Nonlinear Solid Mechanics A Continuum Approach for Engineering
Gerhard A. Holzapfel Technical University Graz, Austria
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Contents
Preface
ix
Acknowledgements
xiii
1 Introduction to Vectors and Tensors
1
....
1.1
Algebra of Vectors
1.2
Algebra of Tensors
1.3
Higher-order Tensors .
1.4
Eigenvalues, Eigenvectors of Tensors
1.5
Transformation Laws for Basis Vectors and Components
1.6
General Bases
1.7
Scalar, Vector, Tensor Functions .
1.8
Gradients and Related Operators .
1.9
Integral Theorems .
1
...... ....
. ....
20
....
...
24 28
.........
...
9
32
40 . . . .
.
.... ....
44 52
55
2 Kinematics 2.1
Configurations, and Motions of Continuum Bodies
56
2.2
Displacement, Velocity, Acceleration Fields . .
61
2.3
Material, Spatial Derivatives
64
2.4
Deformation Gradient
70
2.5
Strain Tensors
76
2.6
Rotation, Stretch Tensors .
85
2.7
Rates of Deformation Tensors
95
2.8
Lie Time Derivatives . . . . .
106
. . . .
v
Contents
vi
109
3 The Concept of Stress 3.1
Traction Vectors, and Stress Tensors . .
3.2
Extremal Stress Values . . .
3.3
Examples of States of Stress
3.4
Alternative Stress Tensors .
109 119 123 127
131
4 Balance Principles 4.1 4.2 4.3 4.4 4.5 4.6
4.7
. ...... 131 138 Reynolds' Transport Theorem .... Momentum Balance Principles . . .... 141 Balance of Mechanical Energy . . ..... 152 Balance of Energy in Continuum Thermodynamics . . 161 Entropy Inequality Principle .... 166 . ... 174 Master Balance Principle . . ..... . ... Conservation of Mass
..... . . . . . .
179
5 Some Aspects of Objectivity 5.1
Change of Observer, and Objective Tensor Fields .
5.2
Superimposed Rigid-body Motions . . . . . . . .
5.3
Objective Rates . . . . . . . . . . . . . .
179 187 192
5.4
Invariance of Elastic Material Response . .
196
6 Hyperelastic Materials
205
.... . . . . . ....
6.1
General Remarks on Constitutive Equations . .
6.2
Isotropic Hyperelastic Materials . . . . .
6.3
Incompressible Hyperelastic Materials .
6.4
Compressible Hyperelastic Materials
6.5
Some Forms of Strain-energy Functions .
6.6
Elasticity Tensors . . . . . . . . .
6.7
Transversely Isotropic Materials .
..
........
206 212 222 227 235 252 265
vii
Contents
6.8
Composite Materials with Two Families of Fibers .
6. 9
Constitutive Models with Internal Variables
. . . .
6.10 Viscoelastic Materials at Large Strains . . . . 6.11
Hyperelastic Materials with Isotropic Damage
7 Thermodynamics of Materials
272 278 282 295
305
7.1
Physical Preliminaries . . . . .
7.2
Thermoelasticity of Macroscopic Networks
7.3
Thermodynamic Potentials . . . . . . . .
7.4
Calorimetry . . . . . . . . . . . . . . . . .
7.5
Isothermal, Isentropic Elasticity Tensors . .
7.6
Entropic Elastic Materials . . . . . . . . .
7.7
Thermodynamic Extension of Ogden's Material Model ..
7.8 7.9
Simple Tension of Entropic Elastic Materials Thermodynamics with Internal Variables . . .
306 311 321 325 328 333 337 343 357
371
8 Variational Principles 8.1
Virtual Displacements, Variations
8.2
Principle of Virtual Work . . . . .
8.3
Principle of Stationary Potential Energy .
8.4
Linearization of the Principle of Virtual Work .
372 377 386 392
8.5
Two-field Variational Principles .
402
8.6
Three-field Variational Principles
. . . . .
