; Fredlund and Rahardjo, 1993). However, the measurement in unsaturated soils is more difficult to perform than in saturated soils because the hydraulic conductivity needs to be determined under controlled water saturation or matric suction conditions. A general expression for the hydraulic conductivity k of unsaturated soils can be written as k ⫽ krK
g ⫽ kr ks
(9.29)
where ks is the saturated conductivity, K is the intrinsic permeability of the medium (L2) such as given by Eq. (9.18), is the density of the permeating fluid (ML⫺3), g is the acceleration of gravity (LT⫺2), is the dynamic viscosity of the permeating fluid (MT⫺1L⫺1), and ks is the conductivity under the condition that the pores are fully filled by the permeating fluid (i.e., full saturation). The dimensionless parameter kr is called the relative permeability, and the values range from 0 (⫽ zero per-
Copyright © 2005 John Wiley & Sons
meability, no interconnected path for the permeating fluid) to 1 (⫽ permeating fluid at full saturation). The equation can be used for a nonwetting fluid (e.g., air) by substituting the values of and of the nonwetting fluid. The data in Fig. 9.11a and 9.11b can be replotted as the relative permeability against matric suction in Fig. 9.11c and against saturation ratio in Fig. 9.11d. The two different curves in Fig. 9.11d clearly show that kr ⫽ S3 derived from Eq. (9.20) is not universally applicable. At very low water contents, the water in the pores becomes disconnected as described in Chapter 7. Careful experiments show that the movement of water exists even at moisture contents of a few percent, but vapor transport becomes more important at this dry state (Grismer et al., 1986). Therefore, Eq. (9.20) is not suitable for low saturations. One reason for this discrepancy is that soil contains pores of various sizes rather than the assumption of uniform pore sizes used to derive Eq. (9.20). Considering that the soil contains pores of random sizes, Marshall (1958) derived the following equation
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9
CONDUCTION PHENOMENA
for hydraulic conductivity as a function of pore sizes for an isotropic material: K⫽
n2 r 21 ⫹ 3r 22 ⫹ 5r 23 ⫹ ⫹ (2m ⫺ 1)r 2m m2 8 (9.30)
Co py rig hte dM ate ria l
in which K is the specific hydraulic conductivity (permeability) (L2), n is the porosity, m is the total number of pore classes, and ri is the mean radius of the pores in pore class i. Pore sizes can be measured from data on the amount of water withdrawn as the suction on the soil is progressively increased. Using the capillary equation, the radius of the largest water-filled pore under a suction of (L) is given by 2 r⫽ wg
(9.31)
in which is the surface tension of water, w is the density of water, and g is the acceleration of gravity. As it is usually more convenient to use moisture suction than pore radius, Eq. (9.29) can be rewritten as K⫽
2 n2 ⫺2 [ ⫹ 3 2⫺2 ⫹ 5 ⫺2 3 22wg2 m2 1
(9.32)
The permeability K can be converted to the hydraulic conductivity k by multiplying the unit weight (wg) divided by the dynamic viscosity of water . This gives 2 n2 ⫺2 ⫺2 [ ⫹ 3 ⫺2 2 ⫹ 5 3 2wg m2 1
⫹ ⫹ (2m ⫺ 1) ⫺2 m ]
(9.33)
Following Green and Corey (1971), the porosity n equals the volumetric water content of the saturated condition S, and m is the total number of pore classes between S and zero water content ⫽ 0. A matching factor is usually used in Eq. (9.33) to equate the calculated and measured hydraulic conductivities. Matching at full saturation is preferable to matching at a partial saturation point because it is simpler and gives better results. Rewriting Eq. (9.33) and introducing a matching factor gives k( i) ⫽
ks 2 2S ksc 2wg m2
m S ⫽ l S ⫺ L
冘 [(2j ⫹ 1 ⫺ 2i) l
⫺2 j
]
j⫽1
(i ⫽ 1, 2, . . . , l)
(9.34)
Copyright © 2005 John Wiley & Sons
(9.35)
A constant value of l is used at all water contents, and the value of l establishes the number of pore classes for which ⫺2 terms are included in the calculation at j saturation. Other pore size distribution models for unsaturated soils are available, and an excellent review of these models is given by Mualem (1986). Equation (9.34) can be written in an integration form as (after Fredlund et al., 1994)
冕
ks 2 Sp ksc 2wg
k( ) ⫽
⫹ ⫹ (2m ⫺ 1) ⫺2 m ]
k⫽
in which k( i) is the calculated hydraulic conductivity for a specified water content i; is i the last water content class on the wet end, for example, i ⫽ 1 denotes the pore class corresponding to the saturated water content S, and i ⫽ l denotes the pore class corresponding to the lowest water content L for which hydraulic conductivity is calculated; ks /ksc is the matching factor, defined as the measured saturated hydraulic conductivity divided by the calculated saturated hydraulic conductivity; and l is the total number of pore classes (a pore class is a pore size range corresponding to a water content increment) between ⫽ L and S. Thus
L
⫺x dx 2(x)
(9.36)
where suction is given as a function of volumetric water content , and x is a dummy variable. The hydraulic conductivity for fully saturated condition is calculated by assigning ⫽ S. For generality, the term 2S in Eq. (9.34) is replaced by ps , where p is a constant that accounts for the interaction of pores of various sizes (Fredlund et al., 1994). From Eq. (9.36), the relative permeability kr is a function of water content as follows: kr( ) ⫽
冕
r
冒冕
⫺x dx 2(x)
S
r
⫺x dx 2(x)
(9.37)
Herein, the lowest water content L is assumed to be the residual water content r. If the moisture content –suction relationship (or the soil–water characteristic curve) is known, the relative permeability kr can be computed from Eq. (9.37) by performing a numerical integration. The hydraulic conductivity k is then estimated from Eq. (9.29) with the knowledge of saturated hydraulic conductivity ks. The use of the soil–water characteristic curve to estimate the hydraulic conductivity of unsaturated soils is attractive because it is easier to determine this curve
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THERMAL CONDUCTIVITY
in the laboratory than it is to measure the hydraulic conductivity directly. Apart from Eq. (9.37), the following relative permeability function proposed by Mualem (1976) is often used primarily because of its simplicity: kr( ) ⫽
冊 冉冕 d( )冒冕
冉
⫺ r s ⫺ r
q
s
r
r
d ( )
冊
2
(9.38)
kr( ) ⫽
冉
larger than the vertically infiltrating water flow. However, if the matric suction is reduced by large infiltration, the barrier breaks and water enters into the initially dry coarse layer. Solutions are available to evaluate the amount of water flowing laterally across the capillary barrier interface at the point of breakthrough for a given set of fine and coarse soil hydraulic properties and interface inclination (Ross, 1990; Steenhuis et al., 1991; Selkar, 1997; Webb, 1997). Capillary barriers have received increased attention as a means for isolating buried waste from groundwater flow and as part of landfill cover systems in dry climates (Morris and Stormont, 1997; Selkar, 1997; Khire et al., 2000). The barrier can be used to divert the flow laterally along an interface and/or to store infiltrating water temporarily in the fine layer so that it can be removed ultimately by evaporation and transpiration. Capillary barriers are constructed as simple two-layer systems of contrasting particle size or multiple layers of fine- and coarse-grained soils. If the thickness of the overlying fine layer is too small, capillary diversion is reduced because of the confining flow path in the fine layer. The minimum effective thickness is several times the air-entry head of the fine soil (Warrick et al., 1997; Smersrud and Selker, 2001). Khire et al. (2000) stress the importance of site-specific metrological and hydrological conditions in determining the storage capacity of the fine layer. The soil for the underlying coarse layer should have a very large particle size contrast with the fine soil, but fines migrations into the coarse sand should be avoided. Smesrud and Sekler (2001) suggest the d50 particle size ratio of 5 to be ideal. The thickness of the coarse sand layer does not need to be great, as the purpose of the layer is simply to impede the downward water migration.
Co py rig hte dM ate ria l
where q describes the degree of connectivity between the water-conducting pores. Mualem (1976) states that q ⫽ 0.5 is appropriate based on permeability measurements on 45 soils. van Genuchten et al. (1991) substituted the soil–water characteristic equation (7.52) into Eq. (9.38) and obtained the following closed-form solution4:
冊再 冋 冉
⫺ r S ⫺ r
p
1⫺ 1⫺
冊 册冎
⫺ r S ⫺ r
1/m
m
2
(9.39)
Both Eq. (9.39) as well as Eq. (9.37) using the soil– water characteristic curve by Fredlund and Xing (1994) give good predictions of measured data as shown in Fig. 9.12. The two hydraulic conductivity–matric suction curves shown in Fig. 9.11a cross each other at a matric suction value of approximately 50 kPa (or 5 m above the water table under hydrostatic condition). Below this value, the hydraulic conductivity of sand is larger than that of the clayey sand. However, as the matric suction increases, the water in the sand drains rapidly toward its residual value, giving a very low hydraulic conductivity. On the other hand, the clayey sand holds the pore water by the presence of fines and the hydraulic conductivity becomes larger than that of the sand at a given matric suction. If the sand is overlain by the clayey sand, then the matric suction at the interface is larger than 50 kPa, and the water infiltrating downward through the finer clayey sand cannot enter into the coarser sand layer because the underlying sand layer is less permeable than the overlying clayey sand. The water will instead move laterally along the bedding interface. This phenomenon is called a capillary barrier (e.g., Zaslavsky and Sinai, 1981; Yeh et al., 1985; Miyazaki, 1988). The barrier will be maintained as long as the lateral discharge along the interface (preferably inclined) is
4 m ⫽ 1 ⫺ 1 / n is assumed (van Genuchten et al., 1991). See Eq. (7.52).
Copyright © 2005 John Wiley & Sons
265
9.5
THERMAL CONDUCTIVITY
Heat flow through soil and rock is almost entirely by conduction, with radiation unimportant, except for surface soils, and convection important only if there is a high flow rate of water or air, as might possibly occur through a coarse sand or rockfill. The thermal conductivity controls heat flow rates. Conductive heat flow is primarily through the solid phase of a soil mass. Values of thermal conductivity for several materials are listed in Table 9.2. As the values for soil minerals are much higher than those for air and water, it is evident that the heat flow must be predominantly through the solids. Also included in Table 9.2 are values for the heat capacity, volumetric heat, heat of fusion, and heat of vaporization of water. The heat capacity can be used
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CONDUCTION PHENOMENA
Hydraulic Conductivity, k (cm/day)
1
0.1
0.01
Predicted coefficient of permeability (drying) Predicted coefficient of permeability (wetting)
Co py rig hte dM ate ria l
Hydraulic Conductivity, k × 10(m/s)
10
0.001
Measured coefficient of permeability (drying) Measured coefficient of permeability (wetting)
0.0001 20
30 40 50 Volumetric Water Content (a)
60
Figure 9.12 Comparisons of predicted and measured relationships between hydraulic con-
ductivity and volumetric water content for two soils. (a) By Eq. (9.37) with the measured data for Guelph loam (from Fredlund et al., 1994) and (b) by Eq. (9.39) with the measured data for crushed Bandelier Tuff (van Genuchten et al., 1991).
to compute the volumetric heat using the simple relationships for frozen and unfrozen soil given in the table. Volumetric heat is needed for the analysis of many types of transient heat flow problems. The heat of fusion is used for analysis of ground freezing and thawing, and the heat of vaporization applies to situations where there are liquid to vapor phase transitions. The denser a soil, the higher is its composite thermal conductivity, owing to the much higher thermal conductivity of the solids relative to the water and air. Furthermore, since water has a higher thermal conductivity than air, a wet soil has a higher thermal conductivity than a dry soil. The combined influences of soil unit weight and water content are shown in Fig. 9.13, which may be used for estimates of the thermal conductivity for many cases. If a more soil-specific value is needed, they may be measured in the laboratory using the thermal needle method (ASTM, 2000). More detailed treatment of methods for the measurement of the thermal conductivity of soils are given by Mitchell and Kao (1978) and Farouki (1981, 1982). The relationship between thermal resistivity (inverse of conductivity) and water content for a partly saturated soil undergoing drying is shown in Fig. 9.14. If drying causes the water content to fall below a certain value, the thermal resistivity increases significantly. This may be important in situations where soil is used as either a thermally conductive material, for example,
Copyright © 2005 John Wiley & Sons
to carry heat away from buried electrical transmission cables, or as an insulating material, for example, for underground storage of liquefied gases. The water content below which the thermal resistivity begins to rise with further drying is termed the critical water content, and below this point the system is said to have lost thermal stability (Brandon et al., 1989). The following factors influence the thermal resistivity of partly saturated soils (Brandon and Mitchell, 1989). Mineralogy All other things equal, quartz sands have higher thermal conductivity than sands containing a high percentage of mica. Dry Density The higher the dry density of a soil, the higher is the thermal conductivity. Gradation Well-graded soils conduct heat better than poorly graded soils because smaller grains can fit into the interstitial spaces between the larger grains, thus increasing the density and the mineral-to-mineral contact. Compaction Water Content Some sands that compacted wet and then dried to a lower water content have significantly higher thermal conductivity than when compacted initially at the lower water content. Time Sands containing high percentages of silica, carbonates, or other materials that can develop ce-
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ELECTRICAL CONDUCTIVITY
Table 9.2
Thermal Properties of Materials a
Material
Btu/h/ft2 / F/ft
W/m/K
Air Water Ice Snow (100 kg m⫺3) (500 kg m⫺3) Shale Granite Concrete Copper Soil Polystyrene
0.014 0.30 1.30
0.024 0.60 2.25
0.03 0.34 0.90 1.60 1.0 225 0.15–1.5 (⬇1.0) 0.015–0.035
0.06 0.59 1.56 2.76 1.8 389 0.25–2.5 (⬇1.7) 0.03–0.06
Material
Btu/lb/ F
kJ/kg/K
Co py rig hte dM ate ria l
Thermal Conductivity
Heat Capacity
Volumetric Heat
Heat of Fusion
Heat of Vaporization a
267
Water Ice Snow (100 kg m⫺3) (500 kg m⫺3) Minerals Rocks
1.0 0.5
4.186 2.093
0.05 0.25 0.17 0.20–0.55
0.21 1.05 0.710 0.80–2.20
Material
Btu/ft3 / F
kJ/m3 /K
Unfrozen Soil Soil Frozen soil Snow (100 kg m⫺3) (500 kg m⫺3) Water Soil Water Soil
d (0.17 ⫹ w/100)
d (72.4 ⫹ 427w/100)
d (0.17 ⫹ 0.5w/100)
d (72.4 ⫹ 213w/100)
3.13 15.66 143.4 Btu/lb 143.4(w/100) d Btu/ft3 970 Btu/lb 970(w/100) d Btu/ft3
210 1050 333 kJ/kg 3.40 ⫻ 104(w/100) d kJ/m3 2.26 MJ/kg 230(w /100) d MJ/m3
d ⫽ dry unit weight, in lb/ft3 for U.S. units and in kN/m3 for SI units; w ⫽ water content in percent.
mentation may exhibit an increased thermal conductivity with time. Temperature All crystalline minerals in soils have decreasing thermal conductivity with increasing temperature; however, the thermal conductivity of water increases slightly with increasing temperature, and the thermal conductivity of saturated pore air increases markedly with increasing temperature. The net effect is that the thermal conductivity of moist sand increases somewhat with increasing temperature.
Copyright © 2005 John Wiley & Sons
9.6
ELECTRICAL CONDUCTIVITY
Ohm’s law, Eq. (9.4), in which e is the electrical conductivity, applies to soil–water systems. The electrical conductivity equals the inverse of the electrical resistivity, or e ⫽
1 L (siemens/meter; S/m) RA
(9.40)
where R is the resistance ( ), L is length of sample
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CONDUCTION PHENOMENA
Co py rig hte dM ate ria l
fects particle size, shape, and surface conductance, soil structure, including fabric and cementation, and temperature. Electrical measurements found early applications in the fields of petroleum engineering, geophysical mapping and prospecting, and soil science, among others. The inherent complexity of soil–water systems and the difficulty in characterizing the wide ranges of particle size, shape, and composition have precluded development of generally applicable theoretical equations for electrical conductivity. However, a number of empirical equations and theoretical expressions based on simplified models may provide satisfactory results, depending on the particular soil and conditions. They differ in assumptions about the possible flow paths for electric current through a soil–water matrix, the path lengths and their relative importance, and whether charged particle surfaces contribute to the total current flow.
Figure 9.13 Thermal conductivity of soil (after Kersten,
1949).
Nonconductive Particle Models
Formation Factor The electrical conductivity of clean saturated sands and sandstones is directly proportional to the electrical conductivity of the pore water (Archie, 1942). The coefficient of proportionality depends on porosity and fabric. Archie (1942) defined the formation factor, F, as the resistivity of the saturated soil, T, divided by the resistivity of the saturating solution, W, that is, F⫽
T ⫽ W W T
(9.41)
where W and T are the electrical conductivities of the pore water and saturated soil, respectively. An empirical correlation between formation factor and porosity for clean sands and sandstones is given by F ⫽ n⫺m
Figure 9.14 Typical relationship between thermal resistivity
and water content for a compacted sand.
(m), and A is its cross-sectional area (m2). The value of electrical conductivity for a saturated soil is usually in the approximate range of 0.01 to 1.0 S/m. The specific value depends on several properties of the soil, including porosity, degree of saturation, composition (conductivity) of the pore water, mineralogy as it af-
Copyright © 2005 John Wiley & Sons
(9.42)
where n is porosity, and m equals from 1.3 for loose sands to 2 for highly cemented sandstones. An empirical relation between formation factor at 100 percent water saturation and ‘‘apparent’’ formation factor at saturation less than 100 percent is FatSw⫽1 ⫽ (Sw)p
W T
(9.43)
where p is a constant determined experimentally. Archie suggested a value of p ⫽ 2; however, other published values of p range from 1.4 to 4.6, depending on the soil and
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ELECTRICAL CONDUCTIVITY
269
whether a given saturation is reached by wetting or by drainage. Capillary Model In this and the theoretical models
tance of clayey particles to the total current flow would be small.
that follow, direct current (DC) conductivity is assumed, although they may apply to low-frequency alternating current (AC) models as well. Consider a saturated soil sample of length L and cross-sectional area A. If the pores are assumed to be connected and can be represented by a bundle of tubes of equal radius and length Le and total area Ae, where Ae ⫽ porosity ⫻ A, and Le is the actual length of the flow path, then an equation for the formation factor as a function of the porosity n and the tortuosity T ⫽ Le /L is
Conductive Particle Models
Co py rig hte dM ate ria l T2 n
F⫽
(9.44)
T ⫽ X(W ⫹ s)
For S ⬍ 1, and assuming that the area available for electrical flow is nSA, then F ⫽ T 2 /nS. In principle, if F is measured for a given soil and n is known, a value of tortuosity can be calculated to use in the Kozeny– Carman equation for hydraulic conductivity. Cluster Model As discussed earlier in connection with hydraulic conductivity, the cluster model (Olsen, 1961, 1962) shown in Fig. 9.10 assumes unequal pore sizes. Three possible paths for electrical current flow can be considered: (1) through the intercluster pores, (2) through the intracluster pores, and (3) alternately through inter- and intracluster pores. On this basis the following equations for formation factor as a function of the cluster model parameters can be derived (Olsen, 1961): F ⫽ T2
冉
冊冉 冊
1 ⫹ eT eT ⫺ ec
1 1⫹X
X⫽Y⫹Z
Y⫽
In conductive particle models the contribution of the ions concentrated at the surface of negatively charged particles is taken into account. Two simple mixture models are presented below; other models can be found in Santamarina et al. (2001). Two-Parallel-Resistor Model A contribution of surface conductance is included, and the soil–water system is equivalent to two electrical resistors in parallel (Waxman and Smits, 1968). The result is that the total electrical conductivity T is
(9.46)
[(1 ⫹ eT)/(eT ⫺ ec)]2 1 ⫹ (Tc /T)2 [(1 ⫹ ec)2 /ec(eT ⫺ ec)] Z⫽a
冉
ec
冊冉 冊
eT ⫺ ec
(9.45)
T Tc
(9.47)
2
(9.48)
in which T is the intercluster tortuosity, Tc is the intracluster tortuosity, and a is the effective cluster ‘‘contact area.’’ The cluster contact area is very small except for heavily consolidated systems. This model successfully describes the flow of current in soils saturated with high conductivity water. In such systems, the contribution of the surface conduc-
Copyright © 2005 John Wiley & Sons
(9.49)
in which s is a surface conductivity term, and X is a constant analogous to the reciprocal of the formation factor that represents the internal geometry. This approach yields better fits of T versus W data for clay-bearing soils. However, it assumes a constant value for the contribution of the surface ions that is independent of the electrolyte concentration in the pore water, and it fails to include a contribution for the surface conductance and pore water conductance in a series path. Three-Element Network Model A third path is included in this formulation that considers flow along particle surfaces and through pore water in series in addition to the paths included in the two-parallelresistor model. The flow paths and equivalent electrical circuit are shown in Fig. 9.15. Analysis of the electrical network for determination of T gives T ⫽
aWs ⫹ bs ⫹ cW (1 ⫺ e)W ⫹ es
(9.50)
If the surface conductivity s is negligible, the simple formulation proposed by Archie (1942) for sands is obtained; that is, T ⫽ constant ⫻ W. Some of the geometric parameters a, b, c, d, and e can be written as functions of porosity and degree of saturation; others are obtained through curve regression analysis of T versus W data. Soil conductivity as a function of pore fluid conductivity is shown in Fig. 9.16 for a silty clay. The three-element model fits the data well over the full range, the two-element model gives good predictions for the higher values of conductivity, and the simple formation factor relationship is a reasonable average
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9
CONDUCTION PHENOMENA
Co py rig hte dM ate ria l
270
Figure 9.15 Three-element network model for electrical conductivity: (a) current flow paths
and (b) equivalent electrical circuit.
for conductivity values in the range of about 0.3 to 0.6 S/m. Alternating Current Conductivity and Dielectric Constant
The electrical response of a soil in an AC field is frequency dependent owing to the polarizability properties of the system constituents. Several scale-dependent polarization mechanisms are possible in soils, as shown in Fig. 9.17. The smaller the element size the higher the polarization frequency. At the atomic and molecular scales, there are polarizations of electrons [electronic resonance at ultraviolet (UV) frequencies], ions [ionic resonance at infrared (IR) frequencies], and dipolar molecules (orientational relaxation at microwave frequencies). A mixture of components (like water and soil particles) having different polarizabilities and conductivities produces spatial polarization by charge accumulation at interfaces (called Maxwell– Wagner interfacial polarization). The ions in the Stern layer and double layer are restrained (Chapter 6), and hence they also exhibit polarization. This polarization results in relaxation responses at radio frequencies. Further details of the polarization mechanisms are given by Santamarina et al. (2001). The effective AC conductivity eff is expressed as eff ⫽ ⫹ !ⴖ"0
(9.51)
where is the conductivity, !ⴖ is the polarization loss (called the imaginary relative permittivity), " is the
Copyright © 2005 John Wiley & Sons
frequency, and 0 is the permittivity of vacuum [8.85 ⫻ 10⫺12 C2 /(Nm2)]. The frequency-dependent effective conductivities of deionized water and kaolinite–water mixtures at two different water contents (0.2 and 33 percent) are shown in Fig. 9.18a. The complicated interactions of different polarization mechanisms are responsible for the variations shown. A material is dielectric if charges are not free to move due to their inertia. Higher frequencies are needed to stop polarization at smaller scales. The dielectric constant (or the real relative permittivity !5) decreases with increasing frequency; more polarization mechanisms occur at lower frequencies. The frequency-dependent dielectric constants of deionized water and kaolinite–water mixtures are shown in Fig. 9.18b. The value for deionized water is about 79 above 10 kHz. Below this frequency, the values increase with decrease in frequency. This is attributed to experimental error caused by an electrode effect in which charges
5 To describe the out-of-phase response under oscillating excitation, the electrical properties of a material are often defined in the complex plane:
⫽ ⫺ jⴖ
where is the complex permittivity, j is the imaginary number (兹⫺1), and and ⴖ are real and imaginary numbers describing the electrical properties. The permittivity is often normalized by the permittivity of vacuum 0 as !⫽
⫽ ! ⫺ j!ⴖ 0
where ! is called the relative permittivity.
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ELECTRICAL CONDUCTIVITY
271
100
σeff (S/m)
Deionized Water 10–2 33% No Data Available
10–4
102
104 106 Frequency (Hz) (a)
Co py rig hte dM ate ria l
0.2% 10–6 100
106 33%
Figure 9.16 Soil electrical conductivity as a function of pore
fluid conductivity and comparisons with three models.
κ
108
1010
Electrode Effect
104 0.2%
Deionized Water
No Data Available
102
100
accumulate at the electrode–specimen interface (Klein and Santamarina, 1997). Similarly to the observations made for the effective conductivities, the real permittivity values of the mixtures show complex trends of frequency dependency. For analysis of AC conductivity and dielectric constant as a function of frequency in an AC field, Smith and Arulanandan (1981) modified the three-element model shown in Fig. 9.15 by adding a capacitor in parallel with each resistor. The resulting equations can be fit to experimental frequency dispersions of the con-
100
102
104 106 Frequency (Hz) (b)
1010
Figure 9.18 (a) Conductivity and (b) relative permittivity as a function of frequency for deionized water and kaolinite at water contents of 0.2 and 33 percent (from Santamarina et al., 2001).
Figure 9.17 Frequency ranges associated with different polarization mechanisms (from Santamarina et al., 2001).
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108
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CONDUCTION PHENOMENA
ductivity and apparent dielectric constant by computer optimization of geometrical and compositional parameters. The resulting parameter values are useful for characterizing mineralogy, porosity, and fabric. More detailed discussions on electrical models, data interpretation, and correlations with soil properties are given by Santamarina et al. (2001).
DIFFUSION
Anion (1)
D0 ⫻ 1010(m2 /s) (2)
Cation (3)
D0 ⫻ 1010(m2 /s) (4)
OH⫺ F⫺ Cl⫺ Br⫺ I⫺ HCO3⫺ NO3⫺ SO42⫺ CO32⫺ — — — — — — — — — — —
52.8 14.7 20.3 20.8 20.4 11.8 19.0 10.6 9.22 — — — — — — — — — — —
H⫹ Li⫹ Na⫹ K⫹ Rb⫹ Cs⫹ Be2⫹ Mg2⫹ Ca2⫹ Sr2⫹ Ba2⫹ Pb2⫹ Cu2⫹ Fe2⫹a Cd2⫹a Zn2⫹ Ni2⫹a Fe3⫹a Cr3⫹a Al3⫹a
93.1 10.3 13.3 19.6 20.7 20.5 5.98 7.05 7.92 7.90 8.46 9.25 7.13 7.19 7.17 7.02 6.79 6.07 5.94 5.95
Co py rig hte dM ate ria l
9.7
Table 9.3 Self-Diffusion Coefficients for Ions at Infinite Dilution in Water
Chemical transport through sands is dominated by advection, wherein dissolved and suspended species are carried with flowing water. However, in fine-grained soils, wherein the hydraulic flow rates are very small, for example, kh less than about 1 ⫻ 10⫺9 m/s, chemical diffusion plays a role and may become dominant when kh becomes less than about 1 ⫻ 10⫺10 m/s. Fick’s law, Eq. (9.5), is the controlling relationship, and D(L2T⫺1), the diffusion coefficient, is the controlling parameter. Diffusive chemical transport is important in clay barriers for waste containment, in some geologic processes, and in some forms of chemical soil stabilization. Comprehensive treatments of the diffusion process, values of diffusion coefficients and methods for their determination, and applications, especially in relation to chemical transport and waste containment barrier systems, are given by Quigley et al. (1987), Shackelford and Daniel (1991a, 1991b), Shincariol and Rowe (2001) and Rowe (2001). Diffusive flow is driven by chemical potential gradients, but for most applications chemical concentration gradients can be used for analysis. The diffusion coefficient is measured and expressed in terms of chemical gradients. Maximum values of the diffusion coefficient D0 are found in free aqueous solution at infinite dilution. Self-diffusion coefficients for a number of ion types in water are given in Table 9.3. Usually cation–anion pairs are diffusing together, thereby slowing down the faster and speeding up the slower. This may be seen in Table 9.4, which contains values of some limiting free solution diffusion coefficients for some simple electrolytes. Diffusion through soil is slower and more complex than diffusion through a free solution, especially when adsorptive clay particles are present. There are several reasons for this (Quigley, 1989): 1. Reduced cross-sectional area for flow because of the presence of solids 2. Tortuous flow paths around particles 3. The influences of electrical force fields caused by the double-layer distributions of charges
Copyright © 2005 John Wiley & Sons
a
Values from Li and Gregory (1974). Reprinted with permission from Geochimica et Cosmochimica Acta, Vol. 38, No. 5, pp. 703–714. Copyright 1974, Pergamon Press.
4. Retardation of some species as a result of ion exchange and adsorption by clay minerals and organics or precipitation 5. Biodegradation of diffusing organics 6. Osmotic counterflow 7. Electrical imbalance, possibly by anion exclusion
The diffusion coefficient could increase with time of flow through a soil as a result of such processes as (Quigley, 1989): 1. K⫹ fixation by vermiculite, which would decrease the cation exchange capacity and increase the free water pore space 2. Electrical imbalances that act to pull cations or anions 3. The attainment of adsorption equilibrium, thus eliminating retardation of some species
In an attempt to take some of these factors, especially geometric tortuosity of interconnected pores, into account, an effective diffusion coefficient D* is
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DIFFUSION
transient diffusion, that is, the time rate of change of concentration with distance:
Table 9.4 Limiting Free Solution Diffusion Coefficients for Some Simple Electrolytes D0 ⫻ 1010(m2 /s) (2)
Electrolyte (1)
33.36 34.00 13.66 13.77 16.10 16.25 16.14 19.93 20.16 19.99 20.44 13.35 13.85
c 2c ⫽ D* 2 t x
Reported by Shackelford and Daniel, 1991a after Robinson and Stokes, 1959. Reprinted from the Journal of Geotechnical Engineering, Vol. 117, No. 3, pp. 467–484. Copyright 1991. With permission of ASCE.
used. Several definitions have been proposed (Shackelford and Daniel, 1991a) in which the different factors are taken into account in different ways. Although these relationships may be useful for analysis of the importance of the factors themselves, it is sufficient for practical purposes to use D* ⫽ a D0
(9.52)
in which a is an ‘‘apparent tortuosity factor’’ that takes several of the other factors into account, and use values of D* measured under representative conditions. The effective coefficient for diffusion of different chemicals through saturated soil is usually in the range of about 2 ⫻ 10⫺10 to 2 ⫻ 10⫺9 m2 /s, although the values can be one or more orders of magnitude lower in highly compacted clays and clays, such as bentonite, that can behave as semipermeable membranes (Malusis and Shackelford, 2002b). Values for compacted clays are rather insensitive to molding water content or method of compaction (Shackelford and Daniel, 1991b), in stark contrast to the hydraulic conductivity, which may vary over a few orders of magnitude as a result of changes in these factors. This suggests that soil fabric differences have relatively minor influence on the effective diffusion coefficient. Whereas Fick’s first law, Eq. (9.5), applies for steady-state diffusion, Fick’s second law describes
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(9.53)
For transient diffusion with constant effective diffusion coefficient D*, the solution for this equation is of exactly the same form as that for the Terzaghi equation for clay consolidation and that for one-dimensional transient heat flow. An error function solution for Eq. (9.53) (Ogata, 1970; Freeze and Cherry, 1979), for the case of onedimensional diffusion from a layer at a constant source concentration C0 into a layer having a sufficiently low initial concentration that it can be taken as zero at t ⫽ 0, is
Co py rig hte dM ate ria l
HCl HBr LiCl LiBr NaCl NaBr NaI KCl KBr KI CsCl CaCl2 BaCl2
273
C x x ⫽ erfc ⫽ 1 ⫺ erf C0 2兹D*t 2兹D*t
(9.54)
where C is the concentration at any time at distance ⫻ from the source. Curves of relative concentration as a function of depth for different times after the start of chloride diffusion are shown in Fig. 9.19a (Quigley, 1989). An effective diffusion coefficient for chloride of 6.47 ⫻ 10⫺10 m2 /s was assumed. Also shown (Fig. 9.19b) is the migration velocity of the C/C0 front within the soil as a function of time. As chloride is one of the more rapidly diffusing ionic species, Fig. 9.19 provides a basis for estimating maximum probable migration distances and concentrations as a function of time that result solely from diffusion. When there are adsorption–desorption reactions, chemical reactions such as precipitation–solution, radioactive decay, and/or biological processes occurring during diffusion, the analysis becomes more complex than given by the foregoing equations. For adsorption– desorption reactions and the assumption that there is linearity between the amount adsorbed and the equilibrium concentration, Eq. (9.53) is often written as c D* 2c ⫽ t Rd x2
(9.55)
where Rd is termed the retardation factor, and it is defined by Rd ⫽ 1 ⫹
d K d
(9.56)
in which d is the bulk dry density of the soil, is the
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9
CONDUCTION PHENOMENA
tailed discussions of distribution coefficients and their determination are given by Freeze and Cherry (1979), Quigley et al., (1987), Quigley (1989), and Shackelford and Daniel (1991a, b).
9.8 TYPICAL RANGES OF FLOW PARAMETERS
Co py rig hte dM ate ria l
Usual ranges for the values of the direct flow conductivities for hydraulic, thermal, electrical, and diffusive chemical flows are given in Table 9.5. These ranges are for fine-grained soils, that is, silts, silty clays, clayey silts, and clays. They are for full saturation; values for partly saturated soils can be much lower. Also listed in Table 9.5 are values for electroosmotic conductivity, osmotic efficiency, and ionic mobility. These properties are needed for analysis of coupling of hydraulic, electrical, and chemical flows, and they are discussed further later.
9.9 SIMULTANEOUS FLOWS OF WATER, CURRENT, AND SALTS THROUGH SOIL-COUPLED FLOWS
Figure 9.19 Time rate of chloride diffusion (from Quigley,
1989). (a) Relative concentration as a function of depth after different times and (b) velocity of migration of the front having a concentration C / C0 of 0.5.
volumetric water content, that is, the volume of water divided by the total volume (porosity in the case of a saturated soil), and Kd is the distribution coefficient. The distribution coefficient defines the amount of a given constituent that is adsorbed or desorbed by a soil for a unit increase or decrease in the equilibrium concentration in solution. Other reactions influencing the amount in free solution relative to that fixed in the soil (e.g., by precipitation) may be included in Kd, depending on the method for measurement and the conditions being modeled. Distribution coefficients are usually determined from adsorption isotherms, and they may be constants for a given soil–chemical system or vary with concentration, pH, and temperature. More de-
Copyright © 2005 John Wiley & Sons
Usually there are simultaneous flows of different types through soils and rocks, even when only one type of driving force is acting. For example, when pore water containing chemicals flows under the action of a hydraulic gradient, there is a concurrent flow of chemical through the soil. This type of chemical transport is termed advection. In addition, owing to the existence of surface charges on soil particles, especially clays, there are nonuniform distributions of cations and anions within soil pores resulting from the attraction of cations to and repulsion of anions from the negatively charged particle surfaces. The net negativity of clay particles is caused primarily by isomorphous substitutions within the crystal structure, as discussed in Chapter 3, and the ionic distributions in the pore fluid are described in Chapter 6. Because of the small pore sizes in fine-grained soils and the strong local electrical fields, clay layers exhibit membrane properties. This means that the passage of certain ions and molecules through the clay may be restricted in part or in full at both microscopic and macroscopic levels. Owing to these internal nonhomogeneities in ion distributions, restrictions on ion movements caused by electrostatic attractions and repulsions, and the dependence of these interactions on temperature, a variety of microscopic and macroscopic effects may be observed when a wet soil mass is subjected to flow
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SIMULTANEOUS FLOWS OF WATER, CURRENT, AND SALTS THROUGH SOIL-COUPLED FLOWS
Table 9.5
275
Typical Range of Flow Parameters for Fine-Grained Soilsa
Parameter
Symbol
Units
Minimum
Maximum
Porosity Hydraulic conductivity Thermal conductivity Electrical conductivity Electro osmotic conductivity Diffusion coefficient Osmotic efficiencyb Ionic mobility
n kh
— m s⫺1
0.1 1 ⫻ 10⫺11
0.7 1 ⫻ 10⫺6
kt
W m⫺1 K⫺1
0.25
2.5
e
siemens m⫺1
0.01
1.0
m2 s⫺1 V⫺1
1 ⫻ 10⫺9
1 ⫻ 10⫺8
D
m2 s⫺1
2 ⫻ 10⫺10
2 ⫻ 10⫺9
—
0
1.0
m2 s⫺1 V⫺1
3 ⫻ 10⫺9
1 ⫻ 10⫺8
Co py rig hte dM ate ria l ke
"
u
a
The above values of flow coefficients are for saturated soil. They may be much less in partly saturated soil. b 0 to 1.0 is the theoretical range for the osmotic efficiency coefficient. Values greater than about 0.7 are unlikely in most fine-grained materials of geotechnical interest.
gradients of different types. A gradient of one type Xj can cause a flow of another type Ji, according to Ji ⫽ Lij Xj
(9.57)
The Lij are termed coupling coefficients. They are properties that may or may not be of significant magnitude in any given soil, as discussed later. Types of coupled flow that can occur are listed in Table 9.6, along with terms commonly used to describe them.6 Of the 12 coupled flows shown in Table 9.6, several are known to be significant in soil–water systems, at least under some conditions. Thermoosmosis, which is water movement under a temperature gradient, is important in partly saturated soils, but of lesser importance in fully saturated soils. Significant effects from thermally driven moisture flow are found in semiarid and arid areas, in frost susceptible soils, and in expansive soils. An analysis of thermally driven moisture
6
Mechanical coupling also occurs in addition to the hydraulic, thermal, electrical, and chemical processes listed in Table 9.6. A common manifestation of this in geotechnical applications is the development of excess pore pressure and the accompanying fluid flow that result from a change in applied stress. This type of coupling is usually most easily handled by usual soil mechanics methods. A few other types of mechanical coupling may also exist in soils and rocks (U.S. National Committee for Rock Mechanics, 1987).
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flow is developed later. Electroosmosis has been used for many years as a means for control of water flow and for consolidation of soils. Chemicalosmosis, the flow of water caused by a chemical gradient acting across a clay layer, has been studied in some detail recently, owing to its importance in waste containment systems. Isothermal heat transfer, caused by heat flow along with water flow, has caused great difficulties in the creation of frozen soil barriers in the presence of flowing groundwater. Electrically driven heat flow, the Peltier effect, and chemically driven heat flow, the Dufour effect, are not known to be of significance in soils; however, they appear not to have been studied in any detail in relation to geotechnical problems. Streaming current, the term applied to both hydraulically driven electrical current and ion flows, has importance to both chemical flow through the ground (advection) and the development of electrical potentials, which may, in turn, influence both fluid and ion flows as a result of additional coupling effects. The complete roles of thermoelectricity and diffusion and membrane potentials are not yet known; however, electrical potentials generated by temperature and chemical gradients are important in corrosion and in some groundwater flow and stability problems. Whether thermal diffusion of electrolytes, the Soret effect, is important in soils has not been evaluated;
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9
Table 9.6
CONDUCTION PHENOMENA
Direct and Coupled Flow Phenomena Gradient X Hydraulic Head
Fluid
Heat
Current
Ion
Temperature
Hydraulic conduction Darcy’s law Isothermal heat transfer or thermal filtration Streaming current
Thermoosmosis
Electroosmosis
Chemical osmosis
Thermal conduction Fourier’s law Thermoelectricity Seebeck or Thompson effect
Peltier effect
Dufour effect
Streaming current ultrafiltration (also known as hyperfiltration)
Thermal diffusion of electrolyte or Soret effect
Electrophoresis
however, since chemical activity is highly temperature dependent, it may be a significant process in some systems. Finally, electrophoresis, the movement of charged particles in an electrical field, has been used for concentration of mine waste and high water content clays. The relative importance of chemically and electrically driven components of total hydraulic flow is illustrated in Fig. 9.20, based on data from tests on kaolinite given by Olsen (1969, 1972). The theory for description of coupled flows is given later. A practical form of Eq. (9.57) for fluid flow under combined hydraulic, chemical, and electrical gradients is qh ⫽ ⫺kh
H
L
Chemical Concentration
Electrical
Co py rig hte dM ate ria l
Flow J
A ⫹ kc
log(CB /CA)
E A ⫺ ke A (9.58) L L
in which kh, kc, and ke are the hydraulic, osmotic, and electroosmotic conductivities, H is the hydraulic head difference, E is the voltage difference, and CA and CB are the salt concentrations on opposite sides of a clay layer of thickness L. In the absence of an electrical gradient, the ratio of osmotic to hydraulic flows is
冉冊
qhc k log(CB /CA) ⫽⫺ c qh kh
H
( E ⫽ 0)
(9.59)
and, in the absence of a chemical gradient, the ratio of electroosmotic flows to hydraulic flows is
Copyright © 2005 John Wiley & Sons
Electric conduction Ohm’s law
冉冊
qhe ke E ⫽ qh kh H
Diffusion and membrane potentials or sedimentation current Diffusion Fick’s law
( C ⫽ 0)
(9.59a)
The ratio (kc /kh) in Fig. 9.20 indicates the hydraulic head difference in centimeters of water required to give a flow rate equal to the osmotic flow caused by a 10fold difference in salt concentration on opposite sides of the layer. The ratio ke /kh gives the hydraulic head difference required to balance that caused by a 1 V difference in electrical potentials on opposite sides of the layer. During consolidation, the hydraulic conductivity decreases dramatically. However, the ratios kc /kh and ke /kh increase significantly, indicating that the relative importance of osmotic and electroosmotic flows to the total flow increases. Although the data shown in Fig. 9.20 are shown as a function of the consolidation pressure, the changes in the values of kc /kh and ke /kh are really a result of the decrease in void ratio that accompanies the increase in pressure, as may be seen in Fig. 9.20c. These results for kaolinite provide a conservative estimate of the importance of osmotic and electroosmotic flows because coupling effects in kaolinite are usually smaller than in more active clays, such as montmorillonite-based bentonites. In systems containing confined clay layers acted on by chemical and/or electrical gradients, Darcy’s law by itself may be an insufficient basis for prediction of hydraulic flow rates, particularly if the clay is highly plastic and at a very low void ratio. Such conditions can be found in deeply buried clay
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277
Co py rig hte dM ate ria l
QUANTIFICATION OF COUPLED FLOWS
Figure 9.20 Hydraulic, osmotic, and electroosmotic conductivities of kaolinite (data from
Olsen 1969, 1972): (a) consolidation curve, (b) conductivity values, and (c) conductivities as a function of void ratio.
and clay shale and in densely compacted clays. For more compressible clays, the ratios kc /kh and ke /kh may be sufficiently high to be useful for consolidation by electrical and chemical means, as discussed later in this chapter.
9.10
QUANTIFICATION OF COUPLED FLOWS
Quantification of coupled flow processes may be done by direct, empirical determination of the relevant parameters for a particular case or by relationships derived from a theoretical thermodynamic analysis of the complete set of direct and coupled flow equations.
Copyright © 2005 John Wiley & Sons
Each approach has advantages and limitations. It is assumed in the following that the soil properties remain unchanged during the flow processes, an assumption that may not be justified in some cases. The effects of flows of different types on the state and properties of a soil are discussed later in this chapter. However, when properties are known to vary in a predictable manner, their variations may be taken into account in numerical analysis methods. Direct Observational Approach
In the general case, there may be fluid, chemical, electrical, and heat flows. The chemical flows can be sub-
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278
9
CONDUCTION PHENOMENA
qw ⫽ LHH (⫺H) ⫹ LHE (⫺E) ⫹ LHC (⫺C)
(9.60)
I ⫽ LEH (⫺H) ⫹ LEE (⫺E) ⫹ LEC (⫺C)
(9.61)
JC ⫽ LCH (⫺H) ⫹ LCE (⫺E) ⫹ LCC (⫺C)
(9.62)
where qw I Jc H E C Lij
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
ductivity coefficient kh is readily determined.7 The coefficient of electroosmotic hydraulic conductivity is usually determined by measuring the hydraulic flow rate developed in a known DC potential field under conditions of ih ⫽ 0. The electrical conductivity e is obtained from the same experiment through measurement of the electrical current. The main advantage of this empirical, but direct, approach is simplicity. It is particularly useful when only a few of the possible couplings are likely to be important and when some uncertainty in the measured coefficients is acceptable.
Co py rig hte dM ate ria l
divided according to the particular chemical species present. Each flow type may have contributions caused by gradients of another type, and their importance depends on the values of Lij and Xj in Eq. (9.57). A complete and accurate description of all flows may be a formidable task. However, in many cases, flows of only one or two types may be of interest, some of the gradients may not exist, and/or some of the coupling coefficients may be either known or assumed to be unimportant. The matrix of flows and forces then reduces significantly, and the determination of coefficients is greatly simplified. For example, if simple electroosmosis under isothermal conditions is considered, then Eq. (9.57) yields
water flow rate electrical current chemical flow rate hydraulic head electrical potential chemical concentration coupling coefficients; the first subscript indicates the flow type and the second denotes the type of driving force
If there are no chemical concentration differences across the system, then the last terms on the right-hand side of Eqs. (9.60), (9.61), and (9.62) do not exist. In this case, Eqs. (9.60) and (9.61) become, when written in more familiar terms, qw ⫽ khih ⫹ keie
I ⫽ hih ⫹ eie
(9.63)
(9.64)
where kh ⫽ hydraulic conductivity ke ⫽ electroosmotic hydraulic conductivity h ⫽ electrical conductivity due to hydraulic flow e ⫽ electrical conductivity ih ⫽ hydraulic gradient ie ⫽ electrical potential gradient If permeability tests are done in the absence of an electrical potential difference, then the hydraulic con-
Copyright © 2005 John Wiley & Sons
General Theory for Coupled Flows
When several flows are of interest, each resulting from several gradients, a more formal methodology is necessary so that all relevant factors are accounted for properly. If there are n different driving forces, then there will be n direct flow coefficients Lii and n(n ⫺ 1) coupling coefficients Lij(i ⫽ j). The determination of these coefficients is best done within a framework that provides a consistent and correct description of each of the flows. Irreversible thermodynamics, also termed nonequilibrium thermodynamics, offers a basis for such a description. Furthermore, if the terms are properly formulated, then Onsager’s reciprocal relations apply, that is, Lij ⫽ Lji
(9.65)
and the number of coefficients to be determined is significantly reduced. In addition, the derived forms for the coupling coefficients, when cast in terms of measurable and understood properties, provide a basis for rapid assessment of their importance. The theory of irreversible thermodynamics as applied to transport processes in soils is only outlined here. More comprehensive treatments are given by DeGroot and Mazur (1962), Fitts (1962), Katchalsky and Curran (1967), Greenberg, et al. (1973), Yeung and Mitchell (1992), and Malusis and Shackelford (2002a). Irreversible thermodynamics is a phenomenological, macroscopic theory that provides a basis for descrip-
7 Note that unless the ends of the sample are short circuited to prevent the development of a streaming potential, there will be a small electroosmotic counterflow contributed by the keie term in Eq. (9.63). Streaming potentials may be up to a few tens of millivolts in soils. Streaming potential is one of four types of electrokinetic phenomena that may exist in soils, as discussed in more detail in Section 9.16.
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SIMULTANEOUS FLOWS OF WATER, CURRENT, AND CHEMICALS
tion of systems that are out of equilibrium. It is based on three postulates, namely,
in the formulation of the flow equations. And # is also the sum of products of fluxes and driving forces:
1. Local equilibrium, a criterion that is satisfied if local perturbations are not large. 2. Linear phenomenological equations, that is,
冘L X n
Ji ⫽
ij
j
( j ⫽ 1,2, . . . , n)
(9.66)
冘JX n
#⫽
i
i
3. Validity of the Onsager reciprocal relations, a condition that is satisfied if the Ji and Xj are formulated properly (Onsager, 1931a, 1931b). Experimental verification of the Onsager reciprocity for many systems and processes has been obtained and is summarized by Miller (1960). Both the driving forces and flows vanish in systems that are in equilibrium, so the deviations of thermodynamic variables from their equilibrium values provide a suitable basis for their formulation. The deviations of the state parameters Ai from equilibrium are given by i ⫽ Ai ⫺ A 0i
(9.67)
where A 0i is the value of the state parameter at equilibrium and Ai is its value in the disturbed state. Criteria for deriving the forces and flows are then developed on the basis of the second law of thermodynamics, which states that at equilibrium, the entropy S is a maximum, and i ⫽ 0. The change in entropy
S that results from a change in state parameter gives the tendency for a variable to change. Thus S/ i is a measure of the force causing i to change, and is called Xi. The flows Ji, termed fluxes in irreversible thermodynamics, are given by i / t, the time derivative of i. On this basis, the resulting entropy production per unit time becomes ⫽
dS ⫽ dt
冘JX
(9.69)
i⫽1
The units of # are energy per unit time, and it is a measure of the rate of local free energy dissipation by irreversible processes. Application of the thermodynamic theory of irreversible processes requires the following steps:
Co py rig hte dM ate ria l
j⫽1
279
1. Finding the dissipation function # for the flows 2. Defining the conjugated flows Ji and driving forces Xi from Eq. (9.69) 3. Formulating the phenomenological equations in the form of Eq. (9.66) 4. Applying the Onsager reciprocal relations 5. Relating the phenomenological coefficients to measurable quantities
When the Onsager reciprocity is used, the number of independent coefficients Lij reduces from n2 to [(n ⫹ 1)n]/2. Application
The quantitative analysis and prediction of flows through soils, for a given set of boundary conditions, depends on the values of the various phenomenological coefficients in the above flow equations. Unfortunately, these are not always known with certainty, and they may vary over wide ranges, even within an apparently homogeneous soil mass. The direct flow coefficients, that is, the hydraulic, electrical, and thermal conductivities, and the diffusion coefficient, exhibit the greatest ranges of values. Thus, it is important to examine these properties first before detailed analysis of coupled flow contributions. For many problems, it may be sufficient to consider only the direct flows, provided the factors influencing their values are fully appreciated.
n
i
i
(9.68)
i⫽1
The entropy production can be related explicitly to various irreversible processes in terms of proper forces and fluxes (Gray, 1966; Yeung and Mitchell, 1992). If the choices satisfy Eq. (9.68), then the Onsager reciprocity relations apply. It has been found more useful to use # ⫽ T, the dissipation function, in which T is temperature, than
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9.11 SIMULTANEOUS FLOWS OF WATER, CURRENT, AND CHEMICALS
Use of irreversible thermodynamics for the description of coupled flows as developed above is straightforward in principle; however, it becomes progressively more difficult in application as the numbers of driving forces and different flow types increase. This is because of (1) the need for proper specification of the different coupling coefficients and (2) the need for independent
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280
9
CONDUCTION PHENOMENA
Co py rig hte dM ate ria l
methods for their measurement. Thus, the analysis of coupled hydraulic and electrical flows or of coupled hydraulic and chemical flows is much simpler than the analysis of a system subjected to electrical, chemical, and hydraulic gradients simultaneously. Relationships for the volume flow rate of water for several cases and for thermoelectric and thermoosmotic coupling in saturated soils are given by Gray (1966, 1969). The simultaneous flows of liquid and charge in kaolinite and the fluid volume flow rates under hydraulic, electric, and chemical gradients were studied by Olsen (1969, 1972). The theory for coupled salt and water flows was developed by Greenberg (1971) and applied to flows in a groundwater basin (Greenberg et al., 1973) and to chemicoosmotic consolidation of clay (Mitchell et al., 1973). Equations for the simultaneous flows of water, electricity, cations, and anions under hydraulic, electrical, and chemical gradients were formulated by Yeung (1990) using the formalism of irreversible thermodynamics as outlined previously. The detailed development is given by Yeung and Mitchell (1993). The results are given here. The chemical flow is separated into its anionic and cationic components in order to permit determination of their separate movements as a function of time. This separation may be important in some problems, such as chemical transport through the ground, where the fate of a particular ionic species, a heavy metal, for example, is of interest. The analysis applies to an initially homogeneous soil mass that separates solutions of different concentrations of anions and cations, at different electrical potentials and under different hydraulic heads, as shown schematically in Fig. 9.21. Only one anion and one cation species are assumed to be present, and no adsorption or desorption reactions are occurring. The driving forces are the hydraulic gradient (⫺P), the electrical gradient (⫺E), and the concentrationdependent parts of the chemical potential gradients of the cation (cc) and of the anion (ca). The fluxes are the volume flow rate of the solution per unit area Jv, the electric current I, and the diffusion flow rates of the cation Jdc and the anion Jda per unit area relative to the flow of water. These diffusion flows are related to the absolute flows according to
Figure 9.21 Schematic diagram of system for analysis of
simultaneous flows of water, electricity, and ions through a soil.
Jv ⫽ L11 (⫺P) ⫹ L12 (⫺E) ⫹ L13 (⫺cc) ⫹ L14 (⫺ca)
(9.71)
I ⫽ L21 (⫺P) ⫹ L22 (⫺E) ⫹ L23 (⫺cc) ⫹ L24(⫺ca)
(9.72)
Jcd ⫽ L31 (⫺P) ⫹ L32 (⫺E) ⫹ L33 (⫺cc) ⫹ L34 (⫺ca)
(9.73)
Jad ⫽ L41 (⫺P) ⫹ L42 (⫺E) ⫹ L43 (⫺cc) ⫹ L44 (⫺ca)
(9.74)
These equations contain 4 conductivity coefficients Lii and 12 coupling coefficients Lij. As a result of Onsager reciprocity, however, the number of independent coupling coefficients reduces because L12 ⫽ L21 L13 ⫽ L31 L14 ⫽ L41 L23 ⫽ L32
Ji ⫽ Jid ⫹ ci Jv
(9.70)
L24 ⫽ L42 L34 ⫽ L43
in which ci is the concentration of ion i. The set of phenomenological equations that relates the four flows and driving forces is
Copyright © 2005 John Wiley & Sons
Thus there are 10 independent coefficients needed for a full description of hydraulic, electrical, anionic,
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281
SIMULTANEOUS FLOWS OF WATER, CURRENT, AND CHEMICALS
e " w cc ca u* c u* a D* c D* a n R T
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
bulk electrical conductivity of the soil coefficient of osmotic efficiency unit weight of water concentration of cation concentration of anion effective ionic mobility of the cation effective ionic mobility of the anion effective diffusion coefficient of the cation effective diffusion coefficient of the anion soil porosity universal gas constant (8.314 J K⫺1 mol⫺1) absolute temperature (K)
Co py rig hte dM ate ria l
and cationic flows through a system subjected to hydraulic, electrical, and chemical gradients. If three of the four forces can be set equal to zero during a measurement of the flow under the fourth force, then the ratio of the flow rate to that force will give the value of its corresponding Lij. However, such measurements are not always possible or convenient. Accordingly, two forces and one flow are usually set to zero and the appropriate Lij are evaluated by solution of simultaneous equations. For measurements of hydraulic conductivity, electroosmotic hydraulic conductivity, electrical conductivity, osmotic efficiency, and effective diffusion coefficients done in the usual manner in geotechnical and chemical laboratories, the detailed application of irreversible thermodynamic theory led Yeung (1990) and Yeung and Mitchell (1993) to the following definitions for the Lij. It was assumed in the derivations that the solution is dilute and there are no interactions between cations and anions.8 k L L L11 ⫽ h ⫹ 12 21 wn L22
(9.75)
L33 ⫽ cc
L44 ⫽ ca
L12 ⫽ L21 ⫽
ke n
(9.76)
L13 ⫽ L31 ⫽
⫺"cckh L L ⫹ 12 23 wn L22
(9.77)
L14 ⫽ L41 ⫽
⫺"cakh L L ⫹ 12 24 wn L22
(9.78)
L22 ⫽
e n
(9.79)
L23 ⫽ L32 ⫽ ccu* c
(9.80)
L24 ⫽ L42 ⫽ ⫺cau*a
(9.81)
D* c cc RT
(9.82)
L33 ⫽
L34 ⫽ L43 ⫽ 0 L44 ⫽
D* a ca RT
Subsequently, Manassero and Dominijanni (2003) pointed out that the practical equations for diffusion L33 and L44 do not take the osmotic efficiency " (Section 9.13) into account, so Eqs. (9.82) and (9.84) more properly should be
冋 冋
where kh ⫽ hydraulic conductivity as usually measured (no electrical short circuiting) ke ⫽ coefficient of electroosmotic hydraulic conductivity 8
The Lij coefficients in Eqs. (9.75) to (9.84) were derived in terms of the cross-sectional area of the soil voids. They may be redefined in terms of the total cross-sectional area by multiplying each term on the right-hand side by the porosity, n.
Copyright © 2005 John Wiley & Sons
册 册
(1 ⫺ ")D* c k"2 a ⫹ a RT wn
(9.85) (9.86)
This modification becomes important in clays wherein osmotic efficiency, that is, the ability of the clay to restrict the flow of ions, is high. As the flows of ions relative to the soil are of more interest than relative to the water, Eq. (9.70) and Eqs. (9.73) and (9.74) can be combined to give Jc ⫽ (L31 ⫹ ccL11) w (⫺h) ⫹ (L32 ⫹ ccL12) (⫺E) ⫹ (L33 ⫹ ccL13)
RT (⫺cc) cc
⫹ (L34 ⫹ ccL14)
RT (⫺ca) ca
(9.83)
(9.84)
(1 ⫺ ")D* c k"2 c ⫹ c RT wn
(9.87)
Ja ⫽ (L41 ⫹ caL11) w (⫺h) ⫹ (L42 ⫹ caL12) (⫺E) ⫹ (L43 ⫹ caL13)
RT (⫺cc) cc
⫹ (L44 ⫹ caL14)
RT (⫺ca) ca
(9.88)
where (⫺h) is the hydraulic gradient. In Eqs. (9.87) and (9.88) the gradient of the chemical potential has been replaced by the gradient of the concentration according to
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(⫺ci ) ⫽
RT (⫺ci) ci
(9.89)
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These equations reduce to the known solutions for special cases such as chemical diffusion, advection– dispersion, osmotic pressure according to the van’t Hoff equation [see Eq. (9.98)], osmosis, and ultrafiltration. They predict reasonably well the distribution of single cations and anions as a function of time and position in compacted clay during the simultaneous application of hydraulic, electrical, and chemical gradients (Mitchell and Yeung, 1990). The analysis of multicomponent systems is more complex. The use of averaged chemical properties and the assumption of composite single species of anions and cations may yield reasonable approximate solutions in some cases. Malusis and Shackelford (2002a) present a more general theory for coupled chemical and hydraulic flow, based on an extension of the Yeung and Mitchell (1993) formulation, which accounts for multicomponent pore fluids and ion exchange processes occurring during transport.9 The flow equations can be incorporated into numerical models for the solution of transient flow problems. Conservation of mass of species i requires that
At the pore scale level, the fluid particles carrying dissolved chemicals move at different speeds because of tortuous flow paths around the soil grains and variable velocity distribution in the pores, ranging from zero at the soil particle surfaces to a maximum along the centerline of the pore. This results in hydrodynamic dispersion and a zone of mixing rather than a sharp boundary between two flowing solutions of different concentrations. Mathematically, this is accounted for by adding a dispersion term to the diffusion coefficient in the L33 and L44 terms to account for the deviation of actual motion of fluid particles from the overall or average movement described by Darcy’s law. More details can be found in groundwater and contamination textbooks such as Freeze and Cherry (1979) and Dominico and Schwartz (1997). Numerical models are available for groundwater flow and contaminant transport into which the above flow equations can be introduced (e.g., Anderson and Woessner, 1992; Zheng and Bennett, 2002). The most widely used groundwater flow numerical code is MODFLOW developed by the United States Geological Survey (USGS); various updated versions are available (e.g., Harbaugh et al., 2000). To solve single-species contaminant transport problems in groundwater, MT3DMS (Zheng and Wang, 1999) can be used. The code utilizes the flow solutions from MODFLOW. More complex multispecies reactions can be simulated by RT3D (Clement, 1997). POLLUTE (Rowe and Booker, 1997) provides ‘‘one- and onehalf-dimensional’’ solution to the advection–dispersion equation and is widely used in landfill design. A variety of public domain groundwater flow and contaminant transport codes is available from the web sites of the USGS, the U.S. Environmental Protection Agency (U.S. EPA), and the U.S. Salinity Laboratory.
ci ⫽ ⫺ Ji ⫺ Gi t
(9.90)
in which Gi is a source–sink term describing the addition or removal rate of species i from the solution. As commonly used in groundwater flow analyses of contaminant transport, Gi is given by
冋
Gi ⫽ 1 ⫹
册
Kd Kd ci ici ⫹ n n t
(9.90a)
where i is the decay constant of species i, is the bulk dry density of the soil, Kd is the distribution coefficient, and n is the soil porosity. As defined previously, the distribution constant is the ratio of the amount of chemical adsorbed on the soil to that in solution. The quantity in the brackets on the right-hand side of Eq. (9.90) is the retardation factor Rd defined by Eq. (9.56). Advection rather than diffusion is the dominant chemical transport mechanism in coarse-grained soils.
9.12
ELECTROKINETIC PHENOMENA
Coupling between electrical and hydraulic flows and gradients can generate four related electrokinetic phenomena in materials such as fine-grained soils, where there are charged particles balanced by mobile countercharges. Each involves relative movements of electricity, charged surfaces, and liquid phases, as shown schematically in Fig. 9.22. Electroosmosis
9
Malusis and Shackelford (2002a) defined parameters in terms of the total cross-sectional area for flow rather than the cross-sectional area of voids as used in the development of Eqs. (9.75) through (9.84).
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When an electrical potential is applied across a wet soil mass, cations are attracted to the cathode and anions to the anode (Fig. 9.22a). As ions migrate, they
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ELECTROKINETIC PHENOMENA
283
Figure 9.22 Electrokinetic phenomena: (a) electroosmosis, (b) streaming potential, (c) elec-
trophoresis, and (d) migration or sedimentation potential.
carry their water of hydration and exert a viscous drag on the water around them. Since there are more mobile cations than anions in a soil containing negatively charged clay particles, there is a net water flow toward the cathode. This flow is termed electroosmosis, and its magnitude depends on ke, the coefficient of electroosmotic hydraulic conductivity and the voltage gradient, as considered in more detail later. Streaming Potential
When water flows through a soil under a hydraulic gradient (Fig. 9.22b), double-layer charges are displaced in the direction of flow. This generates an electrical potential difference that is proportional to the hydraulic flow rate, called the streaming potential, between the opposite ends of the soil mass. Streaming potentials up to several tens of millivolts have been measured in clays. Electrophoresis
If a DC field is placed across a colloidal suspension, charged particles are attracted electrostatically to one
Copyright © 2005 John Wiley & Sons
of the electrodes and repelled from the other. Negatively charged clay particles move toward the anode as shown in Fig. 9.22c. This is called electrophoresis. Electrophoresis involves discrete particle transport through water; electroosmosis involves water transport through a continuous soil particle network. Migration or Sedimentation Potential
The movement of charged particles such as clay relative to a solution, as during gravitational settling, for example, generates a potential difference, as shown in Fig. 9.22d. This is caused by the viscous drag of the water that retards the movement of the diffuse layer cations relative to the particles. Of the four electrokinetic phenomena, electroosmosis has been given the most attention in geotechnical engineering because of its practical value for transporting water in fine-grained soils. It has been used for dewatering, soft ground consolidation, grout injection, and the containment and extraction of chemicals in the ground. These applications are considered in a later section.
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CONDUCTION PHENOMENA
9.13 TRANSPORT COEFFICIENTS AND THE IMPORTANCE OF COUPLED FLOWS
ui ⫽
Di兩zi兩F RT
(9.91)
in which zi is the ionic valence and F is Faraday’s constant. Similarly to the diffusion coefficients, the ionic mobilities are considerably less in a soil than in a free solution, especially in a fine-grained soil. The importance of coupled flows to fluid, electrical current, and chemical transport through soil under different conditions can be examined by study of the contributions of the different terms in Eqs. (9.71), (9.72), (9.87), and (9.88). For this purpose, the equations have been rewritten in one-dimensional form and in terms of the hydraulic, electrical, and chemical concentration gradients: ih ⫽ ⫺dh/dx, ie ⫽ ⫺dV/dx, and ic ⫽ ⫺dc/ dx, respectively. In addition, the chemical flows have been represented by a single equation. This assumes that all dissolved species are moving together. Terms involving the ionic mobility u do not exist in such a formulation because the cations and anions move together, with the effects of electrical fields assumed to accelerate the slower moving ions and to retard the faster moving ions. Thus there is no net transfer of electric charge due to ionic movement. The Lij coefficients have been replaced by the physical and chemical quantities that determine them, as given by Eqs. (9.74) through (9.85). The resulting equations are the following. For fluid flow: Jv ⫽
冋
冋 册 冋册
e ke w ih ⫹ i n n e
(9.93)
For chemical flow relative to the soil: Jc ⫽
冋
册 冋 册 冋 册
(1 ⫺ ")ckh ck2e w cke ⫹ ih ⫹ i n ne n e ⫹
D* ⫺
"ckh RT ic n w
(9.94)
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To assess conditions where coupled chemical, electrical, and hydraulic flows will be significant relative to direct flows, it is necessary to know the values of the Lij relative to the Lii. Estimates can be made by considering the probable values of the soil state parameters and the several flow and transport coefficients given in Eqs. (9.75) to (9.84). Typical ranges are given in Table 9.5. In Table 9.5 the diffusion coefficients and ionic mobilities for cations and anions are considered together since they lie within similar ranges for most species. Values of ionic mobility for specific ions in dilute solution are given in standard chemical references, for example, Dean (1973), and values of diffusion coefficients are given in Tables 9.3 and 9.4. Ionic mobility is related to the diffusion coefficient according to
I⫽
册 冋册
冋
册
kh k2 k "kh ⫹ e w ih ⫹ e ie ⫹ RT ⫺ i n en n wn c
Coupling Influences on Hydraulic Flow
In the absence of applied electrical and chemical gradients, flow under a hydraulic gradient is given by the first bracketed term on the right-hand side of Eq. (9.92). It contains the quantity k2e w /ne, which compensates for the electroosmotic counterflow generated by the streaming potential, which causes the measured value of kh to be slightly less than the true value of L11. As it is not usual practice to short-circuit between the ends of samples during hydraulic conductivity testing, the second bracketed term on the right-hand side of Eq. (9.92) is not zero. This term represents an electroosmotic counterflow that results from the streaming potential and acts in the direction opposite to the hydraulically driven flow. Analysis based on the values of properties in Table 9.5, as well as the results of measurements, for example, Michaels and Lin (1954) and Olsen (1962) show that this counterflow is negligible in most cases, but it may become significant relative to the true hydraulic conductivity for soils of very low hydraulic conductivity, for example, kh ⬍ 1 ⫻ 10⫺10 m/s. For example, for a value of ke of 5 ⫻ 10⫺9 m2 /s-V, an electrical conductivity of 0.01 mho/m, and a porosity of 35 percent, the counterflow term is 0.7 ⫻ 10⫺10 m/s. In the presence of an applied DC field the second bracketed term on the right-hand side of Eq. (9.92) can be very large relative to hydraulic flow in soils finer than silts, as ke, which typically ranges within only narrow limits, is large relative to kh; that is, kh is less than 1 ⫻ 10⫺8 m/s in these soils. The relative effectiveness of hydraulic and electrical driving forces for water movement can be assessed by comparing gradients needed to give equal flow rates. They will be equal if keie ⫽ khih
(9.95)
(9.92) The hydraulic gradient required to balance the electroosmotic flow then becomes
For electrical current flow:
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TRANSPORT COEFFICIENTS AND THE IMPORTANCE OF COUPLED FLOWS
ih ⫽
ke i kh e
(9.96)
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As the hydraulic conductivity of soils in which electroosmosis is likely to be used is usually of the order of 1 ⫻ 10⫺9 m/s or less, whereas ke is in the range of 1 ⫻ 10⫺9 to 1 ⫻ 10⫺8 m2 /s V, it follows that even small electrical gradients can balance flows caused by large hydraulic gradients. Because of this, and because ke is insensitive to particle size while kh decreases rapidly with decreasing particle size, electroosmosis is effective in fine-grained soils, as discussed further in Section 9.15. Chemically driven hydraulic flow is given by the last term on the right-hand side of Eq. (9.92). It depends primarily on the osmotic efficiency ". Osmotic efficiency has an important influence on the movement of chemicals through a soil, the development of osmotic pressure, and the effectiveness of clay barriers for chemical waste containment. Osmotic Efficiency The osmotic efficiency of clay, a slurry wall, a geosynthetic clay liner (GCL), or other seepage and containment barrier is a measure of the material’s effectiveness in causing hydraulic flow under an osmotic pressure gradient and of its ability to act as a semipermeable membrane in preventing the passage of ions, while allowing the passage of water. The osmotic pressure concept can be better appreciated by rewriting the last term in Eq. (9.92): "
kh k RT c 1 RTic ⫽ " h wn n w x
(9.97)
This form is analogous to Darcy’s law, with the quantity RT c/ w being the head difference. The osmotic efficiency is a measure of the extent to which this theoretical pressure difference actually develops. Theoretical values of osmotic pressure, calculated using the van’t Hoff equation, as a function of concentration difference for different values of osmotic efficiency are shown in Fig. 9.23. The van’t Hoff equation for osmotic pressure is ⫽ kT
冘 (n
iA
⫺ niB) ⫽ RT(ciA ⫺ ciB)
285
(9.98)
where k is the Boltzmann constant (gas constant per molecule), R is the gas constant per molecule, T is the absolute temperature, ni is concentration in particles per unit volume, and ci is the molar concentration. The van’t Hoff equation applies for ideal and relatively dilute solution concentrations (Malusis and Shackelford, 2002c). According to Fritz (1986) the error is low (⬍5%) for 1⬊1 electrolytes (e.g., NaCl, KCl) and concentrations 1.0 M.
Copyright © 2005 John Wiley & Sons
Figure 9.23 Theoretical values of osmotic pressure as a function of concentration difference across a clay layer for different values of osmotic efficiency coefficient, ". (T ⫽ 20C).
Values of osmotic efficiency coefficient, ", or membrane efficiency (" expressed as a percentage), have been measured for clays and geosynthetic clay liners; for example, Kemper and Rollins (1966), Letey et al. (1969), Olsen (1969), Kemper and Quirk (1972), Bresler (1973), Elrick et al. (1976), Barbour and Fredlund (1989), and Malusis and Shackelford (2002b, 2002c). Values of membrane efficiency from 0 to 100 percent have been determined, depending on the clay type, porosity, and type and concentration of salts in solution. The results of many determinations were summarized by Bresler (1973) as shown in Fig. 9.24. The efficiency is shown as a function of a normalizing parameter, the half distance between particles b times the square root of the solution concentration 兹c. To put these relations into more familiar terms for use in geotechnical studies, the half spacings were converted to water contents on the assumption of uniform water layer thicknesses on all particles, using specific surface areas corresponding to different clay types and noting that volumetric water content equals surface area times layer thickness. The relationship between specific surface area and liquid limit (LL) obtained by Farrar and Coleman (1967) for 19 British clays LL ⫽ 19 ⫹ 0.56As (20%)
(9.99)
in which the specific surface area As is in square meters per gram, was then used to obtain the relationships shown in Fig. 9.25. The computed efficiencies shown in Fig. 9.25 should be considered upper bounds because the assumption of uniform water distribution over the full surface area underestimates the effective particle spacing in most cases. In most clays, espe-
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CONDUCTION PHENOMENA
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concentrations on the inside of a lined repository should be greater than on the outside, osmotically driven water flow should be directed from the outside toward the inside. The greater the osmotic efficiency the greater the driving force for this flow. Furthermore, if the efficiency is high, then outward diffusion of contained chemicals is restricted (Malusis and Shackelford, 2002b). In diffusion-dominated containment barriers, the effect of solute restriction on reducing solute diffusion is likely substantially more significant than the effect of osmotic flow (Shackelford et al., 2001). Coupling Influences on Electrical Flow
Figure 9.24 Osmotic efficiency coefficient as a function of b兹c where c is concentration of monovalent anion in nor-
Substitution of values for the parameters in Eq. (9.93) indicates, as would be expected, that electrical current flow is dominated completely by the electrical gradient ie. In the presence of an applied voltage difference, the other terms are of little importance, even if the movements of anions and cations are considered separately and the contributions due to ionic mobility are taken into account. On the other hand, when a soil layer behaves as an open electrical circuit, small electrical potentials, measured in millivolts, may exist if there are hydraulic and/or chemical flows. This may be seen by setting I ⫽ 0 in Eq. (9.93) and solving for ie, which must have value if ih has value. These small potentials and flows are important in such processes as corrosion and electroosmotic counterflow.
mality and 2b is the effective spacing between particle surfaces (from Bresler, 1973).
Coupling Influences on Chemical Flow
cially those with divalent adsorbed cations, individual clay plates associate in clusters giving an effective specific surface that is less than that determined by most methods of measurement. This means that the curves in Fig. 9.25 should in reality be displaced to the left. High osmotic efficiencies are developed at low water contents, that is, in very dense, low-porosity clays, and in dilute electrolyte systems. Malusis and Shackelford (2002a, 2002b, 2002c) found that the osmotic efficiency decreases with increasing solute concentration and attribute this to compression of the diffuse double layers adjacent to the clay particles. Water flow by osmosis can be significant relative to hydraulically driven water flow in heavily overconsolidated clay and clay shale, where the void ratio is low and the hydraulic conductivity is also very low. Such flow may be important in geological processes (Olsen 1969, 1972). Densely compacted clay barriers for waste containment, usually composed of bentonite, possess osmotic membrane properties. As the chemical
Equation (9.94) provides a description of chemical transport relative to the soil. It contains two terms that influence chemical flow under a hydraulic gradient; one for chemical transport under an electrical gradient, and one for transport of chemical under a chemical gradient. The first term in the first bracket of the righthand side of Eq. (9.94) describes advective transport. As would be expected, the smaller the osmotic efficiency, the more chemical flow through the soil is possible. The second term in the same bracket simply reflects the advective flow reduction that would result from electroosmotic counterflow caused by development of a streaming potential. As noted earlier, this flow will be small, and its contribution to the total flow will be small, except in clays of very low hydraulic and electrical conductivities. Advective transport is the dominant means for chemical flow for soils having a hydraulic conductivity greater than about 1 ⫻ 10⫺9 m/s. The importance of an electrical driving force for chemical flow depends on the electrical potential gradient. For a unit gradient, that is, 1 V/m, chemical flow
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TRANSPORT COEFFICIENTS AND THE IMPORTANCE OF COUPLED FLOWS
287
Figure 9.25 Osmotic efficiency of clays as a function of water content.
quantities are comparable to those by advective flow under a unit hydraulic gradient in a clay having a hydraulic conductivity of about 1 ⫻ 10⫺9 m/s. Electrically driven chemical flow is relatively less important in higher permeability soils and more important in soils with lower kh. In cases where the electrically driven chemical transport is of interest, as in electrokinetic waste containment barrier applications, anion, cation, and nonionic chemical flows must be considered separately using expanded relationships such as given by Eqs. (9.87) and (9.88). The last bracketed quantity of Eq. (9.94) represents diffusive flow under chemical gradients. The quantity D*ic gives the normal diffusive flow rate. The second term represents a restriction on this flow that depends on the clay’s osmotic efficiency, "; that is, if the clay acts as an effective semipermeable membrane, diffusive flow of chemicals is restricted. However, even un-
Copyright © 2005 John Wiley & Sons
der conditions where the value of " is low such that the second term in the bracket is negligible, chemical transport by diffusion is significant relative to advective chemical transport in soils with hydraulic conductivity values less than about 1 ⫻ 10⫺9 to 1 ⫻ 10⫺10 m/s for chemicals with diffusion coefficients in the range given by Table 9.7, that is, 2 ⫻ 10⫺10 to 2 ⫻ 10⫺9 m2 /s. This is illustrated by Fig. 9.26 from Shackelford (1988), which shows the relative importance of advective and diffusive chemical flows on the transit time through a 0.91-m-thick compacted clay liner having a porosity of 0.5 acted on by a hydraulic gradient of 1.33. A diffusion coefficient of 6 ⫻ 10⫺10 m2 /s was assumed. The transit time is defined as the time required for the solute concentration on the discharge side to reach 50 percent of that on the upstream side. For hydraulic conductivity values less than about 2 ⫻
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CONDUCTION PHENOMENA
term stability of clay liners are discussed by Mitchell and Jaber (1990). Rigid wall, flexible wall, and consolidometer permeameters are used for compatibility testing in the laboratory. These three types of test apparatus are shown schematically in Fig. 9.27. Tests done in a rigid wall system overestimate hydraulic conductivity whenever chemical–clay interactions cause shrinkage and cracking; however, a rigid wall system is well suited for qualitative determination of whether or not there may be adverse interactions. In the flexible wall system the lateral confining pressure prevents cracks from opening; thus there is risk of underestimating the hydraulic conductivity of some soils. The consolidometer permeameter system allows for testing clays under a range of overburden stress states that are representative of those in the field and for quantitative assessment of the effects of chemical interactions on volume stability and hydraulic conductivity. More details of these permeameters are given by Daniel (1994). The effects of chemicals on the hydraulic conductivity of high water content clays such as used in slurry walls are likely to be much greater than on lower water content, high-density clays as used in compacted clay liners. This is because of the greater particle mobility and easier opportunity for fabric changes in a higher water content system. A high compactive effort or an effective confining stress greater than about 70 kPa can make properly compacted clay invulnerable to attack by concentrated organic chemicals (Broderick and Daniel, 1990). However, it is not always possible to ensure high-density compaction or to maintain high confining pressures, or eliminate all construction defects, so it is useful to know the general effects of different types of chemicals on hydraulic conductivity. The influences of inorganic chemicals on hydraulic conductivity are consistent with (1) their effects on the double-layer and interparticle forces in relation to flocculation, dispersion, shrinkage, and swelling, (2) their effects on surface and edge charges on particles and the influences of these charges on flocculation and deflocculation, and (3) their effects on pH. Acids can dissolve carbonates, iron oxides, and the alumina octahedral layers of clay minerals. Bases can dissolve silica tetrahedral layers, and to a lesser extent, alumina octahedral layers of clay minerals. Removal of dissolved material can cause increases in hydraulic conductivity, whereas precipitation can clog pores and reduce hydraulic conductivity. The most important factors controlling the effects of organic chemicals on hydraulic conductivity are (1) water solubility, (2) dielectric constant, (3) polarity, and (4) whether or not the soil is exposed to the pure organic or a dilute solution. Exposure of clay barriers
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288
Figure 9.26 Transit times for chemical flow through a 0.91-
m-thick compacted clay liner having a porosity of 50 percent and acted on by a hydraulic gradient of 1.33 (from Shackelford, 1988).