409
References
415
Index
435
Preface
My desire in writing this textbook was to show the fascination and beauty of nonlinear solid mechanics and thermodynamics from an engineering computational point of view. My primary goal was not only to offer a modern introductory textbook using the continuum approach to be read with interest, enjoyment and curiosity, but also to offer a reference book that incorporates some of the recent developments in the field. I wanted to stimulate and invite the reader to study this exciting science and take him on a pleasant journey in the wonderful world of nonlinear mechanics, which serves as a solid basis for a surprisingly large variety of problems arising in practical engineering. Linear theories of solid mechanics are highly developed and are in a satisfactory state of completion. Most processes in nature, however, are highly nonlinear. The approach taken has the aim of providing insight in the basic concepts of solid mechanics with particular reference to the nonlinear regime. Once familiar with the main ideas the reader will be able to specialize in different aspects of the subject matter. I felt the need for a self-contained textbook intended primarily for beginners who want to understand the correspondence between nonlinear continuum mechanics, nonlinear constitutive models and variational principles as essential prerequisites for finite element formulations. Of course, no single book can cover all aspects of the broad field of solid mechanics, so that many topics are not discussed here at all. The selection of the material for inclusion is influenced strongly by current curricula, trends in the literature and the author's particular interests in engineering and science. Here, a particular selection and sty le was chosen in order to highlight some of the more inspiring topics in solid mechanics. I hope that my choice, which is of course subjective, will be found to be acceptable. My ultimate intention was to present an introduction to the subject matter in a didactically sound manner and as clearly as possible. I hope that the text provides enough insights for understanding of the terminology used in scientific state-of-the art papers and to find the 'right and straightforward path' in the scientific world through the effective use of figures, which are very important learning tools. They are designed in order IX
x
Preface
to attract attention and to be instructive and helpful to the reader. Necessary mathematics and physics are explained in the text. The approach used in each of the eight chapters will enable the reader to work through the chapters in order of appearance, each topic being presented in a logical sequence and based on the preceding topics. A proper understanding of the subject matter requires knowledge of tensor algebra and tensor calculus. For most of the derivations throughout the text I have used symbolic notation with those clear bold-faced symbols which give the subject matter a distinguished beauty. However, for higher-order tensors and for final results in most of the derivations I have used index notation, which provides the reader with more insight. Terminology is printed in bold-face where it appears for the first time while the notation used in the text is defined at the appropriate point. For those who have not been exposed to the necessary mathematics I have included a chapter on tensor algebra and tensor calculus. It includes the essential ideas of linearization in the form of the concept of the directional derivative. Chapter 1 summarizes elementary properties which are needed for the vector and tensor manipulations performed in all subsequent chapters and which are necessary to many problems that arise frequently in engineering and physics. It is the prime consideration of Chapter 2 to use tensor analysis for the description of the motion and finite deformation of continua. The continuum approach is introduced along with the notion 'Lagrangian' (material) and 'Eulerian' (spatial) descriptions. In a systematic way the most important kinematic tensors are provided and their physical significance explained. The push-forward and pull-back operations for material and spatial quantities and the concept of the Lie time derivative are introduced. The concept of stress is the main topic of Chapter 3. Cauchy's stress theorem is introduced, along with the Cauchy and first Piola-Kirchhoff traction vectors, and the essential stress tensors are defined and their interrelationships discussed. In Chapter 4 attention is focused on the discussion of the balance principles. Both statics and dynamics are treated. Based upon continuum thermodynamics the entropy inequality principle is provided and the general structure of all principles is summarized as the master balance (inequality) principle. Chapter 5 deals with important aspects of objectivity, which plays a crucial part in nonlinear continuum mechanics. A discussion of change of observer and superimposed rigid-body motions is followed by a development of objective (stress) rates and invariance of elastic material response. Chapters 6 and 7 form the central part of the book and provide insight in the construction of nonlinear constitutive equations for the description of the mechanical and thermomechanical behavior of solids. These two chapters show the essential richness of the field. They are written for those who want to gain experience in handling material models and deriving stress relations and the associated elasticity tensors that are fundamental for finite element methods. Several examples and exercises are aimed at enabling the reader to think in terms of constitutive models and to formulate more com-