10⫺9 m/s the transit time in the absence of diffusion would be very long. For diffusion alone the transit time would be about 47 years. Most compacted clay barriers and geosynthetic clay liners are likely to have hydraulic conductivity values in the range of 1 ⫻ 10⫺11 to 1 ⫻ 10⫺9 m/s, with the latter value being the upper limit allowed by the U.S. EPA for most waste containment applications. In this range, diffusion reduces the transit time significantly in comparison to what it would be due to advection alone. This is shown by the curve labeled advection– dispersion in Fig. 9.26. The calculations were done using the well-known advection–dispersion equation (Ogata and Banks, 1961) in which the dispersion term includes both mechanical mixing and diffusion. Mechanical mixing is negligible in low-permeability materials such as compacted clay. 9.14 COMPATIBILITY—EFFECTS OF CHEMICAL FLOWS ON PROPERTIES
Chemical Compatibility and Hydraulic Conductivity
The compatibility between waste chemicals, especially liquid organics, and compacted clay liners and slurry wall barriers constructed to contain them must be considered in the design of waste containment barriers. Numerous studies have been done to evaluate chemical effects on clay hydraulic conductivity because of fears that prolonged exposure may compromise the integrity of the liners and barriers and because tests have shown that under some conditions clay can shrink and crack when permeated by certain classes of chemicals. Summaries of the results of chemical compatibility studies are given by Mitchell and Madsen (1987) and Quigley and Fernandez (1989), and factors controlling the long-
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289
Figure 9.27 Three types of permeameter for compatibility testing: (a) rigid wall, (b) flexible wall, and (c) consolidometer permeameter (from Day, 1984).
to water-insoluble pure or concentrated organics is likely only in the case of spills, leaking tanks, and with dense non-aqueous-phase liquids (DNAPLs) or ‘‘sinkers’’ that accumulate above low spots in liners. Some general conclusions about the influences of organics on the hydraulic conductivity are: 1. Solutions of organic compounds having a low solubility in water, such as hydrocarbons, have no large effect on the hydraulic conductivity. This is in contrast to dilute solutions of inorganic compounds that may have significant effects as a result of their influence on flocculation and dispersion of the clay particles. 2. Water-soluble organics, such as simple alcohols and ketones, have no effect on hydraulic conductivity at concentrations less than about 75 to 80 percent.
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3. Many water-insoluble organic liquids (i.e., nonaquoues-phase liquids, NAPLs) can cause shrinkage and cracking of clays, with concurrent increases in hydraulic conductivity. 4. Hydraulic conductivity increases caused by permeation by organics are partly reversible when water is reintroduced as the permeant. 5. Concentrated hydrophobic compounds (like many NAPLs) permeate soils through cracks and macropores. Water remains within mini- and micropores. 6. Hydrophilic compounds permeate the soil more uniformly than NAPLs, as the polar molecules can replace the water in hydration layers of the cations and are more readily adsorbed on particle surfaces. 7. Organic acids can dissolve carbonates and iron oxides. Buffering of the acid can lead to precip-
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itation and pore clogging downstream. However, after long time periods these precipitates may be redissolved and removed, thus leading to an increase in hydraulic conductivity. 8. Pure bases can cause a large increase in the hydraulic conductivity, whereas concentrations at or below the solubility limit in water have no effect. 9. Organic acids do not cause large-scale dissolution of clay particles.
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The combined effects of confining pressure and concentration, as well as permeant density and viscosity, are illustrated by Fig. 9.28 (Fernandez and Quigley, 1988). The data are for water-compacted, brown Sarnia clay permeated by solutions of dioxane in domestic landfill leachate. Increased hydrocarbon concentration caused a decrease in hydraulic conductivity up to concentrations of about 70 percent, after which the hydraulic conductivity increased by about three orders of magnitude for pure dioxane (Fig. 9.28a), for samples that were unconfined by a vertical stress (v ⫽ 0). On the other hand, the data points for samples maintained under a vertical confining stress of 160 kPa indicated no effect of the dioxane on hydraulic conductivity rel-
ative to that measured with water. The decreases in hydraulic conductivity for dioxane concentrations up to 70 percent can be accounted for in terms of fluid density and viscosity, as may be seen in Fig. 9.28b where the intrinsic values of permeability are shown. As noted earlier in this chapter, the intrinsic permeability is defined by K ⫽ k / . Although many chemicals do not have significant effect on the hydraulic conductivity of clay barriers, this does not mean that they will not be transported through clay. Unless adsorbed by the clay or by organic matter, the chemicals will be transported by advection and diffusion. Furthermore, the actual transit time through a barrier by advection, that is, the time for chemicals moving with the seepage water, may be far less than estimated using the conventional seepage velocity. The seepage velocity is usually defined as the Darcy velocity khih, divided by the total porosity n. In systems with unequal pore sizes the flow is almost totally through mini- and macropores, which comprise the effective porosity ne, which may be much less than the total porosity. Thus effective compaction of clay barriers must break down clods and aggregates to decrease the effective pore size and increase the propor-
Figure 9.28 (a) Hydraulic conductivity and (b) intrinsic permeability of compacted Sarnia clay permeated with leachate–dioxane mixtures. Initial tests run using water (●) followed by leachate–chemical solution (䉱). (from Fernandez and Quigley, 1988). Reproduced with per-
mission from the National Research Council of Canada.
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ELECTROOSMOSIS
tion of the porosity that is effective porosity, thereby increasing the transit time. 9.15
ELECTROOSMOSIS
Helmholtz and Smoluchowski Theory
Table 9.7
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
v
E ⫽
L
10
A derivation using a Poisson–Boltzmann distribution of counterions adjacent to the wall gives the same result.
Coefficients of Electroosmotic Permeability
Material
London clay Boston blue clay Kaolin Clayey silt Rock flour Na-Montmorillonite Na-Montmorillonite Mica powder Fine sand Quartz powder ˚ s quick clay A Bootlegger Cove clay Silty clay, West Branch Dam Clayey silt, Little Pic River, Ontario
(9.100)
or
This theory, based on a model introduced by Helmholtz (1879) and refined by Smoluchowski (1914), is one of the earliest and most widely used. A liquidfilled capillary is treated as an electrical condenser with
No.
charges of one sign on or near the surface of the wall and countercharges concentrated in a layer in the liquid a small distance from the wall, as shown in Fig. 9.29.10 The mobile shell of counterions is assumed to drag water through the capillary by plug flow. There is a high-velocity gradient between the two plates of the condenser as shown. The rate of water flow is controlled by the balance between the electrical force causing water movement and friction between the liquid and the wall. If v is the flow velocity and is the distance between the wall and the center of the plane of mobile charge, then the velocity gradient between the wall and the center of positive charge is v / ; thus, the drag force per unit area is dv /dx ⫽ v / , where is the viscosity. The force per unit area from the electrical field is E/
L, where is the surface charge density and E/ L is the electrical potential gradient. At equilibrium
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The coefficient of electroosmotic hydraulic conductivity ke defines the hydraulic flow velocity under a unit electrical gradient. Measurement of ke is made by determination of the flow rate of water through a soil sample of known length and cross section under a known electrical gradient. Alternatively, a null indicating system may be used or it may be deduced from a streaming potential measurement. From experience it is known that ke is generally in the range of 1 ⫻ 10⫺9 to 1 ⫻ 10⫺8 m2 /s V (m/s per V/m) and that it is of the same order of magnitude for most soil types, as may be seen by the values for different soils and a freshwater permeant given in Table 9.7. Several theories have been proposed to explain electroosmosis and to provide a basis for quantitative prediction of flow rates.
291
Water Content (%)
ke in 10⫺5 (cm2 /s-V)
Approximate kh (cm/s)
52.3 50.8 67.7 31.7 27.2 170 2000 49.7 26.0 23.5 31.0 30.0 32.0 26.0
5.8 5.1 5.7 5.0 4.5 2.0 12.0 6.9 4.1 4.3 20.0–2.5 2.4–5.0 3.0–6.0 1.5
10⫺8 10⫺8 10⫺7 10⫺6 10⫺7 10⫺9 10⫺8 10⫺5 10⫺4 10⫺4 2.0 ⫻ 10⫺8 2.0 ⫻ 10⫺8 1.2 ⫻ 10⫺8 –6.5 ⫻ 10⫺8 2 ⫻ 10⫺5
ke and water content data for Nos. 1 to 10 from Casagrande (1952). kh estimated by authors; no. 11 from Bjerrum et al. (1967); no. 12 from Long and George (1967); no. 13 from Fetzer (1967); no. 14 from Casagrande et al. (1961).
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CONDUCTION PHENOMENA
Co py rig hte dM ate ria l
292
Figure 9.29 Helmholtz–Smoluchowski model for electrokinetic phenomena.
⫽ v
L
E
(9.101)
From electrostatics, the potential across a condenser is given by ⫽
D
(9.102)
where D is the relative permittivity, or dielectric constant of the pore fluid. Substitution for in Eq. (9.102) gives v⫽
冉 冊
D E L
(9.103)
The potential is termed the zeta potential. It is not the same as the surface potential of the double-layer 0 discussed in Chapter 6, although conditions that give high values of 0 also give high values of zeta potential. A common interpretation is that the actual slip plane in electrokinetic processes is located some small, but unknown, distance from the surface of particles; thus should be less than 0. Values of in the range of 0 to ⫺50 mV are typical for clays, with the lowest values associated with high pore water salt concentrations. For a single capillary of area a the flow rate is qa ⫽ va ⫽
D E a L
(9.104a)
and for a bundle of N capillaries within total crosssectional area A normal to the flow direction
Copyright © 2005 John Wiley & Sons
qA ⫽ Nqa ⫽
D E Na L
(9.104b)
If the porosity is n, then the cross-sectional area of voids is nA, which must equal Na. Thus, qA ⫽
D E n A
L
(9.105)
By analogy with Darcy’s law we can write Eq. (9.105) as qA ⫽ keie A
(9.106)
in which ie is the electrical potential gradient E/ L and ke the coefficient of electroosmotic hydraulic conductivity is ke ⫽
D n
(9.107)
According to the Helmholtz–Smoluchowski theory and Eq. (9.107), ke should be relatively independent of pore size, and this is borne out by the values listed in Table 9.7. This is in contrast to the hydraulic conductivity kh, which varies as the square of some effective pore size. Because of this independence of pore size, electroosmosis can be more effective in moving water through fine-grained soils than flow driven by a hydraulic gradient. This is illustrated by the following simple example. Consider a fine sand and a clay of hydraulic conductivity kh of 1 ⫻ 10⫺5 m/s and 1 ⫻ 10⫺10 m/s, respectively. Both have ke values of 5 ⫻ 10⫺9 m2 /s V. For equal hydraulic flow rates khih ⫽ keie, so
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ELECTROOSMOSIS
ih ⫽
ke i kh e
E
L
Schmid Theory
The Helmholtz–Smoluchowski theory is essentially a large-pore theory because it assumes a negligible extension of the counterion layer into the pore. Also, it does not account for an excess of ions over those needed to balance the surface charge. A model that overcomes the first of these problems was proposed by Schmid (1950, 1951). It can be considered a smallpore theory. The counterions are assumed to be distributed uniformly throughout the fluid phase in the soil. The electrical force acts uniformly over the entire pore cross section and gives the same velocity profile as shown by Fig. 9.29. The hydraulic flow rate through a single capillary of radius r is given by Poiseuille’s law: q⫽
r i 8 w h 4
(9.109)
The hydraulic seepage force per unit length causing flow is FH ⫽ r 2 wih
(9.112)
where A0 is the concentration of wall charges in ionic equivalents per unit volume of pore fluid, and F0 is the Faraday constant. Replacement of FH by FE in Eq. (9.111) gives qa ⫽
r 4
E F A A F ⫽ 0 0 r 2iea 8 0 0 L 8
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If an electrical potential gradient of 20 V/m is used, substitution in Eq. (9.108) shows that ih is 0.01 for the fine sand and 1000 for the clay. This means that a hydraulic gradient of only 0.01 can move water as effectively as an electrical gradient of 20 V/m in fine sand. However, for the clay, a hydraulic gradient of 1000 would be needed to offset the electroosmotic flow. However, it does not follow that electroosmosis will always be an efficient means to move water in clays because the above analysis does not take into account the power requirement to develop the potential gradient of 20 V/m or energy losses in the system. These points are considered further later.
so
FE ⫽ A0 F0r 2
(9.108)
293
(9.113)
so for a total cross section of N capillaries and area A qA ⫽
A0 F0r 2 nie A 8
(9.114)
This equation shows that ke should vary as r 2, whereas the Helmholtz–Smoluchowski theory leads to ke independent of pore size, as previously noted. Of the two theories, the Helmholtz gives the better results for soils, perhaps because most clays have a cluster or aggregate structure with electroosmotic flow controlled more by the larger pores than by the intracluster pores. Spiegler Friction Model
A completely different concept for electrokinetic processes takes into account the interactions of the mobile components (water and ions) on each other and of the frictional interactions of these components with pore walls (Spiegler, 1958). This theory provides insight into conditions leading to high electroosmotic efficiency. The assumptions include: 1. Exclusion of coions,11 that is, the medium behaves as a perfect perm-selective membrane, admitting ions of only one sign 2. Complete dissociation of pore fluid ions
The following equation for electroosmotic transport of water across a fine-grained porous material containing adsorbed and free ions can be derived:
(9.110)
⫽ (W ⫺ H) ⫽
C3 C1 ⫹ C3(X34 /X13)
(9.115)
(9.111)
in which is the true electroosmotic water flow (moles/faraday), W is the measured water transport
The electrical force per unit length FE is equal to the charge times the potential, that is,
11 Ions of the opposite sign to the charged surface are termed counterions. Ions of the same sign are termed coions.
q⫽
r2 F 8 H
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CONDUCTION PHENOMENA
opposite sign. The greater the difference between the concentrations of cations and anions, the greater the net drag on the water in the direction toward the cathode. The efficiency and economics of the process depend on the volume of water transported per unit electrical charge passed. If the volume is high, then more water is transported for a given expenditure of electrical energy than if it is low. This volume may vary over several orders of magnitude depending on such factors as soil type, water content, and electrolyte concentration. In a low exchange capacity soil at high water content in a low electrolyte concentration solution, there is much more water per cation than in a high exchange capacity, low water content soil having the same pore water electrolyte concentration. This, combined with cation-to-anion ratio considerations, leads to the predicted water transport–water content–soil type–electrolyte concentration relationships shown schematically in Fig. 9.30, where increasing electrolyte concentration in the pore water results in a much
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(moles/faraday), H is the water transport by ion hydration (moles/faraday), C3 is the concentration of free water in the material (mol/m3), C1 is the concentration of mobile counterions m2, X34 is the friction coefficient between water and the solid wall, and X13 is the friction coefficient between cation and water. Concentrations C1 and C3 are hypothetical and probably less than values measured by chemical analysis because some ions may be immobile. Evaluation of X13 and X34 requires independent measurements of diffusion coefficients, conductance, transference numbers, and water transport. Thus Eq. (9.115) is limited as a predictive equation. Its real value is in providing a relatively simple physical representation of a complex process. From Eq. (9.115), ⫽ (W ⫺ H) ⫽
1 (C1 /C3 ⫹ X34 /X13)
(9.116)
At high water contents and for large pores, X34 /X13 → 0 because X34 becomes negligible. Then X34→0
⫽ C3 /C1
(9.117)
This relationship indicates that a high water-to-cation ratio implies a high rate of electroosmotic flow. At low water contents and for small pores, X34 will not be zero, thus reducing the flow. An increase in C1 reduces the flow of water per faraday of current passed because there is less water per ion. An increase in X13 increases the flow because there is greater frictional drag on the water by the ions. Ion Hydration
Water of hydration is carried along with ions in a direct current electric field. The ion hydration transport H is given by H ⫽ t⫹N⫹ ⫺ t N
(9.118)
where t⫹ and t are the transport numbers, that is, numbers that represent the fraction of current carried by a particular ionic species. The numbers N⫹ and N are the number of moles of hydration water per mole of cation and anion, respectively.
9.16
ELECTROOSMOSIS EFFICIENCY
Electroosmotic water flow occurs if the frictional drag between the ions of one sign and their surrounding water molecules exceeds that caused by ions of the
Copyright © 2005 John Wiley & Sons
Figure 9.30 Schematic prediction of water transport by elec-
troosmosis in various clays according to the Donnan concept (from Gray, 1966).
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ELECTROOSMOSIS EFFICIENCY
R⫽ where
y⫽
2C0 A0
C⫹ 1 ⫹ (1 ⫹ y2)1 / 2 ⫽ C⫺ ⫺1 ⫹ (1 ⫹ y2)1 / 2
(9.119)
A0 ⫽
(CEC)w w
(9.121)
where w is the density of water and w is the water content. The higher R, the greater is the electroosmotic water transport, all other things equal. From Eqs. (9.119) to (9.121) it may be deduced that exclusion of anions is favored by a high exchange capacity (active clay), a low water content, and low salinity in the external solution. However, the concentration of anions in the double layer builds up more
Figure 9.31 Electroosmotic water transport versus concentration of external electrolyte solution for homoionic kaolinite and illite at various water content (from Gray, 1966).
Copyright © 2005 John Wiley & Sons
(9.120)
The concentration C0 is in the external solution, is the mean molar activity coefficient in the external solution, is the mean activity coefficient in the double layer, and A0 is the surface charge density per unit pore volume. The parameter A0 is related to the cation exchange capacity (CEC) by
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greater decrease in efficiency for inactive clay than more plastic, active clay. Tests on sodium kaolinite (inactive clay) and sodium illite (more active clay) gave the results shown in Fig. 9.31, which agree well with the predictions in Fig. 9.30. The slopes and locations of the curves can be explained more quantitatively in the following way. Alternatively to the double-layer theory given in Chapter 6, the Donnan (1924) theory can be used to describe equilibrium ionic distributions in fine-grained materials. The basis for the Donnan theory is that at equilibrium the potentials of the internal and external solutions are equal and that electroneutrality is required in both phases. It may be shown (Gray, 1966; Gray and Mitchell, 1967) that the ratio R of cations to anions in the internal phase for the case of a symmetrical electrolyte (z⫹ ⫽ z⫺) is given by
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296
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CONDUCTION PHENOMENA
E L ⫽ ⫺ EH
P LEE
(9.124)
In electroosmosis P ⫽ 0, so Eq. (9.122) is qh ⫽ LHE E
(9.125)
and Eq. (9.122) becomes I ⫽ LEE E
(9.126)
qh LHE ⫽ I LEE
(9.127)
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rapidly as the salinity of the external solution increases in inactive clays than in active clays. As a result the efficiency, as measured by volume of water per unit charge passed, decreases much more rapidly with increasing electrolyte concentration than in the more active clay. The results of electroosmosis measurements on a number of different materials are summarized in Fig. 9.32, which shows water flow rate as a function of water content. This figure may be used as a guide for prediction of electroosmotic flow rates. The flow rates shown are for open systems, that is, solution was admitted at the anode at the same time it was extracted from the cathode. Electrochemical effects (Section 9.18) and water content changes were minimized in these tests. Thus, the values can be interpreted as upper bounds on the flow rates to be expected in practice. Values of water content, electrolyte concentration in the pore water, and type of clay are required for electroosmosis efficiency estimation. Water content is readily measured, the electrolyte concentration is easily determined using a conductivity cell, and the clay type can be determined from plasticity and grain size information if mineralogical data are not available. Electroosmotic flow rates of 0.03 to 0.06 gal/h/amp are predicted using Fig. 9.32 for soils 11, 13, and 14 in Table 9.7. Electrical treatment for consolidation and ground strengthening was effective in these soils. For soil 12, however, a flow rate of 0.008 to 0.012 gal/h/ amp was predicted, and electroosmosis was not effective. Saxen’s Law Prediction of Electroosmosis from Streaming Potential
Streaming potential can be measured directly during a measurement of hydraulic conductivity by using a high-impedance voltmeter and reversible electrodes. Equivalence between streaming potential and electroosmosis may be derived. Expansion of Eq. (9.57) for coupled hydraulic and current flows gives qh ⫽ LHH P ⫹ LHE E
(9.122)
I ⫽ LEH P ⫹ LEE E
(9.123)
in which qh is the hydraulic flow rate, I is the electric current, LHH and LEE are the direct flow coefficients, LHE and LEH are the coupling coefficients for hydraulic flow due to an electrical gradient and electrical flow due to a hydraulic gradient, P is the pressure drop, and E is the electrical potential drop. In a usual hydraulic conductivity measurement, there is no electrical current flow, so I ⫽ 0, and E is the streaming potential. Equation (9.123) then becomes
Copyright © 2005 John Wiley & Sons
so
By Onsager’s reciprocity theorem LEH ⫽ LHE so
冉冊 qh I
冉 冊
⫽⫺
P⫽0
E
P
(9.128)
I⫽0
This equivalence between streaming potential and electroosmosis was first shown experimentally by Saxen (1892) and is known as Saxen’s law. It has been verified for clay–water–electrolyte systems. Care must be taken to ensure consistency in units. For example, the electroosmotic flow rate in gallons per hour per ampere is equal to 0.0094 times the streaming potential in millivolts per atmosphere. Energy Requirements
The preceding analysis leads to a prediction of the amount of water moved per unit charge passed, for example, gallons or cubic meters of water per hour per ampere or moles per faraday. If this quantity is denoted by ki, then qh ⫽ ki I
(9.129)
Unlike ke, ki varies over a wide range, as may be seen in Fig. 9.32. The power consumption P is P ⫽ E I ⫽
Eqh
ki
(in W)
(9.130)
for E in volts and I in amperes. The power consumption per unit volume of flow is P
E ⫽ ⫻ 10⫺3 qh ki
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(in kWh)
(9.131)
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Figure 9.32 Electroosmotic water transport as a function of water content, soil type, and electrolyte concentration: (a) homoionic kaolinite and illite, (b) illitic clay and collodion membrane, and (c) silty clay, illitic clay, and kaolinite.
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CONDUCTION PHENOMENA
Relationship Between ke and ki
From Eqs. (9.108) and (9.129), the electroosmotic flow rate is given by
equations in place of Darcy’s law in consolidation theory. Assumptions
qh ⫽ ki I ⫽ ke
E A
L
(9.132)
The following idealizing assumptions are made: 1. There is homogeneous and saturated soil. 2. The physical and physicochemical properties of the soil are uniform and constant with time.12 3. No soil particles are moved by electrophoresis. 4. The velocity of water flow by electroosmosis is directly proportional to the voltage gradient. 5. All the applied voltage is effective in moving water.13 6. The electrical field is constant with time. 7. The coupling of hydraulic and electrical flows can be formulated by Eqs. (9.63) and (9.64). 8. There are no electrochemical reactions.
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Because E/I is resistance and L/(resistance ⫻ A) is specific conductivity , Eq. (9.132) becomes ki ⫽
ke
(9.133)
As ke varies within relatively narrow limits, Eq. (9.133) shows that the electroosmotic efficiency, measured by ki, is a sensitive function of the electrical conductivity of the soil. For soils 11, 13, and 14 in Table 9.7, is in the range of 0.02 to 0.03 S. For soil 12, in which electroosmosis was not effective, is 0.25 S. In essence, a high value of electrical conductivity means that the current required to develop the voltage is too high for economical movement of water. In addition, if high current is used, the generation of gas, heat, and electrochemical effects become excessive.
Governing Equations
9.17
for the flow rate per unit area. For radial flow for the conditions shown in Fig. 9.33b and a layer of unit thickness
CONSOLIDATION BY ELECTROOSMOSIS
If, in a compressible soil, electroosmosis draws water to a cathode where it is drained away and no water is allowed to enter at the anode, then consolidation of the soil between the electrodes occurs in an amount equal to the volume of water removed. Water movement away from the anode causes consolidation in the vicinity of the anode. The effective stress must increase concurrently. Because the total stress in the vicinity of the anode remains essentially unchanged, the pore water pressure must decrease. Water drains at the cathode where there is no consolidation. Therefore, the total, effective, and pore water pressures at the cathode remain unchanged. As a result, hydraulic gradient develops that tends to cause water flow from cathode to anode. Consolidation continues until the hydraulic force that drives water back toward the anode exactly balances the electroosmotic force driving water toward the cathode. The usefulness of consolidation by electroosmosis as a means for soil stabilization was established by a number of successful field applications, for example, Casangrande (1959) and Bjerrum et al. (1967). Two questions are important: (1) How much consolidation will there be? and (2) How long will it take? Answers to these questions are obtained using the coupled flow
Copyright © 2005 John Wiley & Sons
For one-dimensional flow between plate electrodes (Fig. 9.33a), Eq. (9.63) becomes k u V qh ⫽ ⫺ h ⫺ ke w x x
k u V qh ⫽ ⫺ h 2r ⫺ ke 2r w r r
(9.134)
(9.135)
Introduction of Eq. (9.134) in place of Darcy’s law in the derivation of the diffusion equation governing consolidation in one dimension leads to kh 2u 2V u ⫹ k ⫽ mv e 2 2 w x x t
(9.136)
and
12
Flow of water away from anodes toward cathodes causes a nonuniform decrease in water content along the line between electrodes. This leads to changes in hydraulic conductivity, electroosmotic hydraulic conductivity, compressibility, and electrical conductivity with time and position. To account for these effects, which are discussed by Mitchell and Wan (1977) and Acar et al. (1990), would greatly complicate the analysis because it would be highly nonlinear. Similar problems arise in classical consolidation theory, but the simple linear theory developed by Terzaghi is adequate for most cases. 13 In most cases some of the electrical energy will be consumed by generation of heat and gases at the electrodes. To account for those losses, an effective voltage can be used (Esrig and Henkel, 1968).
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CONSOLIDATION BY ELECTROOSMOSIS
299
kh u V ⫽ ⫺ke w x x
(9.139)
k du ⫽ ⫺ e w dV kh
(9.140)
or
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The solution of this equation is k u ⫽ ⫺ e w V ⫹ C kh
(9.141)
At the cathode, V ⫽ 0 and u ⫽ 0; therefore, C ⫽ 0, and the pore pressure at equilibrium at any point is given by k u ⫽ ⫺ e w V kh
Figure 9.33 Electrode geometries for analysis of consoli-
dation by electroosmosis: (a) one-dimensional flow and (b) radial flow.
2u ke 2V 1 u ⫹ ⫽ w 2 2 x kh x cv t
(9.137)
where mv is the compressibility and cv is the coefficient of consolidation. For radial flow, the use of Eq. (9.135) gives 2u ke 2V 1 ⫹ ⫹ w 2 2 r kh r r
冉
冊
u k V ⫹ e w r kh r
⫽
1 u cv t
(9.138)
Both V and u are functions of position, as shown in Fig. 9.34; V is assumed constant with time, whereas u varies.
where the values of u and V are those at any point of interest. A similar result is obtained from Eq. (9.135) for radial flow. Equation (9.142) indicates that electroosmotic consolidation continues at a point until a negative pore pressure, relative to the initial value, develops that is proportional to the ratio ke /kh and to the voltage at the point. For conditions of constant total stress, there must be an equal and opposite increase in the effective stress. This increase in effective stress causes the consolidation. For the one-dimensional case, consolidation by electroosmosis is analogous to the loading shown in Fig. 9.35. For a given voltage, the magnitude of effective stress increase that develops depends on ke /kh. As ke only varies within narrow limits for different soils, the total consolidation that can be achieved depends largely on kh. Thus, the potential for consolidation by electroosmosis increases as soil grain size decreases because the finer grained the soil, the lower is kh. However, the amount of consolidation in any case depends on the soil compressibility as well as on the change in effective stress. For linear soil compression with increase in effective stress, the coefficient of compressibility av is
Amount of Consolidation
When the hydraulic gradient that develops in response to the differing amounts of consolidation between the anode and cathode generates a counterflow (kh / w)/ (u/ x) that exactly balances the electroosmotic flow ke(V/ x) in the opposite direction, consolidation is complete. As there then is no flow, qh in Eqs. (9.14) and (9.135) is zero. Thus Eq. (9.134) is
Copyright © 2005 John Wiley & Sons
(9.142)
de de av ⫽ ⫺ ⫽ d du
(9.143)
de ⫽ av du ⫽ ⫺av d
(9.144)
or
in which d is the increase in effective stress.
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CONDUCTION PHENOMENA
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300
Figure 9.34 Assumed variation of voltage with distance during electroosmosis: (a) onedimensional flow and (b) radial flow.
Thus, the more compressible the soil, the greater will be the amount of consolidation for a given stress increase, just as in the case of consolidation under applied loads. It follows, also, that electroosmosis will be of little value in an overconsolidated clay unless the effective stress increases are large enough to bring the material back into the virgin compression range. The consolidation loading of any small element of the soil is isotropic, as it is done by increasing the effective stress through reduction in the pore water pressure. The entire soil mass being treated is not consolidated isotropically or uniformly, however, because the amount of consolidation varies with position, de-
Copyright © 2005 John Wiley & Sons
pendent on the voltage at the point. Accordingly, properties at the end of treatment vary along a line between the anode and cathode, as shown, for example, by the posttreatment variations in shear strength and water content shown in Fig. 9.36. Values of these properties before treatment are also shown for comparison. More uniform property distributions between electrodes can be obtained if the polarity of electrodes is reversed after partial completion of consolidation (Wan and Mitchell, 1976). The results shown in Fig. 9.36 were obtained at a site in Norway where electroosmosis was used for the consolidation of quick clay (Bjerrum et al., 1967). The
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CONSOLIDATION BY ELECTROOSMOSIS
301
Figure 9.35 Consolidation by electroosmosis and by direct loading, one-dimensional case: (a) electroosmosis and (b) direct loading.
variations in strength and water content after treatment are consistent with the patterns to be expected based on the predicted variation of pore pressure decrease and vertical strain stress increase with voltage and position shown in Fig. 9.35.
voltage, and TV is the time factor, defined in terms of the distance between electrodes L and real time t as
Rate of Consolidation
where cv is the coefficient of consolidation, given by
Solutions for Eqs. (9.137) and (9.138) have been obtained for several cases (Esrig, 1968, 1971). For the one-dimensional case, and assuming a freely draining (open) cathode and a closed anode (no flow), the pore pressure is u⫽
ke 2k V V(x) ⫹ e w 2 m kh w kh
n⫽0
cv ⫽
n
冋冉 冊 册 1 2 2 TV 2
(9.145)
where V(x) is the voltage at x, Vm is the maximum
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(9.146)
kh mv w
4 3
冘 ⬁
n⫽0
(9.147)
冋冉 冊 册
(⫺1)n 1 exp ⫺ n ⫹ (n ⫹ 1/2)3 2
2
exp ⫺ n ⫹
cvt L2
The average degree of consolidation U as a function of time is U⫽1⫺
x 冘 (n (⫹⫺1)1/2) sin冋(n ⫹ 1/2) 册 L ⬁
TV ⫽
2
2TV
(9.148)
Solutions for Eqs. (9.145) and (9.148) are shown in Figs. 9.37 and 9.38. They are applied in the same way as the theoretical solution for classical consolidation theory.
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CONDUCTION PHENOMENA
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302
˚ s, Norway Figure 9.36 Effect of electroosmosis treatment on properties of quick clay at A (from Bjerrum et al., 1967): (a) Undrained shear strength, (b) remolded shear strength, (c) water content, and (d) Atterberg limits.
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ELECTROCHEMICAL EFFECTS
Figure 9.39 Average degree of consolidation as a function
of dimensionless time for radial consolidation by electroosmosis (from Esrig, 1968). Reprinted with permission of ASCE.
Figure 9.37 Dimensionless pore pressure as a function of
dimensionless time and distance for one-dimensional consolidation by electroosmosis.
Figure 9.38 Average degree of consolidation versus dimen-
sionless time for one-dimensional consolidation by electroosmosis.
A numerical solution to Eq. (9.138) gives the results shown in Fig. 9.39 (Esrig, 1968, 1971). For the case of two pipe electrodes, a more realistic field condition than the radial geometry of Fig. 9.33b, Fig. 9.39 cannot be expected to apply exactly. Along a straight line between two pipe electrodes, however, the flow pattern is approximately the same as for the radial case for a considerable distance from each electrode. A solution for the rate of pore pressure buildup at the cathode for the case of no drainage (closed cathode) is shown in Fig. 9.40. This condition is relevant
Copyright © 2005 John Wiley & Sons
to pile driving, pile pulling, reduction of negative skin friction, and recovery of buried objects. Special solutions for in situ determination of soil consolidation properties by electroosmosis measurements have also been developed (Banerjee and Mitchell, 1980). One of the most important points to be noted from these solutions is that the rate of consolidation depends completely on the coefficient of consolidation, which varies directly with kh, but is completely independent of ke. Low values of kh, as is the case in highly plastic clays, mean long consolidation times. Thus, whereas a low value of kh means a high value of ke /kh and the potential for a high effective consolidation pressure, it also means longer required consolidation times for a given electrode spacing. The optimum situation is when ke /kh is high enough to generate a large pore water tension for reasonable electrode spacings (2 to 3 m) and maximum voltage (50 to 150 V DC), but kh is high enough to enable consolidation in a reasonable time. The soil types that best satisfy these conditions are silts, clayey silts, and silty clays. Most successful field applications of electroosmosis for consolidation have been in these types of materials. As noted earlier, the electrical conductivity of the soil is also important; if it is too high, as in the case of high-salinity pore water, adverse electrochemical effects and unfavorable economics may preclude use of electroosmosis for consolidation. 9.18
ELECTROCHEMICAL EFFECTS
The measured strength increases in the quick clay at ˚ s, Norway (Fig. 9.36), were some 80 percent greater A
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CONDUCTION PHENOMENA
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304
Figure 9.40 Dimensionless pore pressure at the face of a cylindrical electrode as a function
of dimensionless time for the case of a closed cathode (a swelling condition) (from Esrig and Henkel, 1968).
than can be accounted for solely by reduction in water content. Also, the liquid and plastic limits were changed as a result of treatment. Consolidation alone should have no effect on the Atterberg limits because changes in mineralogy, particle characteristics, and/or pore solution characteristics are needed to do this. In addition to movement of water when a DC voltage field is applied between metal electrodes inserted into a wet soil, the following effects may develop: ion diffusion, ion exchange, development of osmotic and pH gradients, desiccation by heat generation at the electrodes, mineral decomposition, precipitation of salts or secondary minerals, electrolysis, hydrolysis, oxidation, reduction, physical and chemical adsorption, and fabric changes. As a result, continuous changes in soil properties that are not readily accounted for by the simplified theory developed previously must be expected. Some of them, such as electrochemical hardening of the soil that results in permanent changes in plasticity and strength, may be beneficial; others, such as heating and gas generation, may impair the efficiency of electroosmosis. For example, heat and gas generation were so great that a field test of consolidation by electroosmosis for foundation stabilization of the leaning Tower of Pisa was unsuccessful. A simplified mechanism for some of the processes during electroosmosis is as follows. Oxygen gas is evolved at the anode by hydrolysis 2H2O ⫺ 4e⫺ → O2 ↑ ⫹ 4H⫹
(9.149)
Anions in solution react with freed H⫹ to form acids.
Copyright © 2005 John Wiley & Sons
Chlorine may also form in a saline environment. Some of the exchangeable cations on the clay may be replaced by H⫹. Because hydrogen clays are generally unstable, and high acidity and oxidation cause rapid deterioration of the anodes, the clay will soon alter to the aluminum or iron form depending on the anode material. As a result, the soil is usually strengthened in the vicinity of the anode. If gas generation at the anode causes cavitation and heat causes desiccation, cracking may occur. This will limit the negative pore pressure that can develop to a value less than 1 atm, and also the electrical resistance will increase, leading to a loss in efficiency. Hydrogen gas is generated at the cathode 4H2O ⫹ 4e⫺ → 2H2 ↑ ⫹ 4OH⫺
(9.150)
Cations in solution are drawn to the cathode where they combine with (OH)⫺ that is left behind to form hydroxides. The pH may rise to values as high as 12 at the cathode. Some alumina and silica may go into solution in the high pH environment. More detailed information about electrochemical reactions during electroosmosis can be found in Titkov et al. (1965), Esrig and Gemeinhardt (1967), Chilingar and Rieke (1967), Gray and Schlocker (1969), Gray (1970), Acar et al. (1990), and Hamed et al. (1991). Soil strength increases resulting from consolidation by electroosmosis and the concurrent electrochemical hardening have application for support of foundations on and in fine-grained soil. Pile capacity for a bridge
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SELF-POTENTIALS
9.19
9.20
SELF-POTENTIALS
Natural DC electrical potential differences of up to several tens of millivolts exist in the earth. These selfpotentials are generated by differing chemical conditions in adjacent soil layers, fluid flow, subsurface chemical reactions, and temperature differences. The self-potential (SP) method is one of the oldest geophysical methods for characterization of the subsurface (National Research Council, 2000). Self-potentials may be the source of phenomena of importance in geotechnical problems as well. The magnitude of self-potential between different soil layers depends on the contents of oxidizing and reducing substances in the layers (F. Hilbert, in Veder, 1981). These potentials can cause a natural electroosmosis in which water flows in the direction from the higher to the lower potential, that is, toward the cathode. The process is shown schematically in Fig. 9.41. An oxidizing soil layer is positive relative to a reducing layer, thus inducing an electroosmotic water flow toward the interface. If water accumulates at the interface, there can be swelling and loss of strength, leading ultimately to formation of a slip surface.
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foundation in varved clay at a site in Canada was well below the design value and inadequate for support of the structure (Soderman and Milligan, 1961; Milligan, 1994). Electrokinetic treatment using the piles as anodes resulted in sufficient strength increase to provide the needed support. Recently reported model tests by Micic et al. (2003) on the use of electrokinetics in soft marine clay to increase the load capacity of skirt foundations for offshore structures resulted in increases in soil strength and supporting capacity of up to a factor of 3.
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ELECTROKINETIC REMEDIATION
The transport of dissolved and suspended constituents into and out of the ground by electroosmosis and electrophoresis, as well as electrochemical, reactions have become of increasing interest because of their potential applications in waste containment and removal of contaminants from fine-grained soils. The electrolysis reactions at the electrodes described in the preceding section, wherein acid is produced at the anode and base at the cathode, are of particular relevance. After a few days of treatment the pH in the vicinity of the anode may drop to less than 2, and that at the cathode increase to more than 10 (Acar and Alshewabkeh, 1993). Toxic heavy metals are preferentially adsorbed by clay minerals and they precipitate except at low pH. Iron or aluminum cations from decomposing anodes can replace heavy-metal ions from exchange sites, the acid generated at the anode can redissolve precipitated material, and the acid front that moves across the soil can keep the metals in solution until removed at the cathode. Geochemical reactions in the soil pores impact the efficiency of the process. Among them are complexation effects that reverse ion charge and reverse flow directions, precipitation/dissolution, sorption, desorption and dissolution, redox, and immobilization or precipitation of metal hydroxides in the high pH zone near the cathode. Some success has been reported in the removal of organic pollutants from soils, at least in the laboratory, as summarized by Alshewabkeh (2001). However, it is unlikely that large quantities of non-aqueous-phase liquids can be effectively transported by electrokinetic processes, except as the NAPL may be present in the form of small bubbles that move with the suspending water. An in-depth treatment of the fundamentals of electrokinetic remediation and the practical aspects of its implementation are given by Alshewabkeh (2001) and the references cited therein.
Copyright © 2005 John Wiley & Sons
Generation of Self-Potentials in Soil Layers
Soils in an oxidizing environment are usually yellow or tan to reddish brown and are characterized by oxides and hydrates of trivalent iron and a low pH relative to reducing soils, which are usually dark gray to bluegray in color and contain sulfides and oxides and hydroxides of divalent iron. The local electrical potential of the soil depends on the iron concentrations and can be calculated from Nernst’s equation:
Figure 9.41 Natural electroosmosis due to self-potential dif-
ferences between oxidizing and reducing soil layers. The oxidizing soil layer is positive relative to the reducing layer (redrawn from Hilbert, in Veder, 1981).
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CONDUCTION PHENOMENA
⫽ 0.771 ⫹
冉 冊
RT c3⫹ ln Fe F c2⫹ Fe
(9.151)
u ⫽ 50 ⫻ 9.81 ⫻ 0.05 ⬇ 25 kPa is generated, which is not an insignificant value. If water that is driven toward the interface cannot escape or be absorbed by the soil, then the effective stress will be reduced by this amount. If the water is absorbed into the clay layer, then softening will result. Either way, the resistance to sliding along the interface will be reduced.
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in which the concentrations are of Fe in solution in moles/liter pore water. The difference in potentials between two layers gives the driving potential for electroosmosis. Values calculated using the Nernst equation are too high for actual soil systems because it applies for conditions of no current flow, and the flowing current also generates a diffusion potential acting in the opposite direction. Hilbert, in Veder (1981), gives the electrical potential as a function of the in situ pH, that is,
then ke /hh ⫽ 50 m/V. If the self-potential difference is 50 mV, then from Eq. (9.142) a pore pressure value of
⫽ 0.186 ⫺ 0.059 pH
(9.152)
Reasonable agreement has been obtained between measured and calculated values of for different soil layers. The end result is that potential differences of up to 50 mV or so are developed between different layers. Potentials measured in a trench excavated in a slide zone are shown in Fig. 9.42. Excess Pore Pressure Generation by Self-Potentials
The pore pressure that may develop at an interface between two different soil layers is given by Eq. (9.142) in which V is the difference in self-potentials between the layers. For a given value of V, the magnitude of pore pressure depends directly on ke /kh. For example, if ke ⫽ 5 ⫻ 10⫺9 m2 /s V and kh ⫽ 1 ⫻ 10⫺10 m/s,
Landslide Stabilization Using Short-Circuit Conductors
If slope instability is caused by a slip surface between reducing and oxidizing soil layers, then a simple means for stabilization can be used (Veder, 1981). Shortcircuiting conductors, such as steel rods, are driven into the soil so that they extend across the slip surface and about 1 to 2 m into the soil below. The mechanism that is then established is shown in Fig. 9.43. Electric current generated by reduction reactions in the oxidizing soil layer and oxidizing reactions in the reducing layer flows through the conductors. Because of the presence of oxidizing agents such as ferric iron, oxygen, and manganese compounds, in the upper oxidizing layer that take up electrons, electrons pass from the metal conductor to the soil. That is, the introduction of electrons initiates reducing reactions. In the reducing layer, on the other hand, there is already a
Figure 9.42 Electrical potentials measured in a trench cut into a slide (from Veder, 1981). Reprinted with permission of Springer-Verlag.
Copyright © 2005 John Wiley & Sons
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THERMALLY DRIVEN MOISTURE FLOW
307
for use of short-circuiting conductors are (1) intact cohesive soils with a low hydraulic conductivity, (2) shear between oxidizing and reducing clay layers, and (3) a relatively thin, well-defined shear zone.
9.21
THERMALLY DRIVEN MOISTURE FLOW
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Thermally driven flows in saturated soils are rather small. Gray (1969) measured thermoelectric currents on the order of 1 to 10 A/ C cm, with the warm side positive relative to the cold side. Thermoosmotic pressures of only a few tenths of a centimeter water head per degree Celsius were measured in saturated soil. Net flows in different directions have been measured in different investigations, evidently because of different temperature dependencies of chemical activity coefficients. These small thermoelectric and thermoosmotic effects in saturated soils may be of little practical significance in geotechnical problems. On the other hand, thermally driven moisture flows in partly saturated soils can be large, and that these flows can be very important in subgrade stability, swelling soils, and heat transfer and storage problems of various types. Theoretical representations of moisture flow through partly saturated soils based solely on the application of irreversible thermodynamics, such as developed by Taylor and Cary (1964), have not been completely successful. They underestimate the flows substantially, perhaps because of the inability to adequately represent all the processes and interactions. A widely used theory for coupled heat and moisture flow through soils was developed by Philip and De Vries (1957). It accounts for both liquid- and vaporphase flows. Vapor-phase flow depends on the thermal and isothermal vapor diffusivities and is driven by temperature and moisture content gradients. The liquidphase flow depends on the thermal and isothermal liquid diffusivities and is driven by the temperature gradient, the moisture content gradient, and gravity. The two governing equations are:
Figure 9.43 Mechanism for slide stabilization using shortcircuiting conductors (adapted from Veder, 1981).
surplus of electrons. If these pass into the conductor, then the environment becomes favorable for oxidation reactions. Thus, positive charges are generated in the reducing soil layer as the conductor carries electrons away. The oxidizing soil layer then takes up these electrons. Completion of the electrical circuit requires current flow through the soil pore water in the manner shown in Fig. 9.43, where adsorbed cations, shown as Na⫹, plus the associated water, flow away from the soil layer interface. This electroosmotic transport of water reduces the water content in the slip zone. Thus, shortcircuit conductors have three main effects (Veder, 1981):
1. Natural electroosmosis is prevented because the short-circuiting conductors eliminate the potential difference between the two soil layers. 2. Electrochemical reactions produce electroosmotic flow in the opposite direction, thus helping to drain the shear zone. 3. Corrosion of the conductors produces high valence cations that exchange for lower valence adsorbed cations, for example, iron for sodium, which leads to soil strengthening.
Several successful cases of landslide stabilization using short-circuiting conductors have been described by Veder (1981) and the references cited therein. Typically, steel rods about 25 mm in diameter are used, spaced a maximum of 3 to 4 m apart in grid patterns covering the area to be stabilized. Conditions favorable
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For vapor-phase flow:
qvap ⫽ ⫺DTV T ⫺ D V w
(9.153)
and for liquid-phase flow:
qliq ⫽ ⫺DTL T ⫺ D L ⫺ k i w where qvap ⫽ vapor flux density (M/L2 /T) w ⫽ density of water (M/L3)
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(9.154)
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CONDUCTION PHENOMENA
T ⫽ temperature (K) ⫽ volumetric water content (L3 /L3) DTV ⫽ thermal vapor diffusivity (L2 /T/K) D V ⫽ isothermal vapor diffusivity (L2 /T) qliq ⫽ liquid flux density (M/L2 /T) DTL ⫽ thermal liquid diffusivity (L2 /T/K) D L ⫽ isothermal liquid diffusivity (L2 /T) k ⫽ unsaturated hydraulic conductivity (L/T) i ⫽ unit vector in vertical direction
DTV ⫽
1. Hydraulic conductivity as a function of water content 2. Thermal conductivity as a function of water content 3. Volumetric heat capacity (see Table 9.2) 4. Suction head as a function of water content
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The thermal vapor diffusivity is given by
in which is the surface tension of water (F/L). Use of the above equations requires knowledge of four relationships to describe the properties of the soils in the system:
冉冊
冉 冊
D0 d 0 v[a ⫹ ƒ(a) ]h w dT
(9.155)
The isothermal vapor diffusivity is given by D V ⫽
冉 冊 冉 冊冉 冊 hg D0 va 0 w RT
d d
(9.156)
where D0 ⫽ molecular diffusivity of water vapor in air (L2 /T) v ⫽ mass flow factor ⫽ P/(P ⫺ p) P ⫽ total gas pressure in pore space p ⫽ partial pressure of water vapor in pore space ⫽ tortuosity factor a ⫽ volumetric air content (L3 /L3) h ⫽ relative humidity of air in pores ⫽ ratio of average temperature gradient in the air-filled pores to the overall temperature gradient g ⫽ acceleration of gravity (L/T2) R ⫽ gas constant (FL/M/K) 0 ⫽ density of saturated water vapor (M/L3) ⫽ suction head of water in the soil (negative head) (L) ƒ(a) ⫽ a/ak for 0 ⬍ a ⬍ ak ⫽ 1 for a ak ak ⫽ a at which liquid conductivity is lost or at which the hydraulic conductivity falls below some arbitrary fraction of the saturated value The thermal liquid diffusivity is given by DTL ⫽ k
冉 冊冉 冊
d dT
(9.157)
The isothermal diffusivity is given by
冉冊
D L ⫽ k
d d
(9.158)
Copyright © 2005 John Wiley & Sons
The hydraulic conductivity and suction relationships are hysteretic; that is, they depend on whether the soil is wetting or drying. Examples of the variations of the different properties needed for the analysis are shown in Fig. 9.44 as a function of degree of saturation and volumetric water content. The data are for a crushed limestone that is used for a trench backfill around buried electrical transmission cables. This material is used because of its low thermal resistivity, which makes it suitable for effective dissipation of heat from the buried cable, provided the saturation does not fall below about 40 percent. The vapor flow is made up of a flow away from the high-temperature side that is driven by a vapor density gradient and a return flow caused by variation in the pore vapor humidity as reflected by variations in soil suction. At moderate soil suction values, for example, a few meters for sand and several tens of meters for clay, the thermal vapor diffusivity predominates, and moisture is driven away from the heat source (McMillan, 1985). The isothermal diffusivity term only becomes important at very high suction levels. The liquid flow consists of a capillarity-driven flow toward the heat source and an outward liquid flow due to variations in water surface tension with temperature. McMillan’s analysis showed that for both sand and clay the isothermal liquid diffusivity term was 4 to 5 orders of magnitude greater than the thermal liquid diffusivity term. Thus capillarity-driven flow predominates for any significant gradient in the volumetric moisture content. The very small thermal liquid diffusivity is consistent with the observations noted earlier for saturated soils in which measured water flows under thermal gradients are small. The total water flow q in an unsaturated soil under the action of a temperature gradient and its resulting water content gradient equals the sum of the vaporphase and liquid-phase movements. Thus, from Eqs. (9.153) to (9.158),
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THERMALLY DRIVEN MOISTURE FLOW
309
Figure 9.44 Examples of properties used for analysis of thermally driven moisture flow in a partially saturated, compacted, crushed limestone: (a) particle size distribution, (b) suction head as a function of volumetric water content, (c) hydraulic conductivity as a function of degree of saturation and volumetric water content, (d) isothermal liquid diffusivity as a function of degree of saturation and volumetric water content, (e) isothermal vapor diffusivity as a function of degree of saturation and volumetric water content, and (f) Thermal water diffusivity as a function of degree of saturation and volumetric water content. Thermal resistivity as a function of water content for this soil is shown in Fig. 9.14.
q ⫽ ⫺(DTV ⫹ DTL) T ⫺ (D V ⫹ D L ) ⫺ k i w ⫽ ⫺DT T ⫺ D ⫺ k i
in which
(9.159)
D ⫽ DTV ⫹ DTL ⫽ thermal water diffusivity
(9.160)
and
Equation (9.159) is the governing equation for moisture movement under a thermal gradient in unsaturated soils as proposed by Philip and De Vries (1957). Differentiation of this equation and application of the continuity requirement gives the general differential equation for moisture flow: k ⫽ (DT T) ⫹ (D ) ⫹ t z
(9.162)
The heat conduction equation for the soil is
D ⫽ D V ⫹ D L ⫽ isothermal water diffusivity (9.161)
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冉 冊
T k ⫽ t T t C
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(9.163)
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CONDUCTION PHENOMENA
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310
Figure 9.44 (Continued )
where kt ⫽ thermal conductivity C ⫽ volumetric heat capacity
The ratio of thermal conductivity to the volumetric heat capacity is the thermal diffusivity A. Both transient and steady-state temperature distributions computed using the Philip and De Vries theory incorporated into numerical models have agreed well with measured values in a number of cases. The actual moisture movements and distributions have not agreed as well, for example, Abdel-Hadi and Mitchell (1981) and Cameron (1986). The numerical simulations have been done using transform methods, finite difference methods, the finite element method, and the integrated finite difference method. Cameron (1986) reformulated the equations in terms of suction head rather than moisture content and incorporated them into the finite element model of Walker et al. (1981) for solution of two-dimensional problems. 9.22
GROUND FREEZING
Heat conduction in soils and rocks is discussed in Section 9.5, and values for thermal properties are given in
Copyright © 2005 John Wiley & Sons
Table 9.2. Three topics are considered in this section: (1) the depth of frost penetration, which illustrates the application of transient heat flow analysis, (2) frost action in soils, a phenomenon of great practical importance that can be understood through consideration of interactions of the physical and physicochemical properties of the soil, and (3) some effects of freezing on the behavior and properties of the soil after thawing. These topics are also covered in some detail by Konrad (2001) and the references therein. Depth of Frost Penetration
Accurate estimation of the depth of ground freezing during the winter, the depth of thawing in permafrost areas during the summer, and the refrigeration and time requirements for artificial ground freezing for temporary ground stabilization are all problems involving transient heat flow analysis. They differ from the conduction analyses in the preceding sections in that the phase change of water to ice must be taken into account. Prediction of the maximum depth of frost penetration illustrates this type of problem. Theoretical solutions of this problem are based on a mathematical
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GROUND FREEZING
analysis developed by Neumann in about 1860 (Berggren, 1943; Aldrich, 1956; Brown, 1964; Konrad, 2001). The relationship between thermal energy u and temperature T for a soil mass at constant water content is shown in Fig. 9.45. In the absence of freezing or thawing
(9.168)
where a ⫽ kt /C is the thermal diffusivity (L2 /T). Equation (9.168) is the one-dimensional, transient heat flow equation. At the interface between frozen and unfrozen soil, z ⫽ Z, and the equation of heat continuity is
(9.164) Ls
dZ ⫽ q ƒ ⫺ qu dt
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u ⫽C T
T 2T ⫽a 2 t z
311
(9.169)
The Fourier equation for heat flow is T qt ⫽ ⫺kt z
(9.165)
In the absence of freezing or thawing, thermal continuity and conservation of thermal energy require that the rate of change of thermal energy of an element plus the rate of heat transfer into the element equal zero, that is, for the one-dimensional case u q ⫹ ⫽0 t z
(9.166)
Using Eqs. (9.164) and (9.165), Eq. (9.166) may be written C
or
T 2T ⫽ kt 2 t z
(9.167)
where Ls is the latent heat of fusion of water and qƒ ⫺ qu is the net rate of heat flow away from the interface. Equation (9.169) can be written Ls
dZ T T ⫽ kƒ ƒ ⫺ ku u dt z z
(9.170)
where the subscripts u and f pertain to unfrozen and frozen soil, respectively. Simultaneous solution of Eqs. (9.168) and (9.170) gives the depth of frost penetration. Stefan Formula The simplest solution is to assume that the latent heat is the only heat to be removed during freezing and neglect the heat that must be removed to cool the soil water to the freezing point, that is, the thermal energy stored as volumetric heat is neglected. This condition is shown by Fig. 9.46. For this case Eq. (9.168) does not exist, and Eq. (9.170) becomes Ls
dZ T ⫽ kƒ s dt Z
(9.171)
where Ts is the surface temperature. The solution of this equation is
Figure 9.45 Thermal energy as a function of temperature Figure 9.46 Assumed conditions for the Stefan equation.
for a wet soil.
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CONDUCTION PHENOMENA
冉
2kƒ
Z⫽
冊
冕 T dt s
Ls
1/2
(9.172)
冉 冊
Z⫽
2kTst Ls
1/2
⫽
T0 Ts
(9.174)
and the fusion parameter is ⫽
C T Ls s
(9.175)
An averaged value for the volumetric heats of frozen and unfrozen soil can be used for C in Eq. (9.175). In application, the quantity Tst in Eq. (9.173) is replaced by the freezing index, and Ts in (9.175) is given by F/t, where t is the duration of the freezing period. The coefficient corrects the Stefan formula for neglect of volumetric heat. For soils with high water content C is small relative to Ls; therefore, is small and
Figure 9.47 Freezing index in relation to the annual temperature cycle.
Copyright © 2005 John Wiley & Sons
(9.173)
where k is taken as an average thermal conductivity for frozen and unfrozen soil. The dimensionless correction coefficient depends on the two parameters shown in Fig. 9.49. The thermal ratio is given by
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The integral of Ts dt is a measure of freezing intensity. It can be expressed by the freezing index F, which has units of degrees ⫻ time. Index F is usually given in degree-days. It is shown in relation to the annual temperature cycle in Fig. 9.47. Freezing index values are derived from meteorological data. Methods for determination of freezing index values are given by Linell et al. (1963), Straub and Wegmann (1965), McCormick (1971), and others. Maps showing mean freezing index values are available for some areas. It is important when using such data sources to be sure that there are not local deviations from the average values that are given. Different types of ground cover, local topography and vegetation, and solar radiation all influence the net heat flux at the ground surface. The Stefan equation can also be used to estimate the summer thaw depth in permafrost; that is, the thickness of the active layer. In this case the ground thawing index, also in degree-days and derived from meteorological data, is used in Eq. (9.172) in place of the freezing index (Konrad, 2001). Modified Berggren Formula The Stefan formula overpredicts the depth of freezing because it neglects the removal of the volumetric heats of frozen and unfrozen soil. Simultaneous solution of Eqs. (9.168) and
(9.170) has been made for the conditions shown in Fig. 9.48, assuming that the soil has a uniform initial temperature that is T0 degrees above freezing and that the surface temperature drops suddenly to Ts below freezing (Aldrich, 1956). The solution is
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GROUND FREEZING
313
Figure 9.48 Thermal conditions assumed in the derivation of the modified Berggren for-
mula.
the Stefan formula is reasonable. For arctic climates, where T0 is not much above the freezing point, is small, is greater than 0.9, and the Stefan formula is satisfactory. However, in more temperate climates and in relatively dry or well-drained soils, the correction becomes important. A comparison between theoretical freezing depths and a design curve proposed by the Corps of Engineers is shown in Fig. 9.50 for several soil types. The theoretical curves were developed by Brown (1964) using the modified Berggren equation and the thermal properties given in Fig. 9.13. Consideration should be given to the effect of different types of surface cover on the ground surface temperature because air temperature and ground temperature are not likely to be the same, and the effects of thermal radiation may be important. Observed
Copyright © 2005 John Wiley & Sons
depths of frost penetration may be misleading if estimates for a proposed pavement or other structure are needed because of differences in ground surface characteristics and because the pavement or foundation base will be at different water content and density than the surrounding soil. The solutions do not account for flow of water into or out of the soil or the formation of ice lenses during the freezing period. This may be particularly important when dealing with frost heave susceptible soils or when developing frozen soil barriers for the cutoff of groundwater flow. Methods for prediction of frost depth in soils susceptible to ice lens formation and the rate of heave are given by Konrad (2001). The initiation of freezing of flowing groundwater requires that the rate of volumetric and latent heat removal be high enough so that ice can form during the residence time
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CONDUCTION PHENOMENA
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314
Figure 9.49 Correction coefficients for use in the modified Berggren formula (from Aldrich,
1956).
of an element of water moving between the boundaries of the specified zone of solidification. Frost Heaving
Freezing of some soils is accompanied by the formation of ice layers or ‘‘lenses’’ that can range from a millimeter to several centimeters in thickness. These lenses are essentially pure ice and are free from large numbers of contained soil particles. The ground surface may ‘‘heave’’ by as much as several tens of centimeters, and the overall volume increase can be many times the 9 percent expansion that occurs when water freezes. Heave pressures of many atmospheres are common. The freezing of frost-susceptible soils beneath pavements and foundations can cause major distress or failure as a result of uneven uplift during freezing and loss of support on thawing, owing to the presence of large water-filled voids. Ordinarily, ice lenses are oriented normal to the direction of cold-front movement and become thicker and more widely separated with depth. The rate of heaving may be as high as several millimeters per day. It depends on the rate of freezing in
Copyright © 2005 John Wiley & Sons
a complex manner. If the cooling rate is too high, then the soil freezes before water can migrate to an ice lens, so the heave becomes only that due to the expansion of water on freezing. Three conditions are necessary for ice lens formation and frost heave: 1. Frost-susceptible soil 2. Freezing temperature 3. Availability of water
Frost heaving can occur only where there is a water table, perched water table, or pocket of water reasonably close to the freezing front. Frost-Susceptible Soils Almost any soil may be made to heave if the freezing rate and water supply are controlled. In nature, however, the usual rates of freezing are such that only certain soil types are frost susceptible. Clean sands, gravels, and highly plastic intact clays generally do not heave. Although the only completely reliable way to evaluate frost susceptibility is by some type of performance test during freezing, soils that contain more than 3 percent of their particles finer than 0.02 mm are potentially frost susceptible.
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GROUND FREEZING
Figure 9.50 Predicted frost penetration depths compared with the Corps of Engineers’ de-
sign curve (Brown, 1964). Curve a—sandy soil: dry density 140 lb / ft3, saturated, moisture content 7 percent. Curve b—silt, clay: dry density 80 lb / ft3, unsaturated, moisture content 2 percent. Curve c—sandy soil; dry density 140 lb / ft3, unsaturated, moisture content 2 percent. Curve d—silt, clay: dry density 120 lb / ft3, moisture content 10 to 20 percent (saturated). Curve e—silt, clay: dry density 80 lb / ft3 saturated, moisture content 30 percent. Curve f—Pure ice over still water.
Frost-susceptible soils have been classified by the Corps of Engineers in the following order of increasing frost susceptibility:
Group (increasing susceptibility) F1 F2 F3
F4
Soil Types
Gravelly soils with 3 to 20 percent finer than 0.02 mm Sands with 3 to 15 percent finer than 0.02 mm a. Gravelly soils with more than 20 percent finer than 0.02mm sands, except fine silt sands with more than 15 percent finer than 0.02 mm b. Clays with PI greater than 12 percent, except varved clays a. Silts and sandy silts b. Fine silty sands with more than 15 percent finer than 0.02 mm c. Lean clays with PI less than 12 percent d. Varved clays
Copyright © 2005 John Wiley & Sons
A method for the evaluation of frost susceptibility that takes project requirements and acceptable risks and freezing conditions into account as well as the soil type is described by Konrad and Morgenstern (1983). Mechanism of Frost Heave The formation of ice lenses is a complex process that involves interrelationships between the phase change of water to ice, transport of water to the lens, and general unsteady heat flow in the freezing soil. The following explanation of the physics of frost heave is based largely on the mechanism proposed by Martin (1959). Although the Martin (1959) model may not be correct in all details in the light of subsequent research, it provides a logical and instructive basis for understanding many aspects of the frost heave process. The ice lens formation cycle involves four stages: 1. 2. 3. 4.
Nucleation of ice Growth of the ice lens Termination of ice growth Heat and water flow between the end of stage 3 and the start of stage 1 again
In reality, heat and water flows continue through all four stages; however, it is convenient to consider them separately. The temperature for nucleation of an ice crystal, Tn, is less than the freezing temperature, T0. In soils, T0 in
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CONDUCTION PHENOMENA
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pore water is less than the normal freezing point of water because of dissolved ions, particle surface force effects, and negative pore water pressures that exist in the freezing zone. The freezing point decreases with decreasing distance to particle surfaces and may be several degrees lower in the double layer than in the center of a pore. Thus, in a fine-grained soil, there is an unfrozen film on particle surfaces that persists until the temperature drops below 0C. The face of an ice front has a thin film of adsorbed water. Freezing advances by incorporation of water molecules from the film into the ice, while additional water molecules enter the film to maintain its thickness. It is energetically easier to bring water to the ice from adjacent pores than to freeze the adsorbed water on the particle or to propagate the ice through a pore constriction. The driving force for water transport to the ice is an equivalent hydrostatic pressure gradient that is generated by freezing point depression, by removal of the water from the soil at the ice front, which creates a higher effective stress in the vicinity of the ice than away from it, by interfacial tension at the ice–water interface, and by osmotic pressure generated by the high concentration of ions in the water adjacent to the ice front. Ice formation continues until the water tension in the pores supplying water becomes great enough to cause cavitation, or decreased upward water flow from below leads to new ice lens formation beneath the existing lens. The processes of freezing and ice lens formation proceed in the following way with time according to Martin’s theory. If homogeneous soil, at uniform water content and temperature T0 above freezing, is subjected to a surface temperature Ts below freezing, then the variation of temperature with depth at some time is as shown in Fig. 9.51. The rate of heat flow at any point is ⫺kt(dT/dz). If dT/dz at point A is greater than at point B, the temperature of the element will drop. When water goes to ice, it gives up its latent heat, which flows both up and down and may slow or stop changes in the value of dT/dz for some time period, thus halting the rate of advance of the freezing front into the soil. Ground heave results from the formation of a lens at A, with water supplied according to the mechanisms indicated above. The energy needed to lift the overlying material, which may include not only the soil and ice lenses above, but also pavements and structures, is available because ice forms under conditions of supercooling at a temperature T X ⬍ TFP, where TFP is the freezing temperature. The available energy is
F ⫽
L(TFP ⫺ T X) TFP
(9.176)
Copyright © 2005 John Wiley & Sons
Figure 9.51 Temperature versus depth relationships in a
freezing soil.
The quantity L is the latent heat. Supercooling of 1C is sufficient to lift 12.5 kg a distance of 10 mm. Alternatively, the energy for heave may originate from the thin water films at the ice surface (Kaplar, 1970). As long as water can flow to a growing ice lens fast enough, the volumetric heat and latent heat can produce a temporary steady-state condition so that (dT/ dz)A ⫽ (dT/dz)B. For example, silt can supply water at a rate sufficient for heave at 1 mm/h. After some time the ability of the soil to supply water will drop because the water supply in the region ahead of the ice front becomes depleted, and the hydraulic conductivity of the soil drops, owing to increased tension in the pore water. This is illustrated in Fig. 9.52, where hydraulic conductivity data as a function of negative pore water pressure are shown for a silty sand, a silt, and a clay, all compacted using modified AASHTO effort, at a water content about 3 percent wet of optimum. A small negative pore water pressure is sufficient to cause water to drain from the pores of the silty sand, and this causes a sharp reduction in hydraulic conductivity. Because the clay can withstand large negative pore pressures without loss of saturation, the hydraulic conductivity is little affected by increasing reductions in the pore pressure (increasing suction). The small decrease that is observed results from the consolidation needed to carry the increased effective stress required to balance the reduction in the pore pressure. For the silt, water drainage starts when the suction reaches
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GROUND FREEZING
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which have now reduced the distance that water can be from a particle surface. The temperature drop must reach a depth where there is sufficient water available after nucleation to supply a growing lens. The thicker the overlying lens, the greater the distance, thus accounting for the increased spacings between lenses with depth. The greater the depth, the smaller the thermal gradient, as may be seen in Fig. 9.51, where (dT/dz)A ⬎ (dT/dz)A where A is on the temperature distribution curve for a later time t2. Because of this, the rate of heat extraction is slowed, and the temporary steady-state condition for lens growth can be maintained for a longer time, thus enabling formation of a thicker lens. More quantitative analyses of the freezing and frost heaving processes in terms of segregation potential, rates, pressures, and heave amounts are available. The Proceedings of the International Symposia on Ground Freezing, for example, Jones and Holden (1988), Nixon (1991), and Konrad (2001) provide excellent sources of information on these issues. Thaw Consolidation and Weakening
Figure 9.52 Hydraulic conductivity as a function of negative pore water pressure (from Martin and Wissa, 1972).
about 40 kPa; however, a significant continuous water phase remains until substantially greater values of suction are reached. In sand, the volume of water in a pore is large, and the latent heat raises the freezing temperature to the normal freezing point. Hence, there is no supercooling and no heave. Negative pore pressure development at the ice front causes the hydraulic conductivity to drop, so water cannot be supplied to form ice lenses. Thus sands freeze homogeneously with depth. In clay, the hydraulic conductivity is so low that water cannot be supplied fast enough to maintain the temporary steadystate condition needed for ice lens growth. Heave in clay only develops if the freezing rate is slowed to well below that in nature. Silts and silty soils have a combination of pore size, hydraulic conductivity, and freezing point depression that allow for large heave at normal freezing rates in the field. The freezing temperature penetrates ahead of a completed ice lens, and a new lens will start to form only after the temperature drops to the nucleation temperature. The nucleation temperature for a new lens may be less than that for the one before because of reduced saturation and consolidation from the previous flows,
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When water in soil freezes, it expands by about 9 percent of its original volume. Thus a fully saturated soil increases in volume by 9 percent of its porosity, even in the absence of ice segregation and frost heave. The expansion associated with freezing disrupts the original soil structure. When thawed, the water returns to its original volume, the melting of segregated ice leaves voids, and the soil can be considerably more deformable and weaker that before it was frozen. Under drained conditions and constant applied overburden stress, the soil may consolidate to a denser state than it had prior to freezing. The lower the density of the soil, the greater is the amount of thaw consolidation. The total settlement of foundations and pavements associated with thawing is the sum of that due to (1) the phase change, (2) melting of segregated ice, and (3) compression of the weakened soil structure. Testing of representative samples under appropriate boundary conditions is the most reliable means for evaluating thaw consolidation. Samples of frozen soil are allowed to thaw under specified levels of applied stress and under defined drainage conditions, and the decrease in void ratio or thickness is determined. An example of the effects of freezing and thawing on the compression and strength of initially undisturbed Boston blue clay is shown in Fig. 9.53 from Swan and Greene (1998). These tests were done as part of a ground freezing project for ground strengthening to enable jacking of tunnel sections beneath operating rail lines during construction of the recently completed Central Artery/Tunnel Project in Boston. Detailed
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0 2 C1-UF e0 = 1.064
6 C4-FT e0 = 1.171
8 10 12 14
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Vertical Strain, εv (%)
4
16 18 20
22 10
100 1000 Effective Stress, σc (kPa)
10000
(a)
120
Deviator Stress, σ1 – σ3 (kPa)
100
UUC1-UF (σ1–σ3)max = 109.6 kPa ε1 = 2.3% su/σ3cell = 0.36 e0 = 1.02; w = 37.5%
80
60
40
UUC4-FT (σ1– σ3)max = 42.4 kPa ε1 = 12.8% su/σ3cell = 0.14 e0 = 1.13; w = 43.2%
20
0
0
5
10
15
20
25
Axial Strain. % (b)
Figure 9.53 (a) Comparison between the compression behavior of unfrozen (C1-UF) and frozen then thawed (C4-FT) samples of Boston blue clay. (b) Deviator stress vs. axial strain in unconsolidated–undrained triaxial compression of unfrozen (UUC1-UF) and frozen and thawed (UUC4-FT) Boston blue clay (from Swan and Greene, 1998).
analysis of the thaw consolidation process and its analytical representation is given by Nixon and Ladanyi (1978) and Andersland and Anderson (1978). Ground Strengthening and Flow Barriers by Artificial Ground Freezing
Artificial ground freezing has applications for formation of seepage cutoff barriers in situ, excavation support, and other ground strengthening purposes. These appli-
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cations are usually temporary, and they have the advantage that the ground is not permanently altered, except for such property changes as may be caused by the freeze–thaw processes. Returning the ground to its pristine state may be important for environmental reasons where alternative methods for stabilization could permanently change the state and composition of the subsoil. Freezing is usually accomplished by installation of freeze pipes and circulation of a refrigerant. For emer-
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CONCLUDING COMMENTS
9.23
160
120 Second Stage
First Stage
Third Stage
CONCLUDING COMMENTS
Conductivity properties are one of the four key dimensions of soil behavior that must be understood and quantified for success in geoengineering. The other three dimensions are volume change, deformation and strength, and the influences of time. They form the subjects of the following three chapters of this book. Water flows through soils and rocks under fully saturated conditions have been the most studied, and hydraulic conductivity properties, their determination and application for seepage studies of various types, construction dewatering, and the like are central to geotechnical engineering. One objective of this chapter has been to elucidate the fundamental factors that control
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Natural Strain, ε -%
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gency and rapid ground freezing, expendable refrigerants such as liquid nitrogen or carbon dioxide in an open pipe can be used. The thermal energy removal and time requirements for freezing the ground can be calculated using the appropriate thermal conductivity, volumetric heat, and latent heat properties for the ground and heat conduction theory in conjunction with the characteristics of the refrigeration system (Sanger, 1968; Shuster, 1972; Sanger and Sayles, 1979). For many applications the energy required to freeze the ground in kcal/m3 will be in the range of 2200 to 2800 times the water content in percent (Shuster, 1972). However, if the rate of groundwater flow exceeds about 1.5 m/day, it may be difficult to freeze the ground without a very high refrigeration capacity to ensure that the necessary temperature decrease and latent heat removal can be accomplished within the time any element of water is within the zone to be frozen. The long-term strength and stress–strain characteristics of frozen ground depend on the ice content, temperature, and duration of loading. The short-term strength under rapid loading, which can be up to 20 MPa at low temperature, may be 5 to 10 times greater than that under sustained stresses. That is, frozen soils are susceptible to creep strength losses (Chapter 12). The deformation behavior of frozen soil is viscoplastic, and the stress and temperature have significant influence on the deformation at any time. The creep curves in Fig. 9.54 illustrate these effects. The onset of the third stage of creep indicates the beginning of failure. The evaluation of stability of frozen soil masses, the prediction of creep deformation, and the possibility of creep rupture are complex problems because of heterogeneous ground conditions, irregular geometries, and temperature and stress variations throughout the frozen soil mass. Design and implementation considerations for use of ground freezing in construction are given by Donohoe et al. (1998).
319
80
Pa
55
T
=
0
,σ °C
=
M
Temperature Effect
0.