Preface
xi
plex material models. All of the types of constitutive equations presented are accessible for use within finite element procedures. The bulk of Chapter 6 is concerned with finite elasticity and finite viscoelasticity. It includes a discussion of isotropic, incompressible and compressible hyperelastic materials and provides constitutive models for transversely isotropic and composite materials which are suitable for a large number of applications in practical engineering. An approach to inelastic materials with internal variables is given along with instructive examples of hyperelastic materials that involve relaxation and/or creep effects and isotropic damage mechanisms at finite strains. The main purpose of Chapter 7 is to provide an introduction to the thermodynamics of materials. This chapter is devoted not only to the foundation of continuum thermodynamics but also to selected topics of statistical thermodynamics. It starts with a statistical approach by summarizing important physical aspects of the thermoelastic behavior of molecular networks (for example, amorphous solid polymers), based almost entirely on an entropy concept, and continues with a systematic phenomenological approach including finite thermoelasticity and finite thermoviscoelasticity. The stress-strain-temperature response of so-called entropic elastic materials is discussed in more detail and based on a representative example which is concerned with the adiabatic stretching of a rubber band. Typical thermomechanical coupling effects are studied. Chapter 8 is designed to cover the essential features of the most important variational principles that are very useful in formulating approximation techniques such as the finite element method. Although finite elements are not treated in this text, it is hoped that this chapter will be attractive to those who approach the subject from the computational side. It shows the relationship between the strong and weak forms of initial boundary-value problems, presents the classical principle of virtual work in both spatial and material descriptions and its linearized form. Two- and three-field variational principles are also discussed. The present text ends where conventionally a book on the finite element method would begin. There are numerous worked examples adjacent to the relevant text. These have the goal of clarifying and supplementing the subject matter. In many cases they are straightforward, but provide an essential part of the text. The symbol • is used to denote the end of an exercise or a proof. The end-of chapter exercises are for homework. The (almost) 200 exercises provided are designed to supplement the text and to consolidate concepts discussed in the text. Most of them serve the purpose of stimulating the reader to further study and to reinforce and develop practical skills in nonlinear continuum and solid mechanics, towards the direction of computational mechanics. In many cases the solutions of selected exercises are given and frequently used later in further developments. Therefore, it should be instructive for the reader to work through a reasonable number of exercises. Numerous references to supplementary material are suggested and discussed briefly
xii
Preface
throughout the book. However, for a book of this kind it is not possible to give a comprehensive bibliography of the field. Some of the references listed serve as a starting point for more advanced studies. The material in this book is based on a sequence of courses that I have taught at the University of Technology in Graz and Vienna. The mechanics and thermodynamics of solids are relevant to all branches of engineering, to applied mechanics, mathematics, physics and material science, and it is a central field in biomechanics. This book is primarily addressed to graduate students, researchers and practitioners, although it has also proven to be of interest to advanced undergraduates. Although I have tried to provide a textbook that is self-contained and appropriate for self-instruction, it is desirable that the reader has a reasonable background in elementary mechanics and thermodynamics. I feel that Chapters 2-5 and parts of the remaining chapters are well suited for a complete course on nonlinear continuum mechanics lasting two semesters or three quarters. A one semester or one quarter course in the nonlinear mechanics of solids would focus on Chapter 6, while a one quarter course in the thermodynamics of solids could be based on Chapter 7. Chapter I is the core for a course that provides the student with the necessary background in vector and tensor analysis. Chapter 8 is certainly not designed to train specialists in variational principles, but to form a basic one quarter course at the graduate level. I believe that the present textbook, in providing many applications to engineering science, is not too advanced mathematically. Of course, some of the results presented may be derived with the help of more advanced mathematics using theorems and proofs. I hope that this book will help pure engineers to teach nonlinear continuum mechanics and solid mechanics. Naturally, as the author, I take full responsibility for not doing it better. Comments and criticisms will be welcome and greatly appreciated. I have learnt that the spirit of modern continuum mechanics and the underlying mathematics are as important to the design of powerful finite element models as are insights in the theoretical foundation of constitutive models and variational principles. A successful transfer of that combination to the reader would indicate that my objective has been achieved.