40
T=
,σ= –2.2 °C
Pa
0.55 M
Stress effect
T = –2.2 °C, σ = 0.138 MPa
0
0
tf
10
20
30
Time, t (hr)
Figure 9.54 Creep curves for a frozen organic silty clay (from Sanger and Sayles, 1979).
the permeability of soils to water and how this property depends on soil type, especially gradation, and is sensitive to testing conditions, soil fabric, and environmental factors. The understanding of these fundamentals is important, not only because of the insights provided but also because many of the same considerations apply to the several other types of flows that are known to be important—chemical, electrical, and thermal. Knowledge of one is helpful in the understanding and quantification of the other because the mathematical descriptions of the flows follow similar force-flux relationships. At the same time it is necessary to take into account that the flows of fluids of different composition and the application of hydraulic, chemical, electrical, and thermal driving forces to soils can cause changes in compositions and properties, with differing consequences, depending on the situation. Furthermore, as examined in considerable detail in this chapter, flow coupling can be important, especially advective and diffusive chemical transport, electroosmotic water and chemical flow, and thermally driven moisture flow. Considerable impetus for research on these processes has been generated by geoenvironmental needs, including enhanced
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and more economical waste containment and site remediation strategies. Ground freezing, in addition to its importance in engineering and construction in cold regions, is seeing new applications for temporary ground stabilization needed for underground construction in sensitive urban areas. QUESTIONS AND PROBLEMS
7. Two parallel channels, one with flowing water and the other with contaminated water, are 100 ft apart. The surface elevation of the contaminated channel is 99 ft, and the surface elevation of the clean water channel is at 97 ft. The soil between the two channels is sand with a hydraulic conductivity of 1 ⫻ 10⫺4 m/s, a dry unit weight of 100 pcf, and a specific gravity of solids of 2.65. Estimate the time it will take for seepage from the contaminated channel to begin flowing into the initially clean channel. Make the following assumptions and simplifications: a. Seepage is one dimensional. b. The only subsurface reaction is adsorption onto the soil particles. c. The soil–water partitioning coefficient is 0.4 cm3 /g. d. Hydrodynamic dispersion can be ignored.
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1. A uniform sand with rounded particles has a void ratio of 0.63 and a hydraulic conductivity, k, of 2.7 ⫻ 10⫺4 m/s. Estimate the value of k for the same sand at a void ratio of 0.75.
6. How can the effects of incompatibility between chemicals in a waste repository and a compacted clay liner best be minimized?
2. The soil profile at a site that must be dewatered consists of three homogeneous horizontal layers of equal thickness. The value of k for the upper and lower layers is 1 ⫻ 10⫺6 m/s and that of the middle layer is 1 ⫻ 10⫺4 m/s. What is the ratio of the average hydraulic conductivity in the horizontal direction to that in the vertical direction? 3. Consider a zone of undisturbed San Francisco Bay mud free of sand and silt lenses. Comment on the probable effect of disturbance on the hydraulic conductivity, if any. Would this material be expected to be anisotropic with respect to hydraulic conductivity? Why?
4. Assume the specific surface of the San Francisco Bay mud in Question 3 is 50 m2 /g and prepare a plot of the hydraulic conductivity in meters/second as a function of water content over the range of 100 percent decreased to 25 percent by consolidation using the Kozeny–Carman equation. Would you expect the actual variation in hydraulic conductivity as a function of water content to be of this form? Why? Sketch the variation you would expect and explain why it has this form. 5. At a Superfund site a plastic concrete slurry wall was proposed as a vertical containment barrier against escape of liquid wastes and heavily contaminated groundwater. The subsurface conditions consist of horizontally bedded mudstone and siltstone above thick, very low permeability clay shale. The cutoff wall was to extend into the slay shale, which has been shown to be able to serve as a very effective bottom barrier. For the final design and construction, however, a 3-ft-wide gravel trench was used instead of the slurry wall. Sumps and pumps placed in the bottom of the trench are used to collect liquids. Explain how this trench can serve as an effective cutoff and discuss the pros and cons of the two systems.
Copyright © 2005 John Wiley & Sons
8. For the compacted clay waste containment liner shown below and assuming steady-state conditions: a. What is the contaminant transport for pure molecular diffusion? b. What is the contaminant transport rate for pure advection? c. What is the contaminant transport rate for advection plus diffusion? d. Why don’t the answers to parts (a) and (b) add up to (c)?
NOTE: Advection and diffusion are in the same direction; therefore, J ⬎ 0, and the solution will be in the form c ⫽ a1ea2x ⫹ a3
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QUESTIONS AND PROBLEMS
9. One-dimensional flow is occurring by electroosmosis between two electrodes spaced at 3.0 m with a potential drop of 100 V (DC) between them. What should the water flow rate be if the coefficient of electroosmotic permeability, ke, is 5 ⫻ 10⫺9 m2 /s V assuming an open system? If no water is resupplied at the anode, what maximum consolidation pressure should develop at a point midway between electrodes if the hydraulic conductivity of the soil is 1 ⫻ 10⫺8 m/s?
Assume that the water pressure at the top of the leachate collection layer is atmospheric and that the only fluxes across the liner are water and electricity. The characteristics of the compacted clay liner are:
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10. a. A soil has a coefficient of electroosmotic permeability equal to 0.3 ⫻ 10⫺8 m/s per V/m and a hydraulic conductivity of 6 ⫻ 10⫺9 m/s. Starting from the general relationship
321
Hydraulic conductivity
Ji ⫽ Lij Xj
kh ⫽ 1 ⫻ 10⫺7 m/s
Electroosmotic coefficients
derive an expression for the pore water tension that may be developed under ideal conditions for consolidation of the clay by electroosmosis and compute the value that should develop at a point where the voltage is 25 V. Be sure to indicate correct units with your answers. b. In the absence of electrochemical effects or cavitation, would you consider your answer to part (a) to represent an upper or lower bound estimate of the pore water tension? Why? (HINT: Consider the influence of consolidation on the soil properties that are used to predict the pore water tension.) 11. In 1892 Saxen established that there is equivalence between electroosmosis and streaming potential such that the results of a hydraulic conductivity test in which streaming potential is measured can be used to predict the volume flow rate during electroosmosis in terms of the electrical current. Starting with the general equations for coupled electrical and hydraulic flow, derive Saxen’s law. What will be the drainage rate from a soil, in m3 /h amp, if the streaming potential is 25 mV/ atm? What will be the cost of electrical power per cubic meter of water drained if electricity costs $0.10 per kWh and a maximum voltage of 75 V is used? 12. It might be possible to prevent leakage of hazardous and toxic chemicals through waste impoundment and landfill clay or geosynthetic-clay liners by means of an electroosmosis counterflow barrier against hydraulically driven seepage. Consider the impoundment and liner system shown below.
Copyright © 2005 John Wiley & Sons
ke ⫽ 2 ⫻ 10⫺9 m2 /s V
ki ⫽ 0.2 ⫻ 10⫺6 m3 /s amp a. Wire mesh is proposed for use as electrodes. Where would you place the anode and cathode meshes? b. If the waste pond is to be filled to an average depth of 6 m, what voltage drop should be maintained between the electrodes? c. What will the power cost be per hectare of impoundment per year? Power costs $0.09 per kWh. d. Assume that the leachate collection layer is flushed continuously with freshwater and that the liquid waste contains dissolved salts. Write the complete set of equations that would be required to describe all the flows across the liner during electroosmosis. Define all terms. e. Will maintenance of a no hydraulic flow condition ensure that no leachate will escape through the clay liner? Why?
13. a. Estimate the minimum footing depths for structures in a Midwestern city where the freezing index is 750 degree-days and the duration of the freezing index is 100 days. The mean annual air temperature is 50F. The soil is silty clay with a water content of 20 percent and a dry unit weight of 110 lb/ft3. Assume no ice segregation and compare values according to the Stefan and modified Berggren formulas. b. What will be the depth of frost penetration below original ground surface level if a surface heave of 6 inches develops due to ice lens for-
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through the liner as a function of the hydraulic conductivity. Show in the same diagram the proportions of the total that are attributable to diffusion and advection. Assume that the leachate collection layer is fully drained, but for purposes of analysis the fluid level can be considered at the bottom of the clay. Determine the leakage rate through the liner per unit area as a function of the hydraulic conductivity and show it on a diagram. 15. The diagram below shows the cross section of a tunnel and underlying borehole in which waste canisters for spent nuclear fuel are located. Such an arrangement is proposed for deep (e.g., several hundred meters) burial of nuclear waste in crystalline rock. The surrounding rock can be assumed fully saturated, and the groundwater table will be within a few tens of meters of the ground surface. Thermal studies have shown that the temperature of the waste canister will rise to as high as 150C at its surface. A canister life of about 100 years is anticipated using either stainless steel or copper for the material. The surrounding environment must be safe against leakage of radionuclides from the repository for a minimum of 100,000 years.
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mation? Assume a frozen ground temperature of 32F. c. If a pavement is to be placed over the soil, what thickness of granular base course should be used to prevent freezing of the subgrade? The base course will be compacted to a dry density of 125 lb/ft3 at a water content of 15 percent. If the pavement structure is to contain an 8inch-thick Portland cement concrete surface layer, will your result tend to overestimate or underestimate the base thickness required? Why? 14. A compacted fine-grained soil is to be used as a liner for a chemical waste storage area. Free liquid leachate and possibly some heavier than water free phase nonsoluble, nonpolar organic liquids (DNAPLs) may accumulate in some areas as a result of rupturing and corrosion of the drums in which they were stored. Two sources of soil for use in the liner are available. They have the following properties: Property
Soil A
Soil B
Unified class Liquid limit (%) Plastic limit (%) Clay size (%) Silt size (%) Sand size (%) Predominant clay mineral Cation exchange capacity (meg/100 g)
(CH) 90 30 50 30 20 Smectite
(CL) 45 25 30 40 30 Illite
60
20
a. Which of the two soils would be best suited for use in the liner? Why? b. What tests would you use to validate your choice? Why? c. Assume that you have confirmed that it will be possible to compact the soil to states that will have hydraulic conductivities in the range of 1 ⫻ 10⫺8 to 1 ⫻ 10⫺11 m/s. A liner thickness of 0.6 m is proposed. Leachate is likely to accumulate to a depth of 1.0 m above the top of the liner. A leachate collection layer will underlie the liner. d. If the concentration of dissolved salts in the leachate is 1.0 M and the average diffusion coefficient is 5 ⫻ 10⫺10 m2 /s, determine for the steady state the total amount of dissolved chemical per unit area per year that will escape
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QUESTIONS AND PROBLEMS
c. Assess the probable natures and directions of heat and fluid flows that will develop, if any. d. What alterations might occur in the material during the life of the repository if any? Consider the effects of groundwater from the surrounding ground, corrosion of the canister, and the prolonged exposure to high temperature. Would each of these alternations be likely to enhance or impair the effectiveness of the clay pack?
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Clay or a mix of clay with other materials such as sand and crushed rock is proposed for use as the fill both around the canisters and in the tunnel. a. What are the most important properties that the backfill should possess to ensure isolation and buffering of the waste from the outside environment? b. What clay material would you propose for this application and under what conditions would you place it?
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CHAPTER 10
10.1
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Volume Change Behavior
INTRODUCTION
Volume changes in soils are important because they determine settlements due to compression, heave due to expansion, and contribute to deformations caused by shear stresses. Changes in volume cause changes in strength and deformation properties that, in turn, influence stability. Volume changes are induced by changes in applied stresses, chemical and moisture environments, and temperature. The effects of stress changes are generally the most important and have been the most studied. In this chapter, factors contributing to volume change are discussed, and their relative importance is considered. Emphasis is on consolidation and swelling. Shrinkage is a special case of consolidation, wherein the consolidation pressure is developed internally from capillary menisci and the surface tension of water. Reader familiarity with the phenomenological aspects of compression and swelling as ordinarily treated in geotechnical engineering is assumed, as described by the idealized void ratio–effective pressure relationships shown in Fig. 10.1. Unless otherwise noted, the discussion in this chapter is based on the behavior in one-dimensional deformation conditions. Although the mathematics and numerical analyses needed for quantification of volume changes in two or three dimensions are more complex, the phenomena and processes that control the behavior are the same.
10.2 GENERAL VOLUME CHANGE BEHAVIOR OF SOILS
Soil void ratio is normally in the range of about 0.5 to 4.0, as shown in Fig. 10.2. Although the range of pressures of interest in most cases (up to a few hundred kilopascals) is relatively small on a geological scale,
the void ratios encompass virtually the full range from fresh sediments to shale. Mechanical and chemical changes accompany and influence the densification process. In general, the void ratio–effective pressure relationship is related to grain size and plasticity in the manner shown by Fig. 10.2b. Particle size and shape, which together determine specific surface area, are the most important factors influencing both the void ratio at any pressure and the effects that physicochemical and mechanical factors have on consolidation and swelling (Meade, 1964). Particle size and shape are direct manifestations of composition, with increasing colloidal activity and expansiveness associated with decreasing particle sizes. Values of compression index, Cc, defined in Fig. 10.1, from less than 0.2 to as high as 17 for specially prepared sodium montmorillonite under low pressure have been measured, although values less than 2.0 are usual. The compression index for most natural clays is less than 1.0, with a value less than 0.5 in most cases. The swelling index, Cs, is less than the compression index, usually by a substantial amount, as a result of particle rearrangement during compression that does not recur during expansion. After one or more cycles of recompression and unloading accompanied with some irrecoverable volumetric strain, the reloading and swelling indices measured in the preyield region become nearly equal. Swelling index values for three clay minerals, muscovite, and sand are listed in Table 10.1. For undisturbed natural soils the swelling index values are usually less than 0.1 for nonexpansive materials to more than 0.2 for expansive soils. The compressibility of dense sands and gravels is far less than that of normally consolidated clays; nonetheless, volume changes under high pressures may be substantial in granular materials as shown in Fig. 10.3. At low stress levels, the compressibility of sand de325
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VOLUME CHANGE BEHAVIOR
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326
Figure 10.1 Idealized void ratio–effective stress relationships for a compressible soil.
Figure 10.2 Compression curves for several soils (redrawn from Lambe and Whitman,
1969).
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PRECONSOLIDATION PRESSURE
Mineral (1) Kaolinite
Illite
Smectite
Muscovite
Sand
Swelling Index Values for Several Minerals Pore Fluid, Adsorbed Cations, Electrolyte Concentration, in Gram Equivalent Weights per Liter (2)
Void Ratio at Effective Consolidation Pressure of 100 psf (5 kPa) (3)
Water, sodium, 1 Water, sodium, 1 ⫻ 10⫺4 Water, calcium, 1 Water, calcium, 1 ⫻ 10⫺4 Ethyl alcohol Carbon tetrachloride Dry air Water, sodium, 1 Water, sodium, 1 ⫻ 10⫺3 Water, calcium, 1 Water, calcium, 1 ⫻ 10⫺3 Ethyl alcohol Carbon tetrachloride Dry air Water, sodium, 1 ⫻ 10⫺1 Water, sodium, 5 ⫻ 10⫺4 Water, calcium, 1 Water, calcium, 1 ⫻ 10⫺3 Ethyl alcohol Carbon tetrachloride Water Carbon tetrachloride Dry air
0.95 1.05 0.94 0.98 1.10 1.10 1.36 1.77 2.50 1.51 1.59 1.48 1.14 1.46 5.40 11.15 1.84 2.18 1.49 1.21 2.19 1.98 2.29
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Table 10.1
327
Swelling Index (4) 0.08 0.08 0.07 0.07 0.06 0.05 0.04 0.37 0.65 0.28 0.31 0.19 0.04 0.04 1.53 3.60 0.26 0.34 0.10 0.03 0.42 0.35 0.41 0.01 to 0.03
From Olson and Mesri (1970). Reprinted with permission of ASCE.
pends on initial density. However, at higher stress levels, yielding is observed, and the compression curves for a given sand at different initial densities merge into a unique compression line. Particle crushing is the primary cause of the large volumetric strains that occur along the normal compression line. The yield stress is related to particle tensile strength (McDowell and Bolton, 1998; Nakata et al., 2001). Compressibility data for several sands, gravels, and rockfills are shown in Fig. 10.4. At a pressure of 700 kPa (100 psi) a compression of 3 percent is common, and values as high as 6.5 percent have been measured. Interestingly, the compacted shells of a rockfill dam are sometimes more compressible than the compacted clay core. 10.3
Figure 10.3 Compressibility of three sands under high pressure (from Pestana and Whittle, 1995).
Copyright © 2005 John Wiley & Sons
PRECONSOLIDATION PRESSURE
Three different relationships between the present overburden effective stress v0 and the maximum past over-
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VOLUME CHANGE BEHAVIOR
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328
Figure 10.4 Field compressibility of earth and rockfill materials (from Wilson, 1973). Re-
printed with permission from John Wiley & Sons.
burden effective stress vm are possible for the soil at a site:
1. vm ⬍ v0 —Underconsolidated The soil has not yet reached equilibrium under the present overburden owing to the time required for consolidation. Underconsolidation can result from such conditions as deposition at a rate faster than consolidation, rapid drop in the groundwater table, insufficient time since the placement of a fill or other loading for consolidation to be completed, and disturbance that causes a structure breakdown and decrease in effective stress. 2. vm ⫽ v0 —Normally Consolidated The soil is in effective stress equilibrium with the present overburden effective stress. Surprisingly few, if any, deposits have been encountered that are exactly normally consolidated. Most are at least very slightly overconsolidated as a result of processes of the type summarized in Table 10.2. Underconsolidated soil behaves as normally consolidated soil until the end of primary con-
Copyright © 2005 John Wiley & Sons
solidation, and overconsolidated clays become normally consolidated clays when loaded beyond their maximum past pressure. 3. vm ⬎ v0 —Overconsolidated or Preconsolidated The soil has been consolidated, or behaves as if consolidated, under an effective stress greater than the present overburden effective stress. Characteristics, causes, and mechanisms of preconsolidation are summarized in Table 10.2. Cemented or structured soil may behave like an overconsolidated soil; the yield pressure is larger than the maximum past pressure even though the soil has not experienced a pressure greater than the present overburden stress.
Accurate knowledge of the maximum past consolidation pressure is needed for reliable predictions of settlement and to aid in the interpretation of geologic history. If the recompression to virgin compression curve does not show a well-defined break, such as at point B in Fig. 10.1, the preconsolidation pressure is difficult to determine. Gentle curvature of the com-
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329
PRECONSOLIDATION PRESSURE
Table 10.2
Preconsolidation Mechanisms for Horizontal Deposits Under Geostatic Stresses
Category
1. Changes in total vertical stress (overburden, glaciers, etc.) 2. Changes in pore pressure (water table, seepage conditions, etc.) 1. Drying due to evaporation, vegetation, etc.
In situ Stress Condition
Uniform with constant p ⫺ v0
K0, but value at given OCR varies for reload versus unload
Remarks/References Most obvious and easiest to identify
(except with seepage)
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A. Mechanical one dimensional
Description
Stress History Profile
B. Desiccation
C. Drained creep (aging)
D. Physicochemical
2. Drying due to freezing 1. Long-term secondary compression
1. Natural cementation due to carbonates, silica, etc. 2. Other causes of bonding due to ion exchange, thixotropy, ‘‘weathering’’ etc.
Often highly erratic
Can deviate from K0, e.g., isotropic capillary stresses
Drying crusts found at surface of most deposits; can be at depth within deltaic deposits
Uniform with constant p / v0
K0, but not necessarily normally consolidated value
Leonards and Altschaeffl (1964); Bjerrum (1967)
Not uniform
No information
Poorly understood and often difficult to prove. Very pronounced in eastern Canadian clays, e.g., Sangrey (1972), Bjerrum (1973), and Quigley (1980)
After Jamiolkowski et al., 1985.
pression curve over the preconsolidation pressure range is characteristic of sands, weathered clays, heavily overconsolidated clays, and disturbed clays. The rate of loading and time have significant effects on the equilibrium void ratio–effective stress relationship, especially for sensitive structured clays as shown in Fig. 10.5. It is not surprising, therefore, that rate of loading and time influence also the measured preconsolidation pressure. The preconsolidation pressure decreases as the duration of load application increases and as the rate of deformation decreases, as shown by
Copyright © 2005 John Wiley & Sons
Fig. 10.6 from Leroueil et al. (1990). The higher values of apparent preconsolidation pressure associated with the faster rates of loading reflect the influences of the viscous resistance of the soil structure. The ratedependent value of preconsolidation pressure, p can be approximated by (e.g., Leroueil et al., 1985) log( p) ⫽ A ⫹ B log(˙a)
(10.1)
where ˙ a is the vertical strain rate in one-dimensional consolidation, and A and B are fitting parameters. Typ-
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10
VOLUME CHANGE BEHAVIOR
80 70
Constant Rate of Strain Tests at 5 °C Constant Rate of Strain Tests at 25 °C Constant Rate of Strain Tests at 35 ° C
5 °C
Creep Tests at 25 °C
25 °C 35 °C
60 50 40
Conventional Consolidation Test at 25 °C (After 24 Hours of Loading)
Conventional Consolidation Test at 25 °C (At End of Primary Consolidation State)
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Preconsolidation Pressure (kPa)
100 90
30 10 -9
Figure 10.5 Compression curves corresponding to different times after the completion of primary consolidation.
10 -8
10 -7 10 -6 10-5 Volumetric Strain Rate (s-1)
10-4
Figure 10.7 Effect of compression strain rate and temperature on measured preconsolidation pressure of Berthierville clay (from Leroueil and Marques, 1996).
dictions of field behavior are possible only if undisturbed samples or in situ tests are used for determination of properties. The following factors, several of which are treated in more detail in later sections, are important in determining resistance to volume change. Physical Interactions Between Particles Physical interactions include bending, sliding, rolling, and crushing of soil particles. Physical interactions are more important than physicochemical interactions at high pressures and low void ratios. Physicochemical Interactions Between Particles
Figure 10.6 Effect of load duration increment and deformation rate on compression curves (Leroueil et al., 1990). (a) Ottawa clay (data from Crawford, 1964). (b) Ba¨ckebol clay (data from Sa¨llfors, 1975).
ical examples of the fitting for the results of different types of compression tests on Berthierville clay are shown in Fig. 10.7 (Leroueil and Marques, 1996). The effect of temperature on preconsolidation pressure can also be seen, and this is further discussed in Section 10.12. The data in Figs. 10.6 and 10.7 also illustrate the difficulties and uncertainties in determining the true in situ conditions from the results of laboratory tests. 10.4 FACTORS CONTROLLING RESISTANCE TO VOLUME CHANGE
Both compositional and environmental factors influence volume change, so meaningful quantitative pre-
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These interactions depend on particle surface forces that are responsible for double-layer interactions, surface and ion hydration, and interparticle attractive forces. Physicochemical interactions are most important in the formational stages of fine-grained soil deposits when they are at low pressures and high void ratios. Chemical and Organic Environment Chemical precipitates cement particles together. Organic matter influences surface forces and water adsorption properties, which, in turn, increase the plasticity and compressibility. Expansion of pyrite minerals in some shales and other earth materials as a result of oxidation caused by exposure to air and water has been the source of significant structural damage (Bryant et al., 2003). Temperature changes may cause changes in hydration states of some salts leading to volume changes. Mineralogical Detail Small differences in certain characteristics of expansive clay minerals can have major effects on the swelling of a soil. Fabric and Structure Compacted expansive soils with flocculent structures may be more expansive than those with dispersed structures. Figure 10.8 is an ex-
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PHYSICAL INTERACTIONS IN VOLUME CHANGE
a decrease in effective stress. The responses of saturated soils to temperature change are analyzed in Section 10.12. Pore Water Chemistry Any change in the pore solution chemistry that depresses the double layers or reduces the water adsorption forces at particle surfaces reduces swell or swell pressure. An example of this is shown in Fig. 10.8, where increased electrolyte concentration in the water imbibed by a compacted clay resulted in reduced swelling. For soils containing only nonexpansive clay minerals, the pore water chemistry has relatively little effect on the compression behavior after the initial fabric has formed and the structure has stabilized under a moderate effective stress. This is in accordance with the principle of chemical irreversibility of clay fabric, discussed in Section 8.2. The leaching of normally consolidated marine clay at high water content, however, may be sufficient to cause a small reduction in volume owing to changes in interparticle forces (Kazi and Moum, 1973; Torrance, 1974). Stress Path The amount of compression or swelling associated with a given change in stress usually depends on the path followed. Loading or unloading from one stress to another in stages can give considerably different volume change behavior than if the stress change is done in one step. An example for swelling of a compacted sandy clay is shown in Fig. 10.10. Each sample was placed under water after compaction and allowed to swell under different surcharge pressures. Further discussion of the stress path dependency on volume change is given in Section 10.11 and Chapter 11.
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Figure 10.8 Effect of structure and electrolyte concentration
of absorbed solution on swelling of compacted clay (adapted from Seed et al. 1962a).
ample. At pressures less than the preconsolidation pressure, the soil with a flocculent structure was less compressible than the same soil with a dispersed structure. The reverse is generally true for pressures greater than the preconsolidation pressure. Stress History An overconsolidated soil is less compressible but more expansive than the same material initially at the same void ratio but normally consolidated. This is illustrated in Fig. 10.9. If anisotropic stress systems have been applied to a soil in the past, then anisotropic compression and swelling characteristics usually result. Temperature Increase in temperature usually causes a decrease in volume for a fully drained soil. If drainage is prevented, increase in temperature causes
Figure 10.9 Comparison of compressibility and swell characteristics for normally consolidated (compression curve) and overconsolidated (rebound and recompression curves) soil.
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331
10.5 PHYSICAL INTERACTIONS IN VOLUME CHANGE
Physical interactions between particles include bending, sliding, rolling, and crushing. In general, the coarser the gradation, the more important are physical particle interactions relative to chemically induced particle interactions. Deformation resistance developed by particle rolling and sliding is discussed in Chapter 11. Particle bending is important in soils with platy particles. Even small amounts of mica in coarse-grained soils can greatly increase the compressibility. Mixtures of a dense sand having rounded grains with mica flakes can even duplicate the form of the compression and swelling curves of clays, as shown in Fig. 10.11. Chattahoochie River sand with a mica content of 5 percent is twice as compressible as the same sand with no mica (Moore, 1971). On the other hand, a well-graded soil may be little affected in terms of compressibility by the addition of mica. Further discussion of the me-
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VOLUME CHANGE BEHAVIOR
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332
Figure 10.10 Effect of unloading stress path on swelling of a compacted sandy clay (Seed
et al. 1962a).
Figure 10.11 Comparison of compression and swelling curves for several clays and sand–
mica mixtures (from Terzaghi, 1931).
chanical behavior of mica–sand mixtures is given in Chapter 11. Cross-linking adds rigidity to soil fabric, especially clays containing platy particles. Particles and particle groups act as struts whose resistance depends both on their bending resistance and on the strengths of the junctions at their ends. According to van Olphen (1977), cross-linking is important even in ‘‘pure clay’’ systems, where the confining pressure is sometimes in-
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terpreted, probably erroneously, as balanced entirely by interparticle repulsion. The importance of grain crushing increases with increasing particle size and confining stress magnitude. Particle breakage is a progressive process that starts at relatively low stress levels because of the wide dispersion of the magnitudes of interparticle contact forces. The number of contacts per particle depends on gradation and density, and the average contact force in-
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PHYSICAL INTERACTIONS IN VOLUME CHANGE
Studies of compressibility and grain crushing in sands and gravels under isotropic and anisotropic triaxial stresses up to 20 MPa showed the following (Lee and Farhoomand, 1967): 1. Coarse granular soils compress more and have more particle breakage than fine granular soils. A comparison of gradation curves before and after isotropic compression is shown in Fig. 10.12. 2. Soils with angular particles compress more and undergo more particle crushing than soils with rounded particles. 3. Uniform soils compress and crush more than well-graded soils with the same maximum grain size. 4. Under a given stress, compression and crushing continue indefinitely at a decreasing rate. 5. Volume change during compression depends primarily on the major principal stress and is independent of the principal stress ratio. 6. The higher the principal stress ratio (Kc ⫽ 1c / 3c) during consolidation, the greater the amount of grain crushing.
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creases greatly with particle size, as summarized in Table 10.3. Statistical analyses of the probable frequency distribution of contact forces show large deviations from the mean (Marsal, 1973). An example of this obtained from a numerical simulation of a particle assemblage is presented in Chapter 11. Unstressed, or idle particles, can occupy voids between larger particles or particle arches associated with strong force chains, as discussed in Chapter 7. The percentage of idle particles depends on gradation, fabric, void ratio, stress history, and stress level. In soils containing idle particles, particulate mechanics analyses of behavior that depend on such quantities as average number of particles per unit area or per unit volume, average number of contacts per particle, and the like lose their relevance unless the analyses allow for their existence. The resistance to grain crushing or breakage depends on the strength of the particles, which, in turn, depends on mineralogy and the soundness of the grains. Failure may be by compression, shear, or in a split tensile mode. Quartz grains are more resistant than feldspar, but there is greater variability in crushing and splitting resistance with changes in particle size for quartz than for feldspar. The amount of grain crushing to be expected for rockfills and gravels is summarized in Table 10.4. In this table, Bq is the proportion of the solid phase by weight that will undergo breakage, and qi is the concentration of solids [Vs /V ⫽ 1/(1 ⫹ e)].
Table 10.3 Soils
Particle crushing results in increase in fines content with increasing confining pressure. An example of the change in particle size distribution curve with increasing confining pressure is shown in Fig. 10.13 (Fukumoto, 1992). Particle crushing can be quantified by Hardin’s (1985) relative breakage parameter Br, which
Contacts and Contact Forces in Granular
Soil Type
Loose uniform gravel Dense uniform gravel Well-graded gravel, 0.8 mm ⬍ d ⬍ 200 mm Medium sand Gravel Rockfill, d ⫽ 0.7 m
Grain Contacts/ Particle (Range)
Grain Contacts/ Particle (Mean)
4–10
6.1
4–13
7.7
5–1912
5.9
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333
Average Contact Force for ⫽ 1 atm (N)
10⫺2 10 104
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VOLUME CHANGE BEHAVIOR
Table 10.4
Grain Crushing in Rockfills and Gravels
Samples
Grain Size Distribution
Crushing Strength of Grains
Particle Breakage Bqqia
High
0.02–0.10 for 5 1f 80 kg/cm2
El infiernillo silicified conglomerate
Well-graded rockfills and gravels
Pinzandaran sand and gravel San Francisco basalt (gradations 1 and 2) El infiernillo diorite
Somewhat uniform rockfills
High
Well-graded rockfills
Low
Uniform rockfill produced by blasting metamorphic rocks (Cu ⬍ 5)
Low
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334
El granero slate (gradation A) Mica granitic– gneiss (gradation X) Mica granitic– gneiss (gradation Y)
0.10–0.20 for 5 1f 80 kg/cm2
Increases with 1f ⬊ maximum value ⫽ 0.30
Bq is grain breakage parameter; qi is initial concentration of solids; 1f is major principal stress at failure. From Marsal, 1973. Reprinted with permission of John Wiley & Sons. a
Figure 10.12 Comparison of crushing of soils with different initial grain sizes for isotropic compression under 8 MPa (from Lee and Farhoomand, 1967). Reproduced with permission from the National Research Council of Canada.
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FABRIC, STRUCTURE, AND VOLUME CHANGE
335
100 80 60
One dimensional consolidation to σ' v=14000 psi (97MPa) σ' v=8000 psi (55MPa) σ' v=5000 psi (34MPa)
40
σ' v=1000 psi (6.9MPa)
Initial
20
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Percent Finer by Weight
Ottawa Sand : Initial Grading 0.42-0.82 mm
0
0.01
0.1
1
Grain Size (mm)
(a)
Percent Finer by Weight
100 80 60 40
Landstejin Sand : Initial Grading 4-7 mm Isotropically consolidated to 490 kPa and then sheared in triaxial compression to axial strain of 24% Isotropically consolidated to 98 kPa and then sheared in triaxial compression to axial strain of 24% Isotropically consolidated to 980 kPa
Initial
20
0 0.01
0.1
1
10
Grain Size (mm)
(b)
Figure 10.13 Change in particle size distribution curve with increasing confining pressure:
(a) Ottawa sand and (b) Landstejn sand (from Fukumoto, 1992).
is defined in Fig. 10.14. The increase in Br with isotropic compression pressure is shown in Fig. 10.15 for Dog’s Bay carbonate sand (Coop and Lee, 1993). The figure also shows the increase in Br at critical-state failure (discussed further in Chapter 11). A unique particle breakage characteristic at failure is obtained irrespective of shearing conditions (i.e., undrained triaxial, drained triaxial, or constant mean pressure shearing). Aggregates of clay mineral particles are often observed in clays, and intact aggregate clusters of clay particles can be considered as the smallest units controlling the macroscopic mechanical behavior. These aggregate clusters behave in some ways similarly to granular particles (e.g., Barden, 1973, and Collins and McGown, 1974). It can be conceptualized that the con-
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solidation of these clays is related to sequential breakage of clay aggregates into smaller aggregates as consolidation pressure increases (Bolton, 2000).
10.6 FABRIC, STRUCTURE, AND VOLUME CHANGE
Collapse, shrinkage, and compression are due to particle rearrangements from shear and sliding at interparticle contacts, disruption of particle aggregates, and grain crushing. Thus, both the arrangement of particles and particle groups and the forces holding them in place are important. Swelling depends strongly on physicochemical interactions between particles, but fabric also plays a role.
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An illustration of such differences is provided by the data in Table 10.5, where dry void ratios of several undisturbed and remolded clays are listed. In each case, the clay was dried from its natural water content either undisturbed or after thorough remolding. The substantially lower dry void ratios for the remolded samples indicate greater shrinkage than in the undisturbed samples. Structure anisotropy on a macroscale may be reflected by anisotropic shrinkage. For preferred orientation of platy particles parallel to the horizontal, vertical shrinkage on drying is greater than lateral shrinkage. For example, the vertical shrinkage of Seven Sisters clay was three times greater than the horizontal shrinkage (Warkentin and Bozozuk, 1961). Collapse
Figure 10.14 Definition of relative breakage parameter Br
by Hardin (1985).
Collapse, as a result of wetting under constant total stress, is an apparent contradiction to the principal of effective stress discussed in Chapter 7. The addition of water increases the pore water pressure and reduces the effective stress; hence, expansion might be expected. The apparent anomaly of volume decrease under decreased effective stress is because of the application of continuum concepts to a phenomenon that is controlled by particulate behavior at contact levels for unsaturated soils. Collapse requires: 1. An open, low-density, partly unstable, partly saturated fabric 2. A high enough total stress that the structure is metastable 3. A strong enough clay binder or other cementing agent to stabilize the structure when dry
Figure 10.15 Increase in Br with confining pressure under
isotropic compression (NCL) and at critical state (CSL) achieved by standard triaxial compression shearing (both drained and undrained) and constant mean pressure shearing.
Shrinkage
Drying shrinkage of fine-grained soils is caused by particle movements resulting from pore water tensions developed by capillary menisci. If two samples of clay are at the same initial water content but have different fabrics, the one that is the more deflocculated and dispersed shrinks the most. This is because the average pore sizes are smaller in the deflocculated sample, thus allowing greater capillary stresses, and because of easier relative movements of particles and particle groups.
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When water is added to a collapsing soil in which the silt and sand grains are stabilized by clay coatings or buttresses, the effective stress in the clay is reduced, the clay swells, becomes weaker, and contacts fail in shear, thereby allowing the coarser silt and sand particles to assume a denser packing. Thus, compatibility with the principle of effective stress is maintained on a microscale. Compression
Sands In Chapter 8 it was shown that the volume changes during the shear of samples of sand at the same void ratio but with different initial fabrics can be different. Different volume change tendencies for different fabrics developed resulting from different methods of sample preparation have also manifested themselves by differences in liquefaction behavior under undrained cyclic loading (see Fig. 8.22).
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FABRIC, STRUCTURE, AND VOLUME CHANGE
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Table 10.5 Void Ratios of Several Clays After Drying in the Undisturbed and Remolded States
Boston blue Boston blue Fore River, Maine Goose Bay, Labrador Chicago Beauharnois, Quebec St. Lawrence
35.6 37.5 41.5 29.0 39.7 61.3 53.6
Sensitivity
Dry Void Ratio Undisturbed
Dry Void Ratio Remolded
6.8 5.8 4.5 2.0 3.4 5.5 5.4
0.69 0.75 0.65 0.60 0.65 0.76 0.79
0.50 0.53 0.46 0.55 0.55 0.70 0.66
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Clay
Natural Water Content (%)
The compression behavior of a natural intact cemented calcarenite sand is shown in Fig. 10.16 (Cuccovillo and Coop, 1997). Similarly to structured clays, the initial compressibility before yielding is stiff due to cementation. If the cementation is stronger than the particle crushing strength, the compression line will lie to the right of the normal compression line of the uncemented reconstituted sand. If the cementation is weaker than the particle crushing strength, the compression curve will merge gradually toward that of the uncemented sand before yielding (Cuccovillo and Coop, 1999). This highlights the importance of relative
p (kPa)
100
2.20
1000
10000
Intact IB
2.00 1.80
10000
Intact Reconstituted
ν
NCL
1.60 2.40 1.20 4
5
6
7
8 9 In p(kPa)
10
11
12
Figure 10.16 Isotropic compression curves of intact and reconstituted calcarenite sand specimens (from Cuccovillo and Coop, 1997).
Copyright © 2005 John Wiley & Sons
strengths of cementation bonding and particles on the compression behavior of structured soils. Clays Compression curves obtained by odometer tests on undisturbed and remolded Leda (Champlain) clay, illite, and kaolinite are shown in Fig. 10.17. Liquidity index is used as an ordinate, and the sensitivity curves from Fig. 8.49 are superimposed. Curve A is for undisturbed Leda clay at an initial water content corresponding to a liquidity index of 1.82. Because the sensitivity contours were developed for normally consolidated clays, they cannot be used to estimate sensitivity for stresses less than the preconsolidation pressure. After the preconsolidation stress has been exceeded the curve cuts sharply across the sensitivity contours, indicating a large decrease in sensitivity as the structure is broken down by compression. Curve B is for kaolinite remolded at a liquidity index of 2.06. The early part of the consolidation curve is not shown in Fig. 10.17. Immediately after remolding at high water content the effective stress is very low, and the sensitivity is equal to 1. Curve B shows that consolidation results in an increase in sensitivity to a maximum of about 15 to 18, at an effective consolidation pressure of about 20 kPa. At this point, the interparticle and interaggregate shear stresses caused by the applied compressive stress begin to exceed the bond strengths, the degree of structural metastability decreases, and the sensitivity decreases. Curve D is for kaolinite remolded at a liquidity index of 0.98. It differs considerably from curve B. This is consistent with the results of other studies that show that the compression behavior, and therefore also the structure, are different for a given clay remolded at
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VOLUME CHANGE BEHAVIOR
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338
Figure 10.17 Change in sensitivity with consolidation for various clays.
different water contents, for example, Morgenstern and Tchalenko (1967b). Significantly lower sensitivity is developed in the kaolinite of curve D than that of curve B. These observations show that both the concentration of clay in suspension and the rate of sediment accumulation are important in determining the initial structure of clay deposits. At high pressures, both curves tend to merge together, indicating that the initial fabrics have been destroyed. Curve E is for a well-graded illitic clay remolded at a liquidity index of 1.36. The consolidation curve indicates a low sensitivity at all consolidation pressures. Results of strength tests showed that the actual sensitivity ranged from 1.0 to 2.6. Curve C is for Leda clay remolded at a liquidity index of 1.82. The sensitivity increases from 1 to about 8 with reconsolidation, indicating development of metastability after remolding and recompression. The sensitivity decreases at high pressures as convergence with curve A is approached. All of the above findings are consistent with the principles stated in Section 8.13.
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Swelling
The structure influences swelling of fine-grained soils that is initiated by reduction of effective stress by unloading and/or addition of water. For example, an expansive soil that is compacted dry of optimum water content can swell more than if compacted to the same density wet of optimum (Seed and Chan, 1959). This difference cannot be accounted for in terms of differences in initial water content and, therefore, must be ascribed to differences in structure. A swell sensitivity has been observed in some clays wherein the swelling index for the remolded clay is higher than that of the same clay undisturbed. The increased swelling of the disturbed material can result both from the rupture of interparticle bonds that inhibit swelling in the undisturbed state and from differences in fabric. Old, unweathered, overconsolidated clays may be particularly swell sensitive. Swell sensitivities as high as 20 were measured in one case (Schmertmann, 1969).
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OSMOTIC PRESSURE AND WATER ADSORPTION INFLUENCES ON COMPRESSION AND SWELLING
339
10.7 OSMOTIC PRESSURE AND WATER ADSORPTION INFLUENCES ON COMPRESSION AND SWELLING
⫽ kT
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Adsorption of cations on clays, the formation of double layers, and water adsorption on soil surfaces generate repulsive forces between particles as described in Chapters 6 and 7. Calculation of interparticle repulsions due to interacting double layers may be done in more than one way; the osmotic pressure concept is convenient and most widely used. By this approach, the pressure that must be applied to prevent movement of water either in or out of clay is determined as a function of particle spacings expressed in terms of void ratio or water content. The concept of osmotic pressure is illustrated by Fig. 10.18. The two sides of the cell in Fig. 10.18a are separated by a semipermeable membrane through which solvent (water) may pass but solute (salt) cannot. Because the salt concentration in solution is greater on the left side of the membrane than on the right side, the free energy and chemical potential of the water on the left are less than on the right.1 Because solute cannot pass to the right to equalize concentrations due to the presence of the membrane, solvent passes into the chamber on the left. The effect of this is twofold as shown by Fig. 10.18b. First, the solute concentration on the left is reduced and that on the right side is increased, which reduces the concentration imbalance between the two chambers. Second, a difference in hydrostatic pressure develops between the two sides. Since the free energy of the water varies directly with pressure and inversely with concentration, both effects reduce the imbalance between the two chambers. Flow continues through the membrane until the free energy of the water is the same on each side. It would be possible in a system such as that shown by Fig. 10.18a to completely prevent flow through the membrane by applying a sufficient pressure to the solution in the left chamber, as shown by Fig. 10.18c. The pressure needed to exactly stop flow is termed the osmotic pressure , and it may be calculated, for dilute solutions, by the van’t Hoff equation, which was introduced in Section 9.13:
冘(n
iA
⫺ niB) ⫽ RT
冘(c
iA
⫺ ciB)
(10.2)
where k is the Boltzmann constant (gas constant per molecule), R is the gas constant per mole, T is the
1
Formal treatment of the concepts stated here and derivation of Eq. (10.1) are given in standard texts on chemical thermodynamics.
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Figure 10.18 Osmotic pressure: (a) Initial condition: no
equilibrium, (b) final condition: equilibrium, and (c) osmotic pressure equilibrium.
absolute temperature, ni is the concentration (particles per unit volume), and ci is the molar concentration. Thus, the osmotic pressure difference between two solutions separated by a semipermeable membrane is directly proportional to the concentration difference. In a soil, there is no true semipermeable membrane separating regions of high- and low-salt concentration. The effect of a restrictive membrane is created, however, by the influence of the negatively charged clay surfaces on the adsorbed cations. Because of the attraction of adsorbed cations to particle surfaces, the cations are not free to diffuse, and concentration differences responsible for osmotic pressures are developed whenever double layers on adjacent particles
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overlap. The situation is shown schematically in Fig. 10.19. The difference in osmotic pressure midway between particles and in the equilibrium solution surrounding the clay is the interparticle repulsive pressure or swelling pressure Ps. It can be expressed in terms of midplane potentials according to the following equation (see Section 6.11): Ps ⫽ p ⫽ 2n0kT(cosh u ⫺ 1)
where ca is the midplane anion concentration, and c⫹ 0 and c⫺ 0 are the equilibrium solution concentrations of cations and anions. At equilibrium in dilute solutions cc ca ⫽ c0⫹ c0⫺ ⫽ c02 ⫺ because c⫹ 0 ⫽ c0 . Thus Eq. (10.5) becomes
(10.3) Ps ⫽ RTc0
冉
冘(c
ic
⫺ ci 0)
(10.4)
For single cation and anion species of the same valence ⫺ Ps ⫽ RT(cc ⫹ ca ⫺ c⫹ 0 ⫺ c0 )
(10.5)
(10.7)
Midplane concentrations can be determined using the relationships in Chapter 6. Equation (10.7) assumes parallel flat plates and may be written in terms of void ratio for saturated clay. The water content w, in terms of weight of water per unit weight of soil solids, divided by the specific surface of soil solids As gives the average thickness of water layer, which is half the particle spacing or d. Thus,
Figure 10.19 Mechanism of osmotic swelling pressure generation in clay.
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冊
cc c0 ⫹ ⫺2 c0 cc
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where n0 is the concentration in the external solution, and u is the midplane potential function. In terms of midplane cation and equilibrium solution concentrations cc and c0 (Bolt, 1956), Eq. (10.2) becomes Ps ⫽ ⫽ RT
(10.6)
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OSMOTIC PRESSURE AND WATER ADSORPTION INFLUENCES ON COMPRESSION AND SWELLING
d⫽
w w As
(10.8)
For saturated soil the void ratio is related to the water content by e ⫽ Gsw
systems that cover most of the moisture suction or overburden ranges of interest in soil mechanics or soil science are available (Collis-George and Bozeman, 1970). They are suitable for 兩 兩 ⫻ 4 ⫻ 10⫺5
冘c
(10.9)
d⫽
e Gs w As
(10.10)
Bolt (1955, 1956) showed that the double-layer equations (see Chapter 6) can be combined with Eq. (10.10) to give v(c0)1 / 2(x0 ⫹ d) ⫽ 2 ⫻
冕
/2
⫽0
冉冊 c0 cc
1/2
d (1 ⫺ (c0 /cc)2 sin2 )1/2
兩 兩 ⫽
冘c ⫺ 冘c m
0
(10.14)
⫺5
4 ⫻ 10
For homovalent and dication/monoanion systems, 兺cm is found from
(10.11)
in which v is the cation valence and distance x0 equals approximately 0.1/ v nm for illite, 0.2/ v nm for kaolinite, and 0.4/ v nm for montmorillonite. The parameter is given by ⫽ 2F 2 /DRT
(10.13)
where 兩 兩 is the swelling pressure or matric suction (see Section 7.12) measured in centimeters of water. Since the sum of the applied constraint 兩 兩 in concentration units and the external solution concentration must equal the midplane concentration, the pressure or suction is given by
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where Gs is the specific gravity of solids. Substituting Eq. (10.9) into Eq. (10.8) gives
20
0
(10.12)
in which F is the Faraday constant, R is the gas constant, and T is the temperature. Combinations of (Ps /RTc0) and v(c0)1/2(x0 ⫹ e/ Gs w As) that satisfy Eqs. (10.7) and (10.11) are given in Table 10.6. These values may be used to calculate theoretical curves of void ratio versus pressure for consolidation or swelling. For any value of log[Ps /(RTc0)] the swelling pressure may be calculated. The void ratio can be computed from the corresponding value of v(c0)1/2(x0 ⫹ e/Gs w As). For a given soil, Ps depends completely on cc and c0 and those factors that cause cc to be large relative to c0; for example, low c0, low valence of cation, and high dielectric constant, cause high interparticle repulsions, high swelling pressures, and large physicochemical resistance to compression. It is apparent from the values in Table 10.6 that the dominating influence on swelling pressure at any given void ratio is the specific surface area, which is determined mainly by mineralogy and particle size. The preceding relationships were developed for soils containing a single electrolyte, and they assume ideal behavior in accord with the DLVO theory as developed in Chapter 6. Approximate equations for mixed-cation
Copyright © 2005 John Wiley & Sons
v()1/2
冉 冊
e ⫽ Gs As
冪冘c
⫺
2
冉
m
冘c 冊
1/2
–41 %2 ⫹
m
(10.15)
where ⬇ 1.0 ⫻ 1015 cm/mmol at 20C and % is the double-layer charge in meq/cm2. For dilute concentrations in the external solution, Eqs. (10.14) and (10.15) reduce to 兩 兩 ⫽ 0.25 ⫻ 105
2 v2(e/Gs As)2
(10.16)
For mixed-cation heterovalent systems, 兺cm is given by v()1/2
冉 冊
e ⫽ Gs As
冉冘 冊 冉 再 冋冘 冒冉 1/2
cm
⫺cos⫺1 1/a 1⫺
⫻
冘c 冊册
–14 %2⫹
cm
1/2
m
冎冊
1/2
cm
(10.17)
The value of a in Eq. (10.17) is given by a⫽
⫹⫹ ⫹⫹ ⫹ ⫹⫹ 2 1/2 2cm ⫺ (c⫹ m ⫹ cm ) ⫹ [4c mcm ⫹ (cm ⫹ cm ) ] 2cm
(10.18)
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VOLUME CHANGE BEHAVIOR
Table 10.6 Relation Between the Distance Variable Expressed as a Function of the Void Ratio and the Swelling Pressure of Pure Clay Systema
log Ps /(RTc0)
0.050 0.067 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.801 0.902
3.596 3.346 2.993 2.389 2.032 1.776 1.573 1.405 1.258 1.130 1.012
v(c0)1/2 (x0 ⫹ e/ Gs w As)
log Ps /(RTc0)
0.997 1.188 1.419 1.762 2.076 2.362 2.716 3.09 3.57 4.35
0.909 0.717 0.505 0.212 ⫺0.046 ⫺0.301 ⫺0.573 ⫺0.899 ⫺1.301 ⫺1.955
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v(c0)1/2 (x0 ⫹ e/ Gs w As)
v is the cation valence; is 8F /1000 DRT ⬇ 1015 cm/mmol for water at normal T; c0 is the concentration in bulk solution (mmol/cm3); x0 ⫽ 4/ vT ˚ for illite, 2/ v A ˚ for kaolinite, and 4/ v A ˚ for montmorillonite; e is ⬇ 1/ v A the void ratio; Gs w is the density of solids, As is the specific surface area of a
clay; Ps is the swelling pressure; R is the gas constant; T is the absolute temperature; F is the Faraday constant; and D is the dielectric constant. Adapted from Bolt (1956).
where cm is the midplane anion concentration. Since evaluation of Eq. (10.18) requires knowledge of the midplane concentrations of the different ions separately, the application of Eq. (10.17) is not as straightforward as is the case of Eqs. (10.13) and (10.14). Applicability of Osmotic Pressure Concepts
A reasonably clear understanding of how well the osmotic pressure concept can account for the compression and swelling behavior of fine-grained soils has been developed. Homoionic Cation Systems
Early testing of the applicability of the osmotic pressure theory was done using ‘‘pure clays’’ consisting of specially prepared, very fine grained clay minerals. Good agreement between theoretical and experimental values of interparticle spacing and pressure for montmorillonite with particles finer than 0.2 m in 10⫺4 NaCl solution is shown in Fig. 10.20. The first compression curves are above decompression and recompression curves because of cross-linking and nonparallel particle arrangements, that is, fabric effects, which are eliminated during the first compression cycle. Theoretical and experimental compression
Copyright © 2005 John Wiley & Sons
Figure 10.20 Relationship between particle spacing and pressure for montmorillonite (modified from Warkentin et al., 1957).
curves for sodium and calcium montmorillonite in 10⫺3 M electrolyte solutions are compared in Fig. 10.21. Agreement is fairly good as regards the influence of cation valence. However, the experimental curves are substantially above the theoretical curves. This may be caused by ‘‘dead’’ volumes of liquid resulting from terraced particle surfaces (Bolt, 1956).
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OSMOTIC PRESSURE AND WATER ADSORPTION INFLUENCES ON COMPRESSION AND SWELLING
343
Figure 10.21 Compression curves of Na-montmorillonite and Ca-montmorillonite, fraction ⬍2 m, in equilibrium with
10⫺3 M NaCl and CaCl2, respectively. The dashed lines represent the theoretical curves for As ⫽ 800 m2 / g (Bolt, 1956).
Osmotic pressure theory was used successfully for prediction of swelling pressure developed in opalinum shale, a Jurassic clay rock (Madsen and Mu¨llerVonmoos, 1985, 1989). Swelling pressure was predicted using Eq. (10.2) and compared with the measured values, with the results shown in Fig. 10.22. Particle spacings were calculated from specific surface area and water content. Agreement between theory and experiment has not been good for clays containing particles larger than a few tenths of a micrometer. The coarse fraction (0.2 to 2.0 m) of two bentonites gave swelling pressures less than predicted, whereas the fine fraction (⬍0.2 m) gave values close to theoretical, even though the charge densities of the two fractions were the same (Kidder and Reed, 1972). Compression and swelling curves for three size fractions of sodium illite are shown in Fig. 10.23. The discrepancies between theory and experiment are fairly large for the ⬍0.2-m fraction; nonetheless, the experimental curves are in the predicted relative positions (Fig. 10.23a). However, for samples containing coarser particles (Figs. 10.23b and 10.23c), the curves are in reverse order to theoretical prediction. This is because the compression was controlled by initial particle orientations and physical interactions between the larger particles rather than by osmotic repulsive pressures. The concentration of CaCl2 or MgCl2 has essentially no influence on the swelling of a 2-m fraction of illite, and the consolidation is influenced only by how the changes in concentration change the initial structure (Olson and Mitronovas, 1962). Factors in addition to clay particle size may also contribute to failure of the theory in natural soils. The DLVO theory that serves as the basis for determination
Copyright © 2005 John Wiley & Sons
Figure 10.22 Predicted and measured swelling pressures for
Opalinum shale (Madsen and Mu¨ller-Vonmoos, 1989).
of the midplane concentrations suffers from several deficiencies, as discussed in Chapter 6. In addition, physical particle interactions and the effects of interparticle short- and long-range forces such as van der Waals forces are neglected. Mixed-Cation Systems
Most soils contain mixtures of sodium, potassium, calcium, and magnesium in their adsorbed cation complex. Therefore, modifications of the double-layer and osmotic pressure equations for homoionic clays are required. The extent to which the resulting equations may be suitable depends on the structural status of the clay as well as on the particle size. Equations for mixed-cation systems are derived on the assumption that ions of all species are distributed uniformly over the clay surfaces in proportion to the amounts present. However, sodium and calcium ions may separate into distinct regions. This is termed demixing (Glaeser and Mering, 1954; McNeal et al., 1966; McNeal, 1970; Fink et al., 1971).
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VOLUME CHANGE BEHAVIOR
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344
Figure 10.23 Influence of NaCl concentration and particle size on compression and swelling
behavior of Fithian illite.
Observed behavior was good for several cases examined using a demixed ion model (5 out of 6) for values of exchangeable sodium percentage (ESP) less than about 50 (McNeal, 1970). Based on X-ray determinations of interplate spacings in montmorillonite (Fink et al., 1971) it appears that for 1. ESP ⬎ 50 percent, there is random mixing of Na⫹ and Ca2⫹ and unlimited swelling between all plates on addition of water. 2. 10 percent ⬍ ESP ⬍ 50 percent, there is demixing on interlayer exchange sites, with progres˚ sively more sets of plates collapsing to a 20-A repeat spacing with decrease in ESP. 3. ESP ⬍ 10 to 15 percent the interlayer exchange complex is predominantly Ca saturated, with Na ions on external planar and edge sites. Summary
Osmotic pressure (double-layer) theory fails to explain the first compression of most natural clays of the type encountered in geotechnical practice because of phys-
Copyright © 2005 John Wiley & Sons
ical particle interference and fabric factors related to particle size. The behavior is consistent with the principle of chemical irreversibility of clay fabric (Bennett and Hurlbut, 1986), which is discussed in Section 8.2. Nonetheless, when the physical and chemical influences of cation type on fabric and effective specific surface are taken into account, the behavior can be better understood, as illustrated, for example, by Di Maio (1996). For those cases in which fabric changes and interparticle interactions are small, such as swelling from a precompressed state, or for clays with very high specific surface area (very small particles) such as bentonite, the theory gives a reasonable description of swelling, at least qualitatively. Water Adsorption Theory of Swelling
An alternative to the osmotic pressure theory for clay swelling is that swelling is caused by surface hydration (Low, 1987, 1992). Interaction of water with clay surfaces reduces the chemical potential of the water, thereby generating a gradient in the chemical potential that causes additional water to flow into the system.
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INFLUENCES OF MINERALOGICAL DETAIL IN SOIL EXPANSION
The general relationships that describe the water properties as a function of water layer thickness and water content are given in Section 6.5. The swelling pressure (in atmospheres) for pure clays follows the following empirical relationship (Low, 1980): ( ⫹ 1) ⫽ B exp[ /w] ⫽ B exp[ki /(tw)]
(10.19)
ness would correspond to a water content of 400 percent. Thus, a material such as sodium montmorillonite (bentonite) with its very high specific surface would be expected to be expansive over a wide range of water contents, and experience shows clearly that it is. On the other hand, consider an illite or a smectite made up of quasi-crystals so that interlayer swelling is negligible. As both materials have surface structures that are essentially the same, it would be expected that the hydration forces should be similar. Thus, an adsorbed water layer of 5 nm would also be reasonable. However, the specific surface areas of pure illite and nonexpanded smectite are only about 100 m2 /g, which corresponds to a water content of 50 percent. For a pure kaolinite having a specific surface of 15 m2 /g, the water content would be only 7.5 percent for a 5-nmthick adsorbed layer. It is evident, therefore, that the specific surface dominates the amount of water required to satisfy forces of hydration. Except for very heavily overconsolidated clays and those soils that contain large amounts of expandable smectite, there is sufficient water present even at low water contents to satisfy surface hydration forces, and swelling is small. On the other hand, when the clay content is high and particle dissociation into unit layers is extensive, the effective specific surface area is large and swelling can be significant. The tendency for smectite dissociation into unit layers can be evaluated through consideration of double-layer interactions, with those conditions that favor the development of high repulsive forces, as discussed in Chapter 6, leading to greater dissociation.
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in which B and are constants characteristic of the clay, w is the water content, w is the density of water, t is the average thickness of water layers, and ki ⫽ / ( w As), where As is the specific surface. Equation (10.19) shows, as would be expected, that the lower the water content and, therefore, the smaller the water layer thickness, the higher is the swelling pressure. Whereas this approach can explain the swelling of pure clays accurately, the osmotic pressure theory cannot (Low, 1987, 1992). On the other hand, the influences of surface charge density, cation valence, electrolyte concentration, and dielectric constant, which have profound influences on swelling and swelling pressure, as shown in the previous section, are not directly accounted for by the hydration theory unless appropriate adjustments can be made for the influences of these factors on B, , and ki. An explanation that is consistent with both the influences of the double-layer/osmotic pressure theory and the water adsorption theory is as follows. Charge density and cation type influence the relative proportions of fully expandable and partially expandable layers in swelling clay. For example, calcium montmorillonite does not swell to interplate distances greater than about 0.9 nm where the particles stabilize by attractive interactions between the basal planes of the unit layers as influenced by exchangeable cations and adsorbed water (Norrish, 1954; Blackmore and Miller, 1962; Sposito, 1984). In the presence of high electrolyte concentrations or pore fluids of low dielectric constant, interlayer swelling is suppressed, and the effective specific surface is greatly reduced relative to that for the case where interlayer swelling occurs. The amount of water required to satisfy surface hydration is reduced greatly. A hydration water layer thickness on smectite surfaces of about 10 nm is needed to reach a distance beyond which the water properties are no longer influenced by surface forces (see Fig. 6.9), and Low (1980) indicates that the swelling pressure of montmorillonite is about 100 kPa for a water layer thickness of about 5 nm. For a fully expanding smectite having a specific surface area of 800 m2 /g, this latter water layer thick-
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345
10.8 INFLUENCES OF MINERALOGICAL DETAIL IN SOIL EXPANSION
In soils where swelling is attributable solely to the clay content, smectite or vermiculite are the most likely minerals because only these minerals have sufficient specific surface area so that there are unsatisfied water adsorption forces at low water contents. Details of structure and the presence of interlayer materials may have significant effects on the swelling properties of these minerals. In addition, the presence of certain other minerals in soils and shales, such as pyrite and gypsum, as well as geochemical and microbiological factors, may lead to significant amounts of swelling and heave. Details of all the phenomena go well beyond the scope of this book; however, a few examples are given in this section to illustrate their nature and importance.
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VOLUME CHANGE BEHAVIOR
Crystal Lattice Configuration Effects
Hydroxy Interlayering
The occurrence, formation, and properties of hydroxyl–cation interlayers (Fe–OH, Al–OH, Mg–H) have been studied regarding their effects on physical
Table 10.7 Influence of Lattice Charge on Expansion
Mineral Margarite Muscovite
1. Optimum conditions for interlayer formation are: a. Supply of A13⫹ ions b. Moderately acid pH (⬇5) c. Low oxygen content d. Frequent wetting and drying 2. Hydroxyaluminum is the principal interlayer material in acid soils, but Fe–OH layers may be present. 3. Mg(OH)2 is probably the principal interlayer component in alkaline soils. 4. Randomly distributed islands of interlayer material bind adjacent layers together. The degree of interlayering in soils is usually small (10 to 20 percent), but this is enough to fix the basal spac˚. ing of montmorillonite and vermiculite at 14 A 5. The cation exchange capacity is reduced by interlayer formation. 6. Swelling is reduced.
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Greatest swelling is observed for charge deficiencies in silicate layer structures of about one per unit cell as indicated in Table 10.7. Evidently, for layer silicates with sufficient isomorphous substitution to give charge deficiencies greater than 1.0 to 1.5 per unit cell, the balancing cations are so strongly held and organized in the interlayer regions that interlayer swelling is prevented. Within the range of charge deficiencies where swell is observed, there is no consistent relationship between charge, as measured by the cation exchange capacity, and the amount of swell (Foster, 1953, 1955). This finding is more consistent with the surface hydration model for clay swelling than with the osmotic pressure theory. An inverse correlation exists between free swell and the b dimension of the montmorillonite crystal lattice (Davidtz and Low, 1970). Differences in b dimension, which may be caused by differences in isomorphous substitution, evidently cause changes in water hydration forces. Furthermore, as the water content increases, so also does the b dimension, as shown in Fig. 6.5. Swelling ceases when the b dimension reaches 0.9 nm.
properties of expansive clays, for example, Rich (1968). Some aspects of interlayering between the basic sheets in the expansive clay minerals are:
Biotite Paragonite Hydrous mica and illite Vermiculite Montmorillonite Beidellite Nontronite Hectorite Pyrophyllite
Negative Charge per Unit Cell Tendency to Expand 4
None Only with drastic chemical treatment, if at all
2
⬎1.2
1.4–0.9
Expanding
1.0–0.6
Readily expanding
0
None
From Brindley and MacEwen (1953).
Copyright © 2005 John Wiley & Sons
Salt Heave
Some saline soils with high contents of salts can undergo changes in volume associated with hydration– dehydration phenomena. One example is the swelling of some soils containing large amounts of sodium sulfate (Na2SO4) found in and around the Las Vegas area of Nevada. When the temperature falls from above about 32C to below about 10C, the salt hydrates to Na2SO4 10H2O with accompanying increase in volume. This salt heave has been responsible for damage to light structures and is described in more detail by Blaser and Scherer (1969) and Blaser and Arulanandan (1973). Impact of Pyrite
Sulfur occurs in rock and soil as sulfide (S⫺ or S2⫺), sulfate (SO42⫺), and organic sulfur. The sulfide minerals, of which pyrite is one of the most common and easily oxidized (Burkart et al., 1999), are of greatest concern. The amount of sulfide sulfur is a good indicator of the potential for oxidation reactions and weathering that can result in expansion. Sulfideinduced heave has occurred in materials containing as little as 0.1 percent sulfide sulfur (Belgeri and Siegel 1998). Products of pyrite oxidation include sulfate minerals, insoluble iron oxides such as goethite (FeOOH) and hematite (Fe2O3), and sulfuric acid (H2SO4). Sulfuric acid can dissolve other sulfides, heavy metals, carbonates, and the like that are present in the oxidation zone, thus allowing the effects of oxidation to increase as the process builds upon itself.
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INFLUENCES OF MINERALOGICAL DETAIL IN SOIL EXPANSION
Bacterially Generated Heave—Case History
About 1000 wooden houses founded on mudstone sediments in Iwaki City, Fukushima Prefecture, Japan, were damaged by heaving of their foundations (Oyama
Table 10.8 Volume Increases of Selected Mineral Transformations Mineral Transformation Original Mineral Illite Illite Calcite Pyrite Pyrite Pyrite
et al., 1998; Yohta, 1999, 2000). The amount of heave was as much as 480 mm. The cost for repairs was estimated at 10 billion yen (Yohta, 2000). The mudstone at the site contained 5 percent pyrite. Whereas the pH of the sediment was initially 7 to 8 before heave, the pH of the heaved ground was about 3, and it contained acidophilic iron-oxidizing bacteria (Oyama et al., 1998). Yamanaka et al. (2002) further confirmed the presence and effects of sulfate-reducing, sulfur-oxidizing, and acidophilic iron-oxidizing bacteria by means of several series of laboratory culture experiments. Test results presented by Yamanaka et al. (2002), which include electron photomicrographs of the bacteria, showed consistent variations of hydrogen sulfide concentration, pH, Fe3⫹ concentration, Fe2⫹ ⫹ Fe3⫹ concentration, and SO42⫺ concentration over time periods up to 50 days for both the natural mudstone and the mudstone after heat treatment to 121C. The heat treatment prevented or greatly slowed the bacterial activity, whereas very significant changes in concentrations and pH were measured for tests done at 28C. For example, the concentration of H2S increased from 0.3 to 2.2 mM in 20 days, the pH decreased from about 6.5 to 1.3 in 47 days, the concentration of Fe3⫹ increased from about 6 to 125 in 5 days, and the concentration of SO42⫺ increased from less than 1 to about 15 mM in 25 days. Based on their results and observations, Yamanaka et al. (2002) developed the following explanation for the processes leading to the foundation heave. The ground temperature, which had been about 18C at depth, increased to about 25C in the summer after excavation. Initial anaerobic, high water content conditions and the stimulation of sulfate-reducing bacteria generated H2S. As the ground dried and became permeable to air, sulfate-oxidizing bacteria grew and stimulated production of H2SO4, the lowering of pH, and pyrite oxidation. The reaction of H2SO4 with the calcium carbonate present in the mudstone led to formation of gypsum and, with potassium and ferric ions, to formation of jarosite. The foundation heave was associated with the volume increase that accompanied the formation of both gypsum and jarosite crystals.
Co py rig hte dM ate ria l
The relative proportion of sulfate sulfur is indicative of the degree of weathering or oxidation that has already occurred. Sulfate crystals develop in the capillary zone and tend to localize along discontinuities due to reduced stress in these regions. The increase in volume resulting from the growth of sulfate minerals along bedding planes is a dominant factor in the vertical heave that occurs in shales and other materials that have subhorizontal fissility (Kie, 1983; Hawkins and Pinches, 1997). The production of sulfates by pyrite oxidation also increases the potential for further deleterious reactions, such as the formation of gypsum and expansive sulfate minerals (e.g., ettringite). Gypsum (CaSO4 2H2O) is considered to be the primary cause of heave resulting from sulfate expansion. Volume increases associated with several sulfidic chemical weathering reactions are given in Table 10.8. For comparative purposes, these percentages are based on the assumption that the altered rock was initially composed of 100 percent of the original mineral. Sulfide oxidation reactions are usually catalyzed by microbial activity. Gypsum forms when sulfate ions react with calcium in the presence of water, resulting in very large volume increases. The products of pyrite oxidation reactions are significantly less dense than the initial sulfide product (pyrite); for example, the specific gravity of pyrite is 4.8 to 5.1, whereas that of gypsum is only 2.3, and that of calcium is 2.6. Acidity produced by pyrite oxidation can also result in significant quantities of acid mine and rock drainage.
Volume Increase of Crystalline Solids (%)
New Mineral
Alunite Jarosite Gypsum Jarosite Anhydrous ferrous sulfate Melanterite
8 10 60 115 350 536
Data from Fasiska et al. (1974), Shamburger et al. (1975), and Taylor (1988).
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347
Sulfate-Induced Swelling of Cement- and LimeStabilized Soils
Some fine-grained soils, especially in arid and semiarid areas, contain significant amounts of sulfate and carbonate. Sodium sulfate, Na2SO4, and gypsum, Ca SO4 2H20, are the common sulfate forms, and calcium carbonate, CaCO3, and dolomite, MgCO3, are the usual carbonate forms. The dominant clay minerals in these soils are expansive smectites. Delayed expansion fol-
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VOLUME CHANGE BEHAVIOR
Ca(OH)2 ⫹ Na2SO4 → CaSO4 ⫹ 2NaOH
Silica (SiO2) and alumina (Al2O3) dissolve from the clay in the high pH environment and/or they may be present in amorphous form initially. These compounds can then combine with calcium, carbonate, and sulfate to form ettringite, Ca6[Si(OH)6]2(SO4)3 26H2O, and/ or thaumasite, Ca6[Si(OH)6]2(SO4)2(CO3)2 24H2O, which are very expansive materials (Mehta and Hu, 1978). In addition, in the case of lime-treated soil, if the available lime is depleted, the pH will drop and the further dissolution of SiO2 from the clay will stop. As silica is needed for formation of the cement (CSH) that is the desired end product of the pozzolanic lime stabilization reaction, long-term strength gain is prevented. Consequently, when the treated material is given access to water, a large amount of swell may occur. Further details concerning lime–sulfate heave reactions in soils are given in Dermatis and Mitchell (1992).
10.9
fective stress is linear, and properties of the soil do not change during the consolidation process. Deformations in only one dimension, usually vertical, are considered since determinations of settlements caused by loadings from structures or fills are common applications of the theory. In such a case, the relationship between void ratio and vertical stress is as shown in Fig. 10.24a for a normally consolidated clay layer, and that in Fig. 10.24b applies for an overconsolidated clay layer.2 As shown in any basic text on soil mechanics, the amount of vertical settlement H that a homogeneous clay layer of thickness H will undergo if subjected to a vertical stress increase at the surface is given by
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lowing admixture stabilization of these soils using Portland cement and lime has developed at several sites (Mitchell, 1986). Although test programs showed suppression of swelling and substantial strength increase at short times (days) as a result of the incorporation of the stabilizer, subsequent heave of magnitude sufficient to destroy pavements developed after of exposure to water at some later time. The mechanism associated with this process appears to be as follows. When cement or lime is mixed with soil and water, there is a pH increase to about 12.4, some calcium goes into solution and exchanges with sodium on the expansive clay. This ion exchange, along with light cementation by carbonate and gypsum, if present, suppresses the swelling tendency of the clay. The mixed and compacted soil is nonexpansive and has higher strength than the untreated material. If sodium sulfate is present, then available lime is depleted according to
H ⫽
(10.20)
in which e0 is the initial void ratio and e is the decrease in void ratio due to the stress increase from v0 to v1 . For convenience, the change in void ratio is often written in terms of compression index or coefficient of compressibility and change in effective stress as defined in Fig. 10.1. The rate at which consolidation under the stress increases from v0 to v1 is determined using Terzaghi’s solution to the one-dimensional diffusion equation applied to the transient state water flow from the consolidating clay layer. It is assumed in this theory that the rate of volume decrease is controlled totally by hydrodynamic lag, that is, the time required for water to flow out of the consolidating soil under the gradients generated by the applied pressures. The governing equation is u 2u ⫽ cv 2 t z
(10.21)
in which u is the excess pore pressure, t is time, z is distance from a drainage surface, and cv is the coefficient of consolidation. The coefficient is given by cv ⫽
CONSOLIDATION
e H 1 ⫹ e0
kh(1 ⫹ e) av w
(10.22)
Introduction and Simple One-Dimensional Theory
Terzaghi’s (1925b) quantitative description of soil compression and its relation to effective stress and the rate at which it occurs marked the beginning of modern soil mechanics. An ideal homogeneous clay layer is assumed to follow the paths shown in Fig. 10.1 when subjected to compression, unloading, and reloading. Key assumptions for analysis of the consolidation rate according to the Terzaghi theory are that the soil is saturated, the relationship between void ratio and ef-
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where kh is the hydraulic conductivity, av ⫽ ⫺de/d v is the coefficient of compressibility, and w is the unit weight of water.
2 In engineering practice compression and swelling curves are often plotted using settlement ratio, H / H as ordinate rather than void ratio, e, for convenience in settlement computations. Void ratio is used herein because it is more indicative of the state and properties of the soil.
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CONSOLIDATION
Figure 10.24 Idealized compression curves for clay layers: (a) normally consolidated and (b) overconsolidated.
Solutions for Eq. (10.22) for different boundary conditions are given in standard soil mechanics texts in terms of a dimensionless depth z/H (where H is the maximum distance to a drainage boundary) and a dimensionless time factor T ⫽ cvt/H 2 for different boundary conditions. The solution for u ⫽ ƒ(z/H, T) for a layer of thickness 2H that is initially at equilibrium and subjected to a rapidly applied uniform surface loading is shown in Fig. 10.25a. The average degree of consolidation U over the full depth of the clay layer as a function of T for this case is shown in Fig. 10.25b. Ranges of Compressibility and Consolidation Parameters
The curves in Fig. 10.2, as well as the fact that the void ratio of a soil cannot decrease without limit under increasing pressure, mean that the assumption of a linear relationship between void ratio and log of effective consolidation pressure that defines the compression index Cc is simply a useful engineering approximation that applies over a range of stresses and void ratios of practical interest.3 Values for compression index less than 0.2 represent soils of slight to low compressibility; values of 0.2 to 0.4 are for soils of moderate to intermediate compressibility; and a compression index
3 Compression index Cc or swelling index Cs and the coefficient of compressibility av are related as follows:
de C av ⫽ ⫺ ⫽ ln 10 c or d v v
ln 10
Cs v
Hence, av is both stress level and stress history dependent.
Copyright © 2005 John Wiley & Sons
greater than 0.4 indicates high compressibility. Correlations between compression index and compositional and state parameters have been proposed by a number of investigators. Several such relationships for cohesive soils were summarized by Djoenaidi (1985) and quoted by Kulhawy and Mayne (1990), and these relationships are shown in Fig. 10.26. A simple correlation between the compression ratio, defined as Cc /(1 ⫹ e0), where e0 is the initial void ratio, and the natural water content is shown in Fig. 10.27. The large increase in compressibility that occurs when sensitive clay is loaded beyond its maximum prior effective consolidation pressure is shown in Fig. 8.44. Values of compression index for the steepest part of the compression curve as a function of in situ void ratio and sensitivity are shown in Fig. 10.28. The profound influence of structure metastability as represented by high sensitivity is clearly evident. Usual ranges of coefficient of consolidation for finegrained soils are given in Fig. 4.19. Owing to the direct dependence of the coefficient of consolidation cv on hydraulic conductivity and its inverse proportionality to coefficient of compressibility, reliable determination of a representative value in any case is difficult. Both hydraulic conductivity and compressibility are changed by sample disturbance and by consolidation itself. Most settlement predictions are done using average values for coefficient of consolidation. Shortcomings of Simple Theory for Predicting Volume Change and Settlements
In many cases, predictions of the volume changes and settlements and the rates at which they develop, which
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VOLUME CHANGE BEHAVIOR
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Figure 10.25 Solution to the one-dimensional consolidation equation: (a) distribution of excess pore water pressures as a function of dimensionless time and depth for a doubly drained clay layer and (b) average degree of consolidation as a function of time factor.
are based on the above simple theory, are poor. Among the types of deviations between the observed and predicted settlement and pore pressure responses are the following (Crooks et al., 1984; Becker et al., 1984; Tse, 1985; Mitchell, 1986; Duncan, 1993):
1. Differences in predicted and observed initial pore pressure development upon load applications 2. Continued pore pressure buildup after completion of loading 3. Differences between field consolidation rates and those predicted based on the results of laboratory tests
Copyright © 2005 John Wiley & Sons
4. Changes in pore pressure dissipation rates during and following construction 5. Apparent lack of strength gain with consolidation following load application
There are two types of reasons for deviations from the simple theory. In the first category are those that relate to soil behavior and the fact that in general the simple relationships between effective stress shown in Figs. 10.1 and 10.24 are neither unique nor time independent. In the second category are those that relate to the constitutive models and their application and the fact that the simplifying assumptions that may be required are not representative of the real conditions.
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CONSOLIDATION
351
Figure 10.26 Representative values of compression index Cc for cohesive soils (Djoenaidi,
1985).
Soil Behavior Factors Characteristics of the real behavior of fine-grained soils that are important in determining the amount and rate of consolidation include:
Figure 10.27 Compression ratio as a function of natural water content (from Lambe and Whitman, 1969). Reprinted with permission from John Wiley & Sons.
1. Fabric and Structure Resistance to compression is determined by both effective stress and structure. Structural influences that must be considered relate to the initial state, the effects of sample disturbance, structural breakdown associated with consolidation under pressures greater than the maximum past consolidation pressure, and the effects of anisotropic loading. 2. Time and Rate of Loading The relationship between void ratio and effective consolidation pressure is not unique for a fine-grained soil but is influenced by rate of loading and time under a constant load as well. That is, e ⫽ e( , t)
(10.23)
In differential form, Eq. (10.23) can be written
冉 冊
e de ⫽ dt
Figure 10.28 The influence of sensitivity and in situ void ratio on compression index (from Leroueil et al., 1983). Reproduced with permission of the National Research Council of Canada.
Copyright © 2005 John Wiley & Sons
t
冉冊
d e ⫹ dt t
(10.24)
According to this relationship, the total void ratio change at any time is the sum of two components: (1) that due to change in effective stress, or effective stress related compressibility, given by the first term on the right-hand side of Eq. (10.24) and (2) that due to time, or time-related compressibility, given by the second term on the right. The rate at which the total void ratio decreases as a function of time after application of
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352
10
VOLUME CHANGE BEHAVIOR
information about them and about how to account for them can be found in Gibson et al. (1981), Tse (1985), Mesri and Castro (1987), Leroueil et al. (1990), Scott (1989), Duncan (1993), and elsewhere. Generalization of Terzaghi’s one-dimensional consolidation theory to three dimensions was made by Biot (1941). At present, there are finite element and finite difference codes that solve Biot’s consolidation equation incorporating nonlinear stress–stress relationships as well as anisotropic hydraulic conductivity. The hydraulic conductivity can also be a function of void ratio or effective stress. Further details can be found in Lewis and Schrefler (1997) and Coussy (2004). Soil behavior factors are considered further in the remainder of this section.
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a stress increase may be controlled by either how rapidly the water can escape under a hydraulic gradient or by how fast the structure of the soil can deform or creep under a given magnitude of effective stress. Component (1) compression is commonly referred to as primary consolidation. Component (2) compression is commonly referred to as secondary compression. In addition, aging phenomena during time under sustained stress generate additional resistance to further compression. 3. Temperature Owing to differential thermal expansions of soil solids and the pore fluid and changes in interparticle bond strength and resistance to sliding that can result from changes in temperature, temperature-induced changes in effective stress and volume are possible. These effects are considered further in Section 10.12. Modeling Factors The commonly used constitutive models for soil compression and consolidation may not give suitable representations of actual behavior for the following reasons:
1. The relationship between void ratio and effective consolidation pressure is not linear, as is assumed for the Terzaghi consolidation theory. In fact, the use of compression index and swelling index to characterize soil compression and swelling recognize the nonlinear nature of the void ratio– effective stress relationship. 2. Changes in void ratio, compressibility, and hydraulic conductivity during consolidation are neglected or not properly taken into account. 3. Secondary compression, which is creep of the soil skeleton, is often neglected, and models for taking it into account are of uncertain validity. 4. Soil properties differ among the strata making up the soil profile and within the individual strata themselves. 5. Boundary conditions are uncertain or unknown, especially the drainage boundaries. Given that the time for primary consolidation varies as the square of the distance to a drainage layer, errors in definition and location of drainage boundaries have a major impact on settlement rate predictions. 6. Although one-dimensional analyses are often used, two- and three-dimensional effects may be important. 7. The stress increments may not be known with certainty. Analysis of modeling factors of the type listed above is outside the scope of this book; however, additional
Copyright © 2005 John Wiley & Sons
Effects of Sample Disturbance
The effects of sample disturbance on the compression curve of sensitive or structured clay are shown in Fig. 8.44 and include: 1. A lower void ratio under any effective stress. 2. Higher values of recompression index and lower values of the compression index for a disturbed clay than for the undisturbed soil. 3. Less clearly defined stress history; determination of the maximum past consolidation pressure may be difficult and uncertain.
Several methods to estimate the influences of sample disturbance on measured compression properties and strength have been proposed. Among them, Schmertmann’s (1955) procedure is useful for determination of a corrected maximum past pressure and for estimation of more representative values of swelling and recompression indices. The SHANSEP (stress history and normalized soil engineering properties) method (Ladd and Foott, 1974) was developed for more accurate determination of the strength of soft clay. By this method, samples are consolidated beyond the maximum past pressure into the virgin compression range. Provided the structure of the consolidated clay does not differ extensively from that of the undisturbed clay, the relationships between the ratios of shear stress divided by effective consolidation pressure versus strain and pore pressure divided by effective consolidation pressure versus strain are the same for both the original undisturbed clay and the consolidated samples. An uncertainty in this method, however, is the extent of breakdown of a structured soil from its initial state when it is consolidated past its prior maximum past pressure. Evidence indicates that it works well for clays of low-to-medium sensitivity.
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SECONDARY COMPRESSION
10.10
SECONDARY COMPRESSION
solidation.5 Thus, it is convenient to define a coefficient of secondary compression, Ce, according to Ce ⫽ ⫺de/d(log t)
Figure 10.29 Idealized relationship between void ratio and logarithm of time showing primary consolidation and secondary compression.
4
It is commonly assumed that there are no excess hydrostatic pressures during secondary compression. However, water is expelled during secondary compression, and water flow is driven by hydrostatic head differences, so there must be some small hydrostatic pressure difference between the interior and a drainage boundary.
Copyright © 2005 John Wiley & Sons
(10.25)
The value of Ce is usually related to the compression index Cc as shown in Table 10.9, where values are listed for a number of different natural soils. Average values for Ce /Cc are 0.04 0.01 for inorganic clays and silts, 0.05 0.01 for organic clays and silts, and 0.075 0.01 for peats. Similar behavior for a number of clean sands is shown in Fig. 10.30, where it may be seen that Ce /Cc falls in the range of 0.015 to 0.03. A general relationship between void ratio, effective consolidation pressure, and time is shown in Fig. 10.31, with slopes Ce and Cc indicated. When the curves corresponding to different times after the end of primary consolidation are projected onto the void ratio–log effective stress plane, Fig. 10.5 is obtained for the assumption of linearity between void ratio and log . Algebraic manipulation of the secondary compression equation and the primary compression equation shows that the preconsolidation pressure is rate dependent (Soga and Mitchell, 1996), consistent with the data presented in Fig. 10.7. Both laboratory tests and field measurements, as well as theoretical arguments, have been made to establish whether or not (1) the relationship between the end-of-primary consolidation void ratio and effective consolidation pressure is unique and independent of load increment ratio or deformation rate, and (2) whether or not both primary consolidation and secondary compression can occur together or if all primary consolidation must be completed before secondary compression begins. The answers to these questions are important as they impact the usefulness of laboratory odometer test results on thin samples with short drainage paths, in which consolidation times are short, for prediction of the consolidation of thick layers in the field wherein consolidation times are often very long. Detailed discussion of these issues is outside the scope of this book. Among the many important references on these points are Taylor (1942), Murayama and Shibata (1961), Bjerrum (1967), Walker (1969),
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According to the simple consolidation theory, which assumes uniqueness between void ratio and effective stress, consolidation ends when excess hydrostatic pressures within a clay layer are fully dissipated. On this basis, the relationship between degree of consolidation and dimensionless time is as shown in Fig. 10.25b. In reality, however, most soils continue to compress in the manner shown in Fig. 10.29. The reason for secondary compression is that the soil structure is susceptible to a viscous or creep deformation under the action of sustained stress as the fabric elements adjust slowly to more stable arrangements. The rate of secondary compression is controlled by the rate at which the structure can deform, as opposed to the rate of primary consolidation, which is controlled by Darcy’s law, which determines how rapidly water can escape from the pores under a hydraulic gradient.4 The mechanism of secondary compression involves sliding at interparticle contacts, expulsion of water from microfabric elements, and rearrangement of adsorbed water molecules and cations into different positions. The observed behavior is consistent with that of a thermally activated rate process, which involves mechanisms that are discussed in more detail in Section 12.4. The relationship between void ratio and log of time during secondary compression is linear for most soils over the time ranges of interest following primary con-
353
5 There is no reason to believe that secondary compression should continue indefinitely because a final equilibrium of the structure should ultimately develop under a given stress state. In nature, chemical, biological, and climate changes also develop over long time periods. These changes can accelerate the establishment of equilibrium or create new conditions of disequilibrium. However, the assumption of linearity between void ratio and log of time after the end of primary consolidation is sufficiently accurate for most practical cases.
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VOLUME CHANGE BEHAVIOR
Table 10.9 Values of the Ratio of Coefficient of Secondary Compression to Compression Index for Natural Soils Soil Type
Ce / Cc
Inorganic clays and silts
Whangamarino clay Leda clay Soft blue clay Portland sensitive clay San Francisco Bay mud New Liskeard varved clay Silty clay C Near-shore clays and silts Mexico City clay Hudson River silt Norfolk organic silt Calcareous organic silt Postglacial organic clay Organic clays and silts New Haven organic clay silt Amorphous and fibrous peat Canadian muskeg Peat Peat Fibrous peat
0.03–0.04 0.025–0.06 0.026 0.025–0.055 0.04–0.06 0.03–0.06 0.032 0.055–0.075 0.03–0.035 0.03–0.06 0.05 0.035–0.06 0.05–0.07 0.04–0.06 0.04–0.075 0.035–0.083 0.09–0.10 0.075–0.085 0.05–0.08 0.06–0.085
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Grouping
Organic clays and silts
Peats
From Mesri and Godlewski (1977).
Figure 10.30 C / Cc values for clean sands (from Mesri et
al., 1990). Reprinted with permission of ASCE.
Copyright © 2005 John Wiley & Sons
Figure 10.31 General relationship among void ratio, effective stress, and time (from Mesri and Godlewski, 1977). Reprinted with permission of ASCE.
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IN SITU HORIZONTAL STRESS (K0)
10.11
In most cases, the horizontal stress in the ground does not equal the vertical overburden stress. The minimum and maximum possible values can be calculated on the basis of plasticity theories for earth pressure. The actual value, which must fall somewhere between these limiting values, is a proportion of the vertical overburden stress that depends primarily on soil type and stress history. It is often determined (or estimated) on the basis of these two factors using empirical correlations, and, sometimes the results of in situ tests such as the self-boring pressuremeter (Mair and Wood, 1987). The main limitation of in situ measurements is that they invariably cause disturbance and allow lateral deformations of the ground that change the stress being measured. The general ranges of in situ lateral stress for different soil types are summarized, and factors influencing lateral stress are reviewed in this section.
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Aboshi (1973), Mesri (1973), Mesri and Godlewski (1977), Jamiolkowski et al. (1985), Leroueil et al. (1985, 1988), Mesri and Choi (1985), Leroueil (1988), Mesri et al. (1995), Leroueil (1995), and Mesri (2003). In spite of these uncertainties, conventional practice has been to assume that secondary compression does not begin until completion of primary consolidation. This has the advantage of simplicity in that settlement estimates can be made on the basis of degree of consolidation according to the simple theory during times up to the end of primary consolidation. For longer times, the total settlement is taken as the consolidation settlement increased by an amount of secondary compression derived from Eq. (10.25). This is undoubtedly an oversimplification of real behavior, as from the perspective of the soil, there should be no difference between the two types of compression. It compresses just sufficiently to withstand the applied stresses at any time, and the rate at which it occurs in any element depends on whether or not the rate of water flow from the element at that time is controlled by a preexisting hydrostatic excess pressure gradient (primary consolidation) or by the time-dependent generation of small pore pressures owing to structural readjustment (secondary compression). On this basis, it would seem most likely that within a clay layer both primary consolidation and secondary compression may be occurring concurrently in different elements. The major difficulty has been in the formulation of a constitutive model to describe both the hydrodynamic and viscous components of the soil response that is both accurate and that can be readily implemented into analytical or numerical solutions. With recent advances in theory and programs that can be run on personal computers, it is now possible to more properly describe the actual soil response and to make improved settlement rate predictions (Duncan, 1993).
IN SITU HORIZONTAL STRESS (K0)
Terzaghi’s consolidation theory considers compression only in one dimension. The soil model relates the vertical strain to the change in vertical stress, and this defines the volume change under zero horizontal displacement conditions. There is no need to consider the change in horizontal stress to calculate the deformation, even though the actual horizontal stress changes during loading and unloading. However, once soil deformation departs from the one-dimensional condition, it is necessary to consider the state and changes of the stresses in the other directions and the associated volume change behavior.
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355
Development of Horizontal Stress
The relationship between the horizontal effective stress and the vertical effective stress depends on the lateral deformation that accompanies changes in vertical stress. If the vertical stress and strain increase without any deformation in the horizontal directions (i.e., onedimensional compression, as would be the case for an accumulating sediment), the soil is said to be in an atrest state, and the horizontal stress associated with this condition is termed the at-rest pressure. The ratio between the horizontal and vertical effective stresses during initial compression of a soil is a constant, defined by the coefficient of earth pressure at rest K0 (⫽ h / v). Values of K0 for normally consolidated soils are generally in the range of 0.3 to 0.75. Jaky’s equation has been found to give a good estimate for many soils: K0 ⫽ 1 ⫺ sin
(10.26)
in which is the effective stress friction angle measured in triaxial compression tests. Although correlations have been published that suggest unique relationships between K0 and liquid limit or plasticity index, a comprehensive set of data for 135 clay soils indicates little correlation, as shown in Fig. 10.32. This is not surprising since the Atterberg limits depend only on composition, and K0 is a state parameter that is dependent on composition, structure, and stress history. When the vertical stress on a normally consolidated soil is reduced, the horizontal stress does not decrease in the same proportion as the vertical stress. Thus, the value of at-rest earth pressure coefficient for an overconsolidated soil (K0)oc is greater than that for the normally consolidated soil (K0)nc, and it varies with the
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VOLUME CHANGE BEHAVIOR
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356
Figure 10.32 Lack of correlation between coefficient of
earth pressure at rest and plasticity index for normally consolidated soils (from Kulhawy and Mayne, 1990). Reprinted with permission from EPRI.
amount of overconsolidation, as shown schematically in Fig. 10.33, and in Fig. 10.34 for 48 clays. The data in Fig. 10.34 can be approximated by the equation K0 ⫽ (1 ⫺ sin )(OCR)sin
(10.27)
Kulhawy and Mayne (1990) give additional useful correlations for estimation of K0. The complicated stress paths associated with onedimensional compression of four clays are illustrated in Fig 10.35. In the upper plot for each clay the deviator stress is shown as a function of the mean effective stress during one-dimensional compression. Before yielding, the stress path shows larger stress ratios than the K0 ⫽ 1 ⫺ sin line. As the stress state approaches the preconsolidation pressure, the stress path moves to the K0 ⫽ 1 ⫺ sin line. The curvature
Figure 10.33 Variation of horizontal effective stress with
vertical effective stress for loading and unloading.
Copyright © 2005 John Wiley & Sons
Figure 10.34 Dependence of (K0)oc on overconsolidation ratio (from Kulhawy and Mayne, 1990). Reprinted with permission from EPRI.
toward the K0 line coincides with the region of largest compression index (steepest slope on the volumetric strain versus effective mean stress diagrams), implying structural degradation.
Effect of Lateral Yielding on the Coefficient of Earth Pressure
If an element of soil initially under an at-rest stress condition is allowed to yield by compressing in a vertical direction while spreading laterally, for example, triaxial or plane strain compression, then the horizontal earth pressure coefficient decreases until a failure condition is reached. If, on the other hand, the element is compressed in the horizontal direction while being allowed to expand in the vertical direction, triaxial or plane strain extension, then the horizontal earth pressure increases until failure develops. These two conditions and the associated variations in K are shown in Fig. 10.36. The two failure conditions are termed active and passive, respectively, and the corresponding earth pressure coefficients are the coefficient of active earth pressure Ka and the coefficient of passive earth pressure Kp. According to classical theories of earth pressure based on limiting equilibrium of a plastic material hav-
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100
(σa – r)/2(kPa)
(σa – r)/2(kPa)
IN SITU HORIZONTAL STRESS (K0)
50
0
0
50
100 150 (σa + r)/2(kPa)
1000 ure
l fai ak
500 Pe
0
200
0
500
d ture g) truc therin s e a d K o( by we
1000 1500 (σa + r)/2(kPa)
2000
0
0
e0 = 0.69 εv(%)
e0 = 1.97 10
4
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εv(%)
357
8
20
12
(b) Unweathered Keuper marl
200
(σa – r)/2(kPa)
(σa – r)/2(kPa)
(a) Sensitive Canadian clay ilure k fa Pea
100
d)
ture
uc estr
d
K o(
0
0
100 100 100 (σa + r)/2(kPa)
50
25
0
0
100
0
25 50 75 (σa + r)/2(kPa)
0
e0 = 1.04
5
e0 = 0.69
εv(%)
εv (%)
100
10
5
10
15
15
(d) Chalk
(c) Artificially bonded soil
Figure 10.35 Variation in lateral stress with mean stress during one-dimensional consolidation of four clays (from Leroueil and Vaughan, 1990).
ing a friction angle and a cohesion c, the limiting minimum and maximum values of the earth pressure coefficients are
冉 冉
冊 冊
冉 冉
冊 冊
Ka ⫽ tan2 45 ⫺
2c ⫺ tan 45 ⫺ 2 v 2
(10.28)
Kp ⫽ tan2 45 ⫹
2c ⫹ tan 45 ⫹ 2 v 2
(10.29)
These limiting values are for isotropic soil and a horizontal ground surface. Standard soil mechanics texts should be consulted for further details on limiting earth pressure coefficients under sloping ground and the influences of changes in applied loads on in situ lateral stress.
Copyright © 2005 John Wiley & Sons
Under one-dimensional conditions, compression is usually plotted on the e–log v plane, as shown in Fig. 10.1. For three-dimensional stress and deformation conditions, however, the volumetric behavior is often plotted on the e–ln p plane (or v –ln p plane), where p is the mean effective pressure and v is the specific volume (⫽1 ⫹ e). When a specimen is consolidated isotropically, the slope of the normal compression line is defined as ⫽ ⫺de/d ln p(⫽ ⫺dv /d ln p) (Schofield and Wroth, 1968).6 Figure 10.37 shows the change in void ratio with mean effective stress (p) for reconstituted kaolin clay specimens consolidated isotropically at constant stress The swelling (or recompression) line is often called the ! line on e–ln p plane and the slope is defined as the recompression index !(⫽ ⫺de / d ln p) 6
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10
VOLUME CHANGE BEHAVIOR
1.9
q
Stress Paths
1 2
1.7 Void Ratio
3
p⬘
1.5 Stress Path 1: q/p⬘ = 0.375
1.3
Stress Path 2: q/p⬘ = 0.288
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Stress Path 3: q/p⬘ = 0
70
100
200
300
500
1000
Mean Pressure p⬘ (kPa)
Figure 10.37 Effect of stress ratio ( 1 / 3 or q / p) on vol-
umetric compression behavior of reconstituted kaolin clay.
Figure 10.36 Variation of lateral earth pressure coefficient
with deformation of a soil element.
ratios ( 1 / 3 or q/p) as shown by the stress paths in the insert diagram. The compression lines are parallel to each other and therefore they will have the same compression index . Similar behavior is observed in sands; the isotropic compression line and the one-dimensional compression line are parallel to each other. Assuming that K0 is constant during loading, the value of isotropically consolidated specimens will be the same as that of one-dimensionally consolidated specimens.7 Examination of Fig. 10.37 indicates that the volumetric behavior of soils can be separated into two components: (i) one due to compression or swelling by the increase or decrease in mean effective pressure p and (ii) the other due to dilation or contraction by shearing of the soil by the increase in q. Further discussion of deformation behavior under combined volumetric and deviatoric stress loading conditions is given in Chapter 11. Anisotropy
Unless the horizontal earth pressure coefficient is equal to 1.0, which is not the usual case, the stress condition In one dimensional consolidation condition, p ⫽ (1 ⫹ 2K0) v. The relationship between Cc (⫽ ⫺de / d log v) and (⫽ ⫺de / d ln p) is Cc ⫽ ln 10: Cc ⫽ ⫺de / d log v ⫽ ⫺ln 10[de / d(ln v)] ⫽ ⫺ln 10de / {d ln p ⫺ d[ln(1 ⫹ 2K0)]} ⫽ ⫺ln 10de / d ln p) (K0 is constant in normally consolidated state, hence d[ln(1 ⫹ 2K0)] ⫽ 0). On the other hand, it is not possible to relate Cs obtained from the onedimensional consolidation test to ! obtained from the isotropic unloading test. This is because the K0 value changes as the specimen is unloaded and therefore the d[ln(1 ⫹ 2K0)] term in the above equation does not become zero. 7
Copyright © 2005 John Wiley & Sons
in the ground is anisotropic. Furthermore, although it is usually assumed that the in situ stresses are the same in all directions beneath level ground, there are some conditions in which this may not be true. These include situations wherein there is a directional component to the soil fabric that formed during deposition, as might be the case, for example, for an alluvial or beach deposit. Directional variability has been measured at some sites by means of pressure cells, pressure meters that contain multiple sensing arms, and flat plate dilatometers. With the development of new shear wave and tomography methods for the nondestructive and nonintrusive testing of soil layers, it is possible to obtain much more data on the actual lateral stress state and its variability, thus providing new insights into geologic and soil formational history, as well as quantitative values for use in the analysis and prediction of behavior. Time Dependence of Lateral Earth Pressure at Rest
It is usually assumed in conventional geotechnical analyses that the coefficient of lateral earth pressure atrest K0 is a time-invariant constant. Whether or not this is indeed the case is not known with certainty, and there is no clear consensus on how K0 should be expected to vary with time (Schmertmann, 1983). However, if a soil is assumed to remain under a constant effective stress state following consolidation and there are no changes in the compositional or environmental conditions, then slow changes in lateral pressure should occur in any material that is susceptible to creep and stress relaxation. Creep and stress relaxation are analyzed in Section 12.7.
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TEMPERATURE–VOLUME RELATIONSHIPS
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As long as a deviator stress is acting K0 ⫽ 1.0, and a soil element will tend to distort. If the vertical stress is greater than the horizontal stress (K0 ⬍ 1.0), then the element will try to expand laterally, but under onedimensional conditions it cannot, and the horizontal stress increases to restrain it. Conversely, if the horizontal stress is initially greater than the vertical stress (K0 ⬎ 1.0), then the element will try to compress laterally, but under one-dimensional conditions it cannot, so the horizontal stress decreases. Thus, over long periods of time, the coefficient of horizontal earth pressure at rest in normally consolidated soil should increase toward 1.0 and that in heavily overconsolidated soil should decrease toward 1.0. Values of K0 as a function of time, as determined in triaxial cells by Lacerda (1976), for undisturbed samples of soft San Francisco Bay mud, are shown in Fig. 10.38. Also shown is a theoretical relationship between K0 and time that was developed using the general stress–strain–time equations developed in Section 12.9. Thus, both theory and experiment support the above reasoning that K0 should increase with time when K0 is less than 1.0.
10.12 TEMPERATURE–VOLUME RELATIONSHIPS
Temperature changes generate volume and/or effective stress changes in saturated soils. For example, the percentage of the original pore water volume that is drained from a saturated specimen of illite subjected to a temperature increase from 18.9 to 60C followed by cooling to 18.9C while maintaining an isotropic effective stress of 200 kPa is shown in Fig. 10.39. The variation in effective stress 3 under the same temperature changes but with drainage prevented is shown in
359
Figure 10.39 Volume of pore water drained from saturated
illite under an isotropic effective stress of 200 kPa as a function of temperature change.
Fig. 10.40. Temperature effects such as these must be considered relative to their influences on deformation and stability both in the laboratory and the field. Theoretical Analysis
Drained Conditions Increase in temperature causes thermal expansion of mineral solids and pore water. In addition, there can be changes in soil structure. For a temperature change T, the volume change of the pore water is
( Vw) T ⫽ wVw T
(10.30)
where w is the thermal expansion coefficient of soil water, and Vw is the pore water volume. The change in volume of mineral solids is ( Vs) T ⫽ sVs T
(10.31)
where s is the thermal coefficient of cubical expansion of mineral solids, and Vs is the volume of solids. The thermal coefficient of water is approximately 15 times greater than that of the solids (Cui et al., 2000). If a saturated soil is free to drain due to a change in temperature while under constant effective stress, the volume of water drained is
Figure 10.38 K0 as a function of time for San Francisco Bay
mud. The theoretical curve was developed by Kavazanjian and Mitchell (1984) using the general stress–strain–time Eq. (12.43) adapted for zero lateral strain.
Copyright © 2005 John Wiley & Sons
( VDR) T ⫽ ( Vw) T ⫹ ( Vs) T ⫺ ( Vm) T
(10.32)
in which ( Vm) T is the change in total volume due to
T, with volume increases considered positive.
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VOLUME CHANGE BEHAVIOR
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360
Figure 10.40 Effect of temperature changes on the effective stress in saturated illite under constant confining pressure.
In a soil mass with all grains in contact, and assuming the same coefficient of thermal expansion for all soil minerals, the soil grains and the soil mass undergo the same volumetric strain s T. In addition, the change in temperature induces a change in interparticle forces, cohesion, and/or frictional resistance that necessitates some particle reorientations to permit the soil structure to carry the same effective stress. If the volume change due to this effect is ( VST) T , then ( Vm) T ⫽ sVm T ⫹ ( VST ) T and
(10.33)
( VDR) T ⫽ wVw T ⫹ s Vs T
⫺ [s Vm T ⫹ ( VST ) T]
(10.34)
Undrained Conditions The governing criterion for
undrained conditions is that the sum of the separate volume changes of the soil constituents due to both temperature and pressure changes must equal the sum of the volume changes of the soil mass due to both temperature and pressure changes; that is ( Vw) T ⫹ ( Vs) T ⫹ ( Vw) P ⫹ ( Vs) P ⫽ ( Vm) T ⫹ ( Vm) P
(10.35)
where the subscripts T and P refer to temperature and pressure changes, respectively. If mw, ms, and ms refer to the compressibility of water, the compressibil-
Copyright © 2005 John Wiley & Sons
ity of mineral solids under hydrostatic pressure, and the compressibility of mineral solids under concentrated loadings, respectively, then ( Vw) P ⫽ mwVw u
(10.36)
( Vs) P ⫽ msVs u ⫹ msVs
(10.37)
where u is the change in pore water pressure and is the change in effective stress. The term ms Vs is the change in volume of mineral solids due to a change in effective stress, which also manifests itself by changes in forces at interparticle contacts. Also ( Vm) P ⫽ mv Vm
(10.38)
where mv is the compressibility of the soil structure. From Eqs. (10.30), (10.31), (10.36), (10.37), and (10.38), Eq. (10.35) becomes wVw T ⫹ s Vs T ⫺ ( Vm) T ⫽ Mv Vm ⫺ mwVw u ⫺ Vs(ms u ⫹ ms )
(10.39)
For constant total stress during a temperature change
⫽ ⫺ u
Thus, Eq. (10.39) becomes
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(10.40)
TEMPERATURE–VOLUME RELATIONSHIPS
w Vw T ⫹ sVs T ⫺ ( Vm) T ⫽ mv Vm ⫺ mwVw u ⫺ u Vs(ms ⫺ ms)
(10.41)
Since ms and ms are not likely to be significantly different, and both are much less than mv and mw , little error results from assuming ms ⫺ ms ⫽ 0, so Eq. (10.41) can be written wVw T ⫹ sVs T ⫺ ( Vm) T
(10.42)
The left side of Eq. (10.42) is equal to ( VDR) T , and the right side is an equivalent volume change caused entirely by a change in pore pressure. Because Vm ⫽ Vw ⫹ Vs
(10.43)
Eq. (10.42) may be written, after substitution for ( Vm) T by Eq. (10.33), w Vw T ⫺ sVw T ⫺ ( VST) T ⫽ ⫺mv Vm u ⫺mwVw u
(10.44)
Rearrangement of Eq. (10.44) gives the pore pressure change accompanying a temperature change:
u ⫽ ⫽
temperature. The compressibilities mv and mw are negative because an increase in pressure causes a decrease in volume, and ST is negative if an increase in temperature causes a decrease in volume of the soil structure. Volume Change Behavior
Permanent volume decreases occur when the temperature of normally consolidated clay is increased under drained conditions, as shown by Fig. 10.41. Temperature changes in the order indicated were carried out on a sample of saturated, remolded illite after initial consolidation to an effective stress of 200 kPa. Water drains from the sample during increase in temperature and is absorbed during temperature decrease. The shape of the curves is similar to normal consolidation curves for volume changes caused by changes in applied stresses. When the temperature is increased, two effects occur. If the increase is rapid, a significant positive pore pressure develops due to greater volumetric expansion of the pore water than of the mineral solids. The lower the hydraulic conductivity of the soil, the longer the time required for this pore pressure to dissipate. Dissipation of this pressure accounts for the parts of the curves in Fig. 10.41 that resemble primary consolidation. The second effect results because increase in temperature causes a decrease in the shearing resistance at individual particle contacts. As a consequence, there is partial collapse of the soil structure and decrease in void ratio until a sufficient number of additional bonds are formed to enable the soil to carry the stresses at the higher temperature. This effect is analogous to secondary compression under stress increase. When the temperature drops, differential thermal contractions between the soil solids and the pore water cause pressure reduction in the pore water. The soil then absorbs water, as shown by the temperature decrease curves in Fig. 10.41. No secondary volume change effect is observed because the temperature decrease causes a strengthening of the soil structure and no further structural adjustment is required to carry the effective stress. On subsequent temperature increases, the secondary effect is negligible because the structure has already been strengthened in prior cycles. The final height changes and volumes of water drained associated with each temperature change shown in Fig. 10.41 are plotted as a function of temperature in Fig. 10.42, and clay structure volume changes are shown in Fig. 10.43. The forms of these plots are similar to conventional compression curves involving virgin compression, unloading, and reload-
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⫽ ⫺mv Vm u ⫺ mwVw u
n T(s ⫺ w) ⫹ ( VST ) T /Vm mv ⫹ nmw
n T(s ⫺ w) ⫹ ST T mv ⫹ nmw
(10.45)
in which n is the porosity, and ST is the physicochemical coefficient of structural volume change defined by ST ⫽
( VST ) T /Vm
T
(10.46)
Thus, the factors controlling pore pressure changes are the magnitude of T, porosity, the difference between thermal expansion coefficients for soil grains and water, the volumetric strain due to physicochemical effects, and the compressibility of the soil structure. For most soils (but not rocks) mv » mw , so
u ⫽
n(s ⫺ w) T ⫹ ST T mv
(10.47)
Consistency in algebraic signs is required for the application of the above equations. Both s and w are positive and indicate volume increase with increasing
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VOLUME CHANGE BEHAVIOR
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Figure 10.41 Volume of water drained from a saturated clay as a function of time as a result of temperature changes.
ing. An irrecoverable volume reduction after each temperature cycle is noted. Again, the effect of temperature increase is analogous to a pressure increase. The slope of the curves in Fig. 10.43 is the coefficient of thermal expansion for the soil structure
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ST , defined previously by Eq. (10.46). For the cases shown, ST has a value of about ⫺0.5 ⫻ 10⫺4 C⫺1. The effect of temperature on clay compression depends on the pressure range (Campanella and Mitchell, 1968; Plum and Esrig, 1969). Weaker structure at low
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TEMPERATURE–VOLUME RELATIONSHIPS
363
Figure 10.42 Effect of temperature variations on the height and volume change of saturated
illite.
stresses caused by increased temperature causes consolidation to a lower void ratio in order to carry the stress. The weakening effect of higher temperature is compensated by the strengthening effect of lower void ratio. As shown in Fig. 10.44, the compression index Cc is found to be approximately independent of temperature. On the other hand, the isothermal swelling index ! (⫽ ⫺de/d ln p) of reconstituted samples of an illitic clay measured under isotropic confining stress conditions is found to be temperature dependent as shown in Fig. 10.45. The preconsolidation pressure of a natural soft clay depends on temperature as illustrated in Fig. 10.7. Figure 10.46 shows the normalized preconsolidation pressure (⫽ preconsolidation pressure at temperature T/ preconsolidation pressure at 20C) with temperature
Copyright © 2005 John Wiley & Sons
(Leroueil and Marques, 1996). The data show that there is approximately 1 percent decrease in preconsolidation pressure per one 1C temperature increase between 5 and 40C and somewhat less at higher temperatures (Leroueil and Hight, 2002). Stress history or overconsolidation ratio has a major influence on the volume change caused by increase in temperature (Hueckel and Baldi, 1990). For normally consolidated to moderately overconsolidated clay, irrecoverable volume reduction was observed by structure degradation and the shear strength increased. Volume expansion was observed in heavily overconsolidated clay, and the expansion rate increased with OCR. The effect of heating followed by cooling at two stages in a consolidation test is shown in Fig. 10.47.
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0.08
κT
0.06
0.04 䉭 䉭 䉭
0.02
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0.00 10
䉭 䉭
20
40
60 80 100
200
Figure 10.45 Effect of temperature on swelling index of isotropically consolidated illitic clay specimens. The clay contained small amounts of Kaolin, chlorite and quartz and had a liquid limit of 30 percent (after Graham et al., 2001).
Figure 10.43 Volume changes in clay structure caused by
temperature change.
Figure 10.46 Effect of temperature on preconsolidation
pressure. The preconsolidation pressure at temperature T is normalized by the preconsolidation pressure at 20C (after Leroueil and Marques, 1996).
The effect is remarkably similar to the development of an apparent precompression due to aging and creep under a sustained stress as discussed in Chapter 12. Pore Pressure Behavior
Figure 10.44 Effect of temperature on isotropic consolida-
tion behavior of saturated illite (Campanella and Mitchell, 1968).
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Pore pressure changes in saturated soils caused by temperature changes are reasonably well predicted by Eq. (10.47). The most important factors are the thermal expansion of the pore water, the compressibility of the soil structure, and the initial effective stress. The appropriate value of the compressibility mv depends on the rebound and recompression characteristics of the soil. When temperature increases, pore pressure increases, and effective stress decreases, which is a condition analogous to unloading. When temperature decreases, pore pressure decreases, and effective stress
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CONCLUDING COMMENTS
pore water pressure and effective stress than for the buckshot clay. The parameter F is approximately the same for different clays (Table 10.10). Knowledge of F values allows determination of laboratory temperature control to assure accurate pore pressure measurements in undrained testing of soil samples. For example, if it were desired to keep pore pressure fluctuations within 5 kPa for one of the clays in Table 10.10, the required temperature control would be about 0.5C for a sample at an effective stress of 500 kPa. The preceding analyses indicate that the overall volume changes that result from changes in temperature may not be large. However, the structural weakening and pore pressure changes that occur may be significant in terms of their influences on shear deformation and strength.
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Figure 10.47 Effect of heating and cooling on void ratio versus pressure relationship of illite (Plum and Esrig, 1969).
10.13
increases. As the previous temperature history caused permanent volume decrease at the higher temperature, the condition is analogous to recompression. Thus, the appropriate value of mv is based on the slope of the rebound or recompression curves, both of which are approximately the same, and can be defined by (mv)R ⫽
Vm /Vm 0.435 Cs ⫽
(1 ⫹ e0)
(10.48)
where Cs is the swelling index, e0 is the initial void ratio, and is the effective stress at which (mv)R is to be evaluated. A pore pressure–temperature parameter F may be defined as the change in pore pressure per unit change in temperature per unit effective stress, or alternatively, the change in unit effective stress per unit change in temperature, that is, F⫽
u/ T
/ e [( ⫺ w) ⫹ ST /n] ⫽⫺ ⫽ 0 s
T 0.435Cs
(10.49)
Some values of F are given in Table 10.10. The values listed for are averages for the indicated temperature ranges. The influence of effective stress on change in pore pressure can be seen for the data for Vicksburg buckshot clay and for the saturated sandstone. The greater change in pore pressure for a given T for a higher initial effective stress is predicted by this theory. Also, the much lower compressibility of the sandstone is responsible for a much higher temperature sensitivity of
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CONCLUDING COMMENTS
Knowledge of volume changes to be expected in a soil mass as a result of changes in confinement, loading, exposure to water and chemicals, changes in temperature and the like is one of the four dimensions of soil behavior that must be understood for success in geoengineering, the other three being fluid and energy conduction properties, deformation and strength properties, and the influences of time. The nature and influences of different factors on volume change have been the subject of this chapter. Soil compression and consolidation under applied stress have been the most studied owing to their essential role in estimation of settlements, and this was one of the first motivations for development of soil mechanics. The mechanical aspects of compression and swelling are far better understood and quantified than are those generated by physicochemical, geochemical, and microbiological factors, although interest and research on the latter is intensifying. Although analysis of volume change is typically done through consideration of a soil mass as a continuum, the processes that determine it are at the particulate level and involve discreet particle movements required to produce a new equilibrium following changes in stress and environmental conditions. Important aspects of colloidal type interactions involving interparticle forces, water adsorption phenomena, and soil fabric effects were analyzed in this chapter. Discreet particle movements and their relationships to macroscopic volumetric and deviatoric behavior are discussed in more detail in Chapters 11 and 12. Soil swelling, sometimes referred to as ‘‘the hidden disaster’’ owing to the very large economic, but unspectacular, damages (several billion dollars in the
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Table 10.10 Temperature-Induced Pore Pressure Changes Under Undrained Condtions
Soil Type
T (C)
u (kN/m2)
200 150
21.1–43.4 21.1–43.4
⫹58 ⫹50
0.013 0.015
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Illite (grundite) San Francisco Bay mud Weald Claya Kaolinite Vicksburg buckshot clayb Saturated sandstone (porous stone)
(kN/m2)
F ( u/ T) (C⫺1)
a b
710 200 100 650
25.0–29.0 21.1–43.4 20.0–36.0 20.0–36.0
⫹51 ⫹78 ⫹28 ⫹190
0.018 0.017 0.017 0.018
250 580
5.3–15.0 5.3–15.0
⫹190 ⫹520
0.079 0.092
From Henkel and Sowa (1963). From Ladd (1961) Fig. VIII-6.
U.S.) to pavements, structures, and utilities each year, is attributable to both double layer repulsions and water adsorption in soils that contain significant amounts of high plasticity clay minerals. Other causes of soil and rock expansion have been identified as well, such as pyrite related mineral transformations and sulfate reactions, often mediated by microorganisms. QUESTIONS AND PROBLEMS
1. What is the single most important property or characteristic controlling the consolidation and swelling behavior of a soil? Why?
2. If two samples of the same sand have the same relative density and are confined under the same effective stress, can they have different volume change properties? Why? 3. In what soil types and under what conditions do physical particle interactions dominate in determining the compression and swelling behavior? In what soil types and under what conditions do physicochemical factors dominate?
4. Provide an explanation for the differences in amount of swelling associated with expansion following the different stress paths shown in Fig. 10.10. 5. Consider the following soil profile beneath a level ground surface:
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Depth Range (m)
Soil Type
Unit Weight (kN/m3)
0–5 5–10 10–18 18–30 ⬎30
Surcharge fill Rubble fill Clean sand Soft clay Bedrock
19.0 17.0 18.0 16.0 —
The water table is at a depth of 8 m. a. Show profiles of vertical total, effective, and water pressure as a function of depth below the ground surface before placement of the surcharge fill. Assume that each layer is normally consolidated. b. Show profiles of vertical total, effective, and water pressure as a function of depth immediately after placement of the surcharge fill. Indicate if the clay layer is normally consolidated, overconsolidated, or underconsolidated at this time. c. Show profiles of vertical total, effective, and water pressure as a function of depth at a long time after the placement of the surcharge fill. Are the sand and clay layers normally consolidated, underconsolidated, or overconsolidated? d. Show profiles of vertical total, effective, and water pressure as a function of depth immediately after removal of the surcharge fill. Are the sand and clay layers normally consolidated, underconsolidated, or overconsolidated?
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QUESTIONS AND PROBLEMS
e. Show profiles of vertical total, effective, and water pressure as a function of depth at a long time after removal of the surcharge fill. Are the sand and clay layers normally consolidated, under-consolidated, or overconsolidated? f. Show depth profiles and approximate values of the horizontal coefficient of earth pressure at rest for the conditions in parts (a) through (e).
a. Sodium montmorillonite in 0.002 M NaCl b. Sodium montmorillonite in 0.2 M NaCl c. Sodium illite in 0.002 M NaCl d. Sodium illite in 0.2 M NaCl Assume any quantities needed but not stated. 11. Consider the real behavior of sediments formed from montmorillonite and illite in waters of the above concentrations. Approximately what void ratios would you expect to find after normal consolidation to a pressure of 1.0 atm? If different than the values you calculated in the preceding problem, state why?
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6. Two near-surface strata of the same soft clay are to be consolidated. In one the consolidation is to be done by placement of a surcharge fill at the ground surface. In the other, the consolidation is to be effected by lowering the water table to the bottom of the clay layer and evaporation of water from the ground surface, which will cause shrinkage of the clay. The ground water table is initially at the top of the clay stratum. Show profiles of effective stress and water pressure versus depth for each stratum corresponding to the condition where the vertical effective stress is the same in each at middepth. Will the clay structure be the same in each stratum at this depth at this time? Why?
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7. Describe and contrast the compression, consolidation, and swelling potential properties of the following soil types. Assume their initial states (water content, overburden pressure, environmental chemistry) to be representative of the indicated soil type as ordinarily encountered in nature. a. Loess b. Varved clay c. Carbonate sand d. Quick clay e. Tropical andisol f. Glacial moraine g. Torrential stream deposit or mudflow h. Sand hydraulic fill i. Compacted clay liner of an earth dam 8. Prepare a schematic diagram of liquidity index versus log effective consolidation pressure. Show the positions of normally consolidated and heavily overconsolidated samples of a given clay on this diagram.
9. Discuss the strengths and weaknesses of the osmotic pressure and water adsorption theories for clay swelling in terms of their adequacy to explain the influences of mineralogical and compositional factors on the swelling of fine-grained soils. 10. Calculate the equilibrium void ratios at a pressure of 1.0 atm for the following systems assuming that the DLVO and osmotic pressure theories are valid:
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12. A normally consolidated, saturated marine clay is sampled without structural disturbance from beneath the seafloor and sealed to prevent water movement in or out. The temperature of the clay in situ is 5C. The effective stress at the time of sampling is 200 kPa and the void ratio of the clay is 0.90. The sealed sample is taken immediately to the shipboard laboratory where the original in situ confining stress is immediately reapplied. a. What will be the subsequent effective stress in the laboratory at a temperature of 20C? The clay has a compression index of 0.5 and a swelling index of 0.05. Other properties are as follows: • Compressibility of water ⫽ ⫺4.83 ⫻ 10⫺5 cm2 /kg • Coefficient of thermal expansion of solid mineral particles ⫽ 0.35 ⫻ 10⫺4 C⫺1 • Coefficient of thermal expansion of water ⫽ 2.07 ⫻ 10⫺4C⫺1 • Coefficient of thermal expansion of the soil structure ⫽ 0.5 ⫻ 10⫺4C⫺1 b. How does the change in effective stress computed in part (a) compare with the value estimated on the basis of Table 10.10 in the text? c. If the same confining stress is maintained but drainage of the sample is then allowed, how much water, expressed as a percentage of the original sample volume, will move in or out of the clay? d. Illustrate the changes accompanying the operation in parts (a) and (c) on a diagram of void ratio versus log effective consolidation pressure.
13. Identify and discuss some possible consequences of seawater intrusion into a freshwater sand aquifer overlying a compressible clay stratum which, in turn, overlies another freshwater aquifer.
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(consolidating) a highly plastic clay slurry that is initially at a liquidity index considerably greater than 1.0. Explain how each of the methods that you have identified works.
15. Volume and temperature stability over long periods of time (thousands of years) is a very important consideration in the utilization of earth materials as containment barriers for various types of chemical and radioactive waste. What mineral types, gradations, and placement conditions would you specify for this application? Why?
17. Comment on the mechanisms of primary consolidation and secondary compression in terms of the rate-controlling factors, influences of and effects on soil structure, whether they occur sequentially or concurrently, and the suitability of our usual procedures for quantifying them for geoengineering analysis.
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14. What is a collapsing soil? What conditions can initiate collapse? What factors determine the magnitude and rate of collapse? Is the process compatible with the principle of effective stress? Why?
16. Suggest possible methods other than direct loading using surcharge fills for reducing the water content
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18. Suggest possible methods for preventing or reducing swelling on the exposure of expansive soil to water and explain the mechanisms involved.
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CHAPTER 11
11.1
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Strength and Deformation Behavior
INTRODUCTION
All aspects of soil stability—bearing capacity, slope stability, the supporting capacity of deep foundations, and penetration resistance, to name a few—depend on soil strength. The stress–deformation and stress– deformation–time behavior of soils are important in any problem where ground movements are of interest. Most relationships for the characterization of the stress–deformation and strength properties of soils are empirical and based on phenomenological descriptions of soil behavior. The Mohr–Coulomb equation is by far the most widely used for strength. It states that ff ⫽ c ⫹ ff tan
(11.1)
ff ⫽ c ⫹ ff tan
(11.2)
where ff is shear stress at failure on the failure plane, c is a cohesion intercept, ff is the normal stress on the failure plane, and is a friction angle. Equation (11.1) applies for ff defined as a total stress, and c and are referred to as total stress parameters. Equation (11.2) applies for ff defined as an effective stress, and c and are effective stress parameters. As the shear resistance of soil originates mainly from actions at interparticle contacts, the second equation is the more fundamental.
In reality, the shearing resistance of a soil depends on many factors, and a complete equation might be of the form Shearing resistance ⫽ F(e, c, , , C, H, T, , ˙ , S) (11.3)
in which e is the void ratio, C is the composition, H is the stress history, T is the temperature, is the strain, ˙ is the strain rate, and S is the structure. All parameters in these equations may not be independent, and the functional forms of all of them are not known. Consequently, the shear resistance values (including c and ) are determined using specified test type (i.e., direct shear, triaxial compression, simple shear), drainage conditions, rate of loading, range of confining pressures, and stress history. As a result, different friction angles and cohesion values have been defined, including parameters for total stress, effective stress, drained, undrained, peak strength, and residual strength. The shear resistance values applicable in practice depend on factors such as whether or not the problem is one of loading or unloading, whether or not short-term or long-term stability is of interest, and stress orientations. Emphasis in this chapter is on the fundamental factors controlling the strength and stress–deformation behavior of soils. Following a review of the general characteristics of strength and deformation, some re369
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11.2 GENERAL CHARACTERISTICS OF STRENGTH AND DEFORMATION Strength
1. In the absence of chemical cementation between grains, the strength (stress state at failure or the ultimate stress state) of sand and clay is approximated by a linear relationship with stress: ff ⫽ ff tan
or
Shear Stress τ or Stress Ratio τ/σ
) ⫽ (1ff ⫹ 3ff )sin (1ff ⫺ 3ff
(11.4)
(11.5)
where the primes designate effective stresses 1ff and 3ff are the major and minor principal effective stresses at failure, respectively. 2. The basic contributions to soil strength are frictional resistance between soil particles in contact and internal kinematic constraints of soil particles associated with changes in the soil fabric. The magnitude of these contributions depends on the effective stress and the volume change tendencies of the soil. For such materials the stress–strain curve from a shearing test is typically of the form shown in Fig. 11.1a. The maximum or peak strength of a soil (point b) may be greater than the critical state strength, in which the soil deforms under sustained loading at constant volume (point c). For some soils, the particles align along a localized failure plane after large shear strain or shear displacement, and the strength decreases even further to the residual strength (point d). The corresponding three failure envelopes can be defined as shown in Fig. 11.1b, with peak, critical, and residual friction angles (or states) as indicated. 3. Peak failure envelopes are usually curved in the manner shown in Fig. 4.16 and schematically in Fig. 11.1b. This behavior is caused by dilatancy suppression and grain crushing at higher stresses. Curved failure envelopes are also observed for many clays at residual state. When
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lationships among fabric, structure, and strength are examined. The fundamentals of bonding, friction, particulate behavior, and cohesion are treated in some detail in order to relate them to soil strength properties. Micromechanical interactions of particles in an assemblage and the relationships between interparticle friction and macroscopic friction angle are examined from discrete particle simulations. Typical values of strength parameters are listed. The concept of yielding is introduced, and the deformation behavior in both the preyield (including small strain stiffness) and post-yield regions is summarized. Time-dependent deformations and aging effects are discussed separately in Chapter 12. The details of strength determination by means of laboratory and in situ tests and the detailed constitutive modeling of soil deformation and strength for use in numerical analyses are outside the scope of this book.
Secant Peak Strength Envelope
Peak
b
Shear Stress τ
c
At Large Strains
Critical state Strength Envelope
Tangent Peak Strength Envelope
Peak Strength
φpeak
d
φcritical state
b, c
Critical State
b
Residual Strength Envelope
φ residual
c
Residual
d
d
a
a Normal effective stress σ
a
Strain
Dense or Overconsolidated
(a)
Loose or Normally Consolidated
(b)
Figure 11.1 Peak, critical, and residual strength and associated friction angle: (a) a typical
stress–strain curve and (b) stress states.
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GENERAL CHARACTERISTICS OF STRENGTH AND DEFORMATION
4. The peak strength of cohesionless soils is influenced most by density, effective confining pressures, test type, and sample preparation methods. For dense sand, the secant peak friction angle (point b in Fig. 11.1b) consists in part
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expressed in terms of the shear strength normalized by the effective normal stress as a function of effective normal stress, curves of the type shown in Fig. 11.2 for two clays are obtained.
Figure 11.2 Variation of residual strength with stress level (after Bishop et al., 1971): (a) Brown London clay and (b) Weald clay.
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STRENGTH AND DEFORMATION BEHAVIOR
resistance depends only on composition and effective stress. The basic concept of the critical state is that under sustained uniform shearing at failure, there exists a unique combination of void ratio e, mean pressure p, and deviator stress q.1 The critical states of reconstituted Weald clay and Toyoura sand are shown in Fig. 11.4. The critical state line on the p –q plane is linear,2 whereas that on an e-ln p (or e-log p) plane tends to be linear for clays and nonlinear for sands. 7. At failure, dense sands and heavily overconsolidated clays have a greater volume after drained shear or a higher effective stress after undrained shear than at the start of deformation. This is due to its dilative tendency upon shearing. At failure, loose sands and normally consolidated to moderately overconsolidated clays (OCR up to about 4) have a smaller volume after drained shear or a lower effective stress after undrained shear than they had initially. This is due to its contractive tendency upon shearing. 8. Under further deformation, platy clay particles begin to align along the failure plane and the shear resistance may further decrease from the critical state condition. The angle of shear resistance at this condition is called the residual friction angle, as illustrated in Fig. 11.1b. The postpeak shearing displacement required to cause a reduction in friction angle from the critical state value to the residual value varies with the soil type, normal stress on the shear plane, and test conditions. For example, for shale mylonite3 in contact with smooth steel or other polished hard surfaces, a shearing displacement of only 1 or 2 mm is sufficient to give residual strength.4 For soil against soil, a slip along the
e ff
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Void Ratio e Water Content w
of internal rolling and sliding friction between grains and in part of interlocking of particles (Taylor, 1948). The interlocking necessitates either volume expansion (dilatancy) or grain fracture and/or crushing if there is to be deformation. For loose sand, the peak friction angle (point b in Fig. 11.1b) normally coincides with the critical-state friction angle (point c), and there is no peak in the stress–strain curve. 5. The peak strength of saturated clay is influenced most by overconsolidation ratio, drainage conditions, effective confining pressures, original structure, disturbance (which causes a change in effective stress and a loss of cementation), and creep or deformation rate effects. Overconsolidated clays usually have higher peak strength at a given effective stress than normally consolidated clays, as shown in Fig. 11.3. The differences in strength result from both the different stress histories and the different water contents at peak. For comparisons at the same water content but different effective stress, as for points A and A, the Hvorslev strength parameters ce and e are obtained (Hvorslev, 1937, 1960). Further details are given in Section 11.9. 6. During critical state deformation a soil is completely destructured. As illustrated in Fig. 11.1b, the critical state friction angle values are independent of stress history and original structure; for a given set of testing conditions the shearing
Normally Consolidated Virgin Compression
A
A
Shear Stress τ
Rebound Overconsolidated
τ
σff
In three-dimensional stress space ⫽ ( x, y, z, xy, yz, zx) or the equivalent principal stresses ( 1, 2, 3), the mean effective stress p, and the deviator stress q is defined as 1
σe
p ⫽ (x ⫹ y ⫹ z) / 3 ⫽ (1 ⫹ 2 ⫹ 3) / 3
Peak Strength Envelope
φcrit
2 2 2 兹(x ⫺ y)2 ⫹ (y ⫺ z)2 ⫹ (z ⫺ x)2 ⫹ 6 xy ⫹ 6 yz ⫹ 6 zx
Overconsolidated
A
A
φe
Hvorslev Envelope
ce
Normally Consolidated
0
σff
q ⫽ (1 / 兹2)
Normal Effective Stress σ
Figure 11.3 Effect of overconsolidation on effective stress
strength envelope.
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⫽ (1 / 兹2)兹(1 ⫺ 2)2 ⫹ (2 ⫺ 3)2 ⫹ (3 ⫺ 1)2
For triaxial compression condition ( 1 ⬎ 2 ⫽ 3), p ⫽ ( 1 ⫹ 2 2) / 3, q ⫽ 1 ⫺ 2 2 The critical state failure slope on p–q plane is related to friction angle , as described in Section 11.10. 3 A rock that has undergone differential movements at high temperature and pressure in which the mineral grains are crushed against one another. The rock shows a series of lamination planes. 4 D. U. Deere, personal communication (1974).
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GENERAL CHARACTERISTICS OF STRENGTH AND DEFORMATION
4 Deviator Stress q (MPa)
Critical State Line 400 300 200
Overconsolidated Normally Consolidated
100 0
0
Critical State Line 3
2
1
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Deviator Stress q (kPa)
500
100
0
200 300 400 500 600 Mean Pressure p(kPa)
0
(a-1) p versus q
0.7
1 2 3 Mean Pressure p(MPa) (b-1) p versus q
Critical State Line
Critical State Line
0.95
0.5
0.4
Isotropic Normal Compression Line
Overconsolidated
0.3
Initial State
0.90
Void ratio e
Void ratio e
0.6
4
0.85 0.80 0.75
Normally Consolidated
100
200
300 400 500
Mean Pressure p (kPa) (a-2) e versus lnp (a)
0.02
0.05 0.1 0.5 1 Mean Pressure p(MPa)
5
(b-2) e versus logp (b)
Figure 11.4 Critical states of clay and sand: (a) Critical state of Weald clay obtained by drained triaxial compression tests of normally consolidated () and overconsolidated (●) specimens: (a-1) q–p plane and (a-2) e–ln p plane (after Roscoe et al., 1958). (b) Critical state of Toyoura sand obtained by undrained triaxial compression tests of loose and dense specimens consolidated initially at different effective stresses, (b-1) q–p plane and (b-2) e– log p plane (after Verdugo and Ishihara, 1996).
shear plane of several tens of millimeters may be required, as shown by Fig. 11.5. However, significant softening can be caused by strain localization and development of shear bands, especially for dense samples under low confinement. 9. Strength anisotropy may result from both stress and fabric anisotropy. In the absence of chemical cementation, the differences in the strength
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of two samples of the same soil at the same void ratio but with different fabrics are accountable in terms of different effective stresses as discussed in Chapter 8. 10. Undrained strength in triaxial compression may differ significantly from the strength in triaxial extension. However, the influence of type of test (triaxial compression versus extension) on the effective stress parameters c and is relatively
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STRENGTH AND DEFORMATION BEHAVIOR
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374
Figure 11.5 Development of residual strength with increasing shear displacement (after
Bishop et al., 1971).
small. Effective stress friction angles measured in plane strain are typically about 10 percent greater than those determined by triaxial compression. 11. A change in temperature causes either a change in void ratio or a change in effective stress (or a combination of both) in saturated clay, as discussed in Chapter 10. Thus, a change in temperature can cause a strength increase or a strength decrease, depending on the circumstances, as illustrated by Fig. 11.6. For the tests on kaolinite shown in Fig. 11.6, all samples were prepared by isotropic triaxial consolidation at 75F. Then, with no further drainage allowed, temperatures were increased to the values indicated, and the samples were tested in unconfined compression. Substantial reductions in strength accompanied the increases in temperature. Stress–Strain Behavior
1. Stress–strain behavior ranges from very brittle for some quick clays, cemented soils, heavily overconsolidated clays, and dense sands to ductile for insensitive and remolded clays and loose sands, as illustrated by Fig. 11.7. An increase in
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Figure 11.6 Effect of temperature on undrained strength of kaolinite in unconfined compression (after Sherif and Burrous, 1969).
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375
GENERAL CHARACTERISTICS OF STRENGTH AND DEFORMATION
(a) Typical Strain Ranges in the Field
Stiffness G or E
Retaining Walls Foundations Tunnels
Linear Elastic Nonlinear Elastic Preyield Plastic
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Full Plastic
10-4
10-3
10-2
Figure 11.7 Types of stress–strain behavior.
10-1
100
101
Strain %
Dynamic Methods
Local Gauges
confining pressure causes an increase in the deformation modulus as well as an increase in strength, as shown by Fig. 11.8. 2. Stress–strain relationships are usually nonlinear; soil stiffness (often expressed in terms of tangent or secant modulus) generally decreases with increasing shear strain or stress level up to peak failure stress. Figure 11.9 shows a typical stiffness degradation curve, in terms of shear modulus G and Young’s modulus E, along with typical strain levels developed in geotechnical construction (Mair, 1993) and as associated with different laboratory testing techniques used to measure the stiffness (Atkinson, 2000). For example, Fig. 11.10 shows the stiffness degradation of sands and clay subjected to increase in shear strain. As illustrated in Fig. 11.9, the stiffness degradation curve can be separated into
Figure 11.8 Effect of confining pressure on the consolidated-drained stress–strain behavior of soils.
Copyright © 2005 John Wiley & Sons
Conventional Soil Testing
(b) Typical Strain Ranges for Laboratory Tests
Figure 11.9 Stiffness degradation curve: stiffness plotted
against logarithm of strains. Also shown are (a) the strain levels observed during construction of typical geotechnical structures (after Mair, 1993) and (b) the strain levels that can be measured by various techniques (after Atkinson, 2000).
four zones: (1) linear elastic zone, (2) nonlinear elastic zone, (3) pre-yield plastic zone, and (4) full plastic zone. 3. In the linear elastic zone, soil particles do not slide relative to each other under a small stress increment, and the stiffness is at its maximum. The soil stiffness depends on contact interactions, particle packing arrangement, and elastic stiffness of the solids. Low strain stiffness values can be determined using elastic wave velocity measurements, resonant column testing, or local strain transducer measurements. The magnitudes of the small strain shear modulus (Gmax) and Young’s modulus (Emax) depend on applied confining pressure and the packing conditions of soil particles. The following empirical equations are often employed to express these dependencies: Gmax ⫽ AG FG(e)pnG
(11.6)
Ei(max) ⫽ AE FE(e)i nE
(11.7)
where FG(e) and FE(e) are functions of void ratio, p is the mean effective confining pressure, i is the effective stress in the i direction, and the other parameters are material constants.
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11
STRENGTH AND DEFORMATION BEHAVIOR
140 120
Confining Pressure
PSC
78.4 kPa
Toyoura Sand
49 kPa
Ticino Sand
100 80 60 40 20 10-4
Confining Pressures
120
σc = 400 kPa
100
σc = 200 kPa
80 60
σc = 100 kPa
40
σc = 30 kPa
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Secant Shear Modulus G (MPa)
TC
Secant Shear Modulus G (MPa)
376
10-3
10-2
10-1
100
20
10-5
10-4
10-3
10-2
10-1
100
Shear Strain (%)
Shear Strain (%) (a)
(b)
Figure 11.10 Stiffness degradation curve at different confining pressures: (a) Toyoura and
Ticino sands (TC: triaxial compression tests, PSC: plain strain compression tests) (after Tatsuoka et al., 1997) and (b) reconstituted Kaolin clay (after Soga et al., 1996).
plastic soils at low confining pressure conditions to greater than 5 ⫻ 10⫺2 percent at high confining pressure or in soils with high plasticity (Santamarina et al., 2001). 5. Irrecoverable strains develop in the pre-yield plastic zone. The initiation of plastic strains can be determined by examining the onset of permanent volumetric strain in drained conditions or residual excess pore pressures in undrained conditions after unloading. Available experi-
104 103 102
Undisturbed Remolded Remolded with CaCO3 nG = 0.13
nG = 0.65
nG = 0.63
101
100 100
101 102 103 104 Confining pressure, p⬘ (kPa) (a)
Vertical Young's Modulus Evmax/FE(e) (MPa)
Shear Modulus,Gmax MPa
Figure 11.11 shows examples of the fitting of the above equations to experimental data. 4. The stiffness begins to decrease from the linear elastic value as the applied strains or stresses increase, and the deformation moves into the nonlinear elastic zone. However, a complete cycle of loading, unloading, and reloading within this zone shows full recovery of strains. The strain at the onset of the nonlinear elastic zone ranges from less than 5 ⫻ 10⫺4 percent for non-
500
At each vertical effective stress, horizontal effective stress σh⬘ (kPa) was varied between 98 kPa and 196 kPa
450 400
nE = 0.49
350 300 250
100
150 200
250
300
Vertical Effective Stress,σv⬘ (kPa) (b)
Figure 11.11 Small strain stiffness versus confining pressure: (a) Shear modulus Gmax of cemented silty sand measured by resonant column tests (from Stokoe et al. 1995) and (b) vertical Young’s modulus of sands measured by triaxial tests (after Tatsuoka and Kohata, 1995).
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GENERAL CHARACTERISTICS OF STRENGTH AND DEFORMATION
mental data suggest that the strain level that initiates plastic strains ranges between 7 ⫻ 10⫺3 and 7 ⫻ 10⫺2 percent, with the lower limit for uncemented normally consolidated sands and the upper limit for high plasticity clays and cemented sands. 6. A distinctive kink in the stress–strain relationship defines yielding, beyond which full plastic strains are generated. A locus of stress states that initiate yielding defines the yield envelope. Typical yield envelopes for sand and natural clay are shown in Fig. 11.12. The yield envelope expands, shrinks, and rotates as plastic strains develop. It is usually considered that expansion is related to plastic volumetric strains; the surface expands when the soil compresses and shrinks when the soil dilates. The two inner envelopes shown in Fig. 11.12b define the boundaries between linear elastic, nonlinear elastic, and pre-yield zones. When the stress state moves in the pre-yield zone, the inner envelopes move with the stress state. This multienvelope concept allows modeling of complex deformations observed for different stress paths (Mroz, 1967; Pre´vost, 1977; Dafalias and Herrman, 1982; Atkinson et al., 1990; Jardine, 1992). 7. Plastic irrecoverable shear deformations of saturated soils are accompanied by volume
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changes when drainage is allowed or changes in pore water pressure and effective stress when drainage is prevented. The general nature of this behavior is shown in Figs. 11.13a and 11.13b for drained and undrained conditions, respectively. The volume and pore water pressure changes depend on interactions between fabric and stress state and the ease with which shear deformations can develop without overall changes in volume or transfer of normal stress from the soil structure to the pore water. 8. The stress–strain relation of clays depends largely on overconsolidation ratio, effective confining pressures, and drainage conditions. Figure 11.14 shows triaxial compression behavior of clay specimens that are first normally consolidated and then isotropically unloaded to different overconsolidation ratios before shearing. The specimens are consolidated at the same confining pressure p0, but have different void ratios due to the different stress history (Fig. 11.14a). Drained tests on normally consolidated clays and lightly overconsolidated clays show ductile behavior with volume contraction (Fig. 11.14b). Heavily overconsolidated clays exhibit a stiff response initially until the stress state reaches the yield envelope giving the peak strength and volume dilation. The state of the
Yield State Pre-yield State
Initial Condition
q = σ⬘a-σ⬘r MPa Failure Line 0.8
q = σ⬘a-σ⬘r
Yield State
Initial State Surrounded by Linear Elastic Boundary
MPa
Stress Path
0.6
Yield Envelope
Yield Envelope
0.6
0.4
Preyield Boundary
0.4
0.2
0.2
Linear Elastic Boundary
MPa
0.0
0.2
0.4
0.6
0.8
1.0 p = (σa + 2σr)/3
-0.2 -0.4
0.0
0.2
MPa
0.4
0.6 p = (σa + 2σr)/3
-0.2
Failure Line
(a)
(b)
Figure 11.12 Yield envelopes: (a) Aoi sand (Yasufuku et al., 1991) and (b) Bothkennar clay (from Smith et al., 1992).
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STRENGTH AND DEFORMATION BEHAVIOR
Same Initial Confining Pressure
Same Initial Confining Pressure
Dense Soil
Critical State Dense Soil
Metastable Fabric
Deviator Stress
Deviator Stress
Cavitation
Critical State Loose Soil
Loose Soil
Critical State
Metastable Fabric Axial or Deviator Strain
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Axial or Deviator Strain
Dense Soil
Dense Soil
+ΔV/V0
-Δu
0
0
Cavitation
Loose soil
Loose Soil
+Δu
-ΔV/V0
Metastable Fabric
Metastable Fabric
(a)
(b)
Figure 11.13 Volume and pore pressure changes during shear: (a) drained conditions and (b) undrained conditions.
Initial State Failure at Critical State (D: Drained, U: Undrained)
Void Ratio
Deviator Stress
3 Heavily Overconsolidated
Deviator 2 Lightly Stress Overconsolidated
U3
2 Lightly Overconsolidated U2
D Critical State
Virgin Compression Line
3 Heavily Overconsolidated
1 Normally Consolidated
U1
1 Normally Consolidated
1 Normally consolidated
U1
Axial or Deviatoric Strain
Axial or Deviatoric Strain
2 Lightly Overconsolidated
U2
D
U3
+ΔV/V0
3 Heavily Overconsolidated
-Δu
3 Heavily Overconsolidated
3 Heavily Overconsolidated
Critical State Line
p0
-ΔV/V0
2 Lightly Overconsolidated
2 Lightly Overconsolidated
+Δu
1 Normally Consolidated
log p
1 Normally Consolidated
(a)
(b)
(c)
Figure 11.14 Stress–strain relationship of normally consolidated, lightly overconsolidated,
and heavily overconsolidated clays: (a) void ratio versus mean effective stress, (b) drained tests, and (c) undrained tests.
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FABRIC, STRUCTURE, AND STRENGTH
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soil then progressively moves toward the critical state exhibiting softening behavior. Undrained shearing of normally consolidated and lightly overconsolidated clays generates positive excess pore pressures, whereas shear of heavily overconsolidated clays generates negative excess pore pressures (Fig. 11.14c). 9. The magnitudes of pore pressure that are developed in undrained loading depend on initial consolidation stresses, overconsolidation ratio, density, and soil fabric. Figure 11.15 shows the undrained effective stress paths of anisotropically and isotropically consolidated specimens (Ladd and Varallyay, 1965). The difference in undrained shear strength is primarily due to different excess pore pressure development associated with the change in soil fabric. At large strains, the stress paths correspond to the same friction angle. 10. A temperature increase causes a decrease in undrained modulus; that is, a softening of the soil. As an example, initial strain as a function of stress is shown in Fig. 11.16 for Osaka clay
Figure 11.16 Effect of temperature on the stiffness of Osaka clay in undrained triaxial compression (Murayama, 1969).
Failure Line in Triaxial Compression
(MPa) 0.3
tested in undrained triaxial compression at different temperatures. Increase in temperature causes consolidation under drained conditions and softening under undrained conditions.
σr/σa = 0.54
Deviator Stress q = σa + σrσ
0.2
0.1
11.3
0.0
0.1
0.2
0.3
0.4
(MPa)
Mean Pressure p = (σa + 2σr )/3
-0.1
-0.2
σr/σa = 1.84
-0.3 Initial
Failure Line in Triaxial Extension
At Failure Anisotropically Consolidated σr/σa = 0.54 Isotropically Consolidated
Anisotropically Consolidated σr/σa = 1.84
Figure 11.15 Undrained effective stress paths of anisotrop-
ically and isotropically consolidated specimens (after Ladd and Varallyay, 1965).
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379
FABRIC, STRUCTURE, AND STRENGTH
Fabric Changes During Shear of Cohesionless Materials
The deformation of sands, gravels, and rockfills is influenced by the initial fabric, as discussed and illustrated in Chapter 8. As an illustration, fabric changes associated with the sliding and rolling of grains during triaxial compression were determined using a uniform sand composed of rounded to subrounded grains with sizes in the range of 0.84 to 1.19 mm and a mean axial length ratio of 1.45 (Oda, 1972, 1972a, 1972b, 1972c). Samples were prepared to a void ratio of 0.64 by tamping and by tapping the side of the forming mold. A delayed setting water–resin solution was used as the pore fluid. Samples prepared by each method were tested to successively higher strains. The resin was then allowed to set, and thin sections were prepared. The differences in initial fabrics gave the markedly different stress–strain and volumetric strain curves shown in Fig. 11.17, where the plunging method refers to
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STRENGTH AND DEFORMATION BEHAVIOR
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380
Figure 11.17 Stress–strain and volumetric strain relationships for sand at a void ratio of
0.64 but with different initial fabrics (after Oda, 1972a). (a) Sample saturated with water and (b) sample saturated with water–resin solution.
tamping. There is similarity between these curves and those for Monterey No. 0 sand shown in Fig. 8.23. A statistical analysis of the changes in particle orientation with increase in axial strain showed: 1. For samples prepared by tapping, the initial fabric tended toward some preferred orientation of long axes parallel to the horizontal plane, and the intensity of orientation increased slightly during deformation. 2. For samples prepared by tamping, there was very weak preferred orientation in the vertical direction initially, but this disappeared with deformation.
Shear deformations break down particle and aggregate assemblages. Shear planes or zones did not appear until after peak stress had been reached; however, the distribution of normals to the interparticle contact planes E() (a measure of fabric anisotropy) did change with strain, as may be seen in Fig. 11.18. This figure shows different initial distributions for samples prepared by the two methods and a concentration of contact plane normals within 50 of the vertical as deformation progresses. Thus, the fabric tended toward greater anisotropy in each case in terms of contact plane orientations. There was little additional change in E() after the peak stress had been reached, which implies that particle rearrangement was proceeding without significant change in the overall fabric.
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As the stress state approaches failure, a direct shearinduced fabric forms that is generally composed of regions of homogeneous fabric separated by discontinuities. No discontinuities develop before peak strength is reached, although there is some particle rotation in the direction of motion. Near-perfect preferred orientation develops during yield after peak strength is reached, but large deformations may be required to reach this state. Compaction Versus Overconsolidation of Sand
Specimens at the same void ratio and stress state before shearing, but having different fabrics, can exhibit different stress–strain behavior. For example, consider a case in which one specimen is overconsolidated, whereas the other is compacted. The two specimens are prepared in such a way that the initial void ratio is the same for a given initial isotropic confining pressure. Coop (1990) performed undrained triaxial compression tests of carbonate sand specimens that were either overconsolidated or compacted, as illustrated in Fig. 11.19a. The undrained stress paths and stress– strain curves for the two specimens are shown in Figs. 11.19b and 11.19c, respectively. The overconsolidated sample was initially stiffer than the compacted specimen. The difference can be attributed to (i) different soil fabrics developed by different stress paths prior to shearing and (ii) different degrees of particle crushing prior to shearing (i.e., some breakage has occurred dur-
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FABRIC, STRUCTURE, AND STRENGTH
381
Figure 11.18 Distribution of interparticle contact normals as a function of axial strain for sand samples prepared in two ways (after Oda, 1972a): (a) specimens prepared by tapping and (b) specimens prepared by tamping.
ing the preconsolidation stage for the overconsolidated specimen). Therefore, overconsolidation and compaction produced materials with different mechanical properties. However, at large deformations, both specimens exhibited similar strengths because the initial fabrics were destroyed. Effect of Clay Structure on Deformations
The high sensitivity of quick clays illustrates the principle that flocculated, open microfabrics are more rigid but more unstable than deflocculated fabrics. Similar behavior may be observed in compacted fine-grained soils, and the results of a series of tests on structuresensitive kaolinite are illustrative of the differences (Mitchell and McConnell, 1965). Compaction conditions and stress–strain curves for samples of kaolinite compacted using kneading and static methods are shown in Fig. 11.20. The high shear strain associated with kneading compaction wet of optimum breaks down flocculated structures, and this accounts for the
Copyright © 2005 John Wiley & Sons
much lower peak strength for the sample prepared by kneading compaction. The recoverable deformation of compacted kaolinite with flocculent structure ranges between 60 and 90 percent, whereas the recovery of samples with dispersed structures is only of the order of 15 to 30 percent of the total deformation, as may be seen in Fig. 11.21. This illustrates the much greater ability of the braced-box type of fabric that remains after static compaction to withstand stress without permanent deformation than is possible with the broken-down fabric associated with kneading compaction. Different macrofabric features can affect the deformation behavior as illustrated in Fig. 11.22 for the undrained triaxial compression testing of Bothkennar clay, Scotland (Paul et al., 1992; Clayton et al., 1992). Samples with mottled facies, in which the bedding features had been disrupted and mixed by burrowing mollusks and worms (bioturbation), gave the stiffest response, whereas samples with distinct laminated features showed the softest response.
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STRENGTH AND DEFORMATION BEHAVIOR 1.0 Overconsolidated 0.8
Normal Compression Line
q (MPa)
2
0.6 0.4 Compacted 0.2
Overconsolidated Sample
0
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Void 1.5 Ratio
0
0.2 0.4 p (MPa)
0.6
(b)
q (MPa)
Compacted Sample
1.0
1
Compacted
0.75 0.5
0.1 1 Mean Pressure p
Overconsolidated
(MPa)
0.25
(a)
0
0
4
8 12 16 Axial strain ε a (%)
20
(c)
Figure 11.19 Undrained response of compacted specimen and overconsolidated specimen
of carbonate sand: (a) stress path before shearing, (b) undrained stress paths during shearing, and (c) stress–strain relationships (after Coop, 1990).
If slip planes develop at failure, platy and elongated particles align with their long axes in the direction of slip. By then, the basal planes of the platy clay particles are enclosed between two highly oriented bands of particles on opposite sides of the shear plane. The dominant mechanism of deformation in the displacement shear zone is basal plane slip, and the overall thickness of the shear zone is on the order of 50 m. Fabrics associated with shear planes and zones have been studied using thin sections and the polarizing microscope and by using the electron microscope (Morgenstern and Tchalenko, 1967a, b and c; Tchalenko, 1968; McKyes and Yong, 1971). The residual strength associated with these fabrics is treated in more detail in Section 11.11. Structure, Effective Stresses, and Strength
The effective stress strength parameters such as c and are isotropic properties, with anisotropy in undrained strength explainable in terms of excess pore pressures developed during shear. The undrained strength loss associated with remolding undisturbed
Copyright © 2005 John Wiley & Sons
clay can also be accounted for in terms of differences in effective stress, provided part of the undisturbed strength does not result from cementation. Remolding breaks down the structure and causes a transfer of effective stress to the pore water. An example of this is shown in Fig. 11.23, which shows the results of incremental loading triaxial compression tests on two samples of undisturbed and remolded San Francisco Bay mud. In these tests, the undisturbed sample was first brought to equilibrium under an isotropic consolidation pressure of 80 kPa. After undrained loading to failure, the triaxial cell was disassembled, and the sample was remolded in place. The apparatus was reassembled, and pore pressure was measured. Thus, the effective stress at the start of compression of the remolded clay at the same water content as the original undisturbed clay was known. Stress–strain and pore pressure–strain curves for two samples are shown in Figs. 11.23a and 11.23b, and stress paths for test 1 are shown in Fig. 11.23c. Differences in strength that result from fabric differences caused by thixotropic hardening or by different compaction methods can be explained in the same
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FRICTION BETWEEN SOLID SURFACES
Figure 11.21 Ratio of recoverable to total strain for samples of kaolinite with different structure.
Stress-Strain Relationships
Stress Paths
0.6
0.4
Facies Mottled Bedded Laminated
0.2
(σa – σr)/2 σao
(σa – σr)/2 σao
0.6
0.0
Figure 11.20 Stress–strain behavior of kaolinite compacted
by two methods.
0.4
0.0
0
2
4
Facies Mottled Bedded Laminated
0.2
0.4
0.6
0.8
1.0
(σa + σr)/2σao
Axial Strain (%)
Figure 11.22 Effect of macrofabric on undrained response
way. Thus, in the absence of chemical or mineralogical changes, different strengths in two samples of the same soil at the same void ratio can be accounted for in terms of different effective stress.