Graz, Austria August 1999
Gerhard A. Ho/zap/el
Acknowledgements
When I was a postdoctoral student at Stanford University I worked with the late Juan C. Simo, Professor of Mechanical Engineering, to whom I owe my deepest thanks. He stimulated, influenced and focused my study and writing in recent years; his friendship, versatility and dedication to scientific excellence provided a unique learning experience for me.
I am particularly indebted to Ray W. Ogden, Professor of Mathematics at the University of Glasgow, who spent a lot of time in reading the entire manuscript and rectifying certain ineptnesses. His outstanding expertise in the field made working with him an inspiring pleasure. His detailed scientific criticism and suggestions for improvements of the text were of immeasurable help. Many others have contributed to the book. Here I mention my collaborators, whose gentle encouragement and support during the course of the preparation of the manuscript I gratefully acknowledge. In particular, the inspiring and detailed comments of D1: Christian A.J. Schulze-Bauer, from a background in physics and medicine, were extremely helpful. He let me filter this text through his sharp mind. Also, Christian T. Gasser, whose background is in mechanical engineering, has suggested a number of valuable improvements to the substance of the text. His profound remarks in class prevented me from getting away with anything. Also~ Elisabeth Pernkopf, a mathematician, deserves special thanks for her generous assistance. She gave much helpful advice on the preparation of this text and offered many suggestions for improving it. Special thanks are due to Michael Stadler for his productive discussions and to Mario 0. Pa/Ii for his patience in preparing the figures. I am grateful to each of these individuals without whose contributions the book would not have taken this shape. I want to thank the Department of Civil Engineering, Graz University of Technology, for providing an environment in which this project could be completed. My gratitude goes to Gernot Beer, Professor of Civil Engineering, for his outspoken support. I also wish to acknowledge the Austrian Science Foundation, which has influenced my scientific agenda through the financial support of several grants over the past eight years.
xm
xiv
Acknowledgements
The enjoyment I experienced in writing this textbook would not have been the same without the moral support of numerous friends. My thanks belong to all of them who tolerated my absence when I disappeared for many evenings and weekends in order to bring the ideas of this fascinating field to you, the reader.
1 Introduction to Vectors and Tensors The aim of this chapter is to present the fundamental rules and standard results of tensor algebra and tensor calculus permanently used in nonlinear continuum mechanics. Some readers may prefer to pass directly to Chapter 2 leaving the present part for reference as needed. Many of the statements are given without proofs. For a more detailed exposition see the standard texts by HALMOS [1958], TRUESDELL and NOLL [1992] or the textbooks by CHADWICK [1976], GURTIN [1981a], SIMMONDS [1994], DANIELSON [1997] and OGDEN [ 1997] among many other references on vectors and tensors. To recall the elements of linear algebra see, for example, the book by STRANG [ 1988a]. In this text we use lowercase Greek letters for scalars, lowercase bold-face Latin letters for vectors, uppercase bold-face Latin letters for second-order tensors, uppercase bold-face calligraphic letters for third-order tensors and uppercase blackboard Latin letters for fourth-order tensors; for example,
a, {3, (, . . . (scalars) A, B, C, . . . (2nd-order tensors)
a, b, c, . . . (vectors)
A
B, C, ... (3rd-order tensors)
A, Jm, C, . . . (4th-order tensors)
(1.1)
.
For equations such as u
= av =
{Ja = 1b ,
(1.2)
we agree that (1.2)2 refers to u = {Ja. Often the derivations of formulas need relations introduced previously. If this is the case we refer to these relations in a particular order, reflecting the consecutive steps necessary for deriving the formula in question.
1.1
Algebra of Vectors
A physical quantity, completely described by a single real number, such as temperature, density or mass, is called a scalar designated by a, /3, ')', ... A vector designated 1
2
1 Introduction to Vectors and Tensors
by u, v, w, ... (or in other texts frequently designated by ''i', ~' 1!!' ... , or fi, 'V, 'lV, ... ), is a directed line element in space. It is a model for physical quantities having both direction and length, for example,force, velocity or acceleration. Two vectors that have the same direction and length are said to be equal. The sum of vectors yields a new vector, based on the parallelogram law of addition. The following properties
u+v=v+u,
(I .3)
(u + v) + w = u + (v + w) ,
( 1.4) (1.5)
u+o=u, u+ (-u) == o
(1.6)
hold, where o denotes the unique zero vector with unspecified direction and zero length. Let u be a vector and a be a real number (a scalar). Then the scalar multiplication au produces a new vector with the same direction as u if a > 0 or with the opposite direction to u if n < 0. Further properties are:
(n,B)u (er+ f3)u
Dot product.
= o{Bu) , = nu+ {Ju
(1.7)
,
(1.8)
a(u+v)=au+o:v.
( 1.9)
The dot (or scalar or inner) product of u and v, denoted by u · v (or
(u, v} ), is Oo
,
')
u-
= u. u
.
( 1.15)
1.1
Algebra of Vectors
3
A vector e is called a unit vector if lel = 1. A nonzero vector u is said to be orthogonal (or perpendicular) to a nonzero vector v if
O(u,v) =7r/2 .
with
( 1.16)
Thus, using ( l .10) we find the projection of a vector u along the direction of a vector e with unit length, i.e.
u · e == jujcosO(u, e) .
( 1.17)
For a geometrical interpretation of eq. ( 1. I 7) see Figure I. I .
u
\0
-~
-""------
l
_____~------·:
e Figure 1.1 Projection of u along a unit vector e.
Index notation. So far algebra has been presented in symbolic (or direct or absolute) notation exclusively employing bold-face letters. It represents a very convenient and concise tool to manipulate most of the relations used in continuum mechanics. It will be the preferred representation in this text. However, particularly in computational mechanics, it is essential to refer vector (and tensor) quantities to a basis. Additionally, to gain more insight in some quantities and to carry out mathematical operations among (higher-order) tensors more readily (see next section) it is often helpful to refer to components. In order to present coordinate (or component) expressions relative to a righthanded (or dextral) and orthonormal system we introduce a fixed set of three basis vectors e 1 , e2 , c:i (sometimes introduced as i,j, k), called a (Cartesian) basis, with properties
e 1 • e.,..,
= e 1 • e..3 = e?- · e..3 = 0
'
These vectors of unit length which are mutually orthogonal form a so-called orthonormal system.
4
1 Introduction to Vectors and Tensors
Then any vector u in the three-dimensional Euclidean space is represented uniquely by a linear combination of the basis vectors e 1 , e2 , e3 , i.e. ( 1.19) where the three real numbers u 1 , 'U 2 , u 3 are the uniquely determined Cartesian (or rectangular) components of vector u along the given directions e1 , e2 , e:1, respectively (see Figure 1.2). The components of e 1 , e 2 , e3 are (1, 0, 0), (0, 1, 0), (0, 0, 1), respectively. u
'lL]
Figure 1.2 Vector u with its Cartesian components u 1 , u 2 , u 3 •
Using index (or subscript or suffix) notation, relation (l.19) can be written as u = E~=t uiei or, in an abbreviated form by leaving out the summation symbol, simply as (sum over i
= 1, 2, 3)
,
( 1.20)