11.4
FRICTION BETWEEN SOLID SURFACES
The friction angle used in equations such as (11.1), (11.2), (11.4), and (11.5) contains resistance contributions from several sources, including sliding of grains in contact, resistance to volume change (dilatancy), grain rearrangement, and grain crushing. The
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of Bothkennar clay in Scotland (after Hight and Leroueil, 2003).
true friction coefficient is shown in Fig. 11.24 and is represented by ⫽
T ⫽ tan N
(11.8)
where N is the normal load on the shear surface, T is the shear force, and , the intergrain sliding friction angle, is a compositional property that is determined by the type of soil minerals.
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STRENGTH AND DEFORMATION BEHAVIOR
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384
Figure 11.23 (a) and (b) Effect of remolding on undrained strength and pore water pressure in San Francisco Bay mud. (c) Stress paths for triaxial compression tests on undisturbed and remolded samples of San Francisco Bay mud.
Basic ‘‘Laws’’ of Friction
Two laws of friction are recognized, beginning with Leonardo da Vinci in about 1500. They were restated by Amontons in 1699 and are frequently referred to as Amontons’ laws. They are:
1. The frictional force is directly proportional to the normal force, as illustrated by Eq. (11.8) and Fig. 11.24. 2. The frictional resistance between two bodies is independent of the size of the bodies. In Fig.
Copyright © 2005 John Wiley & Sons
11.24, the value of T is the same for a given value of N regardless of the size of the sliding block.
Although these principles of frictional resistance have long been known, suitable explanations came much later. It was at one time thought that interlocking between irregular surfaces could account for the behavior. On this basis, would be given by the tangent of the average inclination of surface irregularities on the sliding plane. This cannot be the case, however, because such an explanation would require that de-
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FRICTION BETWEEN SOLID SURFACES
385
Figure 11.23 (Continued )
contacting surfaces. He observed that the actual area of contact is very small because of surface irregularities, and thus the cohesive forces must be large. The foundation for the present understanding of the mobilization of friction between surfaces in contact was laid by Terzaghi (1920). He hypothesized that the normal load N acting between two bodies in contact causes yielding at asperities, which are local ‘‘hills’’ on the surface, where the actual interbody solid contact develops. The actual contact area Ac is given by Ac ⫽
where y is the shearing strength assumed to have force that can be
N y
(11.9)
yield strength of the material. The of the material in the yielded zone is a value m. The maximum shearing resisted by the contact is then T ⫽ Acm
Figure 11.24 Coefficient of friction for surfaces in contact.
(11.10)
The coefficient of friction is given by T/N,
crease as surfaces become smoother and be zero for perfectly smooth surfaces. In fact, the coefficient of friction can be constant over a range of surface roughness. Hardy (1936) suggested instead that static friction originates from cohesive forces between
Copyright © 2005 John Wiley & Sons
⫽
T Acm m ⫽ ⫽ N Acy y
(11.11)
This concept of frictional resistance was subsequently further developed by Bowden and Tabor (1950,
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STRENGTH AND DEFORMATION BEHAVIOR
1964). The Terzaghi–Bowden and Tabor hypothesis, commonly referred to as the adhesion theory of friction, is the basis for most modern studies of friction. Two characteristics of surfaces play key roles in the adhesion theory of friction: roughness and surface adsorption. Surface Roughness
Because of unsatisfied force fields at the surfaces of solids, the surface structure may differ from that in the interior, and material may be adsorbed from adjacent phases. Even ‘‘clean’’ surfaces, prepared by fracture of a solid or by evacuation at high temperature, are rapidly contaminated when reexposed to normal atmospheric conditions. According to the kinetic theory of gases, the time for adsorption of a monolayer tm is given by
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The surfaces of most solids are rough on a molecular scale, with successions of asperities and depressions ranging from 10 nm to over 100 nm in height. The slopes of the nanoscale asperities are rather flat, with individual angles ranging from about 120 to 175 as shown in Fig. 11.25. The average slope of asperities on metal surfaces is an included angle of 150; on rough quartz it may be over 175 (Bromwell, 1966). When two surfaces are brought together, contact is established at the asperities, and the actual contact area is only a small fraction of the total surface area. Quartz surfaces polished to mirror smoothness may consist of peaks and valleys with an average height of about 500 nm. The asperities on rougher quartz surfaces may be about 10 times higher (Lambe and Whitman, 1969). Even these surfaces are probably smoother than most soil particles composed of bulky minerals. The actual surface texture of sand particles depends on geologic history as well as mineralogy, as shown in Fig. 2.12. The cleavage faces of mica flakes are among the smoothest naturally occurring mineral surfaces. Even in mica, however, there is some waviness due to rotation of tetrahedra in the silica layer, and surfaces usually contain steps ranging in height from 1 to 100 nm, reflecting different numbers of unit layers across the particle. Thus, large areas of solid contact between grains are not probable in soils. Solid-to-solid contact is through asperities, and the corresponding interparticle contact stresses are high. The molecular structure and composition in the contacting asperities determine the magnitude of m in Eq. (11.11).
Surface Adsorption
Figure 11.25 Contact between two smooth surfaces.
Copyright © 2005 John Wiley & Sons
tm ⫽
1 SZ
(11.12)
where is the area occupied per molecule, S is the fraction of molecules striking the surface that stick to it, and Z is the number of molecules per second striking a square centimeter of surface. For a value of S equal to 1, which is reasonable for a high-energy surface, the relationship between tm and gas pressure is shown in Fig. 11.26. The conclusion to be drawn from this figure is that adsorbed layers are present on the surface of soil particles in the terrestrial environment,
Figure 11.26 Monolayer formation time as a function of atmospheric pressure.
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FRICTION BETWEEN SOLID SURFACES
and contacts through asperities involve adsorbed material, unless it is extruded under the high pressure.5 Adhesion Theory of Friction
cles. The asperities, caused by surface waviness, are more regular but not as high as those for the bulky minerals. Thus, it can be postulated that for a given number of contacts per particle, the load per asperity decreases with decreasing particle size and, for particles of the same size, is less for platy minerals than for bulky minerals. Because should increase as the normal load per asperity increases, and it is reasonable to assume that the adsorbed film strength is less than the strength of the solid material (c ⬍ m), it follows that the true friction angle () is less for small and platy particles than for large and bulky particles. In the event that two platy particles are in face-to-face contact and the surface waviness is insufficient to cause direct solid-tosolid contact, shear will be through the adsorbed films, and the effective value of will be zero, again giving a lower value of . In reality, the behavior of plastic junctions is more complex. Under combined compression and shear stresses, deformation follows the von Mises–Henky criterion, which, for two dimensions, is
Co py rig hte dM ate ria l
The basis for the adhesion theory of friction is in Eq. (11.10), that is, the tangential force that causes sliding depends on the solid contact area and the shear strength of the contact. Plastic and/or elastic deformations determine the contact area at asperities. Plastic Junctions If asperities yield and undergo plastic deformation, then the contact area is proportional to the normal load on the asperity as shown by Eq. (11.9). Because surfaces are not clean, but are covered by adsorbed films, actual solid contact may develop only over a fraction of the contact area as shown in Fig. 11.27. If the contaminant film strength is c, the strength of the contact will be T ⫽ Ac[m ⫹ (1 ⫺ )c]
(11.13)
Equation (11.13) cannot be applied in practice because and c are unknown. However, it does provide a possible explanation for why measured values of friction angle for bulky minerals such as quartz and feldspar are greater than values for the clay minerals and other platy minerals such as mica, even though the surface structure is similar for all the silicate minerals. The small particle size of clays means that the load per particle, for a given effective stress, will be small relative to that in silts and sands composed of the bulky minerals. The surfaces of platy silt and sand size particles are smoother than those of bulky mineral parti-
2 ⫹ 3 2 ⫽ y2
sorbed surface films.
5
Conditions may be different on the Moon, where ultrahigh vacuum exists. This vacuum produces cleaner surfaces. In the absence of suitable adsorbate, clean surfaces can reduce their surface energy by cohering with like surfaces. This could account for the higher cohesion of lunar soils than terrestrial soils of comparable gradation.
Copyright © 2005 John Wiley & Sons
(11.14)
For asperities loaded initially to ⫽ y, the application of a shear stress requires that become less than y. The only way that this can happen is for the contact area to increase. Continued increase in leads to continued increase in contact area. This phenomenon is called junction growth and is responsible for cold welding in some materials (Bowden and Tabor, 1964). If the shear strength of the junction equals that of the bulk solid, then gross seizure occurs. For the case where the ratio of junction strength to bulk material strength is less than 0.9, the amount of junction growth is small. This is the probable situation in soils. Elastic Junctions The contact area between particles of a perfectly elastic material is not defined in terms of plastic yield. For two smooth spheres in contact, application of the Hertz theory leads to d ⫽ (NR)1 / 3
Figure 11.27 Plastic junction between asperities with ad-
387
(11.15)
where d is the diameter of a plane circular area of contact; is a function of geometry, Poisson’s ratio, and Young’s modulus6; and R is the sphere radius. The contact area is
6
For a sphere in contact with a plane surface ⫽ 12(1 ⫺ 2) / E.
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STRENGTH AND DEFORMATION BEHAVIOR
Ac ⫽
(NR)2 / 3 4
(11.16)
If the shear strength of the contact is i, then T ⫽ i Ac
(11.17)
and T ⫽ i (R)2 / 3 N⫺1 / 3 N 4
(11.18)
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⫽
According to these relationships, the friction coefficient for two elastic asperities in contact should decrease with increasing load. Nonetheless, the adhesion theory would still apply to the strength of the junction, with the frictional force proportional to the area of real contact. If it is assumed that the number of contacting asperities in a soil mass is independent of particle size and effective stress, then the influences of particle size and effective stress on the frictional resistance of a soil with asperities deforming elastically may be analyzed. For uniform spheres arranged in a regular packing, the gross area covered by one sphere along a potential plane of sliding is 4R 2. The normal load per contacting asperity, assuming one asperity per contact, is N ⫽ 4R2
(11.19)
Using Eq. (11.16), the area per contact becomes Ac ⫽
(4R3)2 / 3 4
(11.20)
and the total contact area per unit gross area is (Ac)T ⫽
frictional resistance (Rowe, 1962). The residual friction angles of quartz, feldspar, and calcite are independent of normal stress as shown in Fig. 11.28. On the other hand, a decreasing friction angle with increasing normal load up to some limiting value of normal stress is evident for mica and the clay minerals in Fig. 11.28 and has been found also for several clays and clay shales (Bishop et al., 1971), for diamond (Bowden and Tabor, 1964), and for solid lubricants such as graphite and molybdenum disulfide (Campbell, 1969). Additional data for clay minerals show that frictional resistance varies as ()⫺1 / 3 as predicted by Eq. (11.22) up to a normal stress of the order of 200 kPa (30 psi), that is, the friction angle decreases with increasing normal stress (Chattopadhyay, 1972). There are at least two possible explanations of the normal stress independence of the frictional resistance of quartz, feldspar, and calcite:
冉 冊
1 2 R (4)2 / 3 ⫽ (4)2 / 3 4R2 4 16
(11.21)
The total shearing resistance of is equal to the contact area times i, so ⫽
(4)2 / 3 i ⫽ iK()⫺1 / 3 16
(11.22)
where K ⫽ (4)2 / 3 /16. On this basis, the coefficient of friction should decrease with increasing , but it should be independent of sphere radius (particle size). Data have been obtained that both support and contradict these predictions. A 50-fold variation in the normal load on assemblages of quartz particles in contact with a quartz block was found to have no effect on
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1. As the load per particle increases, the number of asperities in contact increases proportionally, and the deformation of each asperity remains essentially constant. In this case, the assumption of one asperity per contact for the development of Eq. (11.22) is not valid. Some theoretical considerations of multiple asperities in contact are available (Johnson, 1985). They show that the area of contact is approximately proportional to the applied load and hence the coefficient of friction is constant with load. 2. As the load per asperity increases, the value of in Eq. (11.13) increases, reflecting a greater proportion of solid contact relative to adsorbed film contact. Thus, the average strength per contact increases more than proportionally with the load, while the contact area increases less than proportionally, with the net result being an essentially constant frictional resistance.
Quartz is a hard, brittle material that can exhibit both elastic and plastic deformation. A normal pressure of 11 GPa (1,500,000 psi) is required to produce plastic deformation, and brittle failure usually occurs before plastic deformation. Plastic deformations are evidently restricted to small, highly confined asperities, and elastic deformations control at least part of the behavior (Bromwell, 1965). Either of the previous two explanations might be applicable, depending on details of surface texture on a microscale and characteristics of the adsorbed films. With the exception of some data for quartz, there appears to be little information concerning possible variations of the true friction angle with particle size. Rowe (1962) found that the value of for assemblages of quartz particles on a flat quartz surface de-
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389
Figure 11.28 Variation in friction angle with normal stress for different minerals (after
Kenney, 1967).
creased from 31 for coarse silt to 22 for coarse sand. This is an apparent contradiction to the independence of particle size on frictional resistance predicted by Eq. (11.22). On the other hand, the assumption of one asperity per contact may not have been valid for all particle sizes, and additionally, particle surface textures on a microscale could have been size dependent. Furthermore, there could have been different amounts of particle rearrangement and rolling in the tests on the different size fractions. Sliding Friction
The frictional resistance, once sliding has been initiated, may be equal to or less than the resistance that had to be overcome to initiate movement; that is, the coefficient of sliding friction can be less than the coefficient of static friction. A higher value of static friction than sliding friction is explainable by timedependent bond formations at asperity junctions. Stick–slip motion, wherein varies more or less erratically as two surfaces in contact are displaced, appears common to all friction measurements of minerals involving single contacts (Procter and Barton, 1974). Stick–slip is not observed during shear of assemblages of large numbers of particles because the slip of individual contacts is masked by the behavior of the mass as a whole. However, it may be an important mechanism of energy dissipation for cyclic loading at very small strains when particles are not moving relative to each other.
11.5
FRICTIONAL BEHAVIOR OF MINERALS
Evaluation of the true coefficient of friction and friction angle is difficult because it is very difficult to
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do tests on two very small particles that are sliding relative to each other, and test results for particle assemblages are influenced by particle rearrangements, volume changes, surface preparation factors, and the like. Some values are available, however, and they are presented and discussed in this section. Nonclay Minerals
Values of the true friction angle for several minerals are listed in Table 11.1, along with the type of test and conditions used for their determination. A pronounced antilubricating effect of water is evident for polished surfaces of the bulky minerals quartz, feldspar, and calcite. This apparently results from a disruptive effect of water on adsorbed films that may have acted as a lubricant for dry surfaces. Evidence for this is shown in Fig. 11.29, where it may be seen that the presence of water had no effect on the frictional resistance of quartz surfaces that had been chemically cleaned prior to the measurement of the friction coefficient. The samples tested by Horn and Deere (1962) in Table 11.1 had not been chemically cleaned. An apparent antilubrication effect by water might also arise from attack of the silica surface (quartz and feldspar) or carbonate surface (calcite) and the formation of silica and carbonate cement at interparticle contacts. Many sand deposits exhibit ‘‘aging’’ effects wherein their strength and stiffness increase noticeably within periods of weeks to months after deposition, disturbance, or densification, as described, for example, by Mitchell and Solymar (1984), Mitchell (1986), Mesri et al. (1990), and Schmertmann (1991). Increases in penetration resistance of up to 100 percent have been measured in some cases. The relative importance of chemical factors, such as precipitation at
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Table 11.1 Mineral Quartz
STRENGTH AND DEFORMATION BEHAVIOR
Values of Friction Angle () Between Mineral Surfaces Type of Test Block over particle set in mortar
Conditions
(deg)
Dry Moist Water saturated Water saturated
6 24.5 24.5 21.7
Three fixed particles over block
Quartz
Block on block
Quartz Quartz
Particles on polished block Block on block
Quartz
Particle–particle
Saturated
26
Feldspar
Particle–plane Particle–plane Block on block
Saturated Dry Dry Water saturated Water saturated
Feldspar Feldspar
Free particles on flat surface Particle–plane
Calcite
Block on block
Muscovite
Along cleavage faces
Phlogopite
Biotite
Chlorite
Reference
Dried over CaCl2 before testing
Tschebotarioff and Welch (1948)
Normal load per particle increasing from 1 to 100 g Polished surfaces
Hafiz (1950)
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Quartz
Comments
Along cleavage faces
Along cleavage faces
Along cleavage faces
Dry Water saturated Water saturated
7.4 24.2 22–31
Horn and Deere (1962)
decreasing with increasing particle size Depends on roughness and cleanliness Single-point contact
Rowe (1962)
22.2 17.4 6.8 37.6 37
Polished surfaces
Horn and Deere (1962)
25–500 sieve
Lee (1966)
Saturated
28.9
Single-point contact
Dry Water saturated Dry
8.0 34.2 23.3
Polished surfaces
Procter and Barton (1974) Horn and Deere (1962)
Oven dry
Horn and Deere (1962)
Dry Saturated Dry
16.7 13.0 17.2
Air equilibrated
Dry Saturated Dry
14.0 8.5 17.2
Air equilibrated
Dry Saturated Dry
14.6 7.4 27.9
Air equilibrated
Dry Saturated
19.3 12.4
Air equilibrated
Variable
0–45
interparticle contacts, changes in surface characteristics, and mechanical factors, such as time-dependent stress redistribution and particle reorientations, in causing the observed behavior is not known. Further details of aging effects are given in Chapter 12.
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Oven dry
Oven dry
Oven dry
Bromwell (1966) Procter and Barton (1974)
Horn and Deere (1962)
Horn and Deere (1962)
Horn and Deere (1962)
As surface roughness increases, the apparent antilubricating effect of water decreases. This is shown in Fig. 11.29 for quartz surfaces that had not been cleaned. Chemically cleaned quartz surfaces, which give the same value of friction when both dry and wet,
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FRICTIONAL BEHAVIOR OF MINERALS
391
Figure 11.29 Friction of quartz (data from Bromwell, 1966 and Dickey, 1966).
show a loss in frictional resistance with increasing surface roughness. Evidently, increased roughness makes it easier for asperities to break through surface films, resulting in an increase in [Eq. (11.13) and Fig. 11.27]. The decrease in friction with increased roughness is not readily explainable. One possibility is that the cleaning process was not effective on the rough surfaces. For soils in nature, the surfaces of bulky mineral particles are most probably rough relative to the scale in Fig. 11.29, and they will not be chemically clean. Thus, values of ⫽ 0.5 and ⫽ 26 are reasonable for quartz, both wet and dry. On the other hand, water apparently acts as a lubricant in sheet minerals, as shown by the values for muscovite, phlogopite, biotite, and chlorite in Table 11.1. This is because in air the adsorbed film is thin, and surface ions are not fully hydrated. Thus, the adsorbed layer is not easily disrupted. Observations have shown that the surfaces of the sheet minerals are scratched when tested in air (Horn and Deere, 1962). When the surfaces of the layer silicates are wetted, the mobility of the surface films is increased because of their increased thickness and because of greater surface ion hydration and dissociation. Thus, the values of listed in Table 11.1 for the sheet minerals under saturated conditions (7 –13) are probably appropriate for sheet mineral particles in soils.
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Clay Minerals
Few, if any, directly measured values of for the clay minerals are available. However, because their surface structures are similar to those of the layer silicates discussed previously, approximately the same values would be anticipated, and the ranges of residual friction angles measured for highly plastic clays and clay minerals support this. In very active colloidal pure clays, such as montmorillonite, even lower friction angles have been measured. Residual values as low as 4 for sodium montmorillonite are indicated by the data in Fig. 11.28. The effective stress failure envelopes for calcium and sodium montmorillonite are different, as shown by Fig. 11.30, and the friction angles are stress dependent. For each material the effective stress failure envelope was the same in drained and undrained triaxial compression and unaffected by electrolyte concentration over the range investigated, which was 0.001 N to 0.1 N. The water content at any effective stress was independent of electrolyte concentration for calcium montmorillonite, but varied in the manner shown in Fig. 11.31 for sodium montmorillonite. This consolidation behavior is consistent with that described in Chapter 10. Interlayer expansion in calcium montmorillonite is restricted to a c-axis spacing of 1.9 nm, leading to formation of domains or layer aggregates of several unit layers. The interlayer spac-
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STRENGTH AND DEFORMATION BEHAVIOR
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Figure 11.30 Effective stress failure diagrams for calcium and sodium montmorillonite (af-
ter Mesri and Olson, 1970).
Figure 11.31 Shear and consolidation behavior of sodium
montmorillonite (after Mesri and Olson, 1970).
ing of sodium montmorillonite is sensitive to doublelayer repulsions, which, in turn, depend on the electrolyte concentration. The influence of the electrolyte concentration on the behavior of sodium montmorillonite is to change the water content, but not the strength, at any effective consolidation pressure. This suggests that the strength generating mechanism is independent of the system chemistry. The platelets of sodium montmorillonite act as thin films held apart by high repulsive forces that carry the effective stress. For this case, if it is assumed that there is essentially no intergranular contact, then Eq. (7.29) becomes i ⫽ ⫹ A ⫺ u0 ⫺ R ⫽ 0
(11.23)
Since ⫺ u0 is the conventionally defined effective stress , and assuming negligible long-range attractions, Eq. (11.23) becomes ⫽ R
(11.24)
This accounts for the increase in consolidation pressure required to decrease the water content, while at
Copyright © 2005 John Wiley & Sons
the same time there is little increase in shear strength because the shearing strength of water and solutions is essentially independent of hydrostatic pressure. The small friction angle that is observed for sodium montmorillonite at low effective stresses can be ascribed mainly to the few interparticle contacts that resist particle rearrangement. Resistance from this source evidently approaches a constant value at the higher effective stresses, as evidenced by the nearly horizontal failure envelope at values of average effective stress greater than about 50 psi (350 kPa), as shown in Fig. 11.30. The viscous resistance of the pore fluid may contribute a small proportion of the strength at all effective stresses. An hypothesis of friction between fine-grained particles in the absence of interparticle contacts is given by Santamarina et al. (2001) using the concept of ‘‘electrical’’ surface roughness as shown in Fig. 11.32. Consider two clay surfaces with interparticle fluid as shown in Fig. 11.32b. The clay surfaces have a number of discrete charges, so a series of potential energy wells exists along the clay surfaces. Two cases can be considered: 1. When the particle separation is less than several nanometers, there are multiple wells of minimum energy between nearby surfaces and a force is required to overcome the energy barrier between the wells when the particles move relative to each other. Shearing involves interaction of the molecules of the interparticle fluid. Due to the multiple energy wells, the interparticle fluid molecules go through successive solidlike pinned states. This stick–slip motion contributes to frictional resistance and energy dissipation. 2. When the particle separation is more than several nanometers, the two clay surfaces interact only by the hydrodynamic viscous effects of the interparticle fluid, and the frictional force may be estimated using fluid dynamics.
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PHYSICAL INTERACTIONS AMONG PARTICLES
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disorder of particles, (i.e., local spatial fluctuations of coordination number, and positions of neighboring particles) produce packing constraints and disorder. This leads to inhomogeneous but structured force distributions within the granular system. Deformation is associated with buckling of these force chains, and energy is dissipated by sliding at the clusters of particles between the force chains. Discrete particle numerical simulations, such as the discrete (distinct) element method (Cundall and Strack, 1979) and the contact dynamics method (Moreau, 1994), offer physical insights into particle interactions and load transfers that are difficult to deduce from physical experiments. Typical inputs for the simulations are particle packing conditions and interparticle contact characteristics such as the interparticle friction angle . Complete details of these numerical methods are beyond the scope of this book; additional information can be found in Oda and Iwashita (1999). However, some of the main findings are useful for developing an improved understanding of how stresses are carried through discrete particle systems such as soils and how these distributions influence the deformation and strength properties.
Figure 11.32 Concept of ‘‘electrical’’ surface roughness ac-
cording to Santamarina et al., (2001): (a) electrical roughness and (b) conceptual picture of friction in fine-grained particles.
The aggregation of clay plates in calcium montmorillonite produces particle groups that behave more like equidimensional particles than platy particles. There is more physical interference and more intergrain contact than in sodium montmorillonite since the water content range for the strength data shown in Fig. 11.30 was only about 50 to 97 percent, whereas it was about 125 to 450 percent for the sodium montmorillonite. At a consolidation pressure of about 500 kPa, the slope of the failure envelope for calcium montmorillonite was about 10, which is in the middle of the range for nonclay sheet minerals (Table 11.1).
11.6 PHYSICAL INTERACTIONS AMONG PARTICLES
Continuum mechanics assumes that applied forces are transmitted uniformly through a homogenized granular system. In reality, however, the interparticle force distributions are strongly inhomogeneous, as discussed in Chapter 7, and the applied load is transferred through a network of interparticle force chains. The generic
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Strong Force Networks and Weak Clusters
Examples of the computed normal contact force distribution in a granular system are shown in Figs. 11.33a for an isotropically loaded condition and 11.33b for a biaxial loaded condition (Thornton and Barnes, 1986). The thickness of the lines in the figure is proportional to the magnitude of the contact force. The external loads are transmitted through a network of interparticle contact forces represented by thicker lines. This is called the strong force network and is the key microscopic feature of load transfer through the granular system. The scale of statistical homogeneity in a two-dimensional particle assembly is found to be a few tens of particle diameters (Radjai et al., 1996). Forces averaged over this distance could therefore be expected to give a stress that is representative of the macroscopic stress state. The particles not forming a part of the strong force network are floating like a fluid with small loads at the interparticle contacts. This can be called the weak cluster, which has a width of 3 to 10 particle diameters. Both normal and tangential forces exist at interparticle contacts. Figure 11.34 shows the probability distributions (PN and PT) of normal contact forces N and tangential contact forces T for a given biaxial loading condition. The horizontal axis is the forces normalized by their mean force value (⬍N⬎ or ⬍T⬎), which depend on particle size distribution (Radjai et al., 1996). The individual normal contact forces can be as great
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STRENGTH AND DEFORMATION BEHAVIOR
Co py rig hte dM ate ria l
tric compression of a dense granular assembly (Thornton, 2000). The strong force network carries most of the whole deviator load as shown in Fig. 11.36 and is the load-bearing part of the structure. For particles in the strong force networks, the tangential contact forces are much smaller than the interparticle frictional resistance because of the large normal contact forces. In contrast, the numerical analysis results show that the tangential contact forces in the weak clusters are close to the interparticle frictional resistance. Hence, the frictional resistance is almost fully mobilized between particles in the weak clusters, and the particles are perhaps behaving like a viscous fluid. Buckling, Sliding, and Rolling
Figure 11.33 Normal force distributions of a twodimensional disk particle assembly: (a) isotropic stress condition and (b) biaxial stress condition with maximum load in the vertical direction (after Thornton and Barnes, 1986).
as six times the mean normal contact force, but approximately 60 percent of contacts carry normal contact forces below the mean (i.e., weak cluster particles). When normal contact forces are larger than their mean, the distribution law of forces can be approximated by an exponentially decreasing function; Radjai et al. (1996) show that PN ( ⫽ N/ ⬍N⬎) ⫽ ke1.4(1⫺ ) fits the computed data well for both two-and three-dimensional simulations. The exponent was found to change very slightly with the coefficient of interparticle friction and to be independent of particle size distributions. Simulations show that applied deviator load is transferred exclusively by the normal contact forces in the strong force networks, and the contribution by the weak clusters is negligible. This is illustrated in Fig. 11.35, which shows that the normal contact forces contribute greater than the tangential contact forces to the development of the deviator stress during axisymme-
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As particles begin to move relative to each other during shear, particles in the strong force network do not slide, but columns of particles buckle (Cundall and Strack, 1979). Particles in the strong force network collapse upon buckling, and new force chains are formed. Hence, the spatial distributions of the strong force network are neither static nor persistent features. At a given time of biaxial compression loading, particle sliding is occurring at almost 10 percent of the contacts (Kuhn, 1999) and approximately 96 percent of the sliding particles are in the weak clusters (Radjai et al., 1996). Over 90 percent of the energy dissipation occurs at just a small percentage of the contacts (Kuhn, 1999). This small number of sliding particles is associated with the ability of particles to roll rather than to slide. Particle rotations reduce contact sliding and dissipation rate in the granular system. If all particles could roll upon one another, a granular assembly would deform without energy dissipation.7 However, this is not possible owing to restrictions on particle rotations. It is impossible for all particles to move by rotation, and sliding at some contacts is inevitable due to the random position of particles (Radjai and Roux, 1995).8 Some frictional energy dissipation can therefore be considered a consequence of disorder of particle positions. As deformation progresses, the number of particles in the strong force network decreases, with fewer particles sharing the increased loads (Kuhn, 1999). Figure
7 This assumes that the particles are rigid and rolling with a singlepoint contact. In reality, particles deform and exhibit rolling resistance. Iwashita and Oda (1998) state that the incorporation of rolling resistance is necessary in discrete particle simulations to generate realistic localized shear bands. 8 For instance, consider a chain loop of an odd number of particles. Particle rotation will involve at least one sliding contact.
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PHYSICAL INTERACTIONS AMONG PARTICLES
395
Figure 11.34 Probability distributions of interparticle contact forces: (a) normal forces and
(b) tangential forces. The distributions were obtained for contact dynamic simulations of 500, 1024, 1200 and 4025 particles. The effect of number of particles in the simulation on probability distribution appears to be small (after Radjai et al., 1996).
forces to the evolution of the deviator stress during axisymmetric compression of a dense granular assembly (after Thornton, 2000).
Figure 11.36 Contributions of strong and weak contact forces to the evolution of the deviator stress during axisymmetric compression of a dense granular assembly (after Thornton, 2000).
11.37 shows the spatial distribution of residual deformation, in which the computed deformation of each particle is subtracted from the average overall deformation (Williams and Rege, 1997). A group of interlocked particles that instantaneously moves as a rigid body in a circular manner can be observed. The outer
boundary of the group shows large residual deformation, whereas the center shows very small residual deformation. The rotating group of interlocked particles, which can be considered as a weak cluster, becomes more apparent as applied strains increase toward failure. The bands of large residual deformation [termed
Figure 11.35 Contributions of normal and tangential contact
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STRENGTH AND DEFORMATION BEHAVIOR
Stress Ratio q/p
A B C
Triaxial Compression
Contact Plane Normals in Initial State: 1.5 More in Vertical Direction 1.0 Same in All Directions More in Horizontal Direction
0.5
-8
-6
-4
-2
2 -0.5
4
6 8 10 Axial Strain (%)
-1.0
Triaxial Extension
Co py rig hte dM ate ria l
(a)
Fabric Anisotropy A
A
B C
Contact Plane Normals in Initial State: 0.4 More in Vertical Direction 0.3 Same in All Directions 0.2 More in Horizontal Direction
0.1
-8
Figure 11.37 Spatial distribution of residual deformation observed in an elliptic particle assembly at an axial strain level of (a) 1.1%, (b) 3.3%, (c) 5.5%, (d) 7.7%, (e) 9.8%, and (ƒ ) 12.0% (after Williams and Rege, 1997).
microbands by Kuhn (1999)] are where particle translations and rotations are intense as part of the strong force network. Kuhn (1999) reports that their thicknesses are 1.5D50 to 2.5D50 in the early stages of shearing and increase to between 1.5D50 and 4D50 as deformation proceeds. This microband slip zone may eventually become a localized shear band. Fabric Anisotropy
The ability of a granular assemblage of particles to carry deviatoric loads is attributed to its capability to develop anisotropy in contact orientations. An initial isotropic packing of particles develops an anisotropic contact network during compression loading. This is because new contacts form in the direction of compression loading and contacts that orient along the direction perpendicular to loading direction are lost. The initial state of contact anisotropy (or fabric) plays an important role in the subsequent deformation as illustrated in Fig. 11.18. Figure 11.38 shows results
Copyright © 2005 John Wiley & Sons
-6
-4
-2
2 -0.1 -0.2 -0.3 -0.4 -0.5
4
6 8 10 Axial Strain (%)
(b)
Figure 11.38 Discrete element simulations of drained triaxial compression and extension tests of particle assemblies prepared at different initial contact fabrics: (a) stress–strain relationships and (b) evolution of fabric anisotropy parameter A (after Yimsiri, 2001).
of discrete particle simulations of particle assemblies prepared at different states of initial contact anisotropy under an isotropic stress condition (Yimsiri, 2001). The initial void ratios are similar (e0 ⬇ 0.75 to 0.76) and both drained triaxial compression and extension tests were simulated. Although all specimens are initially isotropically loaded, the directional distributions of contact forces are different due to different orientations of contact plane normals (sample A: more in the vertical direction; sample B: similar in all directions; sample C: more in the horizontal direction). As shown in Fig. 11.38a, both samples A and C showed stiffer response when the compression loading was applied in the preferred direction of contact forces, but softer response when the loading was perpendicular to the preferred direction of contact forces. The response of sample B, which had an isotropic fabric, was in between the two. Dilation was most intensive when the contact forces were oriented preferentially in the di-
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PHYSICAL INTERACTIONS AMONG PARTICLES
Fabric Anisotropy Parameter A
0.1
0.05
0.0
1
-0.05
-0.1
forces categorized by their magnitudes when the specimen is under a biaxial compression loading condition (Radjai, 1999). The direction of contact anisotropy of the weak clusters (N/ ⬍N⬎ less than 1) is orthogonal to the direction of compression loading, whereas that of the strong force network (N/⬍ N⬎ more than 2) is parallel. Figure 11.40 shows an example of fabric evolution with strains in biaxial loading (Thornton and Antony, 1998). The fabric anisotropy is separated into that in the strong force networks (N/⬍ N⬎ of more than 1) and that in the weak clusters (N/ ⬍N⬎ less than 1). Again the directional evolution of the fabric in the weak clusters is opposite to the direction of loading. Therefore, the stability of the strong force chains aligned in the vertical loading direction is obtained by the lateral forces in the surrounding weak clusters.
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rection of applied compression; and experimental data presented by Konishi et al. (1982) shows a similar trend. Figure 11.38b shows the development of fabric anisotropy with increasing strain. The degree of fabric anisotropy is expressed by a fabric anisotropy parameter A; the value of A increases with more vertically oriented contact plane normals and is negative when there are more horizontally oriented contact plane normals.9 The fabric parameter gradually changes with increasing strains and reaches a steady-state value as the specimens fail. The final steady-state value is independent of the initial fabric, indicating that the inherent anisotropy is destroyed by the shearing process. The final fabric anisotropy after triaxial extension is larger than that after triaxial compression because the additional confinement by a larger intermediate stress in the extension tests created a higher degree of fabric anisotropy. Close examination of the contact force distribution for the strong force network and weak clusters gives interesting microscopic features. Figure 11.39 shows the values of A determined for the subgroups of contact
397
2
3
4
5
Changes in Number of Contacts and Microscopic Voids
At the beginning of biaxial loading of a dense granular assembly, more contacts are created from the increase in the hydrostatic stress, and the local voids become smaller. As the axial stress increases, however, the local voids tend to elongate in the direction of loading as shown in Fig. 11.41. Consequently particle contacts are lost. As loading progresses, vertically elongated local voids become more apparent, leading to dilation in
6 N/
Figure 11.39 Fabric anisotropy parameter A for different levels of contact force when the specimen is under biaxial compression loading conditions (after Radjai et al., 1996).
9 The density of contact plane normals E( ) with direction is fitted with the following expression (Radjai, 1999):
E( ) ⫽
c {1 ⫹ A cos 2( ⫺ c)}
where c is the total number of contacts, c is the direction for which the maximum E is reached, and the magnitude of A indicates the amplitude of anisotropy. When the directional distribution of contact forces is independent of , the system has an isotropic fabric and A ⫽ 0.
Copyright © 2005 John Wiley & Sons
Figure 11.40 Evolution of the fabric anisotropy parameters of strong forces and weak clusters when the specimen is under biaxial compression loading conditions (after Thornton and Antony, 1998).
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STRENGTH AND DEFORMATION BEHAVIOR
Co py rig hte dM ate ria l
398
Figure 11.41 Simulated spatial distribution of local microvoids under biaxial loading (after Iwashita and Oda, 2000): (a) 11 ⫽ 1.1% (before failure), (b) 11 ⫽ 2.2% (at failure), (c) 11 ⫽ 4.4% (after failure), and (d ) 11 ⫽ 5.5% (after failure).
terms of overall sample volume (Iwashita and Oda, 2000). Void reduction is partly associated with particle breakage. Thus, there is a need to incorporate grain crushing in discrete particle simulations to model the contractive behavior of soils (Cheng et al., 2003). Normal contact forces in the strong force network are quite high, and, therefore, particle asperities, and even particles themselves, are likely to break, causing the force chains to collapse. Local voids tend to change size even after the applied stress reaches the failure stress state (Kuhn, 1999). This suggests that the degrees of shearing required for the stresses and void ratio to reach the critical state are different. Numerical simulations by Thornton (2000) show that at least 50 percent axial
Copyright © 2005 John Wiley & Sons
strain is required to reach the critical state void ratio. Practical implication of this is discussed further in Section 11.7. Macroscopic Friction Angle Versus Interparticle Friction Angle
Discrete particle simulations show that an increase in the interparticle friction angle results in an increase in shear modulus and shear strength, in higher rates of dilation, and in greater fabric anisotropy. Figure 11.42 shows the effect of assumed interparticle friction angle on the mobilized macroscopic friction angle of the particle assembly (Thornton, 2000; Yimsiri, 2001). The macroscopic friction angle is larger than the interparticle friction angle if the interparticle friction angle is smaller than 20. As the interparticle friction becomes
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PHYSICAL INTERACTIONS AMONG PARTICLES
399
50
30
Co py rig hte dM ate ria l
Macroscopic Friction Angle (degrees)
40
20
Drained (Thornton, 2000)
Drained Triaxial Compression (Yimsiri, 2001)
Undrained Triaxial Compression (Yimsiri, 2001)
10
Drained Triaxial Extension (Yimsiri, 2001)
Undrained Triaxial Extension (Yimsiri, 2001) Experiment (Skinner, 1969)
0
0
10
20
30
40
50
60
70
80
90
Interparticle Friction Angle (degrees)
Figure 11.42 Relationships between interparticle friction angle and macroscopic friction
angle from discrete element simulations. The macroscopic friction angle was determined from simulations of drained and undrained triaxial compression (TC) and extension (TE) tests. The experimental data by Skinner (1969) is also presented (after Thornton, 2000, and Yimsiri, 2001).
more than 20, the contribution of increasing interparticle friction to the macroscopic friction angle becomes relatively small; the macroscopic friction angle ranges between 30 and 40, when the interparticle friction angle increases from 30 to 90.10 The nonproportional relationship between macroscopic friction angle of the particle assembly and interparticle friction angle results because deviatoric load is carried by the strong force networks of normal forces and not by tangential forces, whose magnitude is governed by interparticle friction angle. Increase in interparticle friction results in a decrease in the percentage of sliding contacts (Thornton, 2000). The interparticle friction therefore acts as a kinematic constraint of the strong force network and not as the direct source of macroscopic resistance to shear. If the interparticle friction were zero, strong force chains could not develop, and the particle assembly will be-
Reference to Table 11.1 shows that actually measured values of for geomaterials are all less than 45. Thus, numerical simulations done assuming larger values of appear to give unrealistic results.
have like a fluid. Increased friction at the contacts increases the stability of the system and reduces the number of contacts required to achieve a stable condition. As long as the strong force network can be formed, however, the magnitude of the interparticle friction becomes of secondary importance. The above findings from discrete particle simulations are partially supported by the experimental data given by Skinner (1969), which are also shown in Fig. 11.42. He performed shear box tests on spherical particles with different coefficients of interparticle friction angle. The tested materials included glass ballotini, steel ball bearings, and lead shot. Use of glass ballotini was particularly attractive since the coefficient of interparticle friction increases by a factor of between 3.5 and 30 merely by flooding the dry sample. Skinner’s data shown in Fig. 11.42 indicate that the macroscopic friction angle is nearly independent of interparticle friction angle. Effects of Particle Shape and Angularity
10
Copyright © 2005 John Wiley & Sons
A lower porosity and a larger coordination number are achieved for ellipsoidal particles compared to spherical
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STRENGTH AND DEFORMATION BEHAVIOR
particles (Lin and Ng, 1997). Hence, a denser packing can be achieved for ellipsoidal particles. Ellipsoid particles rotate less than spherical particles. An assembly of ellipsoid particles gives larger values of shear strength and initial modulus than an assemblage of spherical particles, primarily because of the larger coordination number for ellipsoidal particles. Similar findings result for two-dimensional particle assemblies. Circular disks give the highest dilation for a given stress ratio and the lowest coordination number compared to elliptical or diamond shapes (Williams and Rege, 1997). An assembly of rounded particles exhibits greater softening behavior with fabric anisotropy change with strain, whereas an assembly of elongated particles requires more shearing to modify its initial fabric anisotropy to the critical state condition (Nouguier-Lehon et al., 2003).
Deviator Stress
σa
M (triaxial compression)
σr
σr
Co py rig hte dM ate ria l
Mean Pressure p Critical State Line
After large shear-induced volume change, a soil under a given effective confining stress will arrive ultimately at a unique final water content or void ratio that is independent of its initial state. At this stage, the interlocking achieved by densification or overconsolidation is gone in the case of dense soils, the metastable structure of loose soils has collapsed, and the soil is fully destructured. A well-defined strength value is reached at this state, and this is often referred to as the critical state strength. Under undrained conditions, the critical state is reached when the pore pressure and the effective stress remain constant during continued deformation. The critical state can be considered a fundamental state, and it can be used as a reference state to explain the effect of overconsolidation ratios, relative density, and different stress paths on strength properties of soils (Schofield and Wroth, 1968).
The basic concept of the critical state is that, under sustained uniform shearing, there exists a unique relationships among void ratio ecs (or specific volume vcs ⫽ 1 ⫹ ecs), mean effective pressure pcs, and deviator stress qcs as shown in Fig. 11.43. An example of the critical state of clay was shown in Fig. 11.4a. The critical state of clay can be expressed as qcs ⫽ Mpcs
(11.25)
vcs ⫽ 1 ⫹ ecs ⫽ % ⫺ cs ln pcs
(11.26)
where qcs is the deviator stress at failure, pcs is the mean effective stress at failure, and M is the critical
Copyright © 2005 John Wiley & Sons
M (triaxial extension)
(a)
Specific Volume v
Compression Lines of Constant Stress Ratio q/p
Γ
11.7 CRITICAL STATE: A USEFUL REFERENCE CONDITION
Clays
Critical State Line
q = σa – σr
λ
λ cs
Isotropic Compression Line
Critical State Line ln p
1
(b)
Figure 11.43 Critical state concept: (a) p–q plane and (b) v–ln p plane.
state stress ratio. The critical state on the void ratio (or specific volume)–mean pressure plane is defined by two material parameters: cs, the critical state compression index and %, the specific volume intercept at unit pressure (p ⫽ 1). The compression lines under constant stress ratios are often parallel to each other, as shown in Fig. 11.43b. Parameter M in Equation (11.25) defines the critical state stress ratio at failure and is similar to for the Mohr–Coulomb failure line. However, Equation (11.25) includes the effect of intermediate principal stress 2 because p ⫽ 1 ⫹ 2 ⫹ 3, whereas the Mohr–Coulomb failure criterion of Eq. (11.4) or (11.5) does not take the intermediate effective stress into account. In triaxial conditions, a ⬎ r ⫽ r and r ⫽ r ⬎ a for compression and extension, respectively (see Fig. 11.43).11 Hence, Eqs. (11.4) and (11.25) can be related to each other for these two cases as follows:
11
a is the axial effective stress and r is the radial effective stress.
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CRITICAL STATE: A USEFUL REFERENCE CONDITION
M⫽
6 sin crit for triaxial compression 3 ⫺ sin crit
(11.27)
M⫽
6 sin crit for triaxial extension 3 ⫹ sin crit
(11.28)
and C for drained triaxial compression. Hence, the deviator stress at critical state is smaller for the undrained case than for the drained case. On the other hand, when the initial state of the soil is overconsolidated from D (Fig. 11.44b), the critical state becomes E for undrained loading and F for drained triaxial compression. The deviator stress at critical state is smaller for the drained case compared to the undrained case. It is important to note that the soil state needs to satisfy both state equations [Eqs. (11.25) and (11.26)] to be at critical state. For example, point G in Fig. 11.44b satisfies pcs and qcs, but not ecs; therefore, it is not at the critical state. Converting the void ratio in Eq. (11.26) to water content, a normalized critical state line can be written using the liquidity index (see Fig. 11.45).
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These equations indicate that the correlation between M and crit is not unique but depends on the stress conditions. Neither is a fundamental property of the soil, as discussed further in Section 11.12. Nonetheless, both are widely used in engineering practice, and, if interpreted properly, they can provide useful and simple phenomenological representations of complex behavior. The drained and undrained critical state strengths are illustrated in Figs. 11.44a and 11.44b for normally consolidated clay and heavily overconsolidated clay, respectively. The initial mean pressure–void ratio state of the normally consolidated clay is above the critical state line, whereas that of the heavily overconsolidated clay is below the critical state line. When the initial state of the soil is normally consolidated at A (Fig. 11.44a), the critical state is B for undrained loading
Critical State Line
Deviator Stress q
M
LIcs ⫽
wcs ⫺ wPL ln(pPL /p) ⫽ wLL ⫺ wPL ln(pPL /pLL)
Deviator Stress q
Critical State Line M
Drained Peak Strength
Undrained Strength
C
E
Drained Strength
F
G
3
B
3
1
A
D
1
Mean pressure p
Specific Volume v
D
Mean pressure p
Specific Volume v
Γ
Γ
B
A
F
Isotropic Compression Line
C
G
D
D
Isotropic Compression Line
λ
λ cs
Critical State Line
1
E
λ
λ cs
Critical State Line
1
ln p
ln p (b)
(a)
Figure 11.44 Drained and undrained stress–strain response using the critical state concept:
(a) normally consolidated clay and (b) overconsolidated clay.
Copyright © 2005 John Wiley & Sons
(11.29)
where wcs is the water content at critical state when the effective mean pressure is p. pLL and pPL are the
Drained Strength
Undrained Strength
401
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STRENGTH AND DEFORMATION BEHAVIOR
Water Content
Critical State Line
Liquidity Index
Isotropic Compression Line
Liquid Limit wLL
Critical State Line
LI = 1 (ln(p), w)
(ln(p’), LI)
LI
w
LIeq-1 LICS
Plastic Limit wPL
LI = 0
Co py rig hte dM ate ria l
wcs
ln(pLL)
ln(p) ln(pPL)
ln(pLL)
Mean pressure
(a)
ln(p) ln(pPL) Mean Pressure (b)
Figure 11.45 Normalization of the critical state line: (a) water content versus mean pressure and (b) liquidity index versus mean pressure.
mean effective pressure at liquid limit (wLL) and plastic limit (wPL), respectively; pLL ⬇ 1.5 to 6 kPa and pPL ⬇ 150 to 600 kPa are expected considering the undrained shear strengths at liquid and plastic limits are in the ranges suLL ⫽ 1 to 3 kPa and suPL ⫽ 100 to 300 kPa, respectively12 (see Fig. 8.48). Using Eq. (11.29), a relative state in relation to the critical state for a given effective mean pressure (i.e., above or below the critical state line) can be defined as (see Fig. 11.45) LIeq ⫽ LI ⫺ LIcs ⫹ 1 ⫽ LI ⫹
log(p /pLL) log(pPL /pLL )
(11.30)
where LIeq is the equivalent liquidity index defined by Schofield (1980). When LIeq ⫽ 1 (i.e., LI ⫽ LIcs) and q/p ⫽ M, the clay has reached the critical state. Figure 11.46 gives the stress ratio when plastic failure (or fracture) initiates at a given water content. When LIeq ⬎ 1 (the state is above the critical state line), and the soil in a plastic state exhibits uniform contractive behavior. When LIeq ⬍ 1 (the state is below the critical state line), and the soil in a plastic state exhibits localized dilatant rupture, or possibly fracture, if the stress ratio reaches the tensile limit (q/p ⫽ 3 for triaxial compression and ⫺1.5 for triaxial extension; see Fig. 11.46b). Hence, the critical state line can be used as a reference to characterize possible soil behavior under plastic deformation.
Sands
The critical state strength and relative density of sand can be expressed as qcs ⫽ Mpcs
DR,cs ⫽
emax ⫺ ecs 1 ⫽ emax ⫺ emin ln(c /p)
A review by Sharma and Bora (2003) gives average values of suLL ⫽ 1.7 kPa and suPL ⫽ 170 kPa.
Copyright © 2005 John Wiley & Sons
(11.32)
where ecs is the void ratio at critical state, emax and emin are the maximum and minimum void ratios, and c is the crushing strength of the particles.13 The critical state line based on Eq. (11.32) is plotted in Fig. 11.47. The plotted critical state lines are nonlinear in the e– ln p plane in contrast to the linear relationship found for clays. This nonlinearity is confirmed by experimental data as shown in Fig. 11.4b. At high confining pressure, when the effective mean pressure becomes larger than the crushing strength, many particles begin to break and the lines become more or less linear in the e–ln p plane, similar to the
13
Equation (11.32) is derived from Eq. 11.36 proposed by Bolton (1986) with IR ⫽ 0 (zero dilation). Bolton’s equation is discussed further in Section 11.8. Other mathematical expressions to fit the experimentally determined critical state line are possible. For example, Li et al. (1999) propose the following equation for the critical state line (ecs versus p): ecs ⫽ e0 ⫺ s
12
(11.31)
冉冊 p pa
where e0 is the void ratio at p ⫽ 0, pa is atmospheric pressure, and s and are material constants.
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CRITICAL STATE: A USEFUL REFERENCE CONDITION
Fracture
Ductile Plastic and Contractive
Dilatant Rupture
q
Tensile Fracture
q/p
403
Triaxial Compression
3 MTC
3
Triaxial Compression
MTC
0.5 Triaxial Extension
1.0
LIeq
p
Co py rig hte dM ate ria l
MTE
1
2
-1.5
MTE
3
Fracture
Dilatant Rupture (a)
Tensile Fracture
Ductile Plastic and Contractive
Triaxial Extension
(b)
Figure 11.46 Plastic state of clay in relation to normalized liquidity index: (a) stress ratio
when plastic state initiates for a given LIeq and (b) definition of stress ratios used in (a) (after Schofield, 1980).
DR,cs =
emax – ecs 1 emax – emin = In (σc/p)
0.2
0.2
Relative Density Dr
emax 0
Relative Density Dr
e max 0
0.4 0.6 0.8
e min 1
1.2 0.001
0.4 0.6 0.8
emin 1 1.2
p/σc
0.2 0.3 p/σc
(a)
(b)
0.01
0.1
1
0
0.1
0.4
0.5
Figure 11.47 Critical state line derived from Eq. (11.32): (a) e–log p plane and (b) e–p
plane.
behavior of clays. Coop and Lee (1993) found that there is a unique relationship between the amount of particle breakage that occurred on shearing to a critical state and the value of the mean normal effective stress. This implies that sand at the critical state would reach a stable grading at which the particle contact stresses
Copyright © 2005 John Wiley & Sons
would not be sufficient to cause further breakage. Coop et al. (2004) performed ring shear tests (see Section 11.11) on a carbonate sand to find a shear strain required to reach the true critical state (i.e., constant particle grading). They found that particle breakage continues to very large strains, far beyond those
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STRENGTH AND DEFORMATION BEHAVIOR
11.8
Early Studies
The important role of volume change during shear, especially dilatancy, was recognized by Taylor (1948). From direct shear box testing of dense sand specimens, he calculated the work at peak shear stress state and showed that the energy input is dissipated by the friction using the following equation: peak dx ⫺ n dy ⫽ n dx
STRENGTH PARAMETERS FOR SANDS
Many factors and phenomena act together to determined the shearing resistance of sands, for example, mineralogy, grain size, grain shape, grain size distribution, (relative) density, stress state, type of tests and stress path, drainage, and the like [see Eq. (11.3)]. In this section, the ways in which these factors have become understood and have been quantified over the last several decades are summarized. Several correlations are given to provide an overview and reference for typical values and ranges of strength parameters for sands and the influences of various factors on these parameters.14
14
A number of additional useful correlations are given by Kulhawy and Mayne (1990).
Copyright © 2005 John Wiley & Sons
(11.33)
where peak is the applied shear stress at peak, n is the confining normal (effective) stress on the shear plane, dx is the incremental horizontal displacement at peak, dy is the incremental vertical displacement (expansion positive) at peak stress, and is the friction coefficient. The energy dissipated by friction (the component in the right-hand side) is equal to the sum of the work done by shearing (first component in the left-hand side) and that needed to increase the volume (the second component in the left-hand side). The latter component is referred to as dilatancy. Rearranging Eq. (11.33),
Co py rig hte dM ate ria l
reached in triaxial tests. Figure 11.48a shows the volumetric strains measured for a selection of their tests, which were carried out at vertical stress levels in the range of 650 to 860 kPa. A constant volumetric strain is reached at a shear strain of around 2000 percent. For specimens at lower stress levels, more shear strains (20,000 percent or more) were required. Similar findings were made for quartz sand (Luzzani and Coop, 2002). Figure 11.48b shows the degree of particle breakage with shear strains in the logarithmic scale. The amount of breakage is quantified by Hardin’s (1985) relative breakage parameter Br defined in Fig. 10.14. At very large strains, the value of Br finally stabilizes. The strain required for stabilization depends on applied stress level. Interestingly, less shear strain was needed for the mobilized friction angle to reach the steady-state value (Fig. 11.48c) than for attainment of the constant volume condition, (Fig. 11.48a). The critical state friction angle was unaffected by the particle breakage. In summary, the critical state concept is very useful to characterize the strength and deformation properties of soils when it is used as a reference state. That is, a soil has a tendency to contract upon shearing when its state is above the critical state line, whereas it has a tendency to dilate when it is below the critical state line. Various normalized state parameters have been proposed to characterize the difference between the actual state and the critical state line, as illustrated in Fig. 11.49. These parameters have been used to characterize the stiffness and strength properties of soils. Some of them are introduced later in this chapter.
冉冊
peak dy ⫽ tan m ⫽ ⫹ dx
(11.34)
Thus, the peak shear stress ratio or the peak mobilized friction angle m consists of both interlocking (dy/dx) and sliding friction between grains (). This equation that relates stress to dilation is called the stress– dilatancy rule, and it is an important relationship for characterizing the plastic deformation of soils, as further discussed in Section 11.20. Rowe (1962) recognized that the mobilized friction angle m must take into account particle rearrangements as well as the sliding resistance at contacts and dilation. Particle crushing, which increases in importance as confining pressure increases and void ratio decreases, should also be added to these components. The general interrelationships among the strength contributing factors and porosity can be represented as shown in Fig. 11.50. In this figure, f is the friction angle corrected for the work of dilation. It is influenced by particle packing arrangements and the number of sliding contacts. The denser the packing, the more important is dilation. As the void ratio increases, the mobilized friction angle decreases. The critical state is defined as the condition when there is no volume change by shearing [i.e., (dy/dx) ⫽ 0 in Eq. (11.34)]. The corresponding mobilized friction angle m is crit . The ‘‘true friction’’ in the figure is associated with the resistance to interparticle sliding.
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STRENGTH PARAMETERS FOR SANDS
Shear Strain (%) 50,000 100,000
150,000
RS3
0
RS5 RS7 RS8
(a)
Co py rig hte dM ate ria l
Volumetric strain (%)
0
20
RS13 RS15
40
Luzzani & Coop, 805 kPa 650-930 kPa
1.0
248-386 kPa 60-97 kPa
Relative Breakage
0.8
RS7
0.6
RS8
(b)
0.4 0.2
?
0.0
800 kPa unsheared
10
100
1000
10,000
100,000 1,000,000
Shear Strain
Mobilized Friction Angle (degrees)
50 40
RS3
30
RS7
20
RS8
(c)
RS9
10
RS10 RS15
0
1
10
100
1000
10,000
100,000
Shear Strain (%)
Figure 11.48 Ring shear test results of carbonate sand: (a) volumetric strain versus shear
strain, (b) the degree of particle breakage with shear strains, and (c) mobilized friction angle versus shear strains (after Coop et al., 2004).
Copyright © 2005 John Wiley & Sons
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STRENGTH AND DEFORMATION BEHAVIOR
1. State parameter (Been and Jefferies, 1985)
Void ratio e
Ψ = e – ec
Critical state line (pc , e c)
Loose Sand
ecD
Loose sand Ψ = eL – ecL (>0)
( pL, eL )
2. State index (Ishihara et al., 1998)
Ψ0 Dense sand
Is = (e 0 – ec )/(e0 – e) Loose sand Is = (e0 – ecL)/(e0 – eL) (>1) Dense sand Is = (e 0 – e cD)/(e0 – e0) ( σh2 > σv
α = 90°
v
B
F Parallel to the Bedding Plane
α
σ11
Co py rig hte dM ate ria l
Loading Direction of Major Principal Stress in Relation to the Bedding Plane
σh1
α
α σσ3
σσ22 σ1v> σ2 > σ3
E
σv
v
σh1
Perpendicular to the Bedding Plane α = 0°
σv
σv
σh2 σvv> σh1 = σh2
0 A Triaxial Compression
σh1
σh2 σvv> σh1 > σh2
σh2
σh1
σvv= σh1 > σh2
C
b - Value
D 1 Triaxial Extension
Figure 11.83 Effects of inherent anisotropy and the intermediate stress.
The influences of shear strain magnitude and number of load cycles are shown in Fig. 11.86. The densification results from the disruption of the initial soil fabric caused by the repeated shear strains followed by repositioning of the soil grains into more efficient packing. The higher the initial void ratio and the greater the number of cycles, the greater the effect. Undrained Behavior
When saturated soil is subjected to repeated cycles of loading, and provided the shear stresses are of sufficient magnitude, the structure begins to break down, and part of the confining stress is transferred to the pore water, with a concurrent reduction of effective stress and strength. This, in turn, leads to increase in shear strain under constant stress cyclic loading or a decrease in the cyclic stress required to cause a given value of cyclic strain. The deformation and failure behavior of sands in undrained cyclic loading depends on initial void ratio, initial effective stress state, and the cyclic shear stress amplitude. The results of an undrained cyclic simple shear test on Monterey sand are shown in Fig. 11.87. Develop-
Copyright © 2005 John Wiley & Sons
ment of shear strains with cyclic loading is called cyclic mobility. Liquefaction is said to have occurred when the pore water pressure has increased to the magnitude of the initial effective confining pressure, at which point the strains become very large. Similar to the undrained response in monotonic loading, the undrained response of sand under cyclic loading depends on density, confining pressure, and soil fabric. The effect of density on cyclic behavior of Toyoura sand under triaxial loading conditions is shown in Fig. 11.88. All sands exhibit increase in pore pressure with increase in number of loading cycles, but the shear strain development for a given number of cycle is smaller for denser specimens. In the loose sand (Fig. 11.88a), when the stress state reaches the collapse surface, the soil softens leading to sudden liquefaction. The medium dense sand (Fig. 11.88b) exhibits quasi-steady state as the stress state reaches the phase transformation line. Some cycles with large stress–strain loops are observed and the specimen finally reaches liquefaction. The dense sand (Fig. 11.88c) never liquefies. Once the stress state reaches the phase transformation line, the stress–strain curve moves back and forth
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Co py rig hte dM ate ria l
RESISTANCE TO CYCLIC LOADING AND LIQUEFACTION
427
Figure 11.84 Effect of and b values on undrained response of dry plluviated Toyoura sand: (a) effect of when b ⫽ 0.5 and (b) effect of b when ⫽ 45 (after Yoshimine et al., 1998).
along and below the steady-state line and shear strain develops gradually. Beneath gently sloping to flat ground, liquefaction may lead to ground oscillation or lateral spread as a consequence of either flow deformation or cyclic mobility (Youd et al., 2001). The liquefaction susceptibility of different types of natural and artificial sedimentary soil deposits is summarized in Table 11.5. As the excess pore pressure developed during liquefaction dissipates, ground settlement is observed. Sand boils can develop through overlying less permeable soils in order to dissipate the excess pore pressures from liquefied soil below. The magnitude of the cyclic shear strains that develop following initial liquefaction decreases with increasing initial relative density and increases with increasing cyclic shear stress. The general relationship
Copyright © 2005 John Wiley & Sons
between cyclic shear stress and number of load cycles to initial liquefaction depends on the relative density, and is of the form shown in Fig. 11.89. In this figure the cyclic shear stress applied by a simple shear apparatus is normalized by dividing by the initial effective confining pressure 0, and the ratio is often called the cyclic resistance ratio (CRR). Methods for determination of the liquefaction susceptibility of a specific site are given by Kramer (1996) and by Youd et al. (2001). In reality, generation of pore pressure is a result of the breakdown of soil structure and a tendency for the soil to densify, and this is caused by shear deformations, so liquefaction is more fundamentally controlled by shear strain than by shear stress. Furthermore, there is a level of shear strain, or threshold shear strain below which no pore pressure is generated. This is illus-
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STRENGTH AND DEFORMATION BEHAVIOR
Stress Ratio q/p Toyoura Sand 2 Air Pluviated Initial Void Ratio = 0.845 p = 98 kPa = constant
_2
Stress Ratio q/p Toyoura Sand 2 Air Pluviated Initial Void Ratio = 0.653 p = 98 kPa = constant
2 Shear Strain γ (%)
_2
2 Shear Strain γ (%) _1.2
Co py rig hte dM ate ria l
_1.2
(a-1) Stress Ratio – Shear Strain Relation
(b-1) Stress Ratio – Shear Strain Relation Volumetric Strain εv 0.6
Volumetric Strain εv 3
_2
_2
0
2 Shear Strain γ (%)
(a-2) Volumetric Strain – Shear Strain Relation
(a)
0
2
Shear Strain γ (%)
_0.6
(b-2) Volumetric Strain – Shear Strain Relation
(b)
Figure 11.85 Cyclic behavior of Toyoura sand in drained conditions: (a) loose sand and (b) dense sand (after Pradhan and Tatsuoka, 1989).
Figure 11.86 Effect of shear strain and number of load cycles on the reduction in void ratio of Ottawa sand (from Youd, 1972). Reprinted with permission of ASCE.
Copyright © 2005 John Wiley & Sons
trated by Fig. 11.90 in which the pore pressure ratio as a function of cyclic shear strain is shown for Monterey No. 0 sand at three relative densities. The mechanics of pore pressure generation during cyclic loading can be understood by reference to Fig. 11.91 from Seed and Idriss (1982) and by Fig. 8.20. In Fig. 11.91, point A represents a soil specimen in its initial state. Under cyclic loading it would, if allowed to drain and compress, decrease in void ratio to point B in order to be able to continue to sustain effective pressure 0. However, since the soil cannot drain, the collapsing soil structure generates a pore pressure denoted in Fig. 11.91 by u. The magnitude of the pore pressure depends on the slope of the rebound curve B– C, as discussed below. From laboratory cyclic simple shear tests on several sands, the general relationship between pore pressure ratio (i.e., the generated pore pressure divided by the initial effective confining pressure) and the cycle ratio as shown in Fig. 11.92 has been determined. The cycle ratio is defined by the number of load cycles Ne divided by the number of load cycles to cause liquefaction Nl.
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RESISTANCE TO CYCLIC LOADING AND LIQUEFACTION
429
Co py rig hte dM ate ria l
mined from testing at different densities under the same confining stress condition, whereas the constant density contour lines move downward if the confining pressure increases, as illustrated in Fig. 11.93 for Aio sand samples prepared at the same relative density. This is because a soil at a given void ratio behaves as if relatively looser or more compressible at higher confining pressure. There are many other factors that impact the actual value of CRR to use in practice; the major ones are the confining pressure, the initial shear stresses under static condition, sample preparation methods, and the mode of shearing (Seed, 1979; Seed and Harder, 1990). Additional information and data can be found in Youd et al. (2001), Vaid et al. (2001), Boulanger (2003), and Hosono and Yoshimine (2004). Residual Strength after Liquefaction
Figure 11.87 Results of an undrained cyclic simple shear test on loose Monterey sand (Seed and Idriss, 1982): (a) pore water pressure response, (b) shear strain response, and (c) applied cyclic shear stress.
Using the slope of the rebound curve (Fig. 11.91) and the densification that would occur if drainage was permitted, it is possible to compute the induced pore pressure by
u ⫽ Er rd
(11.51)
where Er is the rebound modulus and rd is the volumetric strain that would occur if drainage were permitted. Martin et al. (1975) give procedures to evaluate these two parameters from the results of static rebound tests in a consolidation ring and cyclic load tests on dry sand, respectively. Finn (1981) reported good agreement between predicted and measured values using the proposed method. The liquefaction resistance depends not only on cyclic stress amplitude and density but also on the initial effective stress state. For example, Fig. 11.89 is deter-
Copyright © 2005 John Wiley & Sons
The residual strength of sands, silty sands, and silts following liquefaction is a subject of continuing study owing to its importance in the analysis of postearthquake stability and deformation of embankments, dams, and structures. Detailed discussion of this topic is outside the scope of this book; however, two approaches have been used to estimate the residual strength, one based on steady state strength determined by laboratory tests as described in Section 11.8 and the other on the Standard Penetration Test (SPT) N value (Seed, 1987; Seed and Harder, 1990). A correlation between the residual strength and the preearthquake SPT N value is shown in Fig. 11.94. The strength values shown in this figure were determined by back analysis of liquefaction-induced slides; thus, they avoid problems related to sampling disturbance effects on strength and are representative of known field behavior. However, there is some uncertainty relating to how well the measured N values are representative of the zone in which the failure developed. The selection of a particular value within the range of strengths shown for any given N value, and variability in the N values that are measured, add additional uncertainty. Excess pore pressures can be generated either internally as described above or externally by transient seepage flow from an adjacent liquefied region. For example, if there is a less permeable layer above a sand layer, excess pore pressures can develop under the impermeable layer leading to softening of the soil. Hence, local heterogeneity plays an important role in liquefaction-induced soil deformation and failure, which requires careful site investigation to identify any low permeable layers. The strength degradation of clays due to cyclic loading follows similar patterns to that of sand, but it is much less for clays than for cohesionless and slightly
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430
11
STRENGTH AND DEFORMATION BEHAVIOR
80
20 10 0
-15
-10
-5
-10 0 -2 0
Reaching Collapse -30 Surface after Several -40 Cycles
5
10
60
Deviator Stress q (MPa)
Liquefaction Failure
100
Initial Cyclic Loops Before Failure
15
Liquefaction Failure
Reaching Collapse Surface after Several Cycles
40 20 0 -20 0
50
100
150
200
250
-40
Co py rig hte dM ate ria l
Deviator Stress q (MPa)
50
Dr = 30% 40 Initial p = 200 kPa Cyclic Stress Δq = 40 kPa 30
-60
Liquefaction
Initial State
-80
-50 Axial Strain (%)
-100
Mean Pressure p(MPa)
(a)
100
Phase Transformation
80
Deviator Stress q (MPa)
Deviator Stress q (MPa)
100
Dr = 50% 80 Initial p = 200 kPa 60 Cyclic Stress Δq = 60 kPa 40
Liquefaction Failure
20
0
-1 5
-10
-5
-20 0
5
10
15
-40 -60
Reaching Collapse Surface after Several Cycles
-80
Initial State
40 20
0 -20 0
50
100
150
200
250
-40 -60 -80
-100 Axial Strain (%)
Phase Transformation
60
Reaching Collapse Surface after Several Cycles
Liquefaction
-100
Mean Pressure p(MPa)
(b)
50
Increasing Cycles
20 10 0
-15
-10
Increasing Cycles
-5
-10 0
Phase Transformation
40
Deviator Stress q (MPa)
Deviator Stress q (MPa)
50
Dr = 70% 40 Initial p = 100 kPa Cyclic Stress Δq = 40 kPa30
5
10
15
-20 -30 -40
30 20 10
0 -10 0
40
60
80
-30 -50
Phase Transformation
Mean Pressure p(MPa)
(c)
Figure 11.88 Cyclic behavior of Toyoura sand in undrained conditions: (a) loose sand, (b) medium dense sand, and (c) dense sand (after Yamamoto, 1998).
Copyright © 2005 John Wiley & Sons
100
120
-20 -40
-50 Axial Strain (%)
20
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Initial State
431
RESISTANCE TO CYCLIC LOADING AND LIQUEFACTION
Table 11.5
Liquefaction Susceptibility of Soil Deposits
Type of Deposit (1)
⬍500 yr
Holocene (4)
Pleistocene (5)
a. Continental Deposits Locally variable Very High Locally variable High Widespread Moderate Widespread —
High Moderate Low Low
Low Low Low Very low
Very Very Very Very
Widespread Variable Variable Widespread Widespread Variable Variable Rare Widespread Rare Locally variable
Moderate Moderate Moderate Low Moderate High Low Low High Low Moderate
Low Low Low Very low Low High Very low Very low ? Very low Low
Very low Very low Very low Very low Very low Unknown Very low Very low ? Very low Very low
High Moderate Low Moderate Moderate Moderate
Low Low Very low Low Low Low
Very Very Very Very Very Very
— —
— —
(3)
Co py rig hte dM ate ria l
River channel Floodplain Alluvial fan and plain Marine terraces and plains Delta and fan-delta Lacustrine and playa Colluvium Talus Dunes Loess Glacial till Tuff Tephra Residual soils Sabka
Likelihood That Cohesionless Sediments, When Saturated, Would Be Susceptible to Liquefaction (by Age of Deposit)
General Distribution of Cohesionless Sediments in Deposits (2)
High High High Low High High Low Low High Low High
Prepleistocene (6) low low low low
b. Coastal Zone
Delta Esturine Beach high wave energy Low wave energy Lagoonal Fore shore
Widespread Locally variable Widespread Widespread Locally variable Locally variable
Very high High Moderate High High High
low low low low low low
c. Artificial
Uncompacted fill Compacted fill
Variable Variable
Very high Low
— —
(From Youd and Perkins (1978); reprinted from the Journal of Geotechnical Engineering, ASCE, Vol. 104, No. 4, pp. 433–446. Copyright 1978. With permission of ASCE.
cohesive soils that are susceptible to liquefaction as shown in Fig. 11.95 (Hyodo et al., 1994). An assumption of a strength loss of about 20 percent is sometimes used in practice. Figure 11.96 shows the undrained cyclic shear stress ratio cy /su that brings normally consolidated clays to failure after 10 loading cycles (Andersen, 2004). The data include eight clays with different plasticity indices. In the direct shear tests (Fig. 11.96a), the undrained cyclic shear stress ratio at failure decreases with increase in initial static shear stress a /su and increases with plasticity index. In the
Copyright © 2005 John Wiley & Sons
triaxial tests (Fig. 11.96b), the initial static shear is defined as a /su ⫽ (ac ⫺ rc )/2su, where ac and rc are the axial and radial consolidation stresses, respectively. The values of cy /su ⫽ (a ⫺ r)/2su show peaks at a /su ⫽ 0.2 to 0.3, indicating that the small initial anisotropy gave increased cyclic resistance. However, the undrained cyclic shear stress ratio at failure decreases when the initial static shear stress is higher or in triaxial extension conditions. Evidently in normal clays the magnitude of cyclic shear strain is less than that required to cause complete remolding.
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STRENGTH AND DEFORMATION BEHAVIOR
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432
Figure 11.89 Cyclic stress ratio and number of load cycles to cause initial liquefaction of
a sand at different initial relative densities (from De Alba et al., 1976). Reprinted with permission of ASCE.
Figure 11.90 Pore pressure as a function of cyclic shear
strain illustrating a threshold strain of about 0.01 percent, below which no excess pore pressures are developed (from Dobry et al., 1981). Reprinted with permission of ASCE.
Complete remolding would define an absolute lower bound, and its value is defined by the clay sensitivity. Cyclic stresses could cause sufficient deformations in quick clay to initiate a liquefaction-type flow failure. Some examples are given in Andersen et al. (1988). 11.14
STRENGTH OF MIXED SOILS
The presence of fines in sands can significantly influence the strength behavior. Differing effects can be obtained depending on particle size, shapes, and sample
Copyright © 2005 John Wiley & Sons
Figure 11.91 Mechanism of pore pressure generation during
cyclic loading (Seed and Idriss, 1982).
preparation methods. Figure 11.97 shows different scenarios of intergranular matrix of two different size particles (Thevanayangam and Martin, 2002). Initially the maximum and minimum void ratios of a sand–silt mixture decrease with increase in silt content, but then the
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STRENGTH OF MIXED SOILS
Co py rig hte dM ate ria l
Figure 11.92 Rate of pore pressure buildup in cyclic simple shear tests (from Seed et al., 1976). Reprinted with permission of ASCE.
Itukaichi clay and Toyoura sand (Hyodo et al., 1994).
Cyclic Shear Strength / Static Undrained Shear Strength (τcy / su)
tance ratio (Hyodo et al., 2002).
Figure 11.95 Comparison of the cyclic resistance ratios of
Cyclic Shear Strength / Static Undrained Shear Strength (τcy / suDSS)
Figure 11.93 Effect of confining pressure on cyclic resis-
433
Direct Simple Shear Tests - Strength at 10 Cycles OCR = 1
1.2
Offshore Africa, PI=80 -100% Marlin IIa, PI=50%
1.0
Troll I, PI=37%
0.8
Troll II, PI=20%
Marlin IIb+, PI=45%
0.6
Drammen, PI=27%
0.4 0.2
North Sea GC, PI=16-27% Storebælt, PI=7-12%
0.0 0.0 0.2 0.4 0.6 0.8 1.0 Initial Shear Stress / Static Undrained Shear Strength (τa / s uDSS )
1.0 0.8
Triaxial Tests - Strength at 10 Cycles OCR = 1 Offshore Africa, PI=80-100%
Troll II, PI=20%
Marlin IIb+, PI=45%
Troll I, PI=37%
Marlin IIa, PI=50%
0.6 0.4
Drammen, PI=27%
0.2
Storebælt, PI=7-10%
0.0 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 Initial Shear Stress / Static Undrained Shear Strength (τa / s u)
Figure 11.94 Postliquefaction residual strength as a function
Figure 11.96 Normalized shear stresses that give undrained
of Standard Penetration Test N values (Seed and Harder, 1990).
failure after 10 cycles in (a) direct shear tests and (b) triaxial tests (Andersen, 2004).
Copyright © 2005 John Wiley & Sons
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11
STRENGTH AND DEFORMATION BEHAVIOR
Co py rig hte dM ate ria l
434
Figure 11.97 Granular mix classification (Thevanayangam and Martin, 2002).
void ratios increase when the silt becomes the host soil as shown in Fig. 4.4. In case (i), the fine particles fit in the void space formed by the coarse particles. The mechanical behavior is little affected by the presence of fines because the external forces are transferred through the contacts between coarse particles. In cases (ii) and (iii), the fine particles start to fully occupy some void space and separate the coarse particles and prevent them from touching each other. These fine particles may reinforce the skeleton of coarse particles or they may make the skeleton unstable. As the proportion of fine particles increases, the coarse particles float inside the matrix of fine particles as illustrated as case (iv). The fine grains then dominate the mechanical behavior of the mixed soils, and the coarse grains may or may not contribute to shear resistance as a reinforcing element. Once the mixing scenario reaches case (iv), the void ratio increases with increasing fines content due to increasing specific surface of the mixture. The threshold value to become case (iv) depends on the specific mixture but is usually in the range of 25 to 45 percent fines in most cases (Polito and Martin, 2001). For cases (i) to (iii), the granular void ratio eG defined in Chapter 4 is a useful index for consideration of the effect of fines. If two mixed soils with different
Copyright © 2005 John Wiley & Sons
fines content have the same granular void ratio and the same mechanical properties, the fines are just occupying the void space and are not influencing shear resistance. Most reported cases show that, for a given granular void ratio, the undrained strength and cyclic shear resistance are either independent of or increase with silt content (Shen et al., 1977; Vaid, 1994; Polito and Martin, 2001; Carraro et al., 2003). The undrained response of sand mixed with equidimensional silt particles is shown in Fig. 11.98 (Kuerbis et al., 1988). Specimens of the mixture were created by slurry deposition, and the density was controlled in such a way that all specimens had relatively similar granular void ratios eG, even though the actual void ratio decreased with increasing silt content. Both undrained triaxial compression and extension tests were performed following isotropic consolidation. Increased silt content gave stiffer response in triaxial compression. Apparently, the silts filled the void space and stabilized the soil as shown in Fig. 11.99a. However, the effect was small in triaxial extension. Liquefaction resistance increases with relative density as shown in Fig. 11.89. However, increasing silt content gives scattered relationships between relative density and the CRR at 20 loading cycles, as shown in
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Silt Content %
eG
0.764 0.728 0.669 0.547 0.448
0.764 0.802 0.805 0.784 0.863
100
0.8
435
0% fines 5% fines 10% fines 15% fines
0.6 0.4 0.2 0.0 0.0
0
100
200
300
0.4 0.6 Relative Density
400
tests on silty sand with different mixing ratios (Kuerbis et al., 1988).
0.8
Cyclic Resistance Ratio
Figure 11.98 Undrained triaxial compression and extension
0.8
1.0
0.4
0.3
(a)
(σa + r)/2 (kPa)
–100
0.2
Co py rig hte dM ate ria l
(σa – r)/2 (kPa)
200
0 4 7.5 13.3 22.3
e
Cyclic Resistance Ratio
STRENGTH OF MIXED SOILS
0% fines 5% fines 10% fines 15% fines
0.6 0.4 0.2 0.0
0.8
0.7
0.6 0.5 Void Ratio
(a)
(b)
Cyclic Resistance Ratio
(b)
0.8
0% fines 5% fines 10% fines 15% fines
0.6 0.4 0.2 0.0
0.8
0.7
0.6 0.5 Granular Void Ratio
0.4
(c )
Figure 11.100 Cyclic resistance ratios of silty sands plotted
against (a) relative density, (b) void ratio, and (c) granular void ratio (from Carraro et al., 2003).
(c)
(d)
Figure 11.99 Schematic diagrams of how fine particles are
placed inside coarse-grain matrix: (a) sand–silt mixture with silt filling the void, (b) sand–silt mixture with silts between sands and granular void ratio larger than emax, (c) sand–clay mixture, and (d ) sand–mica mixture.
Fig. 11.100a due to variations in maximum and minimum void ratios with increasing silt content. If the CRR values are plotted in terms of void ratio and cyclic resistance as shown in Fig. 11.100b, the liquefaction resistance at a given void ratio decreases with
Copyright © 2005 John Wiley & Sons
increasing silt content. If the CRR values are plotted as a function of the granular void ratio eG, as shown in Fig. 11.100c, the sand–silt mixtures give higher liquefaction resistance than clean sand, but the resistance of these mixtures was independent of silt content. The above results are applicable when the granular void ratio eG is smaller than the maximum void ratio emax of the host medium (without fines). When fines are added, it is possible to create specimens that have eG larger than emax even though the overall void ratio is smaller than emax (Lade and Yamamuro, 1997; Thevanayagam and Mohan, 2000). This condition can be
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436
11
STRENGTH AND DEFORMATION BEHAVIOR
(MPa) 0.3
(σ’a+ σ’ r)/2
0.2
bridged between the host sand particles (Fig. 11.99d), and increased the overall void ratio as shown in Fig. 11.102. On the other hand, inclusion of smaller silt and clay particles decreased the overall void ratio, as also shown in Fig. 4.4. The open fabric of a sand–mica mixture can give complicated soil deformation and strength properties depending on mica particle orientation and shear mode (Hight et al., 1998). Further increase in fines content leads to sand particles floating in clay or silt as shown by case (iv) in Fig. 11.97. The mixed soil then behaves more like pure clay or silt. The deformation behavior then becomes more clay/silt dominated, and the coarser particles may or may not contribute to the strength properties. For example, Fig. 11.103 shows that the liquefaction resistance of mixtures with fines content greater than 35 percent was independent of silt content and granular void ratio (Polito and Martin, 2001).
Co py rig hte dM ate ria l
achieved if some fines are placed between the coarser particles as shown in Fig. 11.99b. In this case, the structure is metastable, and the strength of the mixed soil is reduced due to fewer sand grain contacts. When smaller particles such as clays are added instead of silt-size particles, the clay fines act as a lubricator at sand particle contacts as shown in Fig. 11.99c and make the soil unstable. Undrained responses of Ham river sand mixed with different kaolin contents are shown in Fig. 11.101 (Georgiannou et al., 1991). Samples were prepared by pluviating the sand into distilled water with suspended kaolin particles so that similar granular void ratios were achieved. Both undrained triaxial compression and triaxial extension tests were performed after consolidating the samples under K0 stress conditions. In triaxial compression, the increase in clay content did not affect the peak stress, but the strain-softening behavior was more pronounced. After the specimen passed the phase transformation line, the stress increased toward the critical state. In triaxial extension, addition of clay led to total liquefaction. The friction angle did not change for clay fractions up to 20 percent. This delayed the development of anisotropic fabric needed to resist the increasing load. The shape of fine particles also influences the stability of the mixed soil. Hight et al. (1998) report the behavior of micaceous sands in connection with flow slides that occurred during construction of the Jamuna Bridge in Bangladesh. The large and platy mica flakes
Triaxial Compression Tests Clay Content = 0%, eG = 0.77 Clay Content = 4.6%, eG = 0.80 Clay Content = 7.6%, eG = 0.80
11.15
COHESION
True cohesion is shear strength in excess of that generated by frictional resistance to sliding between particles, the rearrangement of particles, and particle crushing. That is, true cohesion must result from adherence between particles in the absence of any externally applied or self-weight forces. The existence of tensile or shear strength in the absence of effective compressive stress in the soil skeleton or on the failure plane might be considered true cohesion. However, the particulate nature of soil and the fact that most interparticle contacts are not oriented in the plane of shear mean that the application of directional shear stress will induce normal forces at most interparticle contacts. These forces will, in turn, generate a resistance
Initial Stress State
0.1
(σ’a+ σ’r)/2
0.0
-0.1
0.1
0.5 0.3 0.4 (MPa) Triaxial Extension Tests Clay Content = 0%, e G = 0.77 Clay Content = 3.5%, e G = 0.80
0.2
Clay Content = 7.5%, e G = 0.80
Figure 11.101 Undrained triaxial compression and extension test stress paths of clay–sand mixtures with different mixing ratios but at similar granular void ratios (Georgiannou et al., 1991).
Copyright © 2005 John Wiley & Sons
Figure 11.102 Effect of fines (mica, silt, and kaolin) on void
ratio of a sand (Hight et al. 1998).
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COHESION
0.30
× Yatesville Sand with 50% Silt 夽 Yatesville Sand with 75% 䊉 Silt 100% Silt
䉬 Monterey
Sand with 35% Silt 䊏 Monterey Sand with 50% Silt 䉱 Monterey Sand with 75% Silt
0.20
0.15
0.10
䊉 × 䊉 䉬䉬 䊉
䊏
×
×
×
1.40
1.60
䉱
夽 夽 夽
Co py rig hte dM ate ria l
Cyclic Resistance Ratio CRR
0.25
437
0.05
0.00 0.80
1.00
1.20
1.80
2.00
2.20
Granular Void Ratio eG
Figure 11.103 Variation of cyclic resistance ratio with granular void ratio with silt content
above the threshold value (Polito and Martin, 2001).
to sliding at the contact provided the value of is greater than zero. Confirmation of the existence of a true cohesion and determination of its value from strength tests is difficult because projection of the failure envelope back to ⫽ 0 is uncertain, owing to the curvature of most failure envelopes, unless tests are done at very low effective stresses. Tensile tests cannot be made on most soils. Harison et al. (1994) performed various types of tensile tests on compacted clay specimens but found that the tensile strengths decreased with increase in specimen size due to increase in the number of internal flaws. There is no convenient way to run a triaxial compression test while maintaining the effective stress equal to zero on the potential failure plane. Strength can be measured by direct shear with no applied normal stress . Some examples are given in Bishop and Garga (1969), Graham and Au (1985), and Morris et al. (1992); however, for the reason given in the previous paragraph, the measured strength cannot be attributed specifically to true cohesion. Possible Sources of True Cohesion
Three possible sources for true cohesion between soil particles have been proposed:
1. Cementation Chemical bonding between particles by cementation by carbonates, silica, alumina, iron oxide, and organic compounds is possible. Cementing materials may be derived from the soil minerals themselves as a result of
Copyright © 2005 John Wiley & Sons
solution–redeposition processes, or they may be taken from solution. An analysis of the strength of cemented bonds was given by Ingles (1962) and is summarized in Section 7.4 and Eqs. (7.2) to (7.8). Cohesive strengths of as much as several hundred kilopascals (several tens of pounds per square inch) may result from cementation. Stress–strain curves and peak failure envelopes for cemented sands are shown in Fig. 11.104. These curves show that even relatively small amounts of cement can have very large effects on the deformation properties. Small values of cohesion have a large effect on the stability of a soil and its ability to stand unsupported on steep slopes. However, at large strains when the cementation breaks down, the strengths become similar irrespective of the degree of cementation as shown in Fig. 11.104a. 2. Electrostatic and Electromagnetic Attractions Electrostatic and electromagnetic attractions between small particles are discussed in Sections 6.12 and 7.4. Electrostatic attractions become significant (⬎ 7 kPa or 1 psi) for separation distances ⬍2.5 mm. Electromagnetic attractions or van der Waals forces are a source of tensile strength only between closely spaced particles of very small size (⬍1 m). 3. Primary Valence Bonding and Adhesion When normally consolidated clay is unloaded, thus becoming overconsolidated, the strength does not decrease in proportion to the effective
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11
STRENGTH AND DEFORMATION BEHAVIOR
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438
Figure 11.104 Stress–strain curves and failure envelopes for cemented and uncemented sand at a relative density of 74 percent: (a) stress–strain curves and (b) failure envelopes based on peak strength (from Clough et al., 1981). Reprinted with permission of ASCE.
stress reduction, but a part is retained as shown in Fig. 11.3. Whether or not the higher strength in the overconsolidated clay is because of the lower void ratio or due to the formation of interparticle bonds is not known. However, a ‘‘cold welding’’ or adhesion may be responsible for some of it. This could result from the formation of primary valence bonds at interparticle contacts. Apparent Cohesion
An apparent cohesion can be generated by capillary stresses. Water attraction to particle surfaces combined with surface tension causes an apparent attraction between particles in a partly saturated soil. Equation (7.9) can be used to estimate the magnitude of tensile strength that can be developed by capillary stresses in a soil. This is not a true cohesion; instead, it is a frictional strength generated by the positive effective stress created by the negative pore water pressure. Summary
Several contributions to cohesion are summarized in Fig. 11.105 in terms of the potential tensile strengths that can be generated by each mechanism as a function of particle size. For all the mechanisms except chemical cementation, cohesion is a consequence of normal
Copyright © 2005 John Wiley & Sons
stresses between particles generated by internal attractive forces. The mechanism of shear resistance resulting from these attractions should be the same as if the contact normal stresses were derived from effective compression stresses carried by the soil. It is convenient, therefore, to think of cohesion (except for cementation) as due to interparticle friction derived from interparticle attractions, whereas the friction term in the Mohr–Coulomb equation is developed by interparticle friction caused by applied stresses. Essentially the same concept was suggested by Taylor (1948) where cohesion was attributed to an ‘‘intrinsic pressure.’’ Similarly, Trollope (1960) attributed shear strength to the Terzaghi and Bowden–Tabor adhesion theory, with both applied stresses and interparticle forces contributing to the effective stress that developed the frictional resistance. Present evidence indicates that cohesion due to interparticle attractive forces is quite small in almost all cases, whereas that due to chemical cementation can be significant. 11.16
FRACTURING OF SOILS
Soil fracturing is often observed in geotechnical practice. Tensile cracks develop when there is external tension stress such as at the crest of a landslide or vertical cuttings. In some cases, water can fill the cracks, lead-
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FRACTURING OF SOILS
439
Figure 11.105 Potential contributions of several bonding mechanisms to soil strength
(Ingles, 1962).
ing to further instability. Soil piping can occur in a dam from water flow through cracks causing internal erosion. Hydraulic fracturing results from increase in the pressure at the crack tips. Hydraulic fractures can be created by injecting fluids, grouts, or chemicals and used to control settlements caused by underground construction, to determine the in situ horizontal stress state, to create an impermeable hydraulic barrier, or to inject ground treatment chemicals for soil reinforcement and contaminated ground remediation. Desiccation also causes the development of tensile cracks as the suction in the soil increases by evaporation and causes shrinkage of the soil by increase in effective stresses. Resistance to fracturing depends on tensile strength (or true cohesion) of the material, which is often small in geomaterials except when they are cemented. Fracturing can occur in clays in undrained conditions by rapid increase in external pressure or in sands and clays by fluid permeation. Various mechanisms for fracture initiation are described below.
principal effective stress is equal to the negative value of the tensile strength (t ).19 This criterion can be written as 3 ⫽ ⫺t
(11.52)
When a tensile force is applied to a saturated soil in the direction of minor principal stress, it will be sheared in undrained conditions and the soil cracks if Eq. (11.52) is satisfied. The tensile total stress 3 then becomes 3 ⫽ u0 ⫹ u ⫺ t ⫽ u0
⫹ ( 3 ⫹ A( 1 ⫺ 3)) ⫺ t
(11.53)
where u0 is the initial pore pressure, u is the excess pore pressure generated during the shearing process leading to fracture, A is Skempton’s pore pressure parameter (Section 8.9), and 1 and 3 are the changes
Fracture under Undrained Conditions
If particle contacts cannot carry tension, it is often assumed that the tensile cracking occurs when the minor principal effective stress 3 becomes zero. If the soil is cemented, cracking is generated when the minor
Copyright © 2005 John Wiley & Sons
19
Note that the tensile strength of a soil is defined in terms of effective stress. Unfortunately, many tensile strength values are written in total stress since pore pressure is not measured.
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440
11
STRENGTH AND DEFORMATION BEHAVIOR
in major and minor principal total stresses, respectively.20 Rearrangement of Eq. (11.53) gives 1 A
3 ⫽ ⫺ (3i ⫹ t) ⫹ 1
increases, but the circumferential stress initially decreases as long as the soil behaves linear elastically and does not fail in shear (see Fig. 11.106a). Cracks develop in the radial direction when the effective circumferential stress becomes zero for uncemented soils and equal to the negative value of the tensile strength for cemented soils. Assuming that the clay behaves linear elastically,21 the change in the radial total stress
r (⫽ 1) at the cavity is equal to the negative of the change in the circumferential stress (⫽ 3);
r ⫽ ⫺ . Substituting this condition in Eq. (11.54) under plane strain conditions (A ⬇ –12 )22 gives
(11.54)
Co py rig hte dM ate ria l
where 3i is the initial minor principal effective stress prior to applying the tensile force. Injection of fluid into a cylindrical cavity surrounded by a clay formation can lead to fracture by increase in cavity pressure. Examples of this mechanism are fracture grouting and soil fracturing around driven piezometers (Lefebvre et al., 1981, 1991). According to cavity expansion theory, the radial total stress at the cavity
Pf ⫺ 3i ⫽ r ⫽ 3i ⫹ t or Pf ⫽ 23i ⫺ u0 ⫹ t
A more general case can be written as 3 ⫽ u0 ⫹ ( p ⫹ a q) ⫺ t, where p and q are the changes in mean pressure and deviator stress, and a is the modified pore pressure parameter defined by Wood (1990). 20
21
(11.55)
For simplicity, the undrained behavior of clays is assumed to be linear elastic-perfectly plastic. 22 No change of intermediate principal stress ( 2 ⫽ 0) is assumed.
Tension crack
σr = Pf
σθ
Pf
σθ
-σθ = σt
σ0
σr
σ’0
σθ
−σ t
σθ
Cavity Displacement
Solid Line: Total Stresses Dotted Line: Effective Stresses
Pf
(a)
σr = Pf
Plastic Deformation
σ0
σθ
σ0
Pf
σθ 2s σr u
σθ
σθ Crack
Plastic Instability
2su
Cavity Displacement
Solid Line: Total Stresses Dotted Line: Effective Stresses
(b) Figure 11.106 Fracture mechanisms of injection fluids into a cavity: (a) tensile fracture in
undrained conditions and (b) shear failure in undrained conditions.
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FRACTURING OF SOILS
1981; Yanagisawa and Panah, 1989), and they can be generalized by the following equation: Pf ⫽ m3i ⫹ n
Pf ⫽ 3i ⫹ su
(11.56)
where 3i is the initial total stress prior to shearing and su is the undrained shear strength. The fracture pressure Pf increases with initial confining pressure in direct proportion (i.e., slope of 1). If the plastic zone around the expanding cavity increases before fracture initiates or su increases with initial confining pressure, the fracture pressure Pf would increase from the value given in Eq. (11.56) and, therefore, the linear proportion between Pf and 3i is expected to be larger than 1. Empirical equations to estimate soil fracture under undrained conditions are available (Jaworski et al.,
Copyright © 2005 John Wiley & Sons
(11.57)
where m and n are material constants. Experimental data give values of m varying between 1.5 and 1.8 (Jaworski et al., 1981), whereas data indicating shearinduced fracture give values of m ⫽ 1.05 to 1.085 (Panah and Yanagisawa, 1989). Reported values of fracture pressure as a function of confining pressure for various soils are plotted in Fig. 11.107. The m values of individual data sets are in general bounded by Eqs. (11.55) and (11.56).
Co py rig hte dM ate ria l
where Pf is the injection pressure that causes the clay to fracture. The above mechanism assumes that the tensile fracture occurs when Eq. (11.53) is satisfied in a uniform displacement field at the injection cavity. The fracture pressure Pf increases linearly with the initial total confining pressure 3i with a slope of 2. In reality, deformation around the cavity is not uniform and fracture can initiate at a localized zone at a pressure smaller than the prediction. This leads the slope between Pf and 3i to be smaller than 2. Other considerations for this fracture mechanism include the effect of shear-induced pore pressure and a nonlinear stress– strain relationship (Andersen et al., 1994). As injection pressure increases, the clay at the surface of the cavity may reach undrained shear failure before the circumferential effective stress becomes zero in uncemented soils or reaches the tensile strength in cemented soils. In such cases, the changes in the stress state at the cavity boundary with increasing cavity strain are shown in Fig. 11.106b). Upon shear failure, the difference between the radial and circumferential stresses (both total and effective) remains equal to 2su, and, therefore, the minimum principal effective stress never reaches zero. In such circumstance, it is difficult to see how plastic yielding initiates a fracture. However, there is much field and experimental evidence suggesting that fracture has indeed occurred even though plastic deformation was observed at the cavity due to the low undrained shear strength of the soil (Mori and Tamura, 1987; Panah and Yanagisawa, 1989; Au et al., 2003). A possible explanation is that the increase in plastic shear failure zone created shear bands or an unstable state around the cavity. This leads to a localized microscale crack and the injected fluid can penetrate into the crack to produce local tensile stresses at the crack tips, as illustrated in Fig. 11.106b. A simple cylindrical cavity expansion analysis shows that the cavity pressure required for the cavity boundary to reach the plastic state is
441
Fracture under Drained Conditions
Forced seepage flow into a cavity in permeable soil leads to soil fracture if the effective stress reduces to the negative sign of the tensile strength of the soil. Practical applications of this situation are in situ permeability testing and bore hole stability. To interpret the fracture conditions around a driven piezometer, Bjerrum et al. (1972) developed the following conditions for the initiation of fracture in soils using the equilibrium equation with the assumptions of steady state pore fluid flow from a cylindrical cavity and elastic soil material. Horizontal cracks may develop if the injection pressure exceeds the initial total vertical stress: Pinj ⫽ u0 ⫹ v0
(11.58)
where Pinj is the injection pressure, u0 is the initial pore pressure, and v0 is the initial vertical effective stress. Vertical cracks in the radial direction from the piezometer develop when the circumferential effective stress becomes smaller than the tensile strength of the material. Bjerrum et al. (1972) consider two cases: (i) the piezometer is in contact with the surrounding soil and (ii) the piezometer moves away from the surrounding soil (called ‘‘blow off’’). For the former case, cracks develop when the following condition is satisfied: Pinj ⫽ u0 ⫹
冉 冊
1 ⫺ 1 [t ⫹ (1 ⫺ )h0 ]
(11.59)
where is Poisson’s ratio, t is the tensile strength, h0 is the initial horizontal effective stresses; is a disturbance factor that considers the change in circumferential effective stress due to piezometer installation. Typical values of are given in Table 11.6.
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11
STRENGTH AND DEFORMATION BEHAVIOR
Co py rig hte dM ate ria l
442
Figure 11.107 Increase in fracture pressure with initial confining pressure of different soils.
Table 11.6
Typical Values of Disturbance Factors ␣ and 
Soil Type
Range of Compressibility Ratio E/ h0(1 ⫹ v)
High compressibility Medium compressibility Low compressibility
1–3 3–10 10–70
0.4–0.2 0.2 to ⫺0.2 ⫺0.2 to ⫺1.1
0.5–1.1 1.1–2.0 2.0–4.2
From Bjerrum et al. (1972).
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FRACTURING OF SOILS
In some cases, the radial effective stress in the soil next to the piezometer becomes zero and the soil separates from the piezometer. This occurs when the injection pressure becomes larger than the total radial effective stress: Pinj ⫽ u0 ⫹ h0(1 ⫹ )
443
al., 1992). Soil shrinks by the decrease in pore pressure and increase in effective stress. This decrease in volume generates vertical cracks. On the other hand, the tensile strength that provides the resistance to crack formation increases with increased negativity of pore water pressure.
(11.60) Fracture Propagation
Limited knowledge is available concerning fracture orientation and propagation. Some examples of fractures developed by injection of different fluids are shown in Fig. 11.109. When fluid is injected into the
Co py rig hte dM ate ria l
where is a disturbance factor that considers the change in radial effective stress during piezometer installation. Typical values of are given in Table 11.6. Further increase in injection pressure leads to development of vertical cracks in the radial direction, which occurs when the following condition is satisfied: Pinj ⫽ u0 ⫹ (1 ⫺ )[t ⫹ (2 ⫹ ⫺ )h0 ]
(11.61)
Desiccation Cracks
Reduction in moisture by surface evaporation from clays leads to increase in interparticle contact forces by suction. Soil then shrinks and desiccation cracks may develop. The generation of cracks changes the hydraulic properties from Darcy’s-type homogeneous flow to fracture-dominated flow. This can cause some environmental problems, such as unexpected poor performance of contaminant barrier systems. Figure 11.108 shows the crack patterns observed after desiccation of sensitive clays (Konrad and Ayad, 1997). The cracks can be pentagonal and heptagonal in shape, and their size appears to be uniform. Morris et al. (1992) report that crack depths from 0.5 to 6.0 m are observed in natural soils in Australia and Canada. Unfortunately, the available knowledge for prediction of crack depth and spacing is limited. The decrease in matrix suction resulting from evaporation leads to two counteracting effects (Morris et
(a)
(b) Figure 11.109 Different fracture patterns observed in laboFigure 11.108 Photos of desiccation cracks (Konrad and
Ayad, 1997).
Copyright © 2005 John Wiley & Sons
ratory: (a) vertical and radial fractures hardened by epoxy and (b) horizontal fracture by cement bentonite mixture injection.
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11
STRENGTH AND DEFORMATION BEHAVIOR
11.17
well before failure. A good example is the onedimensional compression behavior discussed in Chapter 10. After the stress state becomes larger than the preconsolidation pressure, the soil has yielded and plastic strains develop. This leads to the concept of yield envelope (sometimes referred as yield surface or limit state curve), which differentiates the state of the soil between elastic and plastic. Examples of the yield envelope of sands and clays were shown in Fig. 11.12. When the stress state reaches the yield envelope, the total strain is governed by the development of plastic strain increments. Unfortunately, for soils, there is no distinct transition from elastic to plastic behavior. Plastic strains do develop inside the yield envelope and the stiffness degrades even at very small strain levels. Figure 11.111 shows a schematic nonlinear stress–strain relationship for a soil subjected to monotonic and cyclic deviator loads. Some experimental data are shown in Figs. 11.85 and 11.88. Under cyclic loading, the relationships are hysteretic, which indicates energy absorption, or damping, during each complete cycle of stress reversal. The shear modulus G and damping ratio are used to characterize the curves in Fig. 11.111, and they are defined by
Co py rig hte dM ate ria l
soil to create hydraulic fracture, a rule of thumb is that vertical fractures are formed when K0 is less than 1 [as given Eq. (11.59)] and horizontal fractures develop when K0 is more than 1 [as given in Eq. (11.60) with ⫽ 0]. However, this assumes injection into a linear elastic infinite soil medium. When multiple grout injections are performed at close distance, horizontal fractures can be observed even though K0 is less than 1 (Soga et al., 2004). Natural bedding also affects fracture orientation. In shallow formations, fractures are often horizontally oriented or gradually dipped (Murdoch and Slack, 2002). Simple criteria presented as Eqs. (11.56) to (11.61) are applied for global stress conditions, where microscale cracks often develop by local tensile stresses at the crack tips. Fracture mechanics have been used with some success to characterize the cracking resistance of the soils and to examine possible crack propagation (Morris et al., 1992; Harison et al., 1994; Murdoch and Slack, 2002). The actual mechanisms of fracture development in a fluid–soil system are more complicated than in the above analyses, as illustrated in Fig. 11.110. They may involve plastic deformation at the crack tip, soil rate effects, penetration of injection fluid into the cracks, and permeation of injection fluid from cracks into the soil medium. If the clay is overconsolidated and saturated, the negative pore pressure generated by shearing in front of the crack could possibly lead to cavitation and dry cracks may develop in front of penetrating injection fluid. DEFORMATION CHARACTERISTICS
Shear Stress
444
Strains are often decomposed into elastic (recoverable) and plastic (irrecoverable) parts. Conventional soil mechanics assumes that plastic strains develop only when the stress state satisfies some failure criterion (e.g., the Mohr–Coloumb criterion). Otherwise, the soil behaves elastically. However, plastic strains usually develop
Permeation
Injection Fluid
Localized Shearing with Dilation and Rate Effect
Cavitation?
Secant Stiffness G1
G2
τ2
Monotonic loading curve
τ1
Cyclic loading curves
γ1
γ2 Shear Strain
Monotonic loading curve
Fluid Penetration Into Crack
Figure 11.110 Possible fracture propagation mechanisms in
Figure 11.111 Monotonic and cyclic load stress–strain re-
soils.
lationships at different strain amplitudes.
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DEFORMATION CHARACTERISTICS
G⫽
c c
mation parameters usually cannot be determined accurately by conventional triaxial testing. With the use of local strain measurement systems (Jardine et al., 1984; Goto et al., 1991; Scholey et al., 1995; Cuccovillo and Coop, 1997; Lo Presti et al., 2001; Yimsiri and Soga, 2002), however, it is now possible to measure the development of stresses from very small strains, which can then be used for accurate prediction of deformations in the field. To characterize nonlinear deformation inside the yield envelope, it is convenient to define four zones in the p –q plane as shown in Figs. 11.112b, 112c and 112d. The initial stress state is considered to be at point O, and the boundaries of the zones are determined by stress probe testing in different stress path directions. The boundaries often associated with strain levels (axial or shear strains), and the corresponding secant stiffness values are illustrated in Fig. 11.112a.
(11.62)
in which c is the applied shear stress and c is the corresponding shear strain, and ⫽
1 E 2 G 2c
(11.63)
dεp/dεt
Stiffness G or E
Co py rig hte dM ate ria l
in which E is the energy dissipated per cycle per unit volume, given by the area within the hysteresis loop. Understanding this pre-yield deformation behavior is very important, as most strains observed in geotechnical construction practice are indeed small (less than 0.1 percent) (Burland, 1989). Site response under earthquake loading is influenced by stiffness degradation and damping characteristics that are associated with relatively small strain levels (Seed and Idriss, 1982). This was illustrated in Fig. 11.9, which shows typical strains observed in various types of geotechnical construction and shows that the necessary defor-
1
I
II
III
State A
IV
1. Zone 1 (True Elastic Region) Soil particles do not slide relative to each other under a small
Critical-State Line
q
Y3 Envelope
State B
State C
IVIV
Y2 Envelope III
II
dεp Plastic Strain Increment dεt Total Strain Increment
Strain
O Y1 Envelope Initial State I
p
Strain ’ p
0
(a)
(c) State B
Critical State Line
q
q
Critical State Line
Expanded Y3 Envelope
Y3 Envelope
IV
II
III
I
Y2 Envelope
II
445
Y1 Envelope
III
O Initial State
Y2 Envelope
Y1 Envelope I
p
(b) State A
(d) State C
Figure 11.112 Four zones of deformation characterization: (a) stiffness degradation and
plastic strain development, (b), (c), and (d) are the stress conditions and the location of the four zones associated with three successive states (modified from Jardine, 1992).
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446
11
STRENGTH AND DEFORMATION BEHAVIOR
Table 11.7
elastic even though microscopically soil particles may not be back to their original locations after the cyclic loading. When the stress state reaches the outer boundary of zone 2 (called the Y2 envelope), plastic strains start to develop. The initiation of plastic strains can be determined by examining the onset of permanent volumetric strain in drained conditions or residual excess pore pressures in undrained conditions after unloading. Hence the strain level that defines the Y2 envelope is called volumetric threshold strain.23 The value of the volumetric threshold strain is generally one order of magnitude higher than that of the elastic threshold strains. The available experimental data suggest that it ranges between 7 ⫻ 10⫺5 and 7 ⫻ 10⫺4 (the lower limit for uncemented normally consolidated sands and the upper limit for high plasticity clays and cemented sands). At this strain level, the stiffness degrades to 60 to 85 percent of the true elastic value (Ishihara, 1996).
Co py rig hte dM ate ria l
stress increment, and the stiffness is at its maximum. The soil stiffness is determined from contact interactions, particle packing arrangement, and elastic stiffness of the solids. The soil stiffness values can be obtained from elastic wave velocity measurements, resonant column testing, and very accurate local strain transducers. Cyclic loading produces only very small hysteresis by stick–slip motions at particle contacts and other mechanisms, producing very small energy dissipation less than 1 percent. The strains at which the stress state reaches the outer boundary of zone 1 (called Y1 envelope) are usually described as elastic limit strains or elastic threshold strains. This state is illustrated as state A in Fig. 11.112b. The elastic limit axial strain depends on soil type, solid stiffness, and confining pressure as shown in Table 11.7 for different geomaterials. Micromechanics analysis by Santamarina et al. (2001) shows that it increases from less than 5 ⫻ 10⫺6 strain, for nonplastic soils at low confining pressure conditions, to greater than 5 ⫻ 10⫺4 strain at high confining pressure conditions or in soils with high plasticity. 2. Zone 2 (Nonlinear Elastic Region) Soil particles start to slide or roll relative to each other in this zone. The stress–strain behavior becomes nonlinear, and the stiffness begins to decrease from the true elastic value as the applied strains or stresses increase. However, a complete cyclic loading (unloading and reloading) shows full recovery of strains and therefore the zone is called
23 Other definitions of the Y2 surface are available. For example, (a) perform undrained cyclic loading test and find the linear relationship between max and p / max, where max is the maximum strain for each cycle and p is the residual strain (Smith et al., 1992); (b) the strain level when excess pore pressures start to accumulate in a sequence of undrained cyclic tests at different strain levels (Vucetic, 1994); (c) change in the direction of strain path in the vol–s space in drained tests (Kuwano, 1999); and (d) change in the slope of the excess pore pressure–vertical effective stress in undrained triaxial compression test (Kuwano, 1999).
Elastic Limit Strain for Various Geomaterials from Triaxial Tests
Material
Elastic Limit Axial Strain
Dogs Bay sand Leighton Buzzard sand Kaolinite Berthieville clay Bothkennar clay Queenborough clay Osaka Bay clay London clay Vallericca clay Calcarenite Sandstone High-density chalk Low-density chalk Cement-treated sandy soil Samamihara mudstone
⬍1 ⫻ 10⫺5 2 ⫻ 10⫺5 ⬍2 ⫻ 10⫺5 ⬍2 ⫻ 10⫺5 ⬍2 ⫻ 10⫺5 –3 ⫻ 10⫺5 ⬍2 ⫻ 10⫺5 1 ⫻ 10⫺5 2 ⫻ 10⫺5 ⬍1 ⫻ 10⫺4 1 ⫻ 10⫺4 2 ⫻ 10⫺4 5 ⫻ 10⫺5 2 ⫻ 10⫺5 –4 ⫻ 10⫺4 1 ⫻ 10⫺4 2 ⫻ 10⫺4
Soil Description
Uniform, angular biogenetic carbonate sand Uniform, subround, quzartz sand Reconstituted clay Soft silty clay Soft marine clay Soft silty clay Overconsolidated marine clay Stiff overconsolidated, fissured clay Weakly cemented overconsolidated clay Weak rock, carbonate sand cemented with calcite Weak rock, quartz grain weakly bonded bny iron oxide Dry density ⫽ 1.94 g/cm3 Dry density ⫽ 1.35 g/cm3 Hard soil/weak rock Weak rock
After Matthews et al. (2000).
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LINEAR ELASTIC STIFFNESS
Examples of experimentally determined boundaries are shown in Fig. 11.12b for Bothkennar clay and Fig. 11.113 for Ham River sand. These zones are not fixed in space when the stress state moves inside the Y3 yield
0.3
Y3 Envelope
0.2
q = σa – σr
0.1
0.0
Initial Stress State
Y1 Envelope
Stress Path to Initial State
0.2
Y2 Envelope 0.3
0.4
p = (σa + 2σr )/3
-0.2
11.18
LINEAR ELASTIC STIFFNESS
Knowledge of soil stiffness in the linear elastic region is important for evaluating soil response under dynamic loadings such as earthquakes, mechanical vibration, and vehicle vibration. It also provides indirect information regarding the state and natural soil structure, and, therefore, stiffness values can be used to assess the quality of soil samples (i.e., the degree of soil disturbance). The linear elastic stiffness of soils is evaluated from measurements of elastic wave velocities or use of local displacement transducers. Theoretical analysis of elastic waves in a particulate assembly is outside the scope of this book, but details can be found in Richart et al. (1970) and Santamarina et al. (2001), among others. The small strain shear modulus (Gmax) depends on the applied confining pressure and packing conditions of soil particles. The following empirical equation (Hardin and Black, 1966) is often used for isotropic stress conditions24: Gmax ⫽ AF(e)pn
0.1
-0.1
envelope as illustrated in Figs. 11.112c and 11.112d. If a stress state is probed in a certain direction within zone 2, the Y1 envelope is dragged with the stress state. When the stress path is reversed inside the Y1 envelope, the soil behaves as truly elastic. Once the stress state reaches the other side of the Y1 envelope, the Y1 envelope is again dragged with the stress state. When the stress state is in zone 3, both Y1 and Y2 envelopes are dragged with the stress state. The movement of these surfaces is therefore kinematic. The stiffness and its degradation are controlled by the new stress path direction in relation to the previous stress path direction (Atkinson et al., 1990). If the soil is allowed to age at a fixed effective stress point, the Y1 and Y2 envelopes may grow in size.
Co py rig hte dM ate ria l
3. Zone 3 (Preyield Plastic Region) As the stress state approaches the yield envelope (Y3 envelope), the ratio of plastic to total strain increases, approaching values close to 1.0 at the yield envelope. This state is illustrated as state B in Figs. 11.112a and 11.112c. Soil particles slide relative to one another, with strong force chains breaking and reforming continuously to accommodate the changing stress conditions. 4. Zone 4 (Full Plastic Region) Once the stress state reaches the Y3 yield envelope, there is a distinct kink in the stress–strain relationship and plastic strains develop fully. This state is illustrated as state C in Figs. 11.112a and 11.112d. The yield envelope expands or shrinks depending on the plastic increments; in general, the yield envelope expands if positive plastic volumetric strain (contraction) develops, whereas it shrinks if negative plastic volumetric strain (dilation) develops. The sizes of Y1 and Y2 surfaces may change with the enlargement or shrinkage of Y3 surface. If the stress state reaches the critical state, the soil is considered to have reached failure.
Y3 Envelope
-0.3
Figure 11.113 Y1, Y2, and Y3 envelopes for Ham River sand
(Jardine et al., 2001).
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447
(11.64)
where F(e) is a void ratio function, p is the mean effective stress, and A and n are material constants. An example of the fitting was shown in Fig. 11.11, and Table 11.8 summarizes some experimental data for different types of soils. Equation (11.64) is dimensionally inconsistent, except when n ⫽ 1. Various theoretical solutions such as the Hertz–Mindlin contact theory are available to re-
24 In practice, Gmax and pare often normalized by pa (reference pressure such as atmospheric pressure) so that the equation appears to be dimensionally consistent. However, there is no physical meaning to this.
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11
Table 11.8
STRENGTH AND DEFORMATION BEHAVIOR
Coefficients Used in Eq. (11.64)
Soil Type
A
F(e)
n
Void Ratio Range
Test Method a
Reference
Sand 6,900
(2.174 ⫺ e)2 1⫹e
0.5
0.3–0.8
RC
Hardin and Richart (1963)
Angular-grain crushed 3,270 quartz Several sands 9,000
(2.973 ⫺ e)2 1⫹e (2.17 ⫺ e)2 1⫹e
0.5
0.6–1.3
RC
0.4
0.6–0.9
RC
Hardin and Richart (1963) Iwasaki et al. (1978)
Toyoura sand
8,000
(2.17 ⫺ e)2 1⫹e
0.5
0.6–0.8
Cyclic TX
Kokusho (1980)
Several cohesionless and cohesive soils
4,500– 140,000
1 0.3 ⫹ 0.7e2
0.5
NA
RC
Hardin and Blandford (1989)
Ticino sand
7,100
(2.27 ⫺ e)2 1⫹e
0.43
0.6–0.9
RC and TS
Lo Presi et al. (1993)
Reconstituted NC kaoline
3,270
(2.973 ⫺ e)2 1⫹e
0.5
0.5–1.5
RC
Hardin and Black (1968)
Several undisturbed NC clays
3,270
(2.973 ⫺ e)2 1⫹e
0.5
0.5–1.7
RC
Hardin and Black (1968)
Reconstituted NC kaolin
4,500
(2.973 ⫺ e)2 1⫹e
0.5
1.1–1.3
RC
Marcuson and Wahls (1972)
Reconstituted NC bentonite
450
(4.4 ⫺ e)2 1⫹e
0.5
1.6–2.5
RC
Marcuson and Wahls (1972)
Several undisturbed silts and clays
893–1,726
(2.973 ⫺ e)2 1⫹e
0.46–0.61
0.4–1.1
RC
Kim and Novak (1981)
Undisturbed NC clay
90
(7.32 ⫺ e)2 1⫹e
0.6
1.7–3.8
Cyclic TX
Kokusho et al. (1982)
Undisturbed Italian clays
4,400–8,100
0.40–0.58
0.6–1.8
RC and BE
Jamiolkowski et al. (1995)b
Several soft clays
5,000
e⫺1.3(average from e⫺x: x ⫽ 1.11–1.43) e⫺1.5
Several soft clays
18,000– 30,000
Clays
Co py rig hte dM ate ria l
Round-grain Ottawa sand
1
0.5
1–5
SCPT
Shibuya and Tanaka (1996)c
0.5
1–6
SCPT
Shibuya et al. (1997)c
(1 ⫹ e)2.4
a
RC: resonant column test, TX: triaxial test, TS: torsional shear test, BE: bender element test, SCPT: seismic cone test. From anisotropic stress condition. c Using v instead of p. After Yimsiri (2001). b
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449
LINEAR ELASTIC STIFFNESS
Table 11.9
Some Analytical Solutions for Shear Modulus Under Isotropic Loading of pⴕ Coordination Number
Packing
Shear Modulus
冉冊 冉冊 冉冊 冉
1/3
6
Gmax 3 ⫽ Gg 2
Body-centered cubic
8
Gmax 1 ⫽9 Gg 6
冉冊 冉冊
(1 ⫺ g)1 / 3 p 2 ⫺ g Gg
1/3
(1 ⫺ g)1 / 3 p 6 ⫺ 5g Gg
1/3
1/3
冉冊
Co py rig hte dM ate ria l
Simple cubic
1/3
(4 ⫺ 3g) p 2/3 (2 ⫺ g)(1 ⫺ g) Gg
Face-centered cubic
12
Gmax 1 3 ⫽ Gg 2 2
Random packing
Cn
兹3cn Gmax 1 ⫽ Gg 5 兹2(1 ⫹ e)
冊
1/3
冉冊
(5 ⫺ 4g) p (2 ⫺ g)(1 ⫺ g)2 / 3 Gg
2/3
1/3
After Santamarina and Cascante (1996).
Cn ⫽ 13.28 ⫺ 8e
(11.65)
By varying compaction effort, sand samples can be prepared to different densities for a given applied confining stress. In this case, a smaller void ratio implies that the number of particle contacts is larger, and, therefore, the small strain stiffness increases. This effect is taken into account in the void ratio function
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F(e). Several expressions are available for the void ratio function as listed in Table 11.8. These functions are empirical and apply for specific ranges of void ratios and, therefore, should be used with caution. Equation (11.64) is derived assuming isotropic stress conditions. Anisotropic stress conditions as well as anisotropic soil fabric give stiffness values that depend on the direction of loading. The shear modulus is a function of the principal effective stresses in the directions of wave propagation and particle motion and is relatively independent of the out-of-plane principal stress. This is shown in Fig. 11.114, in which the variations of measured shear wave velocities propagating in three different directions (Vsxy, Vsyz, and Vszx) are shown as the vertical effective stress z was increased
σz Change in Vertical Effective Stress
S-wave Velocity, Vs (m/s)
late the global elastic stiffness to microscopic properties such as particle stiffness and Poisson’s ratio, number of contacts, void ratio, and contact force directions (see Table 11.9). These solutions suggest that the pressure p and Gmax could be normalized by the shear modulus of the particle itself (Gg). It is noted from Table 11.8 that the values of the exponent n range from 0.4 to 0.6. As shown in Table 11.9, however, classical contact mechanics solutions using the Hertz–Mindlin contact theory predict n ⫽ –13 . This is because the soil particles are assumed to be smooth elastic spheres. If the contacts are considered to be an interaction of rough surfaces, the modification of theory leads to increases in the exponent to values that are closer to the experimental observations given in Table 11.8 (Yimsiri and Soga, 2000). By comparing Eq. (11.64) with the micromechanical model listed at the bottom of Table 11.9, it is possible to relate the void ratio function F(e) to number of contacts per particle (i.e., coordination number) and A to the elastic properties of particle itself. From the analysis of uniform grain fabrics, the coordination number Cn can be related to the porosity n by Eq. (5.1) or to the void ratio e by the following equation. (Chang et al., 1991).
400
Direction of Wave Propagation Particle Motion
Vs-zx
Vs-xy
300
Vs-xy
Vs-yz
Vs-yz
Vs-zx
200 20
40
60
80
100
200
300
Vertical Effective Stress, σ z (kPa)
Figure 11.114 Variation of shear wave velocities in different
directions as a function of anisotropic stresses (Stokoe et al., 1995).
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11
STRENGTH AND DEFORMATION BEHAVIOR
with the horizontal effective stresses x and y being held constant (Stokoe et al., 1995). The shear wave Vsxy, which propagates and has the particle motion in the out-of-plane directions, shows no change in its velocity. This leads to the following empirical equation for stiffness under anisotropic stress conditions (Roesler, 1979; Yu and Richart, 1984; Stokoe et al., 1985, 1991; Hardin and Blandford, 1989): (11.66)
Direction of Wave Propagation Particle Motion Vp-zz Vp-xx Vp-xx
600
500
Vp-zz
Vp-yy Vp-yy
400
300 20
40 60 80 100 200 Vertical Effective Stress, σz (kPa)
Co py rig hte dM ate ria l
Gij(max) ⫽ AGFGi rGj sG OCRk
σz Change in Vertical Effective Stress P-wave Velocity, Vp (m/s)
450
where i is the effective normal stress in the direction of wave propagation, j is the effective normal stress in the direction of particle motion, and AG, rG, sG, and k are material constants. Experimental evidence suggests that rG ⬇ sG. Hence, an alternative equation that relates the stiffness to the mean state of stress on the plane of particle motion is also available:
冉
Gij(max) ⫽ AGFG
冊
i ⫹ j 2
nG
OCRk
(11.67)
Equations (11.66) and (11.67) include the effect of overconsolidation ratio (OCR). Hardin and Black (1968) found that k is a function of plasticity index (k increasing from 0 to 0.5 as PI increases from 0 to more than 100). Viggiani and Atkinson (1995) report k ⫽ 0.3 for reconstituted kaolin and k ⫽ 0.35 for reconstituted and undisturbed London clay. It can be argued that the void ratio function is a redundant factor since the void ratio is a unique function of present effective stress, stress history (OCR), and soil compressibility. However, this argument should be restricted to reconstituted clays and not applied to natural clays. Similar empirical equations are proposed for other elastic constants. P-wave velocity is a function only of the effective stress in the coaxial direction as shown in Fig. 11.115 (Stokoe et al., 1995). Hence, the small strain constrained modulus Mi(max) in the i direction can be expressed as Mi(max) ⫽ AMF(e)inM
(11.68)
where AM and nM are material constants. Similarly to the constrained modulus, the small strain Young’s modulus Ei(max) in the i direction (e.g., vertical or horizontal) is a function of the effective stress in the coaxial direction (i direction) only. The increase in Young’s modulus with stress in the coaxial direction is shown in Fig. 11.116a, whereas no change in the modulus with the increase in the stresses in orthogonal direction is shown in Fig. 11.116b (Hoque
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300
Figure 11.115 Variation of P-wave velocities in different directions as a function of anisotropic stresses (Stokoe et al., 1995).
and Tatsuoka, 1998). This leads to the following empirical equation for small strain Young’s modulus: Ei(max) ⫽ AEF(e)i nE
(11.69)
where AE and nE are material constants. Micromechanics analysis by Yimsiri and Soga (2000) supports this relation when the change in contact fabric anisotropy with applied stress is considered. Limited data are available with respect to Poisson’s ratio, and it is often assumed to be a constant value. The data by Hoque and Tatsuoka (1998) shown in Fig. 11.117 indicate that Poisson’s ratio vh (i.e., horizontal expansion by vertical load) increases with vertical effective stress and decreases with increase in horizontal stress. The following empirical equations are proposed by Horque and Tatsuoka (1998) for Poisson’s ratios: vh ⫽ AvhF(e)(v / h)nvh
(11.70)
hv ⫽ AhvF(e)(h / v)nhv
(11.71)
where Avh, Ahv, nvh, and nhv are material constants. Hoque and Tatsuoka (1998) report the values of nvh and nhv can be assumed to be half of nE given in Eq. (11.69). Small strain stiffness anisotropy originates from (i) anisotropic stress conditions and (ii) anisotropic soil fabric. The former is considered in Eqs. (11.66) to (11.71). For the latter, the material constant A should be directionally dependent reflecting a given anisotropic fabric. The effect of soil fabric on small strain stiffness of reconstituted London clay specimens is shown in Fig. 11.118 where the shear wave velocities in different directions are measured under the same confining pressures, and three different values of stiffness (Gvh, Ghv, and Ghh) are obtained. Results indicate
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LINEAR ELASTIC STIFFNESS
5000
Ev SLB Sand
Ev Toyoura Sand E Ehv Toyoura Toyoura sand Sand
4000
3000
2000
1000
Co py rig hte dM ate ria l
Ev/(F(e)Pa) or Eh/(F(e)Pa)
Eh SLB Sand
451
For Ev (or Eh) measurement, σv/Pa (or σh/Pa) is varied between 1.0 and 2.0 at each σv (or σh)
1.0
1.5 σv/Pa or σh/Pa
2.0
Figure 11.117 Poission’s ratio as a function of anisotropic
stresses (Hoque and Tatsuoka, 1998).
(a) Ev versus σv and Eh versus σh
Ev/(AEF(e)σv)
Eh SLB Sand
1.2
1.0
0.8
0.0
1.0
Eh Toyoura Sand
2.0 3.0 σ’h/P a
4.0
(b) E v versus σh
Figure 11.116 Vertical and horizontal Young’s modulus as
Figure 11.118 Stiffness anisotropy of undisturbed London
a function of anisotropic stresses for Toyoura sand (Hoque and Tatsuoka, 1998).
clay under isotropic stress conditions (Jovicic and Coop, 1998).
that, for a given confining pressure, the values of Ghh are larger than those of Gvh ⬇ Ghv. Hence, the soil is inherently stiffer horizontally than vertically due to its soil fabric. The reported data on clay under isotropic stress conditions consistently show that Ghh is approximately 50 percent larger than Gvh, indicating inherent anisotropic characteristics caused by orientation of platy clays (Pennington et al., 1997; Jovicic and Coop, 1998). The ratios of Ghh /Gvh for six Italian clays measured in onedimensional consolidation tests were between 1.3 and 2.0, and the ratio increased with overconsolidation ratio (Jamiolkowski et al., 1995).
For sands, most studies show that the ratio Ghh /Gvh is greater than 1 (e.g., Lo Presti and O’Neill, 1991; Stokoe et al., 1991; Bellotti et al., 1996). However, reported values for the ratio of Ev /Eh are inconclusive; some sands are stiffer in the vertical direction (Hoque and Tatsuoka, 1998), whereas the others are stiffer in the horizontal direction (Stokoe et al., 1991). Anisotropic properties are related to fabric (contact) anisotropy, and therefore the mixed results obtained may be due to the differences in sample preparation procedures. The experimental data show that the small strain stiffness is rather insensitive to the strain rate and number of loading cycles as long as the loading is within
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11
STRENGTH AND DEFORMATION BEHAVIOR
analysis. For instance, assume that the true elastic axial stiffness of a soil is 100 MPa. Considering that the elastic threshold axial strain is of the order of 10⫺5, the axial stress increment required to reach to this strain level is only 1 kPa. Hence, errors in stiffness of 100 percent result in small differences in the associated stress increments (a few kilopascals). Typical strain levels under working loads are usually in an intermediate level between linear elastic and plastic deformation, and, therefore, the knowledge of nonlinear (zone 2) and irreversible (zone 3) deformation characteristics is more important for evaluating ground movements accurately. Stiffness degradation from small strains to intermediate strains has been recognized in resonant column testing since the 1960s when the soil was subjected to cyclic loading (Hardin and Drnevich, 1972). Nowadays, detailed characterization of deformation properties at intermediate strain levels is possible with the use of local strain measurement systems, as described previously. The shear modulus decreases and the damping increases as the shear strain increases because of structural breakdown that results in a decreasing proportion of elastic deformation and an increasing proportion of plastic strain with increasing shear strain. The shear modulus degradation curves of Ticino sand, obtained by monotonic and cyclic loadings using various testing apparatus (triaxial compression, torsional shear, and resonant column) are shown in Fig. 11.119 (Tastuoka et al., 1997). The small strain stiffness is nearly independent of the test type, but at larger strains, the cyclic loading gives consistently larger shear modulus com-
Co py rig hte dM ate ria l
the true elastic range but that the elastic limit strain increases with strain rate (Shibuya et al., 1992; Tatsuoka et al., 1997). Resonant column tests on clays and sands show that the small strain shear modulus is independent of frequency in the range of 0.05 to 2500 Hz (e.g., Hardin and Richart, 1963; Hardin and Drnevich, 1972; Stokoe et al., 1995). Although conservation of energy may be an issue for true elastic response, experimental evidence indicates that energy is dissipated even at this strain level and damping values are typically 0.35 to 1 percent for sands and 1.0 to 1.5 percent for clays. Similar to the small strain stiffness, the damping at very small strain also depends on confining pressure and the following empirical form is proposed (Hardin, 1965): (11.72)
where B and m are material constants. The reported values of the exponent m range from ⫺0.05 to ⫺0.22 (Santamarina and Cascante, 1996; Stokoe et al., 1999). Although the particles in contact are not moving relative to each other, some microscopic proportion of the contact area can slide or slip, which is known as the stick–slip frictional contact loss. Micromechanical analysis considering the energy dissipation by this behavior gives m ⫽ ⫺–23 . Santamarina and Cascante (1996) attribute the difference to other attenuation mechanisms available in soils. These include chemical interaction of adsorbed layers at contacts, wave scattering, thermal relaxation, and other forms of energy coupling (e.g., mechanoelectromagnetic, mechanoacoustic). The damping is also affected by loading frequency, which is further described in Chapter 12. It has been argued that the use of the empirical equations presented above may produce nonconservative ‘‘elastic’’ response in terms of energy conservation (i.e., it may generate energy during a closed stress loop) (Zytynski et al., 1978). To be thermomechanically consistent, theoretical models for the pressuredependent stiffness of soils are available (e.g., Houlsby, 1985; Hueckel et al., 1992; Borja et al., 1997; Einav and Puzrin, 2004). They show that, if both shear and bulk moduli are to be mean pressure dependent, the stiffness needs to be anisotropic and stress induced. This is important in deformation analysis since the anisotropic stiffness in turn leads to cross dependence between shear behavior and volumetric behavior (Graham and Houlsby, 1983).
Monotonic Triaxial Monotonic Torsional Shear Cyclic Triaxial Cyclic Torsional shear Resonant Column
100
Secant Shear Modulus G G
⫽ Bpm
80
60
40
20
Ticino Sand σ’0 = 49 kPa e = 0.640
0
10-4
11.19 TRANSITION FROM ELASTIC TO PLASTIC STATES
In some cases, accurate evaluation of stiffness values at very small strains may not be crucial in geotechnical
Copyright © 2005 John Wiley & Sons
10-3
10-2
10-1
100
Shear Strain γ (%) Figure 11.119 Stiffness degradation of Ticino sand obtained by monotonic and cyclic loadings using various testing apparatus (Tatsuoka et al., 1997).
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TRANSITION FROM ELASTIC TO PLASTIC STATES
cause the specimens had different stress path histories prior to shearing (AO, BO, CO, and DO) [termed recent stress history by Atkinson et al. (1990)], and stiffer response was obtained when the stress path was reversed (D → O → X). The use of the multisurface concept described in Section 11.17 conveniently explains this complex deformation behavior. Since the small strain elastic stiffness is also influenced by the same factors, the stiffness degradation curves are sometimes normalized by the small strain stiffness; G/Gmax versus log or E/Emax versus log a. A summary of normalized shear modulus degradation curves for a variety of soils are shown in Fig. 11.121 (Kokusho, 1987). The curve for modulus degradation with increasing strain may be somewhat flatter for gravels than that for sands and clays. The curves tend to move to the right as the confining pressure increases; it is possible that the degradation curve at very high confining pressure (in the megapascal range) may lie beyond the bands given in Fig. 11.121 (Laird and Stokoe, 1993).
Co py rig hte dM ate ria l
pared to the monotonic loading at a given strain level. This is because the soil densifies during cyclic loading and the number of loading cycles has an effect on stiffness. As noted earlier, the shear strain level that gives an onset of permanent volumetric strain in drained conditions or residual excess pore pressures in undrained conditions after unloading is called the volumetric threshold strain. The stiffness degradation curve is influenced by many factors such as stress state, stress path, soil type, and soil fabric (i.e., anisotropy). For example, Fig. 11.10 shows the stiffness degradation of sands and clays subjected to increase in shear stress at different confining pressures. The effect of stress path directions on the stiffness degradation curve is shown in Fig. 11.120 (Atkinson et al., 1990). Triaxial tests were performed on reconstituted overconsolidated London clay specimens in such a way as to maintain a constant mean pressure. Different stiffness degradation curves were obtained even though they were sheared along the same stress path (OX in Fig. 11.120a). This is be-
453
Deviator stress q (kPa)
Sands and Gravels
The following relationship can be used for the dynamic shear modulus of sands and gravels at different strain levels (Seed et al., 1984):
X
100
D
0
C
O
100
200
400 Mean Pressure p (kPa)
B
-100
冉冊
G p ⫽ 22.1K2 pa pa
A
(11.73)
where p is the mean effective principle stress, pa is the atmospheric pressure, and K2 is a coefficient that depends primarily on grain size, relative density, and shear strain. The coefficient K2 is generally greater by a factor from about 1.35 to 2.5 for gravels than for sands. Values of K2 vary with relative density and shear
(a)
40
1/2
(A➝)O➝X 20
(C➝)O➝X 10
(B➝)O➝X 0 10-2
10-1 100 Deviator strain γ (%)
101
Shear Modulus Ratio G/Gmax
Shear Modulus G (MPa)
(D➝)O➝X
30
1.0
Clay, 100 kPa
0.5
Sand, 50 kPa
Gravel, 50 ~ 830 kPa
0.0 10-4
(b)
Figure 11.120 Recent stress history effect on stiffness deg-
radation: (a) stress paths and (b) stiffness degradation on OX stress path (from Atkinson et al., 1990).
Copyright © 2005 John Wiley & Sons
10-3
10-2
10-1
100
Shear Strain γ (%)
Figure 11.121 Normalized stiffness degradation curves of various types of soils (Kokusho, 1987).
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11
STRENGTH AND DEFORMATION BEHAVIOR
Clays
Dobry (1991) based on the results of a review of available cyclic load data from 16 different studies. The influences of various compositional and environmental factors on shear modulus and damping ratio of normally consolidated and moderately overconsolidated clays are listed in Table 11.10. Vucetic and Dobry (1991) hypothesized that increasing plasticity influences the degradation curves in the following manner. Increasing plasticity index reflects decreasing particle size and increasing specific surface area. The number of interparticle contacts becomes large, and interparticle electrical and chemical bonding and repulsive forces become large relative to the particle weights in comparison with sands. The many bonds within the microstructure act as a system of relatively flexible linear springs that can resist larger shear strains (up to 0.1 percent before they are broken) than is the case for sands, wherein particle elasticity is practically the only source of linear behavior, and interparticle sliding at contacts may start at strains as low as percent with the onset of plastic deformations. To these ideas might be added the fact that the thin, platy morphology of most clay particles make them able to deform elastically to considerably greater levels
Co py rig hte dM ate ria l
strain approximately as shown in Fig. 11.122 and with void ratio and shear strain as shown in Fig. 11.123. Equation (11.73) assumes that the exponent is –12 . Experimental evidence suggests that the exponent increases with strain level as shown in Fig. 11.124 and reaches 0.8 to 0.9 at a strain level of 1 percent (Jovicic and Coop, 1998; Yamashita et al., 2000). Values of the damping ratio for sands and gravels are about the same, and they are only slightly influenced by grain size and density. The ranges of values as a function of cyclic shear strain are shown in Fig. 11.125. The damping value decreases with increasing number of loading cycles and confining pressure, and much of the decrease occurs in the first 10 cycles (Stokoe et al., 1999).
Although the variation of shear moduli and damping ratio with shear strain is relatively independent of composition for sands and gravels, the same is not the case for cohesive soils. Curves of the type shown in Figs. 11.121 and 11.125 are displaced to the right for clays with increasing plasticity, as shown by Fig. 11.126. These relationships were developed by Vucetic and
Figure 11.122 Shear modulus factor K2 for sands as a function of relative density and shear
strain (Seed et al., 1984).
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455
Co py rig hte dM ate ria l
TRANSITION FROM ELASTIC TO PLASTIC STATES
Figure 11.123 Shear modulus factor K2 for sands as a function of void ratio and shear strain (Seed et al., 1984).
Figure 11.124 Variation of the shear modulus n exponent
value with strains on Dogs Bay sand (Jovicic and Coop, 1997).
Copyright © 2005 John Wiley & Sons
Figure 11.125 Damping ratios for sands and gravels (Seed
et al., 1984).
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11
STRENGTH AND DEFORMATION BEHAVIOR
Figure 11.126 Normalized modulus and damping ratio as a function of cyclic shear strain showing the influence of soil composition as measured by plasticity index (from Vucetic and Dobry, 1991). Reprinted with permission of ASCE.
of strain than is possible for the bulky, stiff granular particles. Furthermore, the several orders-of-magnitude smaller size and greater number of interparticle contacts per unit volume for the cohesive materials mean that even minute elastic distortions at interparticle contacts can give a cumulative strain that is large. For a cohesionless soil to develop such large shear deformations would require much greater displacements at intergrain contacts than could be accommodated without sliding.
11.20
fully plastic state is obtained when the stress state reaches the yield envelope as discussed in Section 11.17. As long as the stress state during and after geotechnical construction is within the yield envelope, the strain generated is elastic dominated. Hence, in order to control ground deformation in overconsolidated clays, it is useful to keep the construction-induced stress paths within the yield envelope. Once the stress state reaches the yield envelope, the generated strain will be plastic dominated. Generation of plastic strains is often unavoidable in normally and lightly overconsolidated clays because the initial stress state is either already on or near the yield envelope. The most important mechanical feature of soil in the plastic state is dilatancy, in which there is coupling between shear and volumetric deformations. That is, dense sands and heavily overconsolidated clays exhibit volume dilation in drained conditions and negative excess pore pressure generation in undrained conditions, whereas loose sands and normally consolidated and lightly overconsolidated clays exhibit volume contraction in drained conditions and positive excess pore pressure generation in undrained conditions. The rule that governs the generation of plastic volumetric strain associated with plastic deviator strain is called the dilatancy (or flow) rule. Some examples of this for dense sands were already presented in Eqs. (11.34) and (11.35), in which the degree of dilatancy [dy/dx in Eq. (11.34) and in Eq. (11.35)] is related to the applied principal stress ratio (or the mobilized friction angle) and the internal friction angle. These observations are important because the incorporation of stress–dilatancy into plasticity theory can lead to a useful form of constitutive modeling for soils. The development of plastic strains is often characterized by the following three aspects of soil behavior: (a) yield envelope, (b) dilatancy rule, and (c) hardening rule, which relates the change in the size of yield envelope to plastic strain increments. By assigning mathematical functions to these three aspects of soil behavior, a plastic constitutive model can be developed. Detailed review and development of all recent plasticity theories and proposed constitutive soil models are beyond the scope of this book. However, some essential aspects of soil behavior observed during plastic deformation are summarized here.
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456
PLASTIC DEFORMATION
Irrecoverable plastic strain initiates at a shear strain level of approximately 10⫺2 percent, and the amount of plastic strain increases with further deformation. A
Copyright © 2005 John Wiley & Sons
Yield Envelope and Hardening
The yield envelope defines the stress state when there is full development of plastic strains. Typical yield envelopes measured for a natural clay consolidated at different confining pressures are shown in Fig. 11.127.
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457
PLASTIC DEFORMATION
Table 11.10 Effect of Various Compositional and Environmental Factors on Maximum Shear Modulus Gmax, Modulus Ratio G /Gmax, and Damping Ratio of Normally Consolidated and Moderately Overconsolidated Clays Increasing Factor (1)
Increases with 0 Decreases with e Increases with tg Increases with c Increases with OCR
(4)
G/ Gmax (3) Stays constant or increases with 0 Increases with e May increase with tg May increase with c Not affected
Stays constant or decreases with 0 Decreases with e Decreases with tg May decrease with c Not affected
Increases with PI
Decreases with PI
Decreases with c G increases with ˙ ; G/ Gmax probably not affected if G and Gmax are measured at same ˙ Decreases after N cycles of large c (Gmax measured before N cycles)
Increases with c Stays constant or may increase with ˙
Co py rig hte dM ate ria l
Confining pressure, 0 (or vc) Void ratio, e Geologic age, tg Cementation, c Overconsolidation, OCR Plasticity index, PI
Gmax (2)
Cyclic strain, c Strain rate, ˙ (frequency of cyclic loading)
Number of loading cycles, N
Increases with PI if OCR ⬎ 1; stays about constant if OCR ⫽ 1 — Increases with ˙
Decreases after N cycles of large c but recovers later with time
Not significant for moderate c and N
From Dobry and Vucetic (1987).
Some observations can be made from this figure as follows: 1. The yield envelope is a function of stress and its size is controlled by stress history variables such as preconsolidation pressure. This is often expressed mathematically as F(, pc, ) ⫽ 0
Figure 11.127 Yield surfaces of Winnipeg clay at different
confining pressures (Graham et al., 1983b).
Copyright © 2005 John Wiley & Sons
(11.74)
where is the effective stresses, pc is the preconsolidation pressure, and is the rotation angle of the yield envelope with respect to the mean pressure axis. The yield envelopes of intact samples are larger than those of remolded (or destructured) samples; geological aging processes and cementation produce large yield envelopes for intact clays as shown in Fig. 11.128. When the cementation bonding breaks down and the soil becomes destructured, the yield envelope can become smaller.
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STRENGTH AND DEFORMATION BEHAVIOR
0.6
tions [i.e., (a) triaxial compression, (b) isotropic, and (c) triaxial extension]. A mathematical form that describes the change in with generation of plastic strains is called the rotational hardening rule.
Intact State
0.4
Magnitude of Plastic Strains and Stress–Dilatancy 0.2
0 0
0.2
0.4 0.6 (σa+ σr)/2σp
0.8
1.0
0.8
1.0
(a)
0.6
Intact State
(σa – σr)/2σp
Destructured State
0.4
0.2
0 0
Once the stress state is on the yield envelope, the soil is in the fully plastic state. The arrows in Fig. 11.129 show the vector magnitude of plastic strains measured for a given stress increment. The vertical component of the arrows is the deviator plastic strain increment d ps (or d p), whereas the horizontal component is the volumetric plastic strain increment dpv.25 Similarly, the plastic strain vectors measured in Winnipeg clay are shown in Fig. 11.130 (Graham et al., 1983b). The vector of the plastic strain increment appears to be a function of the current stress state. This observation leads to the concept of stress–dilatancy. Dilatancy during plastic deformation can be expressed as the ratio of plastic volumetric strain increment dpv to plastic deviatoric increment dsp; D ⫽ dpv /dsp. For clays, the value of D can be expressed as a function of stress ratio and material constants. For instance, the following stress–dilatancy equation can be proposed based on Taylor’s equation (11.34)26:
Co py rig hte dM ate ria l
(σa – σr)/2σp
Destructured State
0.2
0.4
0.6
(σa+σr)/2σp
dpv q ⫽M⫺ ⫹ 0 p ds p
(b)
Figure 11.128 Yield surfaces of intact and destructured soft
clays: (a) Saint Alban clay and (b) Ba¨ckebol clay (Leroueil and Vaughan, 1990).
2. The yield envelope increases in size with increasing preconsolidation pressure pc, which is often associated with the generation of plastic volumetric strain. The size increases as the soil is more densely packed along the normal consolidation line. A mathematical form that describes the change in pc with generation of plastic strains is called the hardening rule. 3. The shape of the yield envelope is often an inclined ellipse in the p –q plane. The inclination is related to the anisotropic consolidation history as well as the anisotropic fabrics. Some yield envelopes of sands are shown in Fig. 11.129 (Yasufuku et al., 1991). The yield envelopes were determined by applying different stress paths and connecting the stress state when the plastic strains initiate for a given stress path. The shape of the yield envelopes resembles a tear drop, and the inclinations of the yield envelopes are clearly affected by the initial anisotropic stress condi-
Copyright © 2005 John Wiley & Sons
(11.75)
where 0 is the initial anisotropy (e.g., Sekiguchi and Ohta, 1977). When 0 ⫽ 0, the equation becomes the stress–dilatancy rule used in the Cam-clay model (Roscoe and Schofield, 1963). Soil exhibits contractive behavior when the dilation angle is negative and q/p is less than M ⫹ 0, whereas the soil exhibits dilative behavior when the dilation angle is positive and q/p is more than M ⫹ 0. Figure 11.131 shows the stress– dilatancy relationship for the data presented in Fig. 11.130. The data follow a similar trend to Eq. (11.75). Other stress–dilatancy rules that are used to derive constitutive models for clays are available. Experimental evidence suggests that the stress– dilatancy relationship for sand depends on confining pressure and density as well as soil fabric, compared to a simpler form used in clays such as Eq. (11.75). Rowe (1962) derived the following stress–dilatancy
In triaxial condition, dpv ⫽ dpa ⫹ 2drp, dsp ⫽ (–23 )(dpa ⫺ dpr ), and d p ⫽ dpa ⫺ dpr , where dpa is the axial plastic strain and dpr is the radial plastic strain. 26 Note that Taylor’s expression was for the peak state only. This equation is applied to all stress state conditions under plastic deformation for both loose and dense cases. 25
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459
PLASTIC DEFORMATION
q (MPa)
q (MPa)
q (MPa)
0.8
0.8
0.8
0.6
0.6
0.4
0.4
0.6 Initial State
0.4
0.2
0.2 0.0
0.2
0.4 0.6
0.8
0.0
1.0 p’
0.2
0.2
(MPa)
0.4 0.6
0.8
(MPa)
0.0
1.0 p’
-0.2
0.2
0.4 0.6
0.8
1.0 p’
-0.2
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-0.2 -0.4
Initial State
(MPa)
Initial State
-0.4
-0.4
(a)
(b)
(c)
Figure 11.129 Yield surfaces of sands with different initial stress histories. Initial states (a)
compression, (b) isotropic, and (c) extension (Yasufuku et al., 1991).
dε p
Cam-clay (Roscoe and Schofield, 1963) dε pv/dε ps = M - q/p
dεsp
0.6
q/σp
0.4
0.2
Stress Ratio q/p'
dε vp
1.4
1.2 Modified Cam-clay (Roscoe and Burland, 1968) dε pv/dε ps = [M2 - (q/p)2]/ 2(q/p) 1 0.8
Modified Cam-clay
Data from Graham et al. (1983). See Fig 11.130
0.6
0.4 0.2
Cam-clay
0 0
0.2
0.4
0.6
0.8
1.0
-2.5
-2
-1.5
-0.5
0 0
0.5
1
1.5
2
2.5
p
Plastic Strain Ratio (-dε pv/dε s)
p/σp
Figure 11.130 Plastic strain vectors at yielding of natural
-1
Figure 11.131 Stress dilatancy relations of natural Winnipeg
Winnipeg clay (Graham et al., 1983b).
clay (Wood, 1991).
rule for sand in triaxial loading based on his experimental data as well as theoretical analysis:
respectively. Equations (11.76) and (11.77) have a similar form to Eq. (11.75), in which the dilation depends on stress ratio and material constants.27 However, Rowe (1962) noted that the material constant c used in Eqs. (11.76) and (11.77) is influenced by the density. Different initial anisotropic stress states give different
冉 冊 冉
冊
a ⫺2dpr c ⫽ tan2 ⫹ p r da 4 2
in triaxial compression
冉 冊 冉
r ⫺dap c ⫽ tan2 ⫹ a 2drp 4 2
in triaxial extension
(11.76)
冊
Equations (11.76) and (11.77) can be rewritten in terms of p, q, d pv, and d p (Pradhan and Tatsuoka, 1989): 27
(11.77)
where dpa and dpr are the axial and radial strain increments, c is the ‘‘characteristic friction angle’’ and a and r are the axial and radial effective stresses,
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冋 冋
册 册
q 3 (2K ⫹ 1)(⫺d pv / d p) ⫹ 2(K ⫺ 1) ⫽ p 2 (K ⫺ 1)(⫺d vp / d p) ⫹ (K ⫹ 2) q 3 (K ⫹ 2)(⫺d pv / d p) ⫺ 2(K ⫺ 1) ⫽ p 2 (1 ⫺ K)(⫺d vp / d p) ⫹ (2K ⫹ 1) where K ⫽ (1 ⫹ sin c) / (1 ⫺ sin c).
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for d pa ⬎ 0 for d pa ⬍ 0
460
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STRENGTH AND DEFORMATION BEHAVIOR
dev dp G ⫽3 e ds dq K
(11.78)
where G is the shear modulus, K is the bulk modulus and dev and dse are the elastic volumetric and deviatoric strains, respectively. The physical mechanisms of elastic deformation and plastic deformation are fundamentally different, that is, stress increment dependent versus stress dependent. Because of this, the same stress increment may give very different strain increments. Careful selection of elastic and/or plastic models is therefore necessary in ground deformation analysis.
Stress Ratio q/p
Case (a)
2.0
Case (b) Case (c)
1.5
in Fig.11.129
1.0
Compression
0.5
-4
-3
11.21
TEMPERATURE EFFECTS
The average ground temperature varies between 7 and 10, whereas laboratory conditions are between 18 and 23. In some situations, the soil can undergo large temperature change, for example, ground freezing, heating of nuclear waste repositories, underground storage reservoirs, and the like. It can be important to recognize the significance of temperature when evaluating strength and model parameters. In general, increase in temperature will result in thermal expansion of soil grains as well as pore fluid. The particle contact properties will also be modified. A change in temperature, therefore, causes either a change in void ratio or a change in effective stress (or a combination of both) in a saturated clay, as described in Section 10.12. In this section some effects of temperature on shear resistance of soils are considered. A change in temperature can cause a strength increase or a strength decrease depending on the circumstances (e.g., temperature variation during initial consolidation or during shearing in drained or undrained conditions), as illustrated by Fig. 11.133. The higher the consolidation temperature, the greater the shear strength at any given test temperature because of the greater decrease in void ratio at the higher consolidation temperatures.28 In Fig. 11.133, Tc represents the temperature at consolidation and Ts the temperature of shear for consolidated undrained direct shear tests on highly plastic alluvial clay. For a given consolidation temperature Tc, the undrained strength decreases in a regular manner with the increasing test temperature. From tests such as these, it has been established that for given initial conditions the undrained strength of normally consolidated saturated clay may decrease by about 10 percent for a temperature increase from 0 to 40C. For overconsolidated clays, the undrained shear strength is less influenced by temperature (Marques et al., 2004). The relative insensitivity of overconsolidated clay to temperature may be due to the compensating effects of increase stiffness and softening of soil structure by volume expansion as described in Section 10.12. Similar to the strain rate effect, the preconsolidation pressure, and hence the size of the yield envelope, decreases with increase in temperature, as illustrated in Fig. 10.46 and Fig. 11.134 for natural clay specimens tested between 10 and 50. Hence, the weakening of
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stress–dilatancy curves as shown in Fig. 11.132. The curves were derived from the data presented in Fig. 11.129 and are presented in terms of stress ratio q/p and plastic strain increment ratio d pv /d p. Hence, the stress–dilatancy relationship of a sand depends not only on stress ratio but also on density, confining pressure and initial anisotropic stress conditions. As noted in Eqs. (11.75) to (11.77) and Figs. 11.131 and 11.132, the development of plastic increments is governed by the current stress state. This is in contrast to elastic deformation, which is related directly to stress increments. For example, for an isotropic elastic model,
-2
-1
0
1
?
-0.5
2
3 4 Strain Increment ratio -dεvp/dγ p
?
-1.0
Extension
Figure 11.132 Stress dilatancy relations of sands with dif-
ferent initial anisotropic stress conditions (Yasufuku et al., 1991).
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For all tests, Ts Tc to prevent further consolidation under a higher temperature, which would result in the strength being about the same as if it had been consolidated under the higher temperature initially.
28
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TEMPERATURE EFFECTS
461
Figure 11.133 Effect of consolidation and test temperatures on the strength of alluvial clay
in direct shear (Noble and Demirel, 1969).
Figure 11.134 Influence of temperature on yield surface of a St-Roch-de-l’Achigan clay, Quebec (Marques et al., 2004).
soil structure by increase in temperature is apparent. On the other hand, the critical state friction angle is found to be independent of temperature (Hueckel and Baldi, 1990; Graham et al., 2001; Marques et al., 2004). Drainage conditions during heating prior to shear are important, as illustrated in Fig. 11.133. If drainage is prevented, the expansion of water controls the expan-
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sion of soil volume because thermal expansion of water is much larger than that of soil particles. This results in generation of positive excess pore pressure and, as a consequence, undrained stiffness and shear strength decrease as shown in Fig. 11.16. If drainage is allowed, the expanding water is free to drain and hence the volume change of the soil is governed by the expansion of soil particles and the
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462
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STRENGTH AND DEFORMATION BEHAVIOR
11.22
a dense state (below critical state) can only be achieved by unloading, and, therefore, the preconsolidation pressure can be used to characterize the peak strength and deformation. For sands, on the other hand, the difference in strength and deformation behavior of normally consolidated dense sand and overconsolidated sand is noted even when they are at the same void ratio and confining pressure. This is because of possible different soil fabrics. The critical friction angle of cohesionless soils contains contributions from particle crushing, particle rearrangement by rolling, as well as from interparticle sliding. The critical state concept can be used to characterize the density effect on peak strength for normally consolidated sand. Rearrangement and rolling are unimportant when the clay content is high enough to prevent granular particle interference. Ideally, the critical state strength or friction angle should be used for design of simple geotechnical structures. Otherwise, a careful selection of safety factor is needed when the peak strength or peak friction angle is used. However, whether it is possible to find the true critical state from conventional triaxial and torsional shear tests is questionable, especially for sands. Because of the great diversity of soil types and the range of environmental conditions to which they may be subjected, evaluations of deformation and strength, their characterization for analyses, and prediction of future behavior will continue as major components of any project. In the majority of geotechnical engineering projects and problems, correct site characterization and property evaluation are the two most critical elements. If they are not done reasonably and reliably, then there cannot be understanding or confidence from subsequent soil mechanics analyses, no matter how sophisticated they may be or how powerful the computer that provides the numerical solutions.
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change in particle contact conditions. Normally consolidated clays often result in decrease in void ratio, and hence the initial stiffness generally increases with temperature. However, it has been reported that the decrease in void ratio in normally consolidated clays cannot be solely accounted for the increase in stiffness (Tsuchida et al., 1991; Kuntiwattanakul et al., 1995). This observation is similar to the aging effect discussed in Chapter 12. Hence, it can be considered that temperature is one of the driving forces in time-dependent deformation of soils, and the rate process theory described in the next chapter conveniently explains much of the observed temperature–time–effective stress behavior of soils.
CONCLUDING COMMENTS
Limit equilibrium and plasticity analyses, as done, for example, in studies of slope stability, lateral pressure, and bearing capacity, depend on accurate representation of soil strength. So also does soil resistance against failure due to earthquakes or other cyclic loadings. The stresses and deformations under subfailure loading conditions depend on stress–strain properties. The factors responsible for and influencing strength have been identified and analyzed. The strength of most uncemented soils is provided by interparticle sliding, dilatancy, particle rearrangements, particle crushing, and true cohesion. Frictional resistance is developed by adhesion between contacting asperities on opposing particle surfaces. Values of true friction angle () range from less than 4 for sodium montmorillonite to more than 30 for feldspar and calcite. In the absence of cementation, true cohesion in soils is small. Results from discrete particle simulations indicate that the deviatoric load applied to a particle assembly is transferred exclusively by the normal contact forces in the strong force networks. The interparticle friction therefore acts as a kinematic constraint of the strong force network and not as the direct source of macroscopic resistance to shear. The residual friction angle depends on gradation, mineralogical composition, and effective stress. The value of residual friction angle for clay may decrease by several degrees for increases in effective stress on the shear surface from 0 to 400 kPa (0 to 60 psi). The shear displacements in one direction required to develop residual strength may be several tens of millimeters. These factors should be taken into account when analyzing stability problems. Loose sands behave like normally consolidated clays. The behavior of dense sand appears to be similar to that of overconsolidated clays. However, for clays,
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QUESTIONS AND PROBLEMS
1. Based on the descriptions given in Section 11.3 and 11.6, summarize microscopic interpretation of overconsolidation, compaction, dilation, peak friction angle, and critical state friction angle. 2. A clay has liquid and plastic limits of 80 and 25, respectively. For the following conditions, find possible plastic failure mechanisms at different confining pressures using Eq. (11.30) and Fig. 11.46. Discuss any practical implications. a. The clay is consolidated to a water content of 65 percent. b. The clay is heavy compacted to a water content of 25 percent.
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QUESTIONS AND PROBLEMS
d. The clay at state (c) is sheared in undrained conditions to the critical state. Also, sketch a possible stress–strain relationship. e. Repeat parts (c) and (d) for other OCR conditions. Comment on the results. 6. The virgin compression curve of a clay was found to be e ⫽ 1.3 ⫺ 0.6 log v from one-dimensional consolidation tests. The swelling index Cs was 0.1. The clay was preconsolidated to v ⫽ 100 kPa prior to shearing. a. Using the Hvorslev parameters of hc ⫽ 0.1 and e ⫽ 15, plot the failure envelope on the – plane. b. Plot shear strength f / v as a function of OCR and compare the results to the data shown in Fig. 11.65.
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3. A quartz sand has minimum and maximum void ratios of 0.35 and 0.75, respectively. The critical state friction angle is 35. a. Using Eqs. (11.31) and (11.32), plot the critical state line on the p –q plane and on the e–log p plane. b. Find the undrained shear strengths at critical state when the void ratios are 0.4 and 0.7. Does the initial effective stress state matter to the computed values? How about the values of excess pore pressure generated during undrained tests? c. Draw the effective stress path of a drained triaxial compression test on the p –q plane. The initial effective isotropic confining pressure is 100 kPa. Find the drained strength and void ratio at critical state. d. Sketch possible stress–axial strain and axial strain–void ratio curves of the drained triaxial compression test considered in part (c). Consider two different initial void ratios: (i) e ⫽ 0.4 and (ii) e ⫽ 0.7. Comment on the results. e. Repeat the calculations of parts (c) and (d) when the initial confining pressure is 1 MPa. Comment on the results. 4. Using the critical state of the sand defined in Question 3, plot void ratio versus peak friction angle at three different confining pressures: (i) 5 kPa, (ii) 500 kPa, and (iii) 5 MPa. To develop the plot, try (i) Eq. (11.37) or (ii) Fig. 11.56. Comment on the results by discussing the relative importance of confining pressure and void ratio on friction angle of soils.
5. A clay was isotropically normally consolidated and the isotropic compression line was found to be e ⫽ 1.5 ⫺ 0.35 ln p. The clay was then unloaded isotropically and the slope of unloading line on a e–ln p diagram was found to be ! ⫽ 0.05. A series of undrained triaxial compression tests were performed on the clay, and the critical state was found to be q ⫽ 0.8p and ecs ⫽ 1.3 ⫺ 0.35 ln p. Plot the stress and state paths on the p –q plane and the e–ln p plane for the following conditions: a. The clay is isotropically consolidated to 400 kPa along the isotropic compression line. b. The clay at state (a) is sheared in undrained conditions to the critical state. Also, sketch a possible stress–strain relationship. c. The clay at state (a) is unloaded isotropically to 200 kPa (OCR ⫽ 2).
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463
7. Why does a sample with shear bands give different strengths depending on sample size? 8. Find a case study that describes the importance of knowing the residual friction angle of clay. Explain (a) the geologic and hydrogeologic conditions, (b) the possible peak, critical, and residual friction angles, and (c) microscopic interpretation of decrease in friction angle at residual state. 9. Consider two saturated samples of the same soil having exactly the same water content, density, temperature, and structure are initially at equilibrium under the same effective stress states. Compare and explain differences in strength, if any, that you would expect if a. One is loaded in triaxial compression and the other in plane strain. b. One is tested in triaxial compression and the other is tested in plane stress. c. One is tested as is and the other is tested after heating with (i) no drainage allowed and (ii) full drainage is allowed. d. One is tested in triaxial compression and the other is tested in triaxial extension.
10. An embankment is to be constructed on a soft clay, and a potential failure surface is shown in the figure below. The clay possesses anisotropic fabric. Considering the intermediate stress effect and anisotropy effects described in Section 11.12, consider possible stress paths from the stress before the construction and discuss what strength values should be used in design for the following locations in the clay: (i) location A, which is located underneath the embankment, (ii) location B, which is at some depth near the toe of the embankment,
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STRENGTH AND DEFORMATION BEHAVIOR
at a depth of 20 m. Consider both fracturing in (i) undrained conditions assuming that the injected fluid has not permeated into the ground and (ii) drained conditions assuming the injection is in a steady state seepage state. 14. Convert some of the stiffness degradation curves plotted in Figs. 11.10 and 11.119 to shear stress versus logarithm of strain. Identify the shear stresses required to reach the boundaries of different zones described in Section 11.17. Discuss which zones are important for what type of geotechnical activities.
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and (iii) location C, which is located some distance away from the embankment.
11. Find a paper that describes the effects of soil fabric on liquefaction resistance of sands. Give the microscopic interpretation of why a sample with a certain soil fabric generates more excess pore pressures than others.
12. Provide physical explanations of how and why the following factors can affect the cyclic resistance ratio (CRR) of sands: a. Confining pressure b. Initial K0 stress condition c. Static shear stress along the sloping ground d. Shear modes (triaxial compression and extension, simple shear, etc.) e. Sample preparation and soil fabric f. Silt fines and clay fines
13. Water is injected into overconsolidated clay with an OCR of approximately 3. Using the correlations and data presented throughout the book, estimate the injection pressure required to fracture the clay
Copyright © 2005 John Wiley & Sons
15. Give physical microscopic explanations of different stiffness degradation curves presented in Fig. 11.120. Why can the multisurface concept presented in Section 11.17 be used to model this complex behavior? 16. Discuss the differences between elastic and plastic deformations of soils as microscopic behavior and macroscopic behavior. 17. The data showing volume reduction with increasing temperature at a given pressure were presented in Fig. 10.44 (Campanella and Mitchell, 1968). If we consider the normal compression curve at 76.5F to be the reference state, the compression curves at the other temperatures can be interpreted to have exhibited temperature-induced creep behavior and hence reached the quasioverconsolidated state. Can the data presented in Fig. 11.133 be explained in such a way using the Hvoslev strength concept for overconsolidated clays?
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CHAPTER 12
12.1
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Time Effects on Strength and Deformation
INTRODUCTION
Virtually every soil ‘‘lives’’ in that all of its properties undergo changes with time–some insignificant, but others very important. Time-dependent chemical, geomicrobiological, and mechanical processes may result in compositional and structural changes that lead to softening, stiffening, strength loss, strength gain, or altered conductivity properties. The need to predict what the properties and behavior will be months to hundreds or thousands of years from now based on what we know today is a major challenge in geoengineering. Some time-dependent changes and their effects as they relate to soil formation, composition, weathering, postdepositional changes in sediments, the evolution of soil structure, and the like are considered in earlier chapters of this book. Emphasis in this chapter is on how time under stress changes the structural, deformation, and strength properties of soils, what can be learned from knowledge of these changes, and their quantification for predictive purposes. When soil is subjected to a constant load, it deforms over time, and this is usually called creep. The inverse phenomenon, usually termed stress relaxation, is a drop in stress over time after a soil is subjected to a particular constant strain level. Creep and relaxation are two consequences of the same phenomenon, that is, time-dependent changes in structure. The rate and magnitude of these time-dependent deformations are determined by these changes.
Time-dependent deformations and stress relaxation are important in geotechnical problems wherein longterm behavior is of interest. These include long-term settlement of structures on compressible ground, deformations of earth structures, movements of natural and excavated slopes, squeezing of soft ground around tunnels, and time- and stress-dependent changes in soil properties. The time-dependent deformation response of a soil may assume a variety of forms owing to the complex interplays among soil structure, stress history, drainage conditions, and changes in temperature, pressure, and biochemical environment with time. Timedependent deformations and stress relaxations usually follow logical and often predictable patterns, at least for simple stress and deformation states such as uniaxial and triaxial compression, and they are described in this chapter. Incorporation of the observed behavior into simple constitutive models for analytical description of time-dependent deformations and stress changes is also considered. Time-dependent deformation and stress phenomena in soils are important not only because of the immediate direct application of the results to analyses of practical problems, but also because the results can be used to obtain fundamental information about soil structure, interparticle bonding, and the mechanisms controlling the strength and deformation behavior. Both microscale and macroscale phenomena are discussed because understanding of microscale processes can provide a rational basis for prediction of macroscale responses. 465
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466 12.2
12
TIME EFFECTS ON STRENGTH AND DEFORMATION
GENERAL CHARACTERISTICS
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1. As noted in the previous section soils exhibit both creep1 and stress relaxation (Fig. 12.1). Creep is the development of time-dependent shear and/or volumetric strains that proceed at a rate controlled by the viscouslike resistance of soil structure. Stress relaxation is a timedependent decrease in stress at constant deformation. The relationship between creep strain and the logarithm of time may be linear, concave upward, or concave downward as shown by the examples in Fig. 12.2. 2. The magnitude of these effects increases with increasing plasticity, activity, and water content
3.
4.
5.
Figure 12.1 Creep and stress relaxation: (a) Creep under
constant stress and (b) stress relaxation under constant strain.
1
of the soil. The most active clays usually exhibit the greatest time-dependent responses (i.e., smectite ⬎ illite ⬎ kaolinite). This is because the smaller the particle size, the greater is the specific surface, and the greater the water adsorption. Thus, under a given consolidation stress or deviatoric stress, the more active and plastic clays (smectites) will be at higher water content and lower density than the inactive clays (kaolinites). Normally consolidated soils exhibit larger magnitude of creep than overconsolidated soils. However, the basic form of behavior is essentially the same for all soils, that is, undisturbed and remolded clay, wet clay, dry clay, normally and overconsolidated soil, and wet and dry sand. An increase in deviatoric stress level results in an increased rate of creep as shown in Fig. 12.1. Some soils may fail under a sustained creep stress significantly less (as little as 50 percent) than the peak stress measured in a shear test, wherein a sample is loaded to failure in a few minutes or hours. This is termed creep rupture, and an early illustration of its importance was the development of slope failures in the Cucaracha clay shale, which began some years after the excavation of the Panama Canal (Casagrande and Wilson, 1951). The creep response shown by the upper curve in Fig. 12.1 is often divided into three stages. Following application of a stress, there is first a period of transient creep during which the strain rate decreases with time, followed by creep at nearly a constant rate for some period. For materials susceptible to creep rupture, the creep rate then accelerates leading to failure. These three stages are termed primary, secondary, and tertiary creep. An example of strain rates as a function of stress for undrained creep of remolded illite is shown in Fig. 12.3. At low deviator stress, creep rates are very small and of little practical importance. The curve shapes for deviator stresses up to about 1.0 kg/cm2 are compatible with the predictions of rate process theory, discussed in Section 12.4. At deviator stress approaching the strength of the material, the strain rates become very large and signal the onset of failure. A characteristic relationship between strain rate and time exists for most soils, as shown, for example, in Fig. 12.4 for drained triaxial compression creep of London clay (Bishop, 1966)
The term creep is used herein to refer to time-dependent shear strains and / or volumetric strains that develop at a rate controlled by the viscous resistance of the soil structure. Secondary compression refers to the special case of volumetric strain that follows primary consolidation. The rate of secondary compression is controlled by the viscous resistance of the soil structure, whereas, the rate of primary consolidation is controlled by hydrodynamic lag, that is, how fast water can escape from the soil.
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6.
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GENERAL CHARACTERISTICS
467
Figure 12.2 Sustained stress creep curves illustrating different forms of strain vs. logarithm
of time behavior.
and Fig. 12.5 for undrained triaxial compression creep of soft Osaka clay (Murayama and Shibata, 1958). At any stress level (shown as a percentage of the strength before creep in Fig. 12.4 and in kg/cm2 in Fig. 12.5), the logarithm of the strain rate decreases linearly with increase in the
Copyright © 2005 John Wiley & Sons
logarithm of time. The slope of this relationship is essentially independent of the creep stress; increases in stress level shift the line vertically upward. The slope of the log strain rate versus log time line for drained creep is approximately ⫺1. Undrained creep often results in a slope be-
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TIME EFFECTS ON STRENGTH AND DEFORMATION
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468
Figure 12.3 Variation of creep strain rate with deviator stress for undrained creep of re-
molded illite.
tween ⫺0.8 and ⫺1 for this relationship. The onset of failure under higher stresses is signaled by a reversal in slope, as shown by the topmost curve in Fig. 12.5. 7. Pore pressure may increase, decrease, or remain constant during creep, depending on the volume change tendencies of the soil structure and whether or not drainage occurs during the deformation process. In general, saturated soft sensitive clays under undrained conditions are most susceptible to strength loss during creep due to reduction in effective stress caused by increase in pore water pressure with time. Heavily overconsolidated clays under drained con-
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ditions are also susceptible to creep rupture due to softening associated with the increase in water content by dilation and swelling. 8. Although stress relaxation has been less studied than creep, it appears that equally regular patterns of deformation behavior are observed, for example, Larcerda and Houston (1973). 9. Deformation under sustained stress ordinarily produces an increase in stiffness under the action of subsequent stress increase, as shown schematically in Fig. 12.6. This reflects the time-dependent structural readjustment or ‘‘aging’’ that follows changes in stress state. It is analogous to the quasi-preconsolidation effect
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GENERAL CHARACTERISTICS
469
Figure 12.4 Strain rate vs. time relationships during drained creep of London clay (data
from Bishop, 1966).
due to secondary compression discussed in Section 8.11; however, it may develop under undrained as well as drained conditions. 10. As shown in Fig. 12.7, the locations of both the virgin compression line and the value of the preconsolidation pressure, p, determined in the laboratory are influenced by the rate of loading during one-dimensional consolidation (Graham et al., 1983a; Leroueil et al., 1985). Thus, estimations of the overconsolidation ratio of clay deposits in the field are dependent on the loading rates and paths used in laboratory tests for determination of the preconsolidation pressure. If it is assumed that the relationship between strain and logarithm of time during compression is linear over the time ranges of interest and that the secondary compression index Ce is constant regardless of load, the rate-dependent preconsolidation pressure p at ˙ 1 can be related to the axial strain rate as follows (Silvestri et al, 1986; Soga and Mitchell, 1996; Leroueil and Marques, 1996):
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冉 冊
p ˙ 1 ⫽ p(ref) ˙ 1(ref)
Ce / (Cc⫺Cr)
⫽
冉 冊 ˙ 1
˙ 1(ref)
(12.1)
where Cc is the virgin compression index, Cr is the recompression index and p(ref) is the preconsolidation pressure at a reference strain rate ˙ 1(ref). In this equation, the rate effect increases with the value of ⫽ Ce /(Cc ⫺ Cr). The variation of preconsolidation pressure with strain rate is shown in Fig. 12.8 (Soga and Mitchell, 1996). The data define straight lines, and the slope of the lines gives the parameter . In general, the value of ranges between 0.011 and 0.094. Leroueil and Marques (1996) report values between 0.029 and 0.059 for inorganic clays. 11. The undrained strength of saturated clay increases with increase in rate of strain, as shown in Figs. 12.9 and 12.10. The magnitude of the effect is about 10 percent for each order of magnitude increase in the strain rate. The strain rate
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TIME EFFECTS ON STRENGTH AND DEFORMATION
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470
Figure 12.5 Strain rate vs. time relationships during undrained creep of Osaka alluvial clay
(Murayama and Shibata, 1958).
effect is considerably smaller for sands. In a manner similar to Eq. (12.1), a rate parameter can be defined as the slope of a log–log plot of deviator stress at failure qƒ at a particular strain rate ˙ 1 relative to qƒ(ref), the strength at a reference strain rate ˙ 1(ref), versus strain rate. This gives the following equation: qƒ
qƒ(ref)
⫽
冉 冊 ˙ 1
˙ 1(ref)
(12.2)
The value of ranges between 0.018 and 0.087, similar to the rate parameter values used to define the rate effect on consolidation pressure in Eq. (12.1). Higher values of are associated with more metastable soil structures (Soga and Mitchell, 1996). Rate dependency decreases with increasing sample disturbance, which is consistent with this finding.
12.3 TIME-DEPENDENT DEFORMATION–STRUCTURE INTERACTION Figure 12.6 Effect of sustained loading on (a) stress–strain
and strength behavior and (b) one-dimensional compression behavior.
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In reality, completely smooth curves of the type shown in the preceding figures for strain and strain rate as a function of time may not exist at all. Rather, as dis-
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TIME-DEPENDENT DEFORMATION–STRUCTURE INTERACTION
Figure 12.7 Rate dependency on one-dimensional compression characteristics of Batiscan
clay: (a) compression curves and (b) preconsolidation pressure (Leroueil et al., 1985).
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471
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TIME EFFECTS ON STRENGTH AND DEFORMATION
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472
Figure 12.8 Strain rate dependence on preconsolidation pressure determined from onedimensional constant strain rate tests (Soga and Mitchell, 1996).
Figure 12.9 Effect of strain rate on undrained strength (Kulhawy and Mayne 1990). Re-
printed with permission from EPRI.
cussed by Ter-Stepanian (1992), a ‘‘jump-like structure reorganization’’ may occur, reflecting a stochastic character for the deformation, as shown in Fig. 12.11 for creep of an undisturbed diatomaceous, lacustrine, overconsolidated clay. Ter-Stepanian (1992) suggests that there are four levels of deformation: (1) the molecular level, which consists of displacement of flow units by
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surmounting energy barriers, (2) mutual displacement of particles as a result of bond failures, but without rearrangement, (3) the structural level of soil deformation involving mutual rearrangements of particles, and (4) deformation at the aggregate level. Behavior at levels 3 and 4 is discussed below; that at levels 1 and 2 is treated in more detail in Section 12.4.
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TIME-DEPENDENT DEFORMATION–STRUCTURE INTERACTION
473
Figure 12.10 Strain rate dependence on undrained shear strength determined using constant
strain rate CU tests (Soga and Mitchell, 1996).
Figure 12.11 Nonuniformity of creep in an undisturbed, diatomaceous, lacustrine, overconsolidated clay (from TerStepanian, 1992).
Time-Dependent Process of Particle Rearrangement
Creep can lead to rearrangement of particles into more stable configurations. Forces at interparticle contacts have both normal and tangential components, even if the macroscopic applied stress is isotropic. If, during the creep process, there is an increase in the proportion of applied deviator stress that is carried by interparticle normal forces relative to interparticle tangential forces, then the creep rate will decrease. Hence, the rate at which deformation level 3 occurs need not be uniform owing to the particulate nature of soils. Instead it will reflect a series of structural readjustments as particles move up, over, and around each other, thus leading to
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the somewhat irregular sequence of data points shown in Fig. 12.11. Microscopically, creep is likely to occur in the weak clusters discussed in Section 11.6 because the contacts in them are at limiting frictional equilibrium. Any small perturbation in applied load at the contacts or time-dependent loss in material strength can lead to sliding, breakage or yield at asperities. As particles slip, propped strong-force network columns are disturbed, and these buckle via particle rolling as discussed in Section 11.6. To examine the effects of particle rearrangement, Kuhn (1987) developed a discrete element model that considers sliding at interparticle contacts to be viscofrictional. The rate at which sliding of two particles relative to each other occurs depends on the ratio of shear to normal force at their contact. The relationship between rate and force is formulated in terms of rate process theory (see Section 12.4), and the mechanistic representations of the contact normal and shear forces are shown in Fig. 12.12. The time-dependent component in the tangential forces model is given as a ‘‘sinhdashpot’’.2 The average magnitudes of both normal and 2 Kuhn (1987) used the following equation for rate of sliding at a contact:
X˙ ⫽
冉 冊 冉
冊
2kT
F ƒt exp ⫺ sinh h RT 2kTn1 ƒn
where n1 is the number of bonds per unit of normal force, ƒt is the tangential force and ƒn is the normal force. The others are parameters related to rate process theory as described in Section 12.4.
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TIME EFFECTS ON STRENGTH AND DEFORMATION
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Figure 12.12 Normal and tangential interparticle force mod-
els according to Kuhn (1987).
tangential forces at individual contacts can change during deformation even though the applied boundary stresses are constant. Small changes in the tangential and normal force ratio at a contact can have a very large influence on the sliding rate at that contact. These changes, when summed over all contacts in the shear zone, result in a decrease or increase in the overall creep rate. A numerical analysis of an irregular packing of circular disks using the sinh-dashpot representation gives creep behavior comparable to that of many soils as shown in Fig. 12.13 (Kuhn and Mitchell, 1993). The creep rate slows if the average ratio of tangential to normal force decreases, whereas it accelerates and may ultimately lead to failure if the ratio increases. In some cases, the structural changes that are responsible for the decreasing strain rate and increased stiffness may cause the overall soil structure to become more metastable. Then, after the strain reaches some limiting value, the process of contact force transfer from decreasing tangential to increasing normal force reverses. This marks the onset of creep rupture as the structure begins to collapse. A similar result was obtained by Rothenburg (1992) who performed discrete particle simulations in which smooth elliptical particles were cemented with a model exhibiting viscous characteristics in both normal and tangential directions.
Copyright © 2005 John Wiley & Sons
Figure 12.13 Creep curves developed by numerical analysis
of an irregular packing of circular disks (from Kuhn and Mitchell, 1993).
Particle Breakage During Creep
Particle breakage can contribute to time-dependent deformation of sands (Leung et al., 1996; Takei et al., 2001; McDowell, 2003). Leung et al. (1996) performed one-dimensional compression tests on sands, and Fig.
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TIME-DEPENDENT DEFORMATION–STRUCTURE INTERACTION
100
夹 䊉
Dry sand RD = 75% Pressure = 15.4 MPa 60
40
夹
20
䊉 0夹 0
Additional insight into the structural changes accompanying the aging of clays is provided by the results of studies by Anderson and Stokoe (1978) and Nakagawa et al. (1995). Figure 12.16 shows changes in shear modulus with time under a constant confining pressure for kaolinite clay during consolidation (Anderson and Stokoe, 1978). Two distinct phases of shear modulus–time response are evident. During primary consolidation, values of the shear modulus increase rapidly at the beginning and begin to level off as the excess pore pressure dissipates. After the end of primary consolidation, the modulus increases linearly with the logarithm of time during secondary compression. The expected change in shear modulus due to void ratio change during secondary compression can be estimated using the following empirical formula for shear modulus as a function of void ratio and confining pressure (Hardin and Black, 1968):
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Percentage Passing (%)
80
After 5 days After Before 290 s Test 䊉 夹 䊉
100
200 Sieve Size (μm)
300
400
Figure 12.14 Changes in particle size distribution of sand before loading and after two different load durations (from Leung, et al., 1996).
12.14 shows the particle size distribution curves for samples before loading and after two different load durations. The amount of particle breakage increased with load duration. Microscopic observations revealed that angular protrusions of the grains were ground off, producing fines. The fines fill the voids between larger particles and crushed particles progressively rearranged themselves with time. Aging—Time-Dependent Strengthening of Soil Structure
The structural changes that occur during creep that is continuing at a decreasing rate cause an increase in soil stiffness when the soil is subjected to further stress increase as shown in Fig. 12.6. Leonards and Altschaeffl (1964) showed that this increase in preconsolidation pressure cannot be accounted for in terms of the void ratio decrease during the sustained compression period. Time-dependent changes of these types are a consequence of ‘‘aging’’ effects, which alter the structural state of the soil. The fabric obtained by creep may be different from that caused by increase in stress, even though both samples arrive at the same void ratio. Leroueil et al. (1996) report a similar result for an artificially sedimented clay from Quebec, as shown in Fig. 12.15a. They also measured the shear wave velocities after different times during the tests using bender elements and computed the small strain elastic shear modulus. Figure 12.15b shows the change in shear modulus with void ratio.
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475
G⫽A
(2.97 ⫺ e)2 0.5 p 1⫹e
(12.3)
where A is a unit dependant material constant, e is the void ratio, and p is the mean effective stress. The dashed line in Fig. 12.16 shows the calculated increases in the shear modulus due to void ratio decrease using Eq. (12.3). It is evident that the change in void ratio alone does not provide an explanation for the secondary time-dependent increase in shear modulus. This aging effect has been recorded for a variety of materials, ranging from clean sands to natural clays (Afifi and Richart, 1973; Kokusho, 1987; Mesri et al., 1990, and many others). Further discussion of aging phenomena is given in Section 12.11. Time-Dependent Changes in Soil Fabric
Changes in soil fabric with time under stress influence the stability of soil structure. Changes in sand fabric with time after load application in one-dimensional compression were measured by Bowman and Soga (2003). Resin was used to fix sand particles after various loading times. Pluviation of the sand produced a horizontal preferred particle orientation of soil grains, and increased vertical loading resulted in a greater orientation of particle long axes parallel to the horizontal, which is in agreement with the findings of Oda (1972a, b, c), Mitchell et al. (1976), and Jang and Frost (1998). Over time, however, the loading of sand caused particle long axes to rotate toward the vertical direction (i.e., more isotropic fabric). Experimental evidence (Bowman and Soga, 2003) showed that large voids became larger, whereas small voids became smaller, and particles group or cluster
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TIME EFFECTS ON STRENGTH AND DEFORMATION
A
A
Normal Consolidation Line
2.6
B
2.4
B
Quasipreconsolidation Pressure
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Void Ratio e
Stiffness Change During the Primary Consolidation Between B and C
2.2
C
Creep for 120 days
C
D
D
Increase in Stiffness during Creep (C-D)
E
Destructuring State
2.0
E
F
F
1.8
4
6
8
10
20
Vertical Effective Stress σv (kPa)
(a )
0.5
1
2
5
Small-strain Stiffness G0 (MPa)
(b )
Figure 12.15 (a) Compression curve and (b) variation of the maximum shear modulus G0
with void ratio for artificially sedimented Jonquiere clay (from Leroueil et al., 1996).
together with time. Based on these particulate level findings, it appears that the movements of particles lead to interlocking zones of greater local density. The interlocked state may be regarded as the final state of any one particle under a particular applied load, due to kinematic restraint. The result, with time, is a stiffer, more efficient, load-bearing structure, with areas of relatively large voids and neighboring areas of tightly packed particles. The increase in stiffness is achieved by shear connections obtained by the clustering. Then, when load is applied, the increased stiffness and strength of the granular structure provides greater resistance to the load and the observed aging effect is seen. The numerical analysis in Kuhn and Mitchell (1993) led to a similar hypothesis for how a more ‘‘braced’’ structure develops with time. For load application in a direction different to that during the aging period, however, the strengthening effect of aging may be less, as the load-bearing particle column direction differs from the load direction. Time-Dependent Changes in Physicochemical Interaction of Clay and Pore Fluid
A portion of the shear modulus increase during secondary compression of clays is believed to result from
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a strengthening of physicochemical bonds between particles. To illustrate this, Nakagawa et al. (1995) examined the physicochemical interactions between clays and pore fluid using a special consolidometer in which the sample resistivity and pore fluid conductivity could be measured. Shear wave velocities were obtained using bender elements to determine changes in the stiffness characteristics of the clay during consolidation. Kaolinite clay mixed with saltwater was used for the experiment, and changes in shear wave velocities and electrical properties were monitored during the tests. The test results showed that the pore fluid composition and ion mobility changed with time. At each load increment, as the effective stress increased with pore pressure dissipation, the shear wave velocities, and therefore the shear modulus, generally increased with time as shown in Fig. 12.17. It may be seen, however, that in some cases, the shear wave velocities at the beginning of primary consolidation decreased slightly from the velocities obtained immediately before application of the incremental load, probably as a result of soil structure breakdown. During the subsequent secondary compression stage, the shear wave velocity again increased. As was the case for the results in Fig.12.16, the increases in shear wave velocity dur-
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50
40
477
Ball Kaolinite
IG = ΔG per log time = 6.2 MPa
Possible Change in G by Void Ratio Decrease Only Estimated Using Eq. (12.3)
30
20
10
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Sample Height Change (mm)
Shear Modulus of less than γ = 10-3% (MPa)
TIME-DEPENDENT DEFORMATION–STRUCTURE INTERACTION
Primary Consolidation 0
0.5
1.0
1.5
10 1
Secondary Compression
10 2 Time (min)
10 3
10 4
Initial Consolidation Pressure = 70 kPa Initial Void Ratio e 0 = 1.1 2.0 10 0
10 1
10 2
10 3
10 4
Time (min)
Figure 12.16 Modulus and height changes as a function of time under constant confining pressure for kaolinite: (a) shear modulus and (b) height change (from Anderson and Stokoe, 1978).
Figure 12.17 Changes in shear wave velocity during primary consolidation and secondary compression of kaolinite. Consolidation pressures: (a) 11.8 kPa and (b) 190 kPa (from Nakagawa et al., 1995).
ing secondary compression are greater than can be accounted for by increase in density. The electrical conductivity of the sample measured by filter electrodes increased during the early stages of consolidation, but then decreased continuously thereafter as shown in Fig. 12.18. The electrical conductivity is dominated by flow through the electrolyte solution in the pores. During the initial compression, a breakdown of structure releases ions into the pore water, increasing the electrical conductivity. With time, the conductivity decreased, suggesting that the released ions are accumulating near particle surfaces. Some of these released ions are expelled from the specimen as consolidation progressed as shown in Fig. 12.18b. A slow equilibrium under a new state of effective stress is hypothesized to develop that involves both small particle rearrangements, associated with decrease in void ratio during secondary compression, and development of increased contact strength as a result of pre-
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cipitation of salts from the pore water and/or other processes. Primary consolidation can be considered a result of drainage of pore water fluid from the macropores, whereas secondary compression is related to the delayed deformation of micropores in the clay aggregates (Berry and Poskitt, 1972; Matsuo and Kamon, 1977; Sills, 1995). The mobility of water in the micropores is restricted due to small pore size and physicochemical interactions close to the clay particle surfaces. Akagi (1994) did compression tests on specially prepared clay containing primarily Ca in the micropores and Na in the macropores. Concentrations of the two ions in the expelled water at different times after the start of consolidation were consistent with this hypothesis.
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TIME EFFECTS ON STRENGTH AND DEFORMATION
sen and Wu (1964), Mitchell (1964), Mitchell et al. (1968, 1969), Murayama and Shibata (1958, 1961, 1964), Noble and Demirel (1969), Wu et al. (1966), Keedwell (1984), Feda (1989, 1992), and Kuhn and Mitchell (1993). Concept of Activation
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The basis of rate process theory is that atoms, molecules, and/or particles participating in a timedependent flow or deformation process, termed flow units, are constrained from movement relative to each other by energy barriers separating adjacent equilibrium positions, as shown schematically by Fig. 12.19. The displacement of flow units to new positions requires the acquisition of an activation energy F of sufficient magnitude to surmount the barrier. The potential energy of a flow unit may be the same following the activation process, or higher or lower than it was initially. These conditions are shown by analogy with the rotation of three blocks in Fig. 12.20. In each case, an energy barrier must be crossed. The assumption of a steady-state condition is implicit in most applications to soils concerning the at-rest barrier height between successive equilibrium positions. The magnitude of the activation energy depends on the material and the type of process. For example, values of F for viscous flow of water, chemical reactions, and solid-state diffusion of atoms in silicates are about 12 to 17, 40 to 400, and 100 to 150 kJ/mol of flow units, respectively.
Figure 12.18 Changes in electrical conductivity of the pore
water during primary consolidation and secondary compression of kaolinite. Consolidation pressures: (a) 95 kPa and (b) 190 kPa (from Nakagawa et al., 1995).
12.4 SOIL DEFORMATION AS A RATE PROCESS
Deformation and shear failure of soil involve timedependent rearrangement of matter. As such, these phenomena are amenable for study as rate processes through application of the theory of absolute reaction rates (Glasstone et al., 1941). This theory provides both insights into the fundamental nature of soil strength and functional forms for the influences of several factors on soil behavior. Detailed development of the theory, which is based on statistical mechanics, may be found in Eyring (1936), Glasstone et al. (1941), and elsewhere in the physical chemistry literature. Adaptations to the study of soil behavior include those by Abdel-Hady and Herrin (1966), Andersland and Douglas (1970), Christen-
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Activation Frequency
The energy to enable a flow unit to cross a barrier may be provided by thermal energy and by various applied potentials. For a material at rest, the potential energy– displacement relationship is represented by curve A in Fig. 12.21. From statistical mechanics it is known that
Figure 12.19 Energy barriers and activation energy.
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SOIL DEFORMATION AS A RATE PROCESS
Figure 12.20 Examples of activated processes: (a) steady-state, (b) increased stability, and
(c) decreased stability.
Figure 12.21 Effect of a shear force on energy barriers.
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TIME EFFECTS ON STRENGTH AND DEFORMATION
the average thermal energy per flow unit is kT, where k is Boltzmann’s constant (1.38 ⫻ 10⫺23 J K⫺1) and T is the absolute temperature (K). Even in a material at rest, thermal vibrations occur at a frequency given by kT/h, where h is Planck’s constant (6.624 ⫻ 10⫺34 J s⫺1). The actual thermal energies are divided among the flow units according to a Boltzmann distribution. It may be shown that the probability of a given unit becoming activated, or the proportion of flow units that are activated during any one oscillation is given by
( →) ⫺ ( ←) ⫽ 2
冉 冊 冉 冊
kT
F ƒ exp ⫺ sinh h RT 2kT
(12.8) Strain Rate Equation
At any instant, some of the activated flow units may successfully cross the barrier; others may fall back into their original positions. For each unit that is successful in crossing the barrier, there will be a displacement . The component of in a given direction times the number of successful jumps per unit time gives the rate of movement per unit time. If this rate of movement is expressed on a per unit length basis, then the strain rate ˙ is obtained. Let X ⫽ F (proportion of successful barrier crossings and ) such that
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冉 冊
The net frequency of activation in the direction of the force then becomes
F p( F) ⫽ exp ⫺ NkT
(12.4)
where N is Avogadro’s number (6.02 ⫻ 1023), and Nk is equal to R, the universal gas constant (8.3144 J K⫺1 mol⫺1). The frequency of activation then is ⫽
冉 冊
⫺ F kT exp h NkT
(12.5)
In the absence of directional potentials, energy barriers are crossed with equal frequency in all directions, and no consequences of thermal activations are observed unless the temperature is sufficiently high that softening, melting, or evaporation occurs. If, however, a directed potential, such as a shear stress, is applied, then the barrier heights become distorted as shown by curve B in Fig. 12.21. If ƒ represents the force acting on a flow unit, then the barrier height is reduced by an amount (ƒ /2) in the direction of the force and increased by a like amount in the opposite direction, where represents the distance between successive equilibrium positions.3 Minimums in the energy curve are displaced a distance from their original positions, representing an elastic distortion of the material structure. The reduced barrier height in the direction of force ƒ increases the activation frequency in that direction to →⫽
冉
冊
kT
F/N ⫺ ƒ /2 exp ⫺ h kT
(12.6)
˙ ⫽ X[( →) ⫺ ( ←)]
Then from Eq. (12.8) ˙ ⫽ 2X
冉
冊
冉 冊 冉 冊
kT
F ƒ exp ⫺ sinh h RT 2kT
(12.10)
The parameter X may be both time and structure dependent. If (ƒ /2kT) ⬍ 1, then sinh(ƒ /2kT) ⬇ (ƒ /2kT), and the rate is directly proportional to ƒ. This is the case for ordinary Newtonian fluid flow and diffusion where ˙ ⫽
1
(12.11)
where ˙ is the shear strain rate, is dynamic viscosity, and is shear stress. For most solid deformation problems, however, (ƒ /2kT) ⬎ 1 (Mitchell et al., 1968), so
冉 冊
sinh
and in the opposite direction, the increased barrier height decreases the activation frequency to
F/N ⫹ ƒ /2 kT ←⫽ exp ⫺ h kT
(12.9)
冉 冊
ƒ 1 ƒ ⬇ exp 2kT 2 2kT
(12.12)
and
(12.7)
Work (ƒ / 2) done by the force ƒ as the flow unit drops from the peak of the energy barrier to a new equilibrium position is assumed to be given up as heat.
3
Copyright © 2005 John Wiley & Sons
˙ ⫽ X
冉 冊 冉 冊
kT
F ƒ N exp ⫺ exp h RT 2RT
(12.13)
Equation (12.13) applies except for very small stress intensities, where the exponential approximation of the hyperbolic sine is not justified. Equations (12.10) and
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BONDING, EFFECTIVE STRESSES, AND STRENGTH
12.5 BONDING, EFFECTIVE STRESSES, AND STRENGTH
Using rate process theory, the results of timedependent stress–deformation measurements in soils can be used to obtain fundamental information about soil structure, interparticle bonding, and the mechanisms controlling strength and deformation behavior. Deformation Parameters from Creep Test Data
If the shear stress on a material is and it is distributed uniformly among S flow units per unit area, then
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(12.13) or comparable forms have been used to obtain dashpot coefficients for rheological models, to obtain functional forms for the influences of different factors on strength and deformation rate, and to study deformation rates in soils. For example, Kuhn and Mitchell (1993) used this form as part of the particle contact law in discrete element modeling as described in the previous section. Puzrin and Houlsby (2003) used it as an internal function of a thermomechanical-based model and derived a rate-dependent constitutive model for soils. Soil Deformation as a Rate Process
S
Although there does not yet appear to be a rigorous proof of the correctness of the detailed statistical mechanics formulation of rate process theory, even for simple chemical reactions, the real behavior of many systems has been substantially in accord with it. Different parts of Eq. (12.13) have been tested separately (Mitchell et al., 1968). It was found that the temperature dependence of creep rate and the stress dependence of the experimental activation energy [Eq. (12.14)] were in accord with predictions. These results do not prove the correctness of the theory; they do, however, support the concept that soil deformation is a thermally activated process.
Displacement of a flow unit requires that interatomic or intermolecular forces be overcome so that it can be moved. Let it be assumed that the number of flow units and the number of interparticle bonds are equal. If D represents the deviator stress under triaxial stress conditions, the value of ƒ on the plane of maximum shear stress is
Arrhenius Equation
so Eq. (12.13) becomes
ƒ⫽
ƒ⫽
Equation (12.13) may be written
冉 冊
kT E ˙ ⫽ X exp ⫺ h RT
where
E ⫽ F ⫺
ƒ N 2
˙ ⫽ X
(12.14)
(12.15)
is termed the experimental activation energy. For all conditions constant except T, and assuming that X(kT/h) ⬇ constant ⫽ A, ˙ ⫽ A exp
冉 冊 ⫺
E RT
(12.16)
Equation (12.16) is the same as the well-known empirical equation proposed by Arrhenius around 1900 to describe the temperature dependence of chemical reaction rates. It has been found suitable also for characterization of the temperature dependence of processes such as creep, stress relaxation, secondary compression, thixotropic strength gain, diffusion, and fluid flow.
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D 2S
冉 冊 冉 冊
kT
F D exp ⫺ exp h RT 4SkT
(12.17)
(12.18)
(12.19)
This equation describes creep as a steady-state process. Soils do not creep at constant rate, however, because of continued structural changes during deformation as described in Section 12.3, except for the special case of large deformations after mobilization of full strength. Thus, care must be taken in application of Eq. (12.19) to ensure that comparisons of creep rates and evaluations of the influences of different factors are made under conditions of equal structure. The time dependency of creep rate and the possible time dependencies of the parameters in Eq. (12.19) are considered in Section 12.8. Determination of Activation Energy From Eq. (12.14) ln(˙ /T) E ⫽⫺ (1/T) R
(12.20)
provided strain rates are considered under conditions of unchanged soil structure. Thus, the value of E can be determined from the slope of a plot of ln(˙ /T) versus (1/T). Procedures for evaluation of strain rates for
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TIME EFFECTS ON STRENGTH AND DEFORMATION
known, /S is calculated as a measure of the number of interparticle bonds.4
soils at different temperatures but at the same structure are given by Mitchell et al. (1968, 1969). Determination of Number of Bonds For stresses large enough to justify approximating the hyperbolic sine function by a simple exponential in the creep rate equation and small enough to avoid tertiary creep, the logarithm of strain rate varies directly with the deviator stress. For this case, Eq. (12.19) can be written
where
Activation energies for the creep of several soils and other materials are given in Table 12.1. The free energy of activation for creep of soils is in the range of about 80 to 180 kJ/mol. Four features of the values for soils in Table 12.1 are significant:
(12.21)
1. The activation energies are relatively large, much higher than for viscous flow of water. 2. Variations in water content (including complete drying), adsorbed cation type, consolidation pressure, void ratio, and pore fluid have no significant effect on the required activation energy. 3. The values for sand and clay are about the same. 4. Clays in suspension with insufficient solids to form a continuous structure deform with an activation energy equal to that of water.
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˙ ⫽ K(t) exp(D)
Activation Energies for Soil Creep
K(t) ⫽ X ⫽
冉 冊
kT
F exp ⫺ h RT
4SkT
(12.22) (12.23)
Parameter is a constant for a given value of effective consolidation pressure and is given by the slope of the relationship between log strain rate and stress. It is evaluated using strain rates at the same time after the start of creep tests at several stress intensities. With
Table 12.1
A procedure for evaluation of from the results of a test at a succession of stress levels on a single sample is given by Mitchell et al. (1969).
4
Activation Energies for Creep of Several Materials
Activation Energy (kJ/ mol)a
Material
1. Remolded illite, saturated, water contents of 30 to 43% 2. Dried illite: samples air-dried from saturation, then evacuated 3. San Francisco Bay mud, undisturbed 4. Dry Sacramento River sand 5. Water 6. Plastics 7. Montmorillonite–water paste, dilute 8. Soil asphalt 9. Lake clay, undisturbed and remolded 10. Osaka clay, overconsolidated 11. Concrete 12. Metals 13. Frozen soils 14. Sault Ste. Marie clay, suspensions, discontinuous structures 15. Sault Ste. Marie clay, Li⫹, Na⫹, K⫹ forms, in H2O and CCl4, consolidated
Reference
105–165
Mitchell, et al. (1969)
155
Mitchell, et al. (1969)
105–135 ⬃105 16–21 30–60 84–109 113 96–113 120–134 226 210⬃ 393 Same as water 117
Mitchell, et al. (1969) Mitchell, et al. (1969) Glasstone, et al. (1941) Ree and Eyring (1958) Ripple and Day (1966) Abdel-Hady and Herrin (1966) Christensen and Wu (1964) Murayama and Shibata (1961) Polivka and Best (1960) Finnie and Heller (1959) Andersland and Akili (1967) Andersland and Douglas (1970) Andersland and Douglas (1970)
The first four values are experimental activation energies, E. Whether the remainder are values of F or E is not always clear in the references cited. a
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BONDING, EFFECTIVE STRESSES, AND STRENGTH
Number of Interparticle Bonds
50 kPa, and then remolded at constant water content. The effective consolidation pressure dropped to 25 kPa as a result of the remolding. The drop in effective stress was accompanied by a corresponding decrease in the number of interparticle bonds. Tests on remolded illite gave comparable results. A continuous inverse relationship between the number of bonds and water content over a range of water contents from more than 40 percent to air-dried and vacuum-desiccated clay is shown in Fig. 12.24. The dried material had a water content of 1 percent on the usual oven-dried basis. The very large number of bonds developed by drying is responsible for the high dry strength of clay. Overconsolidated Clay Samples of undisturbed San Francisco Bay mud were prepared to overconsolidation ratios of 1, 2, 4, and 8 following the stress paths shown in the upper part of Fig. 12.25. The sample represented by the triangular data point was remolded after consolidation and unloading to point d, where it had a water content of 52.3 percent. The undrained compressive strength as a function of consolidation pressure is shown in the middle section of Fig. 12.25, and the number of bonds, deduced from the creep tests, is shown in the lower part of the figure. The effect of overconsolidation is to increase the number of interparticle bonds over the values for normally consolidated clay. Some of the bonds formed during consolidation are retained after removal of much of the consolidation pressure. Values of compressive strength and numbers of bonds from Fig. 12.25 are replotted versus each other in Fig. 12.26. The resulting relationship suggests that strength depends only on the number of bonds and is independent of whether the clay is undisturbed, remolded, normally consolidated, or overconsolidated. Dry Sand Creep tests on oven-dried sand yielded results of the same type as obtained for clay, as shown in Fig. 12.27, suggesting that the strength-generating and creep-controlling mechanisms may be similar for both types of material. Composite Strength-Bonding Relationship Values of S and strength for many soils are combined in Fig. 12.28. The same proportionality exists for all the materials, which may seem surprising, but which in reality should be expected, as discussed further later.
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Evaluation of S requires knowledge of , the separation distance between successive equilibrium positions in the interparticle contact structure. A value of 0.28 nm ˚ ) has been assumed because it is the same as the (2.8 A distance separating atomic valleys in the surface of a silicate mineral. It is hypothesized that deformation involves the displacement of oxygen atoms along contacting particle surfaces, as well as periodic rupture of bonds at interparticle contacts. Figure 12.22 shows this interpretation for schematically. If the above assumption for is incorrect, calculated values of S will still be in the same correct relative proportion as long as remains constant during deformation. Normally Consolidated Clay Results of creep tests at different stress intensities for different consolidation pressures enable computation of S as a function of consolidation pressure. Values obtained for undisturbed San Francisco Bay mud are shown in Fig. 12.23. The open point is for remolded bay mud. An undisturbed specimen was consolidated to 400 kPa, rebounded to
483
Significance of Activation Energy and Bond Number Values
Figure 12.22 Interpretation of in terms of silicate mineral
surface structure.
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The following aspects of activation energies and numbers of interparticle bonds are important in the understanding of the deformation and strength behavior of uncemented soils.
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TIME EFFECTS ON STRENGTH AND DEFORMATION
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484
Figure 12.23 Number of interparticle bonds as a function of consolidation pressure for normally consolidated San Francisco Bay mud.
Figure 12.24 Number of bonds as a function of water con-
tent for illite.
Copyright © 2005 John Wiley & Sons
1. The values of activation energy for deformation of soils are high in comparison with other materials and indicate breaking of strong bonds. 2. Similar creep behavior for wet and dry clay and for wet and dry sand indicates that deformation is not controlled by viscous flow of water. 3. Comparable values of activation energy for wet and dry soil indicate that water is not responsible for bonding. 4. Comparable values of activation energy for clay and sand support the concept that interparticle bond strengths are the same for both types of material. This is supported also by the uniqueness of the strength versus number of bonds relationship for all soils. 5. The activation energy and presumably, therefore, the bonding type are independent of consolidation pressure, void ratio, and water content. 6. The number of bonds is directly proportional to effective consolidation pressure for normally consolidated clays. 7. Overconsolidation leads to more bonds than in normally consolidated clay at the same effective consolidation pressure. 8. Strength depends only on the number of bonds. 9. Remolding at constant water content causes a decrease in the effective consolidation pressure, which means also a decrease in the number of bonds. 10. There are about 100 times as many bonds in dry clay as in wet clay.
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BONDING, EFFECTIVE STRESSES, AND STRENGTH
485
Figure 12.25 Consolidation pressure, strength, and bond numbers for San Francisco Bay
mud.
Although it may be possible to explain these results in more than one way, the following interpretation accounts well for them. The energy F activates a mole of flow units. The movement of each flow unit may involve rupture of single bonds or the simultaneous rupture of several bonds. Shear of dilute montmorillonite–water pastes involves breaking single bonds (Ripple and Day, 1966). For viscous flow of water, the activation energy is approximately that for a single hydrogen bond rupture per flow unit displacement, even though each water molecule may form simultaneously
Copyright © 2005 John Wiley & Sons
up to four hydrogen bonds with its neighbors. If the single-bond interpretation is also correct for soils, then consistency in Eq. (12.10) requires that shear force ƒ pertain to the force per bond. On this basis, parameter S indicates the number of single bonds per unit area. In the event activation of a flow unit requires simultaneous rupture of n bonds, then S represents 1/nth of the total bonds in the system. That the activation energy for deformation of soil is well into the chemical reaction range (40 to 400 kJ/ mol) does not prove that bonding is of the primary
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TIME EFFECTS ON STRENGTH AND DEFORMATION
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Figure 12.26 Strength as a function of number of bonds for San Francisco Bay mud.
Figure 12.28 Composite relationship between shear strength
valence type because simultaneous rupture of several weaker bonds could yield values of the magnitude observed. On the other hand, the facts that (1) the activation energy is much greater than for flow of water, (2) it is the same for wet and dry soils, and (3) it is essentially the same for different adsorbed cations and
and number of interparticle bonds (from Matsui and Ito, 1977). Reprinted with permission from The Japanese Society of SMFE.
Figure 12.27 Strength as a function of number of bonds for dry Antioch River sand.
Copyright © 2005 John Wiley & Sons
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BONDING, EFFECTIVE STRESSES, AND STRENGTH
ever, for equal numbers of contacts per particle, the number per unit volume should vary inversely with the cube of particle size. Thus, the number of clay particles of 1-m particle size should be some nine orders of magnitude greater than for a sand of 1-mm average particle size. Each contact between sand particles would involve many bonds; in clay, the much greater number of contacts would mean fewer bonds per particle. The contact area required to develop bonds in the numbers indicted in Figs. 12.23 to 12.27 is very small. For example, for a compressive strength of 3 kg/cm2 (⬇ 300 kPa) there are 8 ⫻ 1010 bonds/cm2 of shear surface. Oxygen atoms on the surface of a silicate mineral have a diameter of 0.28 nm. Allowing an area 0.30 nm on a side for each oxygen gives 0.09 nm2, or 9 ⫻ 10⫺16 cm2, per bonded oxygen for a total area of 9 ⫻ 10⫺16 ⫻ 8 ⫻ 1010 ⫽ 7.2 ⫻ 10⫺5 cm2 /cm2 of soil cross section.
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pore fluids (Andersland and Douglas, 1970) suggest that bonding is through solid interparticle contacts. Physical evidence for the existence of solid-to-solid contact between clay particles has been obtained in the form of photomicrographs of particle surfaces that were scratched during shear (Matsui et al., 1977, 1980) and acoustic emissions (Koerner et al., 1977). Activation energy values of 125 to 190 kJ/mol are of the same order as those for solid-state diffusion of oxygen in silicate minerals. This supports the concept that creep movements of individual particles could result from slow diffusion of oxygen ions in and around interparticle contacts. The important minerals in both sand and clay are silicates, and their surface layers consist of oxygen atoms held together by silicon atoms. Water in some form is adsorbed onto these surfaces. The water structure consists of oxygens held together by hydrogen. It is not too different from that of the silicate layer in minerals. Thus, a distinct boundary between particle surface and water may not be discernable. Under these conditions, a more or less continuous solid structure containing water molecules that propagates through interparticle contacts can be visualized. An individual flow unit could be an atom, a group of atoms or molecules, or a particle. The preceding arguments are based on the interpretation that individual atoms are the flow units. This is consistent with both the relative and actual values of S that have been determined for different soils. Furthermore, by using a formulation of the rate process equation that enabled calculation of the flow unit volume from creep test data, Andersland and Douglas (1970) obtained a value ˚ 3, which is of the same order as that of of about 1.7 A individual atoms. On the other hand, Keedwell (1984) defined flow units between quartz sand particles as consisting of six O2⫺ ions and six Si4⫹ ions and between two montmorillonite clay particles as consisting of four H2O molecules. If particles were the flow units, not only would it be difficult to visualize their thermal vibrations, but then S would relate to the number of interparticle contacts. It is then difficult to conceive how simply drying a clay could give a 100-fold increase in the number of interparticle contacts, as would have to be the case according to Fig. 12.27. A more plausible interpretation is that drying, while causing some increase in the number of interparticle contacts during shrinkage, causes mainly an increase in the number of bonds per contact because of increased effective stress. At any value of effective stress, the value of S is about the same for both sand and clay. The number of interparticle contacts should be vastly different; how-
487
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Hypothesis for Bonding, Effective Stress, and Strength
Normal effective stresses and shear stresses can be transmitted only at interparticle contacts in most soils.5 The predominant effects of the long-range physicochemical forces of interaction are to control the initial soil fabric and to alter the forces transmitted at contact points from what they would be due to applied stresses alone. Interparticle contacts are effectively solid, and it is likely that both adsorbed water and cations in the contact zone participate in the structure. An interparticle contact may contain many bonds that may be strong, approaching the primary valence type. The number of bonds at any contact depends on the compressive force transmitted at the contact, and the Terzaghi–Bowden and Tabor adhesion theory of friction presented in Section 11.4, can account for strength. The macroscopic strength is directly proportional to the number of bonds. For normally consolidated soils the number of bonds is directly proportional to the effective stress. As a result of particle rearrangements and contacts formed during virgin compression, an overconsolidated soil at a given effective stress has a greater number of bonds and higher strength than a normally consolidated soil. This effect is more pronounced in clays than in sands because the larger and bulky sand grains tend to re-
5 Pure sodium montmorillonite may be an exception since a part of the normal stress can be carried by physicochemical forces of interaction. The true effective stress may be less than the apparent effective stress by R ⫺ A as discussed in Chapter 7.
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TIME EFFECTS ON STRENGTH AND DEFORMATION
12.6 SHEARING RESISTANCE AS A RATE PROCESS
Deformation at large strain can approach a steady-state condition where there is little further structural change with time (such as at critical state). In this case, Eq. (12.19) can be used to describe the shearing resistance as a function of strain rate and temperature. If the maximum shear stress is substituted for the deviator stress D, then ⫽
and
From the relationships in Section 12.5, the following relationship between bonds per unit area and effective stress is suggested. S ⫽ a ⫹ bƒ
˙ ⫽ X
1 ⫺ 3 2
冉 冊 冉 冊
kT
F exp ⫺ exp h RT 2SkT
⫽
冉
冉 冊
kT
F ⫺ ⫹ h RT 2SkT
冉冊
2S 2SkT ˙
F ⫹ ln N B
(12.29)
Equation (12.29) is of the same form as the Coulomb equation for strength: ⫽ c ⫹ ƒ tan
(12.30)
By analogy,
c⫽
2a F 2akT ˙ ⫹ ln N B
(12.31)
tan ⫽
2b F 2bkT ˙ ⫹ ln N B
(12.32)
These equations state that both cohesion and friction depend on the number of bonds times the bond strength, as reflected by the activation energy, and that the values of c and should depend on the rate of deformation and the temperature. Strain Rate Effects
(12.25)
All other factors being equal, the shearing resistance should increase linearly with the logarithm of the rate of strain. This is shown to be the case in Fig. 12.9, which contains data for 26 clays. Additional data for several clays are shown in Fig. 12.29, where shearing resistance as a function of the speed of vane rotation in a vane shear test is plotted. Analysis of the relationship between shearing stress and angular rate of vane rotation " shows that / log " decreases with an increase in water content. This follows directly from Eq. (12.29) because
(12.26)
By assuming X(kT/h) is a constant equal to B (Mitchell, 1964), Eq. (12.26) can be rearranged to give ⫽
冊
2a F 2akT ˙ 2b F 2bkT ˙ ⫹ ln ⫹ ⫹ ln ƒ N B N B
(12.24)
Taking logarithms of both sides of Eq. (12.25) gives ln ˙ ⫽ ln X
(12.28)
where a and b are constants and ƒ is the effective normal stress on the shear plane. Thus, Eq. (12.27) becomes
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cover their original shapes when unloaded, thus rupturing most of the bonds in excess of those needed to resist the lower stress. The strength of the interparticle contacts can vary over a wide range, depending on the number of bonds per contact. The unique relationship between strength and number of bonds for all soils, as indicted by Fig. 12.28, reflects the fact that the minerals comprising most soils are silicates, and they all have similar surface structures. In the absence of chemical cementation, interparticle bonds may form in response to interparticle contact forces generated by either applied stresses, physicochemical forces of interaction, or both. Any bonds existing in the absence of applied effective stress, that is, when ⫽ 0, are responsible for true cohesion. There should be no difference between friction and cohesion in terms of the shearing process. Complete failure in shear involves simultaneous rupture or slipping of all bonds along the shear plane.
(12.27)
Copyright © 2005 John Wiley & Sons
d 2akT 2bkT 2kT ⫽ ⫹ ƒ ⫽ (a ⫹ bƒ) d ln(˙ /B) (12.33)
that is, d /d ln (˙ /B) is proportional to the number of bonds, which decreases with increasing water content.
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CREEP AND STRESS RELAXATION
489
Figure 12.29 Effect of rate of shear on shearing resistance of remolded clays as determined by the laboratory vane apparatus (prepared from the data of Karlsson, 1963).
This interpretation of the data in Figs. 12.9 and 12.29 assumes that the effective stress was unaffected by changes in the strain rate, which may not necessarily be true in all cases. Effect of Temperature
Assumptions of reasonable values for parameters show that the term (˙ /B) is less than one (Mitchell, 1964). Thus the quantity ln(˙ /B) in Eq. (12.29) is negative, and an increase in temperature should give a decrease in strength, all other factors being constant. That this is the case is demonstrated by Fig. 12.30, which shows deviator stress as a function of temperature for samples of San Francisco Bay mud compared under conditions of equal mean effective stress and structure. Other examples of the influence of temperature on strength are shown in Figs. 11.6 and 11.133.
Copyright © 2005 John Wiley & Sons
12.7
CREEP AND STRESS RELAXATION
Although the designation of a part of the strain versus time relationship as steady state or secondary creep may be convenient for some analysis purposes, a true steady state can exist only for conditions of constant structure and stress. Such a set of conditions is likely only for a fully destructured soil, and a fully destructured state is likely to persist only during deformation at a constant rate, that is, at failure. This state is often called ‘‘steady state,’’ in which the soil is deforming continuously at constant volume under constant shear and confining stresses (Castro, 1975; Castro and Poulos, 1977). Otherwise, bond making and bond breaking occur at different rates as a result of different internal timeand strain-dependent phenomena, which might include thixotropic hardening, viscous flows of water and ad-
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TIME EFFECTS ON STRENGTH AND DEFORMATION
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490
Figure 12.30 Influence of temperature on the shearing resistance of San Francisco Bay mud. Comparison is for samples at equal mean effective stress and at the same structure.
sorbed films, chemical, and biological transformations, and the like. Furthermore, distortions of the soil structure and relative movements between particles cause changes in the ratio of tangential to normal forces at interparticle contacts that may be responsible for large changes in creep rate. Because of these time dependencies some of the parameters in Eq. (12.19) may be time dependent. For example, Feda (1989) accounted for the time dependency of creep rate by taking changes in the number of structural bonds into account. Therefore, application of Eq. (12.19) for the determination of the bonding and effective stress relationships discussed in Section 12.5 required comparison of creep rates under conditions of comparable time and structure. The influence of creep stress magnitude on the creep rate at a given time after the application of the stress to identical samples of a soil was shown in Fig. 12.3. At low stresses the creep rates are small and of little practical importance. The curve shape is compatible with the hyperbolic sine function predicted by rate process theory, as given by Eq. (12.10). In the midrange of stresses, a nearly linear relationship is found between logarithm of strain rate and stress, also as predicted by Eq. (12.10) for the case where the argument of the hyperbolic sine is greater than 1. At stresses approaching the strength of the material, the strain rate becomes very large and signals the onset of failure. Other examples of the relationships between logarithm of strain rate and creep stress corresponding to different times after the application of the creep stress are given in Fig. 12.31 for drained tests on London clay
Copyright © 2005 John Wiley & Sons
Figure 12.31 Variation of creep strain rate with deviator
stress for drained creep of London clay (data from Bishop, 1966).
and Fig. 12.32 for undrained tests on undisturbed San Francisco Bay mud. Only values for the midrange of stresses are shown in Figs. 12.31 and 12.32. Effect of Composition
In general, the higher the clay content and the more active the clay, the more important are stress relaxation and creep, as illustrated by Figs. 4.22 and 4.23, where creep rates, approximated by steady-state values, are related to clay type, clay content, and plasticity. Timedependent deformations are more important at high water contents than at low. Deviatoric creep and secondary compression are greater in normally consolidated than overconsolidated soils. Although the magnitude of creep strains and strain rates may be small in sand or dry soil, the form of the
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CREEP AND STRESS RELAXATION
creep tests on sands. The conflicting evidence may be due to the presence or absence of impurities that may lubricate or cement the soil in the presence of water (Human, 1992; Bowman, 2003). Volume Change and Pore Pressures
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Due to the known coupling effects between shearing and volumetric plastic deformations in soils, an increase in either mean pressure or deviator stress can generate both types of deformations. Creep behavior is no exception. Time-dependent shear deformations are usually referred to as deviatoric creep or shear creep. Time-dependent deformations under constant stress referred to as volumetric creep. Secondary compression is a special case of volumetric creep. Deviatoric creep is often accompanied by volumetric creep. The ratio of volumetric to deviatoric creep fol-
E
Drained Triaxial Test
1
2820 min
Volumetric Strain (%)
Confining Pressure σ3 = 414 kPa
0.8
Deviator Stress from 344 to 377 kPa
0.6
D
90 min
0.4
0.2
Figure 12.32 Variation of creep strain rate with deviator
stress for undrained creep of normally consolidated San Francisco Bay mud.
1450 min
20 min C
2 min
0
A B
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Deviator strain (%)
(a)
1.0
Stress ratio q/p
behavior conforms with the patterns described and illustrated above. This is to be expected, as the basic creep mechanism is the same in all inorganic soils.6 Water may ‘‘lubricate’’ the particles and possibly increase the creep rate even though the basic mechanism of creep is the same for dry and wet materials (Losert et al., 2000). Takei et al. (2001) showed that the development of creep strains due to time-dependent breakage of talc specimens increased more for saturated specimens than dry ones. However, a negligible effect of water on creep rate was reported by Ahn-Dan et al. (2001) who performed creep tests on unsaturated and saturated crushed gravel and by Leung et al. (1996) who performed one-dimensional compression
0.8 0.6 0.4 0.2 0
0 1 2 3 4 Strain increment ratio dεv/dεs during creep
(b) Figure 12.33 Dilatancy relationship obtained from drained
6
Volumetric creep and secondary compression of organic soils, peat, and municipal waste fills can develop also as a result of decomposition of organic matter.
Copyright © 2005 John Wiley & Sons
creep tests on kaolinite: (a) development of volumetric and deviatoric strains with time and (b) effect of stress ratio on strain increment ratio d / ds (from Walker, 1969).
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TIME EFFECTS ON STRENGTH AND DEFORMATION
history, with some samples contracting or dilating (Lade and Liu, 1998; Ahn-Dan et al., 2001). Some dense sand samples contract initially but then dilate with time (Bowman and Soga, 2003). Further discussion of the creep behavior of sands in relation to mechanical aging phenomena is given in Section 12.11. The fundamental process of creep strain development is therefore similar to that of time-independent plastic strains, and the same framework of soil plasticity can possibly be used. It can be argued whether it is necessary to separate the deformation into time-dependent and independent components. Rateindependent behavior can be considered as the limiting case of rate-dependent behavior at a very slow rate of loading. Volumetric-deviatoric creep coupling implies that rapid application of a stress or a strain invariably results in rapid change of pore water pressures in a saturated soil under undrained conditions. For a constant total minor principal stress, the magnitude of the pore pressure change depends on the volume change tendencies of the soil when subjected to shear distortions. These tendencies are, in turn, controlled by the void ratio, structure, and effective stress, and can be quantified in terms of the pore pressure parameter A as discussed in Chapters 8 and 10. An example showing pore pressure increase with time for consolidated undrained creep tests on illite at several stress intensities is shown in Fig. 12.34. Figure 12.35 shows a slow decrease in
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lows a plastic dilatancy rule. Walker (1969) investigated the time-dependent change of these two components from incremental drained triaxial creep tests on normally consolidated kaolinite. The increase in shear strains with increase in volumetric strains at different times is shown in Fig. 12.33a. At the beginning of the triaxial test, the deviator stress was instantaneously increased from 344 to 377 kPa and kept constant. After an immediate increase in shear strains at constant volume (AB in Fig. 12.33a), section BD corresponds to primary consolidation that is controlled by the dissipation of pore pressures. After point D, creep occurred, and the ratio of volumetric to deviatoric strains was independent of time. This ratio decreased with increasing stress ratio as shown in Fig. 12.33b. This observation led to the time-dependent flow rule, which is similar to the dilatancy rule described in Section 11.20. Sand deforms with time in a similar manner. Under progressive deviatoric creep, the volumetric creep response is highly dependent on density, the stress level, and the stress path before creep. The rate of both volumetric and deviatoric creep increases with confining pressure, particularly after particle crushing becomes important at high stresses (Yamamuro and Lade, 1993). For dense sand under high deviator stress, dilative creep is observed (Murayama et al., 1984; Mejia et al., 1988). The volumetric response of dense sand and gravel with time is a highly complex function of stress
Figure 12.34 Pore pressure development with time during undrained creep of illite.
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CREEP AND STRESS RELAXATION
493
Figure 12.35 Normalized pore pressure vs. time relationships during creep of kaolinite.
An increase in temperature decreases effective stress, increases pore pressure, and weakens the soil structure. Creep rates ordinarily increase and the relaxation stresses corresponding to specific values of strain decrease at higher temperature. These effects are illustrated by the data shown in Figs. 12.38 and 12.39.
Although the general form of the stress–strain–time and stress–strain rate–time relationships are similar to those shown above for triaxial loading conditions, the actual values may differ considerably. For example, undisturbed Haney clay, a gray silty clay from British Columbia, with a sensitivity in the range of 6 to 10, was tested both in triaxial compression and plane strain (Campanella and Vaid, 1974). Samples were normally consolidated both isotropically and under K0 conditions to the same vertical effective stress. Samples consolidated isotropically were tested in triaxial compression. Coefficient K0 consolidation was used for both K0 triaxial and plane strain tests. The results shown in Fig. 12.40 indicate that the precreep stress history had a significant effect on the deformations. The plane strain and K0 consolidated triaxial samples gave about the same creep behavior under the same deviatoric stress, which suggests that preventing strain in one horizontal direction and/or the intermediate principal stress were not factors of major importance for this soil under the test conditions used.
Effects of Test Type, Stress System, and Stress Path
Interaction Between Consolidation and Creep
Most measurements of time-dependent deformation and stress relaxation in soils have been done on samples consolidated isotropically and tested in triaxial compression or by measurement of secondary compression in oedometer tests. However, most soils in nature have been subjected to an anisotropic stress history, and deformation conditions conform more to plane strain than triaxial compression in many cases. Some investigations of these factors have been made.
Experimental evidence suggests that creep occurs during primary consolidation (Leroueil et al., 1985; Imai and Tang, 1992). Following the initial large change following load application, the pore pressure may either dissipate, with accompanying volume change if drainage is allowed, or change slowly during creep or stress relaxation, if drainage is prevented. The development of complete effective stress and void ratio equilibrium may take a long time. One illustration of
pore pressure during the sustained loading of kaolinite. Similar behavior was demonstrated in the measured stress paths of undrained creep test on San Francisco Bay mud (Arulanandan et al., 1971). As shown in Fig. 12.36, the effective stress states shifted toward the failure line. At higher stress levels, the specimens eventually underwent creep rupture. However, soil strength in terms of effective stresses does not change unless there are chemical, biological, or mineralogical changes during the creep period. This is illustrated by the stress paths shown schematically in Fig. 12.37, where the pre- and postcreep strengths fall on the same failure envelope. Effects of Temperature
Copyright © 2005 John Wiley & Sons
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TIME EFFECTS ON STRENGTH AND DEFORMATION
Normal Undrained Triaxial Compression Test: Effective Stress Path
50 Possible Critical State Line
Total Stress Path of 3 Triaxial Compression Test
40
1
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Deviator Stress q (kPa)
12
30
Effective Stress State After 1,000 min Creep
20
After 20,000 min Creep Effective Stress Path of Undrained Creep
10
0
0
10
20
30
40
50
60
70
80
Mean Pressure p (kPa)
(a)
400
Possible Critical State Line
Normal Undrained Triaxial Compression Test: Effective Stress Path Total Stress Path of Triaxial Compression Test 3
320
Deviator Stress q (kPa)
494
1
240
Effective Stress State After 1,000 min Creep
160
After 20,000 min Creep Effective Stress Path of Undrained Creep
80
0
0
80
160
240 320 400 480 Mean Pressure p (kPa)
560
640
(b)
Figure 12.36 Measured stress paths of undrained creep tests of San Francisco Bay mud.
Initial confining pressure: (a) 49 kPa and (b) 392 kPa (from Arulanandan et al., 1971).
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CREEP AND STRESS RELAXATION
495
Figure 12.37 Effects of undrained creep on the strength of normally consolidated clay.
Figure 12.38 Creep curves for Osaka clay tested at different temperatures—undrained tri-
axial compression (Murayama, 1969).
this is given by Fig. 10.5, where it is shown that the relationship between void ratio and effective stress is dependent on the time for compression under any given stress. Another is given by Fig. 12.41, which shows pore pressures during undrained creep of San Francisco Bay mud. In each sample, consolidation under an effective confining pressure of 100 kPa was allowed for 1800 min prior to the cessation of drainage and the start of a creep test. The consolidation period was greater than that required for 100 percent primary consolidation. The curve marked 0 percent stress level refers to a specimen maintained undrained but not subject to a deviator stress. This curve indicates that each
Copyright © 2005 John Wiley & Sons
of the other tests was influenced by a pore pressure that contained a contribution from the prior consolidation history. The magnitude and rate of pore pressure development if drainage is prevented following primary consolidation depend on the time allowed for secondary compression prior to the prevention of further drainage. This is illustrated by the data in Fig. 12.42, which show pore pressure as a function of time for samples that have undergone different amounts of secondary compression. In summary, creep deformation depends on the effective stress path followed and any changes in stress
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TIME EFFECTS ON STRENGTH AND DEFORMATION
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Figure 12.39 Influence of temperature on the initial and final stresses in stress relaxation
tests on Osaka clay—undrained triaxial compression (Murayama, 1969).
Figure 12.40 Creep curves for isotropically and K0-consolidated samples of undisturbed
Haney clay tested in triaxial and plane strain compression (from Campanella and Vaid, 1974). Reproduced with permission from the National Research Council of Canada.
Copyright © 2005 John Wiley & Sons
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RATE EFFECTS ON STRESS–STRAIN RELATIONSHIPS
Figure 12.41 Pore pressure development during undrained creep of San Francisco Bay mud
after consolidation at 100 kPa for 1800 min (from Holzer et al., 1973). Reproduced with permission from the National Research Council of Canada.
Figure 12.42 Pore pressure development under undrained conditions following different periods of secondary compression (from Holzer et al., 1973). Reproduced with permission from the National Research Council of Canada.
with time. Furthermore, time-dependent volumetric response is governed both by the rate of volumetric creep and by the rate of consolidation. The latter is a complex function of drainage conditions and material properties, especially the permeability and compressibility. Because the effective stress path is controlled by the rate of loading and drainage conditions, the separation of consolidation and creep deformations can be difficult in the early stage of time-dependent deformation as given by section BD in Fig. 12.33a. In some cases, a fully coupled analysis of soil–pore fluid interaction with an appropriate time-dependent constitutive model
Copyright © 2005 John Wiley & Sons
is necessary to reconcile the time-dependent deformations observed in the field and laboratory.
12.8 RATE EFFECTS ON STRESS–STRAIN RELATIONSHIPS
An increase in strain rate during soil compression is manifested by increased stiffness, as was noted in Section 12.3. In essence, the state of the soil jumps to the stress–strain curve that corresponds to the new strain rate. Commonly, this rate-dependent stress–strain
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TIME EFFECTS ON STRENGTH AND DEFORMATION
curve, noted by S˘uklje (1957), is the same as if the soil had been loaded from the beginning at the new strain rate. This phenomenon is often observed in clays. Examples are given in Fig. 12.43a for undrained triaxial compression tests of Belfast and Winnipeg clays (Graham et al., 1983a) and Fig. 12.43b for onedimensional compression tests of Batiscan clay (Leroueil et al., 1985).
0.6 5% /h R 0.5 R 0.5% /h 0.05% /h Belfast Clay 4 m σ1c = σv0
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Yield and Strength Envelopes of Clays
(σ1 – σ3)/2σ1c
0.4
The strain rate is defined as ˙ vs ⫽ 兹˙ 2v ⫹ ˙ s2, where ˙ v is the volumetric strain rate and ˙ s is the deviator strain rate (Leroueil and Marques, 1996). Whether the use of this strain rate measure is appropriate or not remains to be investigated. 7
Copyright © 2005 John Wiley & Sons
0.3
R
16%/h
0.2
1%/h
0.25%/h
Winnipeg Clay 11.5 m σ1c > σv0
0.1
CAU Triaxial Compression Tests Relaxation Tests (R)
0
0
4
8
12
16
20
Axial Strain
0
0
50
Effective Stress σv (kPa) 100 150 200
.
εv2
5
10
Strain εv (%)
The undrained shear strength and apparent preconsolidation pressure of soils decrease with decreasing strain rate or increasing duration of testing. Preconsolidation pressures obtained from one-dimensional consolidation tests and undrained shear strengths obtained from triaxial tests are just two points on a soil’s yield envelope in stress space. For a given metastable soil structure, the degree of rate dependency of preconsolidation pressure is similar to that of undrained shear strength (Soga and Mitchell, 1996). If the apparent preconsolidation pressure depends on the strain rate at which the soil is deforming, then the same analogy can be expanded to the assumption that the size of the entire yield envelope is also strain rate dependent (Tavenas and Leroueil, 1977). Figure 12.44 shows a family of strength envelopes corresponding to constant strain rates7 obtained from drained and undrained creep tests on stiff plastic Mascouche clay from Quebec (Leroueil and Marques, 1996). The effective stress failure line of soil is uniquely defined regardless of the magnitude of the strain rate applied in undrained compression. Figure 12.45a shows the failure line of Haney clay (Vaid and Campanella, 1977). The line represents the stress conditions at the maximum ratio of 1 / 3. The data were obtained by various undrained tests, and a unique failure line can be observed. Figure 12.45b shows the undrained stress paths and the critical state line of reconstituted mixtures of sand and clay with plasticity indices ranging from 10 to 30 (Nakase and Kamei, 1986). A unique critical state line can be observed although the rates of shearing are different. The change in undrained shear strength with strain rate results from a difference in generation of excess pore pressures. A decrease in strain rate leads to larger excess pore pressures at failure due to creep deformation.
.
εv2
.
εv1
.
εv1
.
.
εv2
εv1
.
15
εv1
.
20
25
250
εv2 SP1 test SP2 test . εv1 = 2.70 & 10–6 s–1 . εv2 = 1.05 & 10–7 s–1 . εv3 = 1.34 & 10–5 s–1
.
εv3
.
εv1
.
εv1
30
Figure 12.43 Rate-dependent stress–strain relations of clays: (a) undrained triaxial compression tests of Belfast and Winnipeg clay (Graham et al., 1983a) and (b) onedimensional compression tests of Batiscan clay (Leroueil et al., 1985).
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Co py rig hte dM ate ria l
RATE EFFECTS ON STRESS–STRAIN RELATIONSHIPS
499
Figure 12.44 Influence of strain rate on the yield surface of Masouche clay (from Leroueil and Marques, 1996).
Excess Pore Pressure Generation in Normally Consolidated Clays
Excess pore pressure development depends primarily on the collapse of soil structure. Accordingly, strain is the primary factor controlling pore pressure generation. This is shown in Fig. 12.46 by the undrained stress– strain–pore pressure response of normally consolidated natural Olga clay (Lefebvre and LeBouef, 1987). The natural clay specimens were normally consolidated under consolidation pressures larger than the field overburden pressure and then sheared at different strain rates. Although the deviator stress at any strain increases with increasing strain rate, the pore pressure versus strain curves are about the same at all strain rates. At a given deviator stress, the pore pressure generation was larger at slower strain rates as a result of more creep under slow loading. This is consistent with the observation made in connection with undrained creep of clays as discussed in Section 12.7. Straindriven pore pressure generation was also suggested by Larcerda and Houston (1973) who showed that pore pressure does not change significantly during triaxial
Copyright © 2005 John Wiley & Sons
stress relaxation tests in which the axial strain is kept constant. Overconsolidated Clays
Rate dependency of undrained shear strength decreases with increasing overconsolidation, since there is no contraction or collapse tendency observed during creep of heavily overconsolidated clays. Sheahan et al. (1996) prepared reconstituted specimens of Boston blue clay at different overconsolidation ratios and sheared them at different strain rates in undrained conditions. Figure 12.47 shows that the undrained stress path and the strength were much more strain rate dependent for lightly overconsolidated clay (OCR ⫽ 1 and 2) than for more heavily overconsolidated clay (OCR ⫽ 4 and 8). The results also show that the strength failure envelope is independent of strain rate as discussed earlier. The strain rate effects on stress–strain–pore pressure response of overconsolidated structured Olga clays are shown in Fig. 12.48 (Lefebvre and LeBouef, 1987). The natural samples were reconsolidated to the field
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500
12
TIME EFFECTS ON STRENGTH AND DEFORMATION
0.4 Const. Stress Creep Const. Rate of Strain Shear Const. Rate of Loading Shear Aged Samples Const. Load Creep Step Creep Thixotropic Hardened
0.2
Co py rig hte dM ate ria l
(σ1 – σ3)/2σ1c
0.3
0.1
0
0
0.1
0.2
0.3
0.4 (σ1 + σ3)/2σ1c
0.5
0.6
0.7
0.8
(a)
1.0
0.8
0.8
al-
St
0.2
itic
M-15. ε(%/min) . 7&10-1: ε. 1 -2 7&10 : ε2 . 7&10-3: ε3
0.4
0
Cr
Cr
e
in
-L
K0
q/σvc
St
itic
al-
q/σvc
0.4
ate
0.6
ate
0.6
Lin
Lin
e
e
1.0
M-10 . ε(%/min) . 7&10-1: ε. 1 7&10-2: ε2 . 7&10-3: ε3
0.2 0
Cr
Cr
0
0.2
e
e
Lin
–0.4
Lin
–0.4
ate
ate
St
St
al-
al-
itic
–0.2
itic
–0.2
–0.6
e
in
-L
K0
–0.6
0.4
0.6 p/σvc
0.8
1.0
M15 Soil (Plasticity Index = 15)
(b)
0
0.2
0.4
0.6 p/σvc
0.8
1.0
M10 Soil (Plasticity Index = 10)
Figure 12.45 Strain rate independent failure line: (a) Haney clay (from Vaid and Campa-
nella, 1977) and (b) reconstituted mixtures of sand and clay (from Nakase and Kamei, 1986).
overburden pressure. The deformation is brittle, with strain softening indicating development of localized shear failure planes. Up to the peak stress, the response follows what has been described previously, that is the stress–strain response is rate dependent and the pore pressure generation is strain dependent but independent of rate. However, after the peak, the pore pressure
Copyright © 2005 John Wiley & Sons
generation becomes rate dependent. This is due to local drainage within the specimens as the deformation becomes localized. As the time to failure increased, there is more opportunity for local drainage toward the dilating shear band and the measured pore pressure may not represent the overall behavior of the specimens. The difference in softening due to swelling at the fail-
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501
RATE EFFECTS ON STRESS–STRAIN RELATIONSHIPS
100 80 60 Axial Strain Rate 0.1 %/hr 0.5 %/hr 2.6 %/hr 12.3 %/hr
Excess Pore Pressure Δu (kPa)
20 0 140
1
120 100 80 40 20 0
2
3
4
5
6
7 8 Axial Strain (%)
Axial Strain Rate 0.1 %/hr 0.5 %/hr 2.6 %/hr 12.3 %/hr
60
1
2
3
4
5
6
7 8 Axial Strain (%)
(σa – σr)/2σvm
Figure 12.46 Stress–strain and pore pressure–strain curves for normally consolidated Olga clay (from Lefebvre and LeBouef, 1987).
0.4
0.3
0.2
0.1
Effective Stress State at Peak for Axial Strain Rate = 0.051 %/hr Axial Strain Rate = 0.50 %/hr Axial Strain Rate = 5.0 %/hr Axial Strain Rate = 49 %/hr
OCR=8 Negligible Rate Effect
Initial K0 Consolidation State Effective Stress Path for Axial Strain Rate = 0.50 %/hr
OCR=4
OCR=1 Large Rate Effect
OCR=2
OCR=1
OCR=2
OCR=4
0.0
70
0.2
OCR=8
0.4
0.6
0.8 (σa + σr)/2σvm
-0.1
Figure 12.47 Rate dependency stress path and strength of overconsolidated Boston blue clay (from Sheahan et al., 1996).
Copyright © 2005 John Wiley & Sons
Excess Pore Pressure Δu (kPa)
40
Overconsolidated Olga Clay Undrained Triaxial Compression Tests Initial Isotropic Confining Pressure = 17.6 kPa
60 50 40 30 20
Co py rig hte dM ate ria l
Deviator Stress q (kPa)
120
Deviator Stress q (kPa)
Normally Consolidated Olga Clay Undrained Triaxial Compression Tests Initial Isotropic Confining Pressure =137 kPa
10 0
1
2
3
4
5
6
Axial Strain Rate 0.1 %/hr 0.5 %/hr 2.5 %/hr 12.3 %/hr 7
8
Axial Strain (%)
20
10
0
1
2
3
4
5
6
Axial Strain Rate 0.1 %/hr 0.5 %/hr 2.6 %/hr 12.3 %/hr 7
8
Axial Strain (%)
Figure 12.48 Stress–strain and pore pressure–strain curves for overconsolidated Olga clay (from Lefebvre and LeBouef, 1987).
ure plane results in apparent rate dependency at large strains. Similar observations were made by Atkinson and Richardson (1987) who examined local drainage effects by measuring the angles of intersection of shear bands with very different times of failure. Rate Effects on Sands
Similar rate-dependent stress–strain behavior is observed in sands (Lade et al., 1997), but the effects are quite small in many cases (Tatsuoka et al., 1997; Di Benedetto et al., 2002). An example of time dependency observed for drained plane strain compression tests of Hostun sand is shown in Fig. 12.49 (Matsushita et al., 1999). The stress–strain curves for three different strain rates (1.25 ⫻ 10⫺1, 1.25 ⫻ 10⫺2, and 1.25 ⫻ 10⫺3 %/min) are very similar, indicating very small rate effects when the specimens are sheared at a constant strain rate. On the other hand, the change from one rate to another temporarily increases or decreases the resistance to shear. The influence of acceleration rather than the rate is reflected by the significant creep deformation and stress relaxation of this rateinsensitive material as shown the figure. This is differ-
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12
6.0
TIME EFFECTS ON STRENGTH AND DEFORMATION
Variation of Stress-strain curve by Constant Strain Rate Tests at Axial Strain Rates = 0.125, 0.0125 and 0.00125 %/min. A Very Small Rate Effect Is Observed for Continuous Loading.
Creep CRS at 0.125%/min
CRS at 0.00125%/min
CRS at 0.125%/min
5.0
Creep Stress Relaxation CRS at 0.00125%/min
4.5 Creep
4.0
CRS at 0.125%/min
3.5
Creep
Accidental Pressure Drop Followed by Relaxation Stage
CRS at 0.125%/min
3.0 0
1
Esec = Δq/εa, Eeq = (Δq)SA / (εa)SA NSF-Clay Isotropically Consolidated Emax = 239 MPa p0 = 300 kPa
200
100
Co py rig hte dM ate ria l
Stress Ratio σa /σr
5.5
300 Young's Modulus, Esec or Eeq (MPa)
502
2
3
4
5
6
7
8
Shear Strain γ = εa – εr (%)
0
10-3
10-2
10-1
100
Axial Strain, εa or Single Amplitude of Cyclic Axial Strain, (εa)SA (%)
Figure 12.49 Creep and stress relaxation of Hostun sand
(from Matsushita et al., 1999).
Figure 12.50 Clay stiffness degradation curves at three
strain rates (from Shibuya et al., 1996).
ent from the observations made for clays as shown in Fig. 12.43 in which a unique stress–strain–strain rate relationship was observed. Hence, the modeling of stress–strain–rate behavior of sands appears to be more complicated than that of clays, and further investigation is needed, as time-dependent behavior of sands can be of significance in geotechnical construction as discussed further in Section 12.10.
Although the magnitude is small, the strain rate dependency of the stress–strain relationship is observed even at small strain levels for clays. The stiffness increases less than 6 percent per 10-fold increase in strain rate (Leroueil and Marques, 1996). The rate dependency on stiffness degradation curves measured by monotonic loading of a reconstituted clay is shown in Fig. 12.50 (Shibuya et al., 1996). At different strain levels, the increase in the secant shear modulus with shear strain rate ˙ is often expressed by the following equation (Akai et al., 1975; Isenhower and Stokoe, 1981; Lo Presti et al., 1996; Tatsuoka et al., 1997): G( ) ⫽
G
log ˙ G( , ˙ ref)
(12.34)
where G is the increase in secant shear modulus with increase in log strain rate log ˙ , and G( , ˙ ref) is the secant shear modulus at strain and reference strain rate ˙ ref. The magnitude is large in clays, considerably less in silty and clayey sands, and small in clean sands (Lo Presti et al., 1996; Stokoe et al., 1999). The variation of G with strain is shown in Fig. 12.51 for dif-
Copyright © 2005 John Wiley & Sons
Coefficient of Strain Rate, (γ)
Stiffness at Small and Intermediate Strains
Shear Strain, γ (%)
Figure 12.51 Strain rate parameter G and strain level for
several clays (from Lo Presti et al., 1996).
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MODELING OF STRESS–STRAIN–TIME BEHAVIOR
where ƒ is the frequency and c is the cyclic shear strain amplitude. Using Eq. (12.35), Matesic and Vucetic (2003) report values of G of 2 to 11 percent for clays and 0.2 to 6 percent for sands as the strain rate increases 10-fold. The values of G in general decreased when the applied cyclic shear strain increased from 5 ⫻ 10⫺4 percent to 1 ⫻ 10⫺2 percent. It should be noted that the strain range examined is within the non-linear elastic range (zone 1 to zone 2 in Section 11.17). The monotonic loading data presented in Fig. 12.51 show that the rate effect becomes more pronounced at larger strain, that is, as plastic deformations become more significant. Hence, it is possible that the fundamental mechanisms of rate dependency are different at small elastic strain levels than at larger plastic strains. Small strain damping shows more complex frequency dependency, as shown in Fig. 12.53 (Shibuya et al., 1995; Meng and Rix, 2004). At a frequency of more than 10 Hz, the damping ratio increases with increased frequency, possibly due to pore fluid viscosity effects. As the applied frequency decreases, the damping ratio decreases. However, at a frequency less than 0.1 Hz, the damping ratio starts to increase with decreasing frequency. This may result from creep of the soil (Shibuya et al., 1995).
ferent plasticity clays (Lo Presti et al., 1996). The magnitude of rate dependency increases with strain level, especially for strain levels larger than 0.01 percent, which is within the preplastic region zone 3 described in Section 11.17. Rate Effects During Cyclic Loading
Co py rig hte dM ate ria l
The frequencies of cyclic loading to which a soil is subjected can vary widely. For example, the frequency of sea and ocean waves is in the range of 10⫺2 to 10⫺1 Hz, earthquakes are in the range of 0.1 to a few hertz, and machine foundations are in the range of 10 to 100 Hz. Similarly to monotonic loading, the effect of loading frequency on shear modulus degradation is small in clean, coarse-grained soils (Bolton and Wilson, 1989; Stokoe et al., 1995), but the effect becomes more significant in fine-grained soils (Stokoe et al., 1995; d’Onofrio et al., 1999; Matesic and Vucetic, 2003; Meng and Rix, 2004). An example of frequency effects on a shear modulus degradation curve for a clay obtained from cyclic loading is shown in Fig. 12.50 along with the monotonic data. Figure 12.52 shows the effect of frequency on shear modulus of several soils at very small shear strains (less than 10⫺3 percent) measured by torsional shear and resonant column apparatuses (Meng and Rix, 2004). The effect is 10 percent increase per log cycle at most. At a given frequency of cyclic loading, the strain rate applied to a soil increases with applied shear strain as shown by the equation below: ˙ ⫽ 4ƒ c
12.9 MODELING OF STRESS–STRAIN–TIME BEHAVIOR
Constitutive models are needed for the solution of geotechnical problems requiring the determination of de-
(12.35)
250
Shear Modulus G (MPa)
Vallencca Clay
200
Sandy Elastic Silt
150 100
Kaolin
Kaolin
Sandy Lean Clay
50
Sandy Silty Clay
Kaolin Subgrade Fat Clay
0
10-2
10-1
503
100 Frequency (Hz)
101
102
Figure 12.52 Rate dependency of cyclic small strain stiffness of a sandy elastic silt (from
Meng and Rix, 2004).
Copyright © 2005 John Wiley & Sons
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504
12
TIME EFFECTS ON STRENGTH AND DEFORMATION
6
Clayey Subgrades
5 Sandy Lean Clay
4 Pisa Clay Vallencca Clay
3
Fat Clay Sandy Silty Clay
2
Co py rig hte dM ate ria l
Damping (%)
Augusta Clay
Sandy Elastic Silt
1
Kaolin
10-2
10-1
100
101
102
Frequency (Hz)
Figure 12.53 Effect of strain rate of damping ratio of soils (from Shibuya et al., 1995 and
Meng and Rix, 2004).
formations, displacements, and strength and stability changes that occur over time periods of different lengths. Various approaches have been used, including empirical curve fitting, extensions of rate process theory, rheological models, and advanced theories of viscoelasticity and viscoplasticity. Owing to the complexity of stress states, the many factors that influence the creep and stress relaxation properties of a soil, and the difficulty of accounting for concurrent volumetric and deviatoric deformations in systems that are many times undergoing consolidation as well as secondary compression or creep, it is not surprising that development of general models that can be readily implemented in engineering practice is a challenging undertaking. Nonetheless, some progress has been made in establishing functional forms and relationships that can be applied for simple analyses and comparisons, and one of these is developed in this section. A complete review and development of all recent theories and proposed relationships for creep and stress relaxation is beyond the scope of this book. Comprehensive reviews of many models for representation of the timedependent plastic response of soils are given in Adachi et al. (1996). General Stress–Strain–Time Function
Strain Rate Relationships between axial strain rate ˙ and time t of the type shown in Figs. 12.4 and 12.5
冉冊
ln ˙ ⫽ ln ˙ (t1,D) ⫺ m ln
˙ ˙ (t1,D)
冉冊
⫽ ⫺m ln
t t1
(12.36)
冋 册 ˙ ˙ (t,D0)
ln
Copyright © 2005 John Wiley & Sons
⫽ D
(12.38)
or
ln ˙ ⫽ ln ˙ (t,D0) ⫹ D
(12.39)
in which ˙ (t,D0) is a fictitious value of strain rate at D ⫽ 0, a function of time after start of creep, and is the slope of the linear part of the log strain rate versus stress plot. From Eqs. (12.37) and (12.39)
冉冊
ln ˙ (t1,D) ⫺ m ln
t ⫽ ln ˙ (t,D0) ⫹ D t1
For D ⫽ 0,
or
(12.37)
where ˙ (t1,D) is the axial strain rate at unit time and is a function of stress intensity D, m is the absolute value of the slope of the straight line on the log strain rate versus log time plot, and t1 is a reference time, for example, 1 min. Values of m generally fall in the range of 0.7 to 1.3 for triaxial creep tests; lower values are reported for undrained conditions than for drained conditions. For the development shown here, the stress intensity D is taken as the deviator stress (1 ⫺ 3). A shear stress or stress level could also be used. The same data plotted in the form of Figs. 12.3, 12.31, and 12.32 can be expressed by
can be expressed by ln
t t1
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(12.40)
冉冊
ln ˙ (t,D0) ⫽ ln ˙ (t1,D0) ⫺ m ln
t t1
MODELING OF STRESS–STRAIN–TIME BEHAVIOR
(12.41)
in which ˙ (t1,D0) is the value of strain rate obtained by projecting the straight-line portion of the relationship between log strain rate and deviator stress at unit time to a value of D ⫽ 0. Designation of this value by A and substitution of Eq. (12.41) into Eq. (12.39) gives
冉冊 t t1
vided the variation of strength with water content is known. Since normal strength tests are considerably simpler and less time consuming than creep tests, the uniqueness of the quantity Dmax can be useful because the results of a limited number of tests can be used to predict behavior over a range of conditions. A further generalization of Eq. (12.43) then is
冉冊
m
t ˙ ⫽ A exp(D) 1 t
(12.42)
Co py rig hte dM ate ria l
ln ˙ ⫽ ln A ⫹ D ⫺ m ln
˙ ⫽ Ae
冉冊 t1 t
(12.44)
where
which may be written
D
505
⫽ Dmax
m
(12.43)
This simple three-parameter equation has been found suitable for the description of the creep rate behavior of a wide variety of soils. The parameter A is shown in Fig. 12.54. Since it reflects an order of magnitude for the creep rate under a given set of conditions, it is in a sense a soil property. A minimum of two creep tests are needed to establish the values of A, , and m for a soil. If identical specimens are tested using different creep stress intensities, a plot of log strain rate versus log time yields the value of m, and a plot of log strain rate versus stress for different values of time can be used to find and A from the slope and the intercept at unit time, respectively. The parameter has units of reciprocal stress. If stress is expressed as the ratio of creep stress to strength at the beginning of creep, D/Dmax, then the dimensionless quantity Dmax should be used. For a given soil and test type, values of Dmax do not vary greatly for different water contents, as the change in with water content is compensated by a change in Dmax. Thus the strain rate versus time behavior for any stress at any water content can be predicted from the results of creep tests at any other water content, pro-
Figure 12.54 Influence of creep stress magnitude on the
creep rate at a given time after stress application.
Copyright © 2005 John Wiley & Sons
D⫽
D Dmax
(12.45)
Strain A general relationship between strain and time is obtained by integration of Eq. (12.43). Two solutions are obtained, depending on the value of m. If ⫽ 1 at t ⫽ t1 ⫽ 1, then ⫽ 1 ⫹
A
1⫺m
exp (D)(t1⫺m ⫺ 1)
when m ⫽ 1 (12.46)
and
⫽ 1 ⫹ A exp (D)ln t
when m ⫽ 1
(12.47)
Creep curve shapes corresponding to these relationships are shown in Fig. 12.55. These curves encompass the variety of shapes shown in Fig. 12.2. A similar equation to Eq. (12.46) was developed by Mesri et al. (1981) from Eq. (12.43). The initial time-independent strain was neglected, and the resulting equation is At1
冉冊
t ⫽ exp(D) 1⫺m t1
1⫺m
(12.48)
It may be seen in Fig. 12.56 that this equation describes the uniaxial creep behavior of three clays very well. Data for both drained and undrained creep are shown. Stress Relaxation Stress decay during stress relaxation is approximately linear with logarithm of time until it levels off at some residual stress after a long time. There is equivalency between creep and stress relaxation in that a general phenomenological model that predicts one can be used to predict the other, as shown by Akai et al. (1975), Lacerda (1976), Borja (1992), and others. For example, Eq. (12.44) takes the following form when stress relaxation is started after deformation at constant rate of strain (Lacerda and Houston, 1973):
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12
TIME EFFECTS ON STRENGTH AND DEFORMATION
Co py rig hte dM ate ria l
506
Figure 12.55 Creep curve shapes predicted by the general stress–strain–time function of Eqs. (12.46) and (12.47).
D D t ⫽ ⫽ 1 ⫺ s log D0 D0 t0
(t ⬎ t0)
where s is the slope of the stress relaxation curve, and the zero subscript refers to conditions at the start of stress relaxation. Also s⫽
#
D
t0 ⫽
(12.49)
(12.50)
h0 ˙
(12.52)
where h0 is the strain rate to give a delay time of t0 ⫽ 1 min before stresses begin to relax. The data presented by Lacerda and Houston (1973) indicate that the values of # and h0 increase with increasing plasticity of the soil. Constitutive Models
where
#⫽
2.3(1 ⫺ m)
(12.51)
The validity of this equation has been established for m ⬍ 1.0. Pore pressures decrease slightly during undrained stress relaxation. Stresses may not begin to relax immediately after the strain rate is reduced to zero. The time t0 between the time that the strain rate is reduced to zero and the beginning of relaxation is a variable that depends on the soil type and the prior strain rate. This is shown schematically in Fig. 12.57. The greater the initial rate of strain to a given deformation, the more quickly relaxation begins. This is a direct reflection of the relative differences in equilibrium soil structures during and after deformation. Values of t0 as a function of prior strain rate are shown in Fig. 12.58 for several soils. These curves can be described empirically by
Copyright © 2005 John Wiley & Sons
Different rheological models have been proposed for the mathematical description of the stress–strain–time behavior of soils that are made up of combinations of linear springs, viscous dashpots, and sliders. In the Murayama and Shibata (1958), Christensen and Wu (1964), and Abdel-Hady and Herrin (1966) models, the dashpots are nonlinear, with stress–flow rate response governed by rate process theory. Rheological models are useful conceptually to aid in recognition of elastic and plastic components of deformation. They are helpful for visualization by analogy of viscous flow that accompanies time-dependent change of structure to a more stable state. Mathematical relationships can be developed in a straightforward manner for the description of creep, stress relaxation, steady-state deformation, and the like in terms of the model constants. In most cases, these relationships are complex and necessitate the evaluation of several parameters that may not be valid for different stress intensities or soil states. Only one-dimensional stresses and deformations are
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Co py rig hte dM ate ria l
MODELING OF STRESS–STRAIN–TIME BEHAVIOR
Figure 12.56 Correspondence between creep strain predicted by Eq. (12.48) and measured
values. Diagrams are from Mesri et al. (1981), which were based on analyses by Semple (1973).
Copyright © 2005 John Wiley & Sons
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507
508
12
TIME EFFECTS ON STRENGTH AND DEFORMATION
12.10
CREEP RUPTURE
As discussed in Section 12.2 and shown in Fig. 12.6, the strength of a soil and the stress–strain curve may be changed as a result of creep. In some cases, such as the drained creep of a compressive soil, the strength may be increased. Changes in strength may be as much as 50 percent or more of the strength measured in normal undrained tests prior to creep.
Co py rig hte dM ate ria l
Causes of Strength Loss During Creep
Figure 12.57 Influence of prior strain rate on stress relaxa-
tion.
Figure 12.58 Influence of prior strain rate on the time to start of stress relaxation (adapted from Lacerda and Houston, 1973).
considered. None appears to exist that has the generality and simplicity of the three-parameter creep Eqs. (12.43), (12.46), and (12.47). Both plasticity and creep are controlled by the motion of dislocations or breakage among soil particles, so it may be physically more correct to predict both plastic and creep deformations with one equation. Two particularly promising approaches are based on an extension of the Cam-clay model to take into account time-dependent volumetric and deviatoric deformations (Kavazanjian and Mitchell, 1980; Borja and Kavazanjian, 1985; Kaliakin and Dafalias, 1990; Borja, 1992; Al-Shamrani and Sture, 1998; Hashiguchi and Okayasu, 2000) and on an elasto-viscoplastic equation developed using flow surface theory (Sekiguchi, 1977, 1984; Matsui and Abe, 1985, 1986, 1988; Matsui et al., 1989; Yin and Graham, 1999) and overstress theory (Adachi and Oka, 1982; Katona, 1984: Kutter and Sathialingham, 1992; Rocchi et al., 2003).
Copyright © 2005 John Wiley & Sons
Loss of strength during creep is particularly important in soft clays deformed under undrained conditions and heavily overconsolidated clays in drained shear. Both of these conditions are pertinent to certain types of engineering problems: the former in connection with stability of soft clays immediately after construction, and the latter in connection with problems of long-term stability. The loss of strength as a result of creep may be explained in terms of the following principles of behavior: 1. If a significant portion of the strength of a soil is due to cementation, and creep deformations cause failure of cemented bonds, then strength will be lost. 2. In the absence of chemical or mineralogical changes the strength depends on effective stresses. If creep causes changes in effective stress, then strength changes will also occur. 3. In almost all soils, shear causes changes in pore pressure during undrained deformation and changes in water content during drained deformation. 4. Water content changes cause strength changes.
These processes are illustrated by the stress paths and effective stress envelope shown schematically in Fig. 12.37. Strength loss in saturated, heavily overconsolidated clays tested under undrained conditions has also been reported, for example, Casagrande and Wilson (1951), Goldstein and Ter-Stepanian (1957), and Vialov and Skibitsky (1957). This may be explained as follows. Shear deformations cause dilation and the development of negative pore pressures, which do not develop uniformly throughout the sample but concentrate along planes where the greatest shearing stresses and strains develop. With time during sustained loading, water migrates into zones of high negative pore pressures leading to softening and strength decrease relative to the strength in ‘‘normal’’ undrained strength tests. This leads to shear band formation.
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CREEP RUPTURE
in turn, a function of deformation rates, the hydraulic conductivity, and the surrounding water pressure and drainage conditions. The time to failure of heavily overconsolidated clays in which negative pore water pressures develop as a result of unloading is best estimated on the basis of drained strengths, effective stresses, and consideration of the rate of swelling that is possible for the particular clay and ambient stress and groundwater conditions. An exception would be when strength loss results from the time-dependent rupture of cementing bonds. In this case, sustained load creep tests in the laboratory may allow establishment of a stress level versus time-to-failure relationship. For soils subject to failure during undrained creep, the time to failure is usually a negative exponential function of the stress, for stresses greater than some limiting value below which no failure develops even after very long times.8 The relationship between deviator stress, normalized to the pretest major principal effective stress, and time to failure for Haney clay is shown in Fig. 12.60. These and similar data define cer-
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This process is shown in Fig. 12.59 with reference to an effective stress failure envelope for a heavily overconsolidated clay. The effective stress path is represented by AB, and AC represents the total stress path in a conventional consolidated-undrained (CU) test. The negative pore pressure at failure is CB. If a creep stress DE is applied to the same clay, a negative pore pressure EF is induced. This negative pore pressure dissipates during creep, and the clay in the shear zone swells. At the end of the creep period, the effective stress will be as represented by point E. Further shear starting from these conditions leads to strength G, which is less than the original value at B. It is evident also that if the negative pore water pressure is large enough, and the sustained load is applied long enough, then point E could reach the failure envelope. This appears to have been the conditions that developed in several cuts in heavily overconsolidated brown London clay, which failed some 40 to 70 years after excavations were made (Skempton, 1977).
509
Time to Failure
The time to failure of soils susceptible to strength loss under sustained stresses depends on the rates at which pore pressures develop and at which water can migrate into or out of the critical shear zone. These rates are,
8 This critical stress below which creep rupture does not occur has been termed the upper yield, the lower yield being the stress below which deformations are elastic (Murayama and Shibata 1958, 1964).
Figure 12.59 Stress paths for normal undrained shear and drained creep of heavily overconsolidated clay.
Copyright © 2005 John Wiley & Sons
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TIME EFFECTS ON STRENGTH AND DEFORMATION
tain principles relating to the probability of creep rupture and the time to failure:
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1. Values of the parameter m less than 1.0 in Eqs. (12.43) through (12.46) are indicative of a high potential strength loss during creep and eventual failure (Singh and Mitchell, 1969). 2. The minimum strain rate ˙ min prior to the onset of creep rupture decreases, and the time to failure increases, as the stress intensity decreases, as shown in Fig. 12.61 for Haney clay. The relationship is unique, as may be seen in Fig. 12.62, which shows that tƒ ⫽
Figure 12.60 Time to rupture as a function of creep stress
for Haney clay (Campanella and Vaid, 1972).
C
˙ min
Values of the constant C accurate to about 0.2 log cycles are given in Table 12.2. 3. The strain at failure is a constant independent of stress level, as shown in Fig. 12.63. The failure
Figure 12.61 Creep rate behavior of K0-consolidated, undisturbed Haney clay under axisymmetric loading (Campanella and Vaid, 1972).
Copyright © 2005 John Wiley & Sons
(12.53)
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SAND AGING EFFECTS AND THEIR SIGNIFICANCE
ƒ ⫽ constant ⫽
1 1⫺m
˙ mintƒ ⫽
C 1⫺m
(12.56)
Thus, the constant in Eq. (12.53) is defined by C ⫽ (1 ⫺ m)ƒ
Values of ƒ for Haney clay tested in three ways are shown in Fig. 12.63, and values of C and m are in Table 12.3. The agreement between predicted and measured values of C is reasonable. Predictions of the time to failure under a given stress may be made in the following way. Strain at failure can be determined by either a creep rupture test or by a normal shear or compression test. If a normal strength test is used, then the rate of strain must be slow enough to allow pore pressure equalization or drainage, depending on the conditions of interest, and the stress history and stress system should simulate those in the field. Parameter m can be established from a creep test, and then C can be computed from Eq. (12.57). Values of A and are established from creep tests at two stress intensities. Then, for t1 ⫽ 1,
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Figure 12.62 Relationship between time to failure and min-
imum creep rate (from Campanella and Vaid, 1974). Reproduced with permission from the National Research Council of Canada.
C ⫽ ˙ mintƒ ⫽ A exp(D)t1⫺m ƒ
strain is taken as the strain corresponding to the minimum strain rate. For the case of undrained creep rupture, this is consistent with the concept that pore pressure development is uniquely related to strain and independent of the rate at which it accumulates (Lo, 1969a, 1969b).
The relationship expressed by Eq. (12.53) results directly from the fact that the strain at the point of minimum strain rate is a constant independent of stress or strain rate. The general stress–strain rate–time function [see Eq. (12.43)] describes the strain rate–time behavior until ˙ min is reached. For t1 ⫽ 1 and ⫽ 0 at t ⫽ 0, the corresponding strain–time equation is ⫽
A
1⫺m
exp(D)t1⫺m
1 1⫺m
˙ t mt1⫺m
(12.58)
and corresponding values of D and tƒ can be calculated using Eq. (12.58) rewritten as ln tƒ ⫽
1
1⫺m
冋 冉冊 册 ln
C ⫺ D A
(12.59)
Other constitutive models are available to model the complex time-dependent behavior under various loading conditions. For example, Sekiguchi (1977) developed a viscoplastic model that gives excellent representations of strain rate effects on undrained stress–strain behavior, stress relaxation, and creep rupture of normally consolidated clays. Other models listed in Section 12.9 are able to simulate timedependent behavior in a similar manner.
(12.54)
By setting ⫽ 0 at t ⫽ 0, the assumption is made that there is no instantaneous deformation. Substitution for A exp(D) in Eq. (12.54) gives ⫽
(12.57)
(12.55)
which at the point of minimum strain rate becomes
Copyright © 2005 John Wiley & Sons
12.11 SAND AGING EFFECTS AND THEIR SIGNIFICANCE
Over geological time, lithification and chemical reactions can change sand into sandstone or clay into mudstone or shale. However, even over engineering time, behavior of soils can alter as stresses redistribute after construction (Fookes et al., 1988). As discussed in the previous sections, it is well established that finegrained soils and clays have properties and behavior that change over time as a result of consolidation,
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TIME EFFECTS ON STRENGTH AND DEFORMATION
Table 12.2
Creep Rupture Parameters for Several Clays
Soil Undisturbed Haney clay, N.C.b Undisturbed Haney clay, N.C.b Undisturbed Haney clay, N.C.b Undisturbed Seattle clay, O.C.c Undisturbed Tonegawa loamc Undisturbed Redwood City clay, N.C.c Undisturbed Bangkok mudc Undisturbed Osaka clayc
Test Typea
Creep Rate Parameter, m
C ⫽ (˙min tƒ) (0.2 log cycles)
ICU
0.7
1.2
ACU
0.4
0.2
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512
ACU-PS
0.5
0.3
ICU
0.5
0.6
U
0.8
1.6
ICU
0.75
2.8
ICU
0.70
1.4
1.0
0.07
a
ICU, isotropic consolidated, undrained triaxial; ACU, K0 consolidated, undrained triaxial; ACU-PS, K0 consolidated, plane strain; and U, compression test. b Data from Campanella and Vaid (1974). c Data from Singh and Mitchell (1969).
Figure 12.63 Axial strain at minimum strain rate as a function of creep stress for undisturbed Haney clay (from Campanella and Vaid, 1974). Reproduced with permission from the National Research Council of Canada.
Copyright © 2005 John Wiley & Sons
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SAND AGING EFFECTS AND THEIR SIGNIFICANCE
Table 12.3 Predicted and Measured Values of C for Haney Clay
Test Condition
Creep Rate Parameter m (from Table 12.2)
(from Fig. 12.63)
C Predicted by Eq. (12.57)
C Measured
ICUa ACUb ACU-PSc
0.7 0.4 0.5
2.8 0.3 0.5
0.84 0.18 0.25
1.2 0.20 0.30
shear, swelling, chemical and biological changes, and the like. Until recently it has not been appreciated that cohesionless soils exhibit this behavior as well. Much recent field evidence of the changing properties of granular soils over time is now available and these data suggest that recently disturbed or deposited granular soils gain stiffness and strength over time at constant effective stress—a phenomenon called aging. The evidence includes the time-dependent increase in stiffness and strength of densified sands as measured by cone penetration resistance (Mitchell and Solymer, 1984; Thomann and Hryciw, 1992; Ng et al., 1998) and the setup of displacement piles in granular materials (Astedt et al., 1992; York et al., 1994; Chow et al., 1998; Jardine and Standing, 1999; Axelsson, 2000). Hypotheses to explain this phenomenon include both creep processes and chemical and biological cementation processes. The discussion in this section is focused primarily on granular soils as the relevant aspects for clays are treated in detail throughout other sections of the book. Increase in Shear Modulus with Time
As discussed in Section 12.3, the shear modulus at small strain is known to increase with time under a confining stress, and this is considered to be the consequence of aging. This behavior can be quantified by a coefficient of shear modulus increase with time using the following formula (Anderson and Stokoe, 1978):
ΔG : Modulus Increase in Every 10-fold Time Increase G1000 : Modulus at 1,000 min
0.30
ΔG/G1000 = 0.03PI 0.5
Modulus Increase Ratio ΔG/G1000
Isotropic consolidated, undrained triaxial. Anisotropic, consolidated, undrained triaxial. c Anisotropic consolidated, undrained, plane strain. Data from Campanella and Vaid (1974). b
idation, t2 is some time of interest thereafter, G is the change in small strain shear modulus from t1 to t2, G1000 is the shear modulus measured after 1000 min of constant confining pressure, which must be after completion of primary consolidation, and NG is the normalized shear modulus increase with time. Large increase in stiffness due to aging is represented by large values of IG or NG. In general, the measured NG value for clays ranges between 0.05 and 0.25. The aging effect also increases with an increasing plasticity index as shown in Fig. 12.64 (Kokusho, 1987). The data in the figure have been supplemented by values of G/G for several sands compiled by Jamiolkowski (1996). Mesri et al. (1990) report that NG for sands varies between 0.01 and 0.03 and increases as the soil becomes finer. Jamiolkowski and Manassero (1995) give values of 0.01 to 0.03 for silica sands, 0.039 for sand with 50 percent mica, and 0.05 to 0.12 for carbonate sand. Experimental results show that the rate of increase in stiffness with time for very loose carbonate sand increases as the stress level increases (Howie et al., 2002). Isotropic stress state resulted in a slower rate of increase in stiffness. There is only limited field data that shows evidence of aging effects on stiffness. Troncoso and Garces (2000) measured shear wave velocities using downhole
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a
ƒ
513
0.25
0.20
Clay Marcuson et al. (1972) Afifi et al. (1973) Trudeau et al. (1973) Anderson et al. (1973) Zen et al. (1978) Kokusho et al. (1982) Umehara et al. (1985) Sand Jamiolkowski (1996)
0.15
f
0.10
0.05
a = Ticino Sand (Silica) b = Hokksund Sand (Silica) c = Messina Sand and Gravel (Silica) d = Glauconite Sand (Quartz/Glauconite) e = Quiou Sand (Carbonate) f = Kenya Sand (Carbonate)
e
d c
a, b
IG ⫽ G/log(t2 /t1) NG ⫽ IG /G1000
(12.60) (12.61)
where IG is the coefficient of shear modulus increase with time, t1 is a reference time after primary consol-
Copyright © 2005 John Wiley & Sons
0.00
0
20
40
60
80
100
Plasticity Index PI Figure 12.64 Modulus increase ratio for clays (from Kokusho, 1987), supplemented by the data for sands (from Jamiolkowski, 1996).
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TIME EFFECTS ON STRENGTH AND DEFORMATION
and seepage blanket. Due to large depths of the loose sand deposit requiring densification, a two-stage densification program was performed. The upper 25 m of sand (and a 5- to 10-m-thick sand pad placed by hydraulic filling of the river) was densified using vibrocompaction. Deposits between depths of 25 to 40 m were densified by deep blasting. During the blasting operations, it was observed that the sand exhibited both sensitivity—that is, strength loss on disturbance—and aging effects. A typical example of the initial decrease in penetration resistance after blasting densification and subsequent increase with time is shown in Fig. 12.66. Initially after improvement, there was in some cases a decrease in penetration resistance, despite the fact that surface settlements ranging from 0.3 to 1.1 m were measured. With time (measured up to 124 days after improvement), however, the cone penetration resistance was
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wave propagation tests in low-plasticity silts with fines contents from 50 to 99 percent at four abandoned tailing dams in Chile. The shear modulus normalized by the vertical effective stress is plotted against the age of the deposit in Fig. 12.65. The age of the deposits is expressed as the time since deposition. Although the soil properties vary to some degree at the four sites,9 very significant increase in stiffness at small strains can be observed after 10 to 40 years of aging. The degree to which secondary compression could have contributed to this increase is not known. Time-Dependent Behavior after Ground Improvement
Stiffness and strength of sand increase with time after disturbance and densification by mechanical processes such as blasting and vibrocompaction. Up to 50 percent or more increase in strength has been observed over 6 months (Mitchell and Solymer, 1984; Thomann and Hryciw, 1992; Charlie et al., 1992; Ng et al., 1998; Ashford, et al., 2004) as measured by cone penetration testing. The Jebba Dam project on the Niger River, Nigeria, was an early well-documented field case where aging effects in sands were both significant and widespread (Mitchell and Solymer, 1984). The project involved the treatment of foundation soils beneath a 42-m-high dam
Figure 12.65 Normalized shear modulus as function of aging of tailings (from Troncoso and Garces, 2000).
9
The four sites identified by Troncoso and Garces (2000) are called Barahona, Cauquenes, La Cocinera, and Veta del Agua and the aging times between abandonment and testing were 28, 19, 5, and 2 years, respectively. The tailing deposits at Barahona had a liquid limit of 41 percent and a plastic limit of 14 percent, whereas those at the other three sites had liquid limits of 23 to 29 percent and plastic limits of 2 to 6 percent.
Copyright © 2005 John Wiley & Sons
Figure 12.66 Effect of time on the cone penetration resistance of sand following blast densification at the Jebba Dam site.
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SAND AGING EFFECTS AND THEIR SIGNIFICANCE
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found to increase by approximately 50 to 100 percent of the original values. Similar behavior was found following blast densification of hydraulic fill sand that had been placed for construction of Treasure Island in San Francisco Bay more than 60 years previously (Ashford et al., 2004). Aging effects were also observed after placement of hydraulic fill working platforms in the river at the Jebba Dam site and after densification by vibrocompaction as shown in Figs. 12.67 and 12.68. In the case of vibrocompaction, however, there was considerable variability in the magnitude of aging effects throughout the site. Because of the greater density increase caused by vibrocompaction than by blast densification, no initial decrease in the penetration resistance was observed at the end of the compaction process. Charlie et al. (1992) found a greater rate of aging after densification by blasting for sands in hotter climates than in cooler climates and suggested a correlation between the rate of aging and mean annual air temperature for available field data as shown in Fig. 12.69. In the figure, the increase in the CPT tip resistance (qc) with time is expressed by the following equation:
Figure 12.68 Effect of time on the cone penetration resistance of hydraulic fill sand after placement at the Jebba Dam site.
(12.62)
where N is the number of weeks since disturbance and K expresses the rate of increase in tip resistance in logarithmic time.
Mitchell and Solymer (1984)
1.0
Empirical Constant K
qc (N weeks) ⫽ 1 ⫹ K log N qc (1 week)
515
Schmertmann (1987) and Fordham et al. (1991)
Charlie et al. (1992)
0.1
Jefferies et al. (1988)
0.02 -10
0
10
20
30
40
Temperature (°C)
Figure 12.69 Rate of increase of normalized CPT tip resis-
tance against temperature for different cases of reported aging effects after blasting (by Charlie et al., 1992).
Figure 12.67 Effect of time on the cone penetration resistance of sand following vibrocompaction densification at the Jebba Dam site.
Copyright © 2005 John Wiley & Sons
Schmertmann (1991) postulated that a ‘‘complicated soil structure’’ is present in freshly deposited soil. The structure then becomes more stable by ‘‘drained dispersive movements’’ of soil particles. He suggests that stresses would arch from softer, weaker areas to stiffer zones with time, leading to an increase in K0 with time. Mitchell and Solymar (1984) suggested that the cementation of particles may be the mechanism of aging
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TIME EFFECTS ON STRENGTH AND DEFORMATION
of sands, similar to diagenesis in locked sands and young rocks (Dusseault and Morgenstern, 1979; Barton, 1993) in which grain overgrowth has been observed. However, others have questioned whether significant chemical reactions can occur over the short time of observations. In addition, there is some evidence of aging in dry sands wherein chemical processes would be anticipated to be very slow.
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Setup of Displacement Piles
with time (Axelsson, 2000). Evidence suggests that piles in silts and find sands set up more than those in coarse sands and gravels (York et al., 1994). Both driven and jacked piles exhibit setup, whereas bored piles do not. Hence, the stress–strain state achieved during the construction processes of pile driving have an influence on this time-dependent behavior and various mechanisms have been suggested to explain this (Astedt et al., 1992; Chow et al., 1998; Bowman, 2002). Unfortunately, at present, there is no conclusive evidence to confirm any of the proposed hypotheses. Despite the many field examples and laboratory studies on aging effects, there is still uncertainty about the mechanism(s) responsible for the phenomenon. Understanding the mechanism(s) that cause aging is of direct practical importance in the design and evaluation of ground improvement, driven pile capacity, and stability problems where strength and deformation properties and their potential changes with time are important. Mechanical, chemical, and biological factors have been hypothesized for the cause of aging. Biological processes have so far been little studied; however, mechanical and chemical phenomena have been investigated in more detail, and some current understanding is summarized below.
Much field data indicates that the load-carrying capacity of a pile driven into sand may increase dramatically over several months, long after pore pressures have dissipated (e.g., Chow et al., 1998; Jardine and Standing, 1999). The amount of increase is highly variable, ranging from 20 to 170 percent per log cycle of time as shown in Fig. 12.70 (Chow et al., 1998; Bowman, 2002). Most of the increase in capacity occurs along the shaft of the pile as the radial stress at rest increases
12.12
Figure 12.70 Increase in total and shaft capacity with time
for displacement piles in sand (from Chow et al., 1998 and Bowman, 2003).
Copyright © 2005 John Wiley & Sons
MECHANICAL PROCESSES OF AGING
Creep is hypothesized as the dominant mechanism of aging of granular systems on an engineering timescale by Mesri et al. (1990) and Schmertmann (1991). Increased strength and stiffness does not occur solely from the change in density that occurs during secondary compression. Rather, it is due to a continued rearrangement of particles resulting in the increased macrointerlocking of particles and the increased microinterlocking of surface roughness. This is supported by the existence of locked sands (Barton, 1993; Richards and Barton, 1999), which exhibit a tensile strength even without the presence of binding cement. Some micromechanical explanations of the process are given in Section 12.3. Although no increase in stiffness was detected when glass balls were loaded isotropically (Losert et al., 2000), sand has been found to increase in strength and stiffness under isotropic stress conditions (Daramola, 1980; Human, 1992). These increases develop even under isotropic confinement because the angular particles can lock together in an anisotropic fabric. It has been shown that more angular particles produce materials more susceptible to creep deformations (Mejia et al., 1988, Human, 1992, Leung et al., 1996). Isotropic
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CHEMICAL PROCESSES OF AGING
12.13
Stress State at Creep: p = 600 kPa and q = 800 kPa All Samples Were Prepared With Relative Density of Approximately 70%.
Deviatoric Strain (%)
0.25 0.20 Montpellier Natural Sand
0.15 Glass Ball
0.10
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compression tests by Kuwano (1999) showed that radial creep strains were greater than axial strains in soils with angular particles than in soils with rounded particles due to a more anisotropic initial fabric. Angular particles can result in longer duration of creep and a greater aging effect since they have a larger range of stable contacts and the particles can interlock. As spherical particles rearrange more easily than elongated ones (Oda, 1972a), rounder particles initially creep at a higher rate before settling into a stable state. Hence, any aging effect on rounded particles tends to disappear quickly when the soil is subjected to new stress state. When a constant shear stress is applied to loose sand, large creep accompanied by volumetric contraction is observed (Bopp and Lade, 1997). Higher contact forces due to loose assemblies contribute to increased particle crushing, contributing to contraction behavior. Hence, decrease in volume by soil crushing leads to increase in stiffness and strength. Field data suggest that displacement piles in medium-dense to dense sands set up more than those in loose sand (York et al., 1994). Dense granular materials may dilate with time depending on the applied stress level during creep as shown in Fig. 12.71 (Bowman and Soga, 2003). Initially, the soil contracts with time, but then at some point the creep vector rotates and the dilation follows. Similar observations were made by Murayama et al. (1984) and Lade and Lui (1998). This implies that sands at a high relative density will set up more as more interlock between particles may occur (Bowman, 2002). The laboratory observation of initial contraction followed by dilation conveniently explains the field data of dynamic compaction where the greater initial losses and eventual gains in stiffness and strength of sands are found close to the point of application where larger shear stresses are applied to give dilation (Dowding and Hryciw, 1986; Thomann and Hryciw, 1992; Charlie et al., 1992). Increased strength and stiffness due to mechanical aging occurs predominantly in the direction of previously applied stress during creep (Howie et al., 2002). No increase was observed when the sand was loaded in a direction orthogonal to that of the applied shear stress during creep (Losert et al., 2000).
CHEMICAL PROCESSES OF AGING
Chemical processes are a possible cause of aging. Historically, the most widespread theory used to explain aging effects in sand has involved interparticle bond-
Copyright © 2005 John Wiley & Sons
517
Leighton Buzzard Uniform Silica Sand
0.05 0.00
10 1
Volumetric Strain (%)
0.05
10 2
10 3
10 4
10 5
Montpellier Natural Sand
0.00
-0.05 -0.10
Dilation
Glass Ball
-0.15
Leighton Buzzard Uniform Silica Sand
-0.20 -0.25
-0.30 10 0
10 1
10 2
10 3
10 4
10 5
Time (s)
Figure 12.71 Dilative creep observed in triaxial creep tests
of dense fine sand (by Bowman and Soga, 2003).
ing. Terzaghi originally referred to a ‘‘bond strength’’ in connection with the presence of a quasipreconsolidation pressure in the field (Schmertmann, 1991). Generally, this mechanism has been thought of as type of cementation, which would increase the cohesion of a soil without affecting its friction angle. Denisov and Reltov (1961) showed that quartz sand grains adhered to a glass plate over time. They placed individual sand grains on a vibrating quartz or glass plate and measured the force necessary to move the grains as shown in Fig. 12.72. The dry grains were allowed to sit on the plate for varying times and then the plate was submerged, also for varying times, before vibrating began. It was found that the force required to move the sand grains continued to increase up to about 15 days of immersion in water. The cementating agent was thought to be silica-acid gel, which has an amorphous structure and would form a precipitate at
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TIME EFFECTS ON STRENGTH AND DEFORMATION
(1) Glass or Quartz Plate (2) To the Oscillation Generator 2
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1
particle contacts (Mitchell and Solymer, 1984). The increased strength is derived from crystal overgrowths caused by pressure solution and compaction. Strong evidence of a chemical mechanism being responsible for some aging was obtained by Joshi et al. (1995). A laboratory study was made of the effect of time on penetration resistance of specimens prepared with different sands (river sand and sea sand) and pore fluid compositions (air, distilled water, and seawater). After loading under a vertical stress of 100 kPa, the values of penetration resistance were obtained after different times up to 2 years. Strength and stiffness increases were observed in all cases, and a typical plot of load–displacement curves at various times is shown in Fig. 12.73. The effects of aging were greater for the submerged sand than for the dry specimens. Scanning electron micrographs of the aged specimens in distilled water and seawater showed precipitates on and in between sand grains. For the river sand in distilled water, the precipitates were composed of calcium (the soluble fraction of the sand) and possibly silica. For the river sand in seawater, the precipitates were composed of sodium chloride. However, there are several reported cases in which cementation was an unlikely mechanism of aging, at least in the short term. For example, dry granular soils can show an increase in stiffness and strength with time (Human, 1992; Joshi et al., 1995; Losert et al., 2000). Cementation in dry sand is unlikely, as moisture is required to drive solution and precipitation reactions involving silica or other cementation agents. Mesri et al. (1990) used the triaxial test data from Daramola (1980) to argue against a chemical mecha-
䉭
f ––– 3.0 f0
2.0
(1) Without Soaking (2) 42-hour Soaking (3) 6-day Soaking (4) 14-day Soaking
䊊
䉭
4
䉭 䊊
3
1.0
0
10 min
䉭
䊊
&
&
2 1
&
䊊
&
2h
20h
Time
Figure 12.72 Results of vibrating plate experiment from
Denisov and Reltov (1961). Term ƒ / ƒ0 is a measure of the bonding force between sand and glass or quartz plate.
Figure 12.73 Effect of aging on the penetration resistance of River sand (from Joshi et al.,
1995).
Copyright © 2005 John Wiley & Sons
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CHEMICAL PROCESSES OF AGING
application in denser materials. It is also associated with the microinterlocking occurring during the generation of creep strain. The increase in stiffness and strength is observed in the direction of the applied stresses, but the aging effect disappears rather quickly when loads are applied in other directions. Chemical aging can also occur within days depending on such factors as chemical environment and temperature. Some conditions in natural deposits are not replicated in small-scale laboratory testing. Most laboratory tests are done using clean granular materials, whereas in the field there will be impurities, biological activity, and heterogeneity of void ratio and fabric. Furthermore, the introduction of air and other gases during ground improvement may have consequences that have so far not been fully evaluated. Arching associated with dissipation of blast gases and the redistribution of
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nism responsible for aging effects in sands. Figure 12.74 shows the effects of aging on both the stiffness and shear strength of Ham River sand. Four consolidated drained triaxial tests were performed on samples with the same relative density and confining pressure (400 kPa) but consolidated for different periods of time (0, 10, 30, and 152 days) prior to the start of the triaxial tests. The results showed that the stiffness increased and the strain to failure decreased with increasing time of consolidation. Although increased values of modulus were observed, the strain at failure is approximately 3 percent. Mesri et al. (1990) argue that this large strain would destroy any cementation, and therefore another less brittle mechanism must be responsible for the increase in stiffness. In summary, experimental evidence indicates that mechanical aging behavior is enhanced by shear stress
Figure 12.74 Effect of aging on stress–strain relationship of Ham River sand (from Dara-
mola, 1980).
Copyright © 2005 John Wiley & Sons
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TIME EFFECTS ON STRENGTH AND DEFORMATION
stresses through the soil skeleton may also play a role (Baxter and Mitchell, 2004). The boundary conditions associated with penetration testing in rigid-wall cylinders in the laboratory may prevent detection of timedependent increases in penetration resistance that are measured under the free-field conditions in the field.
CONCLUDING COMMENTS
QUESTIONS AND PROBLEMS
1. Find an article about a problem, project, or issue that involves some aspect of the long-term behavior of a soil as an important component. The article may be from a technical journal or magazine or elsewhere. The only requirement is that it involves consideration of time-dependent ground behavior in some way. a. Prepare a one-page informative abstract of the article. b. Summarize the important geotechnical issues in the article and write down what you believe you would need to know to understand them well enough to solve the problem, resolve the issue, advise a client, and so forth. Do not exceed two pages. c. Identify topics, figures, equations, and other material in Chapter 12, if any, that might be useful in addressing the problems.
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12.14
analyses and predictions can be made for large and complex geotechnical structures.
With exception of settlement rate predictions, most soil mechanics analyses used in geotechnical engineering assume limit equilibrium and are based on the assumption of time-independent properties and deformations. In reality, time-dependent deformations and stress changes that result from the time-dependent or viscous rearrangement of the soil structure may be responsible for a significant part of the total ground response. Rate process theory has proven a particularly fruitful approach for the study of time-dependent phenomena in soils at consistencies of most interest in engineering problems, that is, at water contents from about the plastic limit to the liquid limit. From an analysis of the influences of stress and temperature on deformation rates and other evidence, it has been possible to deduce that interparticle contacts are essentially solid and that clay strength derives from interatomic bonding in these contacts. The strength depends on the number of bonds per unit area, and the constant of proportionality between number of bonds and strength is essentially the same for all silicate minerals, probably because of their similar surface structures. Recognition of the fact that any macroscopic stress applied to a soil mass induces both tangential and normal forces at the interparticle contacts is essential to the understanding of rheological behavior. The results of discrete particle simulations show that changes in creep rate with time can be explained by changes in the tangential and normal force ratio at interparticle contacts that result from particle rearrangement during deformation. The change in microfabric in relation to strong particle networks and weak clusters leads to possible explanation of the mechanical aging process. Time-dependent deformations and stress relaxation follow predictable patterns that are essentially the same for all soil types. Simple constitutive equations can reasonably describe time-dependent behavior under limited conditions. Much remains to be learned, however, about the influences of combined stress states, stress history and transient drainage conditions on creep, stress relaxation, and creep rupture before reliable
Copyright © 2005 John Wiley & Sons
2. The figure below shows relationships between (1) number of interparticle bonds and effective consolidation pressure and (2) compressive strength and number of interparticle bonds for three soils as determined using rate process theory. Determine the angle of internal friction in terms of effective stresses (as determined from CU tests with pore pressure measurements), for each soil. Assume Aƒ ⫽ 0, 0.3, and 0.3 for the sand, illite, and Bay mud, respectively, in the range 0 ⬍ (1 ⫺ 3)ƒ ⬍ 500 kPa, where Aƒ is the ratio of pore pressure at failure to the deviator stress at failure (1 ⫺ 3)ƒ. 40
Number of Bonds - 1010 cm-2
520
30
Sand
20
10
Illite Bay Mud
0 0
100 200 300 400 σc = Effective Consolidation Pressure (kPa)
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500
QUESTIONS AND PROBLEMS
1000
0
5
10
15
20
521
90
25
-2
Number of Bonds - 1010 cm
80
Water Content (%)
600 Sand, Illite, Bay Mud 400
200
0
XRupture
Axial Strain (%)
30 20
Water Content = 60% Dmax = 125 kPa
D = 100 kPa
D = 85 kPa
10
100 Time (min)
1000
40
80 100 200 Compressive Strength (kPa)
400
intensities are shown below, as is the variation of compressive strength with water content. A temporary excavation is planned that will create a slope with an average factor of safety of 1.5. The average water content of the clay in the vicinity of the cut is 50 percent. The excavation is planned to remain open for a period of 4 months. Prepare a plot of strain rate versus time for an element of clay and assess the probability of a creep rupture failure occurring during this period.
5. Given that a. The creep rate of a soil, for times up to the onset of failure, can be expressed by Eq. (12.43), in which D is the deviator stress, and b. The time to failure by creep rupture, tƒ, can be taken as the time corresponding to minimum strain rate, ˙ min, prior to acceleration of deformation and failure, and tests have shown that ˙ mintƒ ⫽ constant
D = 68 kPa
0 1
50
30 20
4. The results of triaxial compression creep tests on samples of overconsolidated Bay mud at three stress
40
60
40
3. Equation (12.43) is a simple three-parameter equation for strain rate during constant stress creep of soils. a. Show the meaning of , D, and m on a clearly labeled sketch. b. Modify Eq. (12.43) and indicate the information needed to permit prediction of creep rates for a given soil at any value of water content and stress intensity from a knowledge of creep rates at a single water content corresponding to different stress intensities. c. Develop a relationship between stress intensity and time to failure for a soil subject to strength loss under the application of a sustained stress.
50
70
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(σ1 – σ3)max(kPa)
800
10,000
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If a test embankment designed at a factor of safety of 1.05 based on shear strength determined in a short-term test fails in creep rupture after 3 months, how long should it be before failure of a prototype embankment having a factor of safety of 1.3? From a plot of deformation rate versus time for the test embankment, it has been found that m ⫽ 0.75. The results of shortterm creep tests have shown also that Dmax ⫽ 6.0. The factor of safety is defined as the strength available divided by the strength that must be mobilized for stability.
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12
TIME EFFECTS ON STRENGTH AND DEFORMATION
6. Would you expect that creep and stress relaxation will be significant contributors to the stress– deformation and long-term strength of soils on the Moon? Why?
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7. List possible causes of sand aging wherein the stiffness and strength (usually as determined by pene-
tration tests) can increase significantly over time periods as short as weeks or months following deposition and/or densification. Outline a test program that might be done to test the validity of one of these causes.
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a a a a a a a ac am at av A A A A A A A A A A A A0 A0 Ac Aƒ Aƒ Ah Ai Ai A0i
Co py rig hte dM ate ria l
List of Symbols
area coefficient for harmonics cross-sectional area of a tube crystallographic axis direction or distance effective cluster contact area volumetric air content thermal diffusivity effective area of interparticle contact coefficient of compressibility with respect to changes in water content coefficient of compressibility with respect to changes in ( ⫺ ua) coefficient of compressibility in one dimensional compression activity area creep rate parameter cross section area normal to the direction of flow Hamaker constant long-range interparticle attractions Skempton’s pore pressure parameter thermal diffusivity van der Waal’s constant short-range attractive stress pore pressure parameter ⫽ u/
(1 ⫺ 3) concentration of charges on pore wall surface charge density per unit pore volume solid contact area area of flow passages pore pressure parameter at failure Hamaker constant state parameter in disturbed state total surface area of the ith grain state parameter at equilibrium
As
˚ A b b b B
B Bq Br c c c c c c c c c0
c0⫹ c0⫺ ca ce, ce cec cic, cc ci 0 cm cm cu cv cw C C C C
specific surface area per unit weight of solids Angstrom unit ⫽ 1 ⫻ 10⫺10 m coefficient of harmonics crystallographic axis direction or distance intermediate stress parameter parameter in rate process equation ⫽ X(kT/h) Bishop’s pore water pressure coefficient grain breakage parameter Hardin’s relative breakage parameter cohesion cohesion intercept in total stress concentration molar concentration crystallographic axis direction or distance undrained shear strength velocity of light cohesion intercept in effective stress equilibrium solution concentration, bulk solution concentration cation equilibrium solution concentration anion equilibrium solution concentration mid-plane anion concentration Hvorslev’s cohesion parameter cation exchange capacity mid-plane cation concentration equilibrium solution concentration mid-plane concentration mid-plane anion concentration undrained shear strength coefficient of consolidation concentration of water capacitance chemical concentration clay content by weight composition 523
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C C C C C C Cc C* c Cl Cn CR CRR Cs Cs Cs Cu Cu CW C, Ce d d d10 d60 dx dy D D D D D D0 D0 D50 Deƒƒ DeV Dmax DR, Dr Ds DTV D* e e e0 e* 100 ec
LIST OF SYMBOLS
electrical capacitance short-range repulsive force between contacting particles soil compressibility speed of light in vacuum or in air, 3 ⫻ 108 m/sec volumetric heat volumetric heat capacity compression index intrinsic compression index compressibility of pore fluid coordination number compression ratio cyclic resistance ratio compressibility of a solid shape coefficient swelling index coefficient of uniformity compressibility of soil skeleton by pore pressure change compressibility of water coefficient of secondary compression diameter distance sieve size that 10% of the particles by weight pass through sieve size that 60% of the particles by weight pass through incremental horizontal displacement at peak incremental vertical displacement at peak diameter of particle dielectric constant, relative permittivity diffusion coefficient deviator stress stress level ⫽ D/Dmax molecular diffusivity of water vapor in air self-diffusion coefficient sieve size that 50% of the particles by weight pass through effective diameter isothermal vapor diffusivity strength at the beginning of creep relative density characteristic grain size thermal vapor diffusivity effective diffusion coefficient electronic charge ⫽ 4.8029 ⫻ 10⫺10 esu ⫽ 1.60206 ⫻ 10⫺10 coulomb void ratio initial void ratio intrinsic void ratio under effective vertical stress of 100 kPa intracluster void ratio
ecs eƒƒ eg, eG eini eL emax emin ep eT E E E E E50
void ratio at critical state void ratio at failure void ratio of the granular phase, granular void ratio initial void ratio void ratio at liquid limit maximum void ratio minimum void ratio intercluster void ratio total void ratio experimental activation energy potential energy Young’s modulus voltage, electrical potential secant modulus at 50 percent of peak strength small strain Young’s modulus rebound modulus exchangeable sodium percentage distribution function for interparticle contact plane normals force acting on a flow unit frequency fraction of particles between two sizes normal force tangential force force of electrostatic attraction formation factor free energy freezing index pressure-temperature parameter tensile strength Faraday constant ⫽ 96,500 coulombs partial molar free energy on adsorption free energy of the double layer per unit area at a plate spacing of 2d free energy of activation electrical force per unit length hydraulic seepage force per unit length causing flow fabric index free energy of a single non-interacting double layer acceleration due to gravity shear modulus source-sink shear modulus measured after 1000 minutes of constant confining pressure shear modulus of grains small strain shear modulus secant shear modulus specific gravity of soil solids specific gravity of clay particles specific gravity of the granular particles
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Emax Er ESP E( )
ƒ ƒ ƒi ƒn ƒt F F F F F F F, F0 F Fd
F FE FH
FI F⬁
g G G G1000 Gg Gmax Gs Gs GSC GSG
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LIST OF SYMBOLS
IR Iv Jc JD Ji Ji Js Jv Jw J 0i k k k k k k k0 kc ke kh ki kr ks k(S) kt k K K
head or head loss relative humidity of air in pores Planck’s constant ⫽ 6.624 ⫻ 10⫺27 erg sec matrix or capillary head osmotic or solute head maximum distance to drainage boundary stress history thickness total head water transport by ion hydration partial molar heat content gradient unit vector chemical gradient electrical gradient hydraulic gradient thermal gradient electrical current intensity stress invariants coefficient of shear modulus increase with time dilatancy index void index chemical flow rate chemical flow rate flux of constituent i value of property i in clay-water system flow rate of salt relative to fixed soil layer volume flow rate of solution flow rate of water value of property i in pure water Boltzmann’s constant ⫽ 1.38045 ⫻ 10⫺23 J/ K hydraulic conductivity, hydraulic permeability mean coordination number of a grain selectivity coefficient thermal conductivity true cohesion in a solid pore shape factor osmotic conductivity electro-osmotic conductivity hydraulic conductivity constant characteristic of a property relative permeability saturated conductivity saturation dependent hydraulic conductivity thermal conductivity unsaturated hydraulic conductivity absolute permeability or intrinsic permeability bulk modulus
K K K K0 Ka Kc Kc Kd Kp Kso K l l l L L Lij
double-layer parameter ⫽ (8n0e2v2 /DRT)1 / 2 pore shape factor rate of increase in tip resistance in logarithmic time coefficient of lateral earth pressure at rest coefficient of active earth pressure principal stress ratio principal stress ratio during consolidation distribution coefficient coefficient of passive earth pressure stress-optical material constant wavelengths of monochromatic radiation length material thickness total number of pore classes latent heat of fusion length coupling coefficient or conductivity coefficient liquidity index equivalent liquidity index liquid limit latent heat of fusion of water slope of relationship between log creep strain rate and log time total mass per unit total volume total number of pore classes mass of clay compressibility of mineral solids under hydrostatic pressure compressibility of mineral solids under concentrated loadings compressibility compressibility of water mass of water constrained modulus or coefficient of volume change metal cations monovalent cation concentration concentration, ions per unit volume harmonic number integer number of grains in an ideal breakage plane porosity total number of pore classes unspecified atomic ratio concentration in external solution number of bonds per unit of normal force effective porosity Refractive index in i direction Avogadro’s number ⫽ 6.0232 ⫻ 1023 mole⫺1
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h h h hm hs H H H H H H i i ic ie ih it I I I1, I2, I3 IG
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LI LIeq LL Ls m m m mc ms
ms
mv mw mw M M M n n n n n n n n0 n1 ne ni N
525
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N N N N N N N N1 Ne NG Ns Nw OCR p p p p p p p po pa pc pcs ps pz P P P P P P P Pc Pˆ c Pƒ PI Pinj PL PN PR Ps PT
LIST OF SYMBOLS
coordination number monovalent cation concentration normal load or force number of moles of hydration water per mole of ion number of particles per cluster in a cluster structure number of weeks since disturbance total number of harmonics number of load cycles to cause liquefaction number of load cycles normalized shear modulus increase with time moles of water per unit volume of sediment moles of salt per unit volume of sediment overconsolidation ratio constant that accounts for the interaction of pores of various sizes hydrostatic pressure matrix or osmotic pressure pressure partial pressure of water vapor in pore space vertical consolidation pressure mean effective pressure present overburden pressure atmospheric pressure preconsolidation pressure mean effective pressure at critical state osmotic or solute pressure gravitational pressure area bond strength per contact zone concentration of divalent cations power consumption total gas pressure in pore space total pressure wetted perimeter capillary pressure capillary pressure at air entry injection pressure that causes clay to fracture plasticity index injection pressure plastic limit probability distribution of normal contact force peak ratio swelling pressure probability distribution of tangential contact force
q q q q qc qcs qƒ qh qhc qhe qi qt qvap qw Q Q r r rk
degree of connectivity between waterconducting pores deviator stress flow rate hydraulic flow rate CPT tip resistance deviator stress at critical state deviator stress at failure hydraulic flow rate osmotic flow rate electro-osmotic flow rate concentration of solids heat flow rate vapor flux density water flow rate electrical charge quantity of heat pore radius radius ratio of horizontal to vertical hydraulic conductivities pore size tube radius coefficient of roundness electrical resistance gas constant ⫽ 1.98726 cal/ K-mole 8.31470 joules/ K-mole 82.0597 cm3 atm/ K-mole long-range repulsion pressure ratio of cations and anions source or sink mass transfer term sphere radius tube radius retardation factor hydraulic radius average particle radius radius at angle slope of stress relaxation curve undrained shear strength entropy fraction of molecules striking a surface that stick to it number of flow units per unit area partial molar entropy saturation specific surface area per unit volume of solids structure swell partial molar entropy specific surface per unit volume of soil particles sodium adsorption ratio
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526
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rp rp R R R
R R R R R Rd RH Rp R( ) s su S S S S S S
S S S S0
SAR
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LIST OF SYMBOLS
t t t t t1 tƒ tm T T T T T0 Tc Tc TFP Ts Ts TV u u u u u u u u* u0 u0 Uƒ U v v v v v vave vc0 vcs vh
V V V V V V
sensitivity undrained shear strength water saturation ratio projected areas of interparticle contact surfaces average thickness tetrahedral coordinations time transport number reference time time to failure time for adsorption of a monolayer intercluster tortuosity shear force temperature time factor initial temperature intracluster tortuosity temperature at consolidation freezing temperature surface temperature temperature of shear for consolidated undrained direct shear tests time factor excess pore pressure ionic mobility midplane potential function pore water pressure pore water pressure in the interparticle zone pressure thermal energy effective ionic mobility initial pore pressure pore water pressure remote from the interparticle zone pore pressure at failure average degree of consolidation flow velocity frequency of activation ionic valance settling velocity specific volume ⫽ 1 ⫹ e average flow velocity specific volume of the pure clay specific volume at critical state apparent water flow velocity area difference in self-potentials electrical potential speed valence voltage
V V0 VA VDR VGS Vm Vp VR Vs Vs Vw Vw w wL, wl wP, wp W W W W W x X X Xi y z z z z Z Z
volume initial volume attractive energy volume of water drained volume of granular solids total volume of soil mass compression wave velocity repulsive energy shear wave velocity volume of solids partial molar volume of water volume of water water content liquid limit plastic limit water content width fluid volume water transport weight distance from the clay surface distance friction coefficient driving force potential function ⫽ ve /kT direction of gravity distance from drainage surface electrolyte ionic valence elevation or elevation head number of molecules per second striking a surface potential function ⫽ 'e 0 /kT angle between b and c crystallographic axes directional parameter disturbance factor geometrical packing parameter inclination of failure plane to horizontal plane slope of the relationship between logarithm of creep rate and creep stress thermal ratio tortuosity factor normalized strain rate parameter thermal expansion coefficient of soil solids thermal expansion coefficient of soil structure thermal expansion coefficient of water angle between a and c crystallographic axes birefringence ratio
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St Su Sw Sx, Sy, Sz
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Z
G s
ST w
527
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0, i p ˙ 0 1 ˙ a ƒ ˙ min rd s ˙ s v ˙ v
E
b c crit e, e ƒ m r repose v , #
LIST OF SYMBOLS
disturbance factor geometrical packing parameter rotation angle of yield envelope constant characteristic of the property and the clay Bishop’s unsaturated effective stress parameter clay plate thickness measured between centers of surface layer atoms deformation parameter in Hertz theory displacement, distance solid fraction of a contact area relative retardation particle eccentricity distance dielectric constant, permittivity porosity strain strain rate permittivity of vacuum, 8.85 ⫻ 10⫺12 C2 /(Nm2) axial strain vertical strain rate in one dimensional consolidation strain at failure minimum strain rate volumetric strain that would occur if drainage were permitted deviator strain deviator strain rate volumetric strain volumetric strain rate energy dissipated per cycle per unit volume friction angle local electrical potential friction angle in effective stress angle defining the rate of increase in shear strength with respect to soil suction characteristic friction angle friction angle at critical state Hvorslev friction parameter friction angle corrected for the work of dilation peak mobilized friction angle residual friction angle angle of repose apparent specific volume of the water in a clay/water system of volume V intergrain sliding friction angle dissipation function activity coefficient angle between a and b crystallographic axes unit weight
˙ c d % % 0 ! ! !ⴖ
shear strain rate applied shear strain or cyclic shear strain amplitude dry unit weight double layer charge specific volume intercept at unit pressure dynamic viscosity fraction of pore pressure that gives effective stress initial anisotropy swelling index real relative permittivity polarization loss, imaginary relative permittivity compression index correction coefficient for frost depth prediction equation damping ratio decay constant pore size distribution index separation distance between successive positions in a structure wave length of X ray wave length of light critical state compression index chemical potential coefficient of friction dipole moment fusion parameter Poisson’s ratio viscosity critical state stress ratio Poisson’s ratio Poisson’s ratio of soil skeleton osmotic or swelling pressure angle of bedding plane relative to the maximum principal stress direction contact angle geometrical packing parameter liquid-to-solid contact angle orientation angle volumetric water content volumetric water content at full saturation residual water content volumetric water content at full saturation bulk dry density charge density mass density bulk dry density resistivity of saturated soil density of water resistivity of soil water area occupied per absorbed molecule on a surface
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cs ( b m r s d T w W
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LIST OF SYMBOLS
1ƒ 1ƒƒ 2 3 3 3c 3ƒƒ a ac as aw c c e e eƒƒ ƒ ƒ ƒƒ ƒƒ h h0 i i i iso max min n p r r rc s s s T T , t
double-layer charge electrical conductivity entropy production normal stress surface tension of water surface charge density total stress effective stress initial effective confining pressure major principal total stress tensile strength of the interface bond major principal effective stress major principal stress during consolidation major principal stress at failure major principal effective stress at failure intermediate principal effective stress minor principal total stress minor principal effective stress minor principal stress during consolidation minor principal effective stress at failure axial effective stress axial consolidation stress interfacial tension between air and solid interfacial tension between air and water crushing strength of particles tensile strength of cement electrical conductivity equivalent consolidation pressure effective AC conductivity partial stress increment for fluid phase effective normal stress on shear plane normal total stress on failure plane normal effective stress on failure plane electrical conductivity due to hydraulic flow initial horizontal effective stress effective stress in the i-direction intergranular stress isotropic consolidation isotropic total stress maximum principal stress minimum principal stress effective normal stress preconsolidation pressure radial total stress radial effective stress radial consolidation stress conductivity of soil surface partial stress increment for solid phase tensile strength of the sphere electrical conductivity of saturated soil tensile strength of cemented soil
v v v0 v0 vm vp W ws y a c c cy d ƒƒ i i m
vertical stress vertical effective stress overburden vertical effective stress overburden effective stress maximum past overburden effective stress vertical preconsolidation stress electrical conductivity of pore water interfacial tension between water and solid yield strength circumferential stress shear strength shear stress surface tension swelling pressure or matric suction undrained shear strength apparent tortuosity factor applied shear stress contaminant film strength undrained cyclic shear stress drained shear strength shear stress at failure on failure plane shear strength shear strength of contact shear strength of solid material in yielded zone applied shear stress at peak initial static shear stress mass flow factor cation valence distance function ⫽ Kx, double-layer theory ratio of average temperature gradient in air filled pores to overall temperature gradient dilation angle electrical potential intrinsic friction angle matric suction surface potential of double layer displacement pressure electrical potential state parameter total potential of soil water electrical potential at the surface gravitational potential matrix or capillary potential gas pressure potential osmotic or solute potential angular velocity frequency osmotic efficiency true electroosmotic flow zeta potential
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0 1 1 1 1c
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peak ' '
0 d 0 s m p s " " "
529
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