where we have adopt the summation convention, invented by Einstein. The summation convention says that whenever an index is repeated (only once) in the same term, then, a summation over the range of this index is implied unless otherwise indicated. We consider only the three-dimensional Euclidean space, which we characterize by means of Latin indices i, j, k ... running over 1, 2, 3. We denote the basis vectors by { ei he{ 1,2 ,3 } collectively. Subsequently, in this text, the braces { •} will stand for a fixed set of basis vectors and the symbol • for any tensor element. The index i that is summed over is said to be a dummy (or summation) index, since a replacement by any other symbol does not affect the value of the sum. An index that is not summed over in a given term is called a free (or live) index. Note that in the same equation an index is either dummy or free. Thus, relations ( 1.18) can be
1.1
5
Algebra of Vectors
written in a more convenient form as if
I
i
if
=J
,
#j
'
( 1.2 l)
which defines the Kronecker delta 8ii. The useful properties
8ii
=3
(1.22)
'
hold. Note that 8ii also serves as a replacement operator; for example, the index on becomes an j when the components 'Ui are multiplied by oii· The projection of a vector u = ·u.iei onto the basis vectors ei yields the j-th component of u. Thus, from eq. ( 1.20) and properties ( 1.13), ( 1.21) and ( l.22h we have
'Ui
( 1.23) Taking the basis {ei} and eqs. (l.13), (l.20), (l.21) and (l.22)2, the component expression for the dot product (I. I 0) gives
u ·v =
=
u;ei · viei
U.(Vjei · ei
= 'l1. I 'VJ + 'll2V2 + 'll:fV3
=
Ui'vi8ii
=
u.ivi
( 1.24)
,
which is commonly used as the definition of the dot product. Thus, we may derive the dot product of u and v without knowledge of the angle between u and v. In an analogous manner, the component expression for the square of the length of u, i.e. ( 1.15), is
lul 2 = U · U = •)
U;Ci •
')
•)
= 'llj + ll~ + 'll3
'lljej =
U(lljc)ij
=
'll(l/.;
.
( 1.25)
Note that in eqs. ( 1.24) and ( 1.25) one index is repeated, indicating summation over 1, 2, 3. In symbolic notation this is indicated by one dot.
Cross product.
The cross (or vector) product of u and v, denoted by u xv (in the
literature also u /\ v), produces a new vector. The cross product is not commutative. It is defined as
u xv= -(v x u) ,
( l.26)
UXV=O (O'.U) X V = U X (O'.V) =
u and v are linearly dependent ft (
U X V) ,
u · (v x w) = v · (w x u) = w · (u x v) , u x (v + w) = (u xv)+ (u x w) = u xv+ u x w .
( l.27) ( 1.28) (I .29)
( l.30)
1 Introduction to Vectors and Tensors
6
If relation ( 1.27) holds with u and v assumed to be nonzero vectors, we say that vector u is parallel to vector v. From eq. ( 1.29) we learn that
=0
u · (u x v)
( l.31)
.
The magnitude of the cross product is defined to be
In x vi
= lnl lvlsinO(u, v)
,
0
< O(u, v) < " .
( 1.32)
It characterizes the area of a parallelogram spanned by the vectors u and v (see Figure 1.3). The right-handed cross product of u and v, i.e. the vector u x v, is perpendicular to the plane spanned by u and v (0 is the angle between u and v). uxv
v
.~
Area=
lox vi
I
I I
I I I
:
u
I I
f -u xv= v x u
Figure 1.3 Cross product of vectors u and v.
In order to express the cross product in terms of components we introduce the permutation (or alternating or Levi-Civita e-) symbol C.ijk' which is defined as
C.ijk
=
I ,
for even permutations of (i, j, k) (i.e. 123, 231, 312) ,
-I '
for odd permutations of (i, j, k) (i.e. 132, 213, 321) ,
0 ,
if there is a repeated index
( 1.33)
,
with the properties Eijk = C.jki == C.kij' C.ijk = -Eiki and Eijk = -c.J'ib respectively. Consider the right-handed and orthonormal basis {ei}, then e 1 x e 1 = e2 x e 2
==
C3 X e:1
=0
,
( 1.34)
1.1
7
Algebra of Vectors
or in a more convenient short-hand notation, with ( 1.33), ( 1.35) With relations ( 1.2 I), ( 1.34), ( 1.35) it is easy to verify that EiJk may be expressed as the determinant of a matrix ( 1.36)
where we have introduced the square brackets [•] for a matrix. The product of the permutation symbols Eijk Epqr is related to the Kronecker delta by the relation
( 1.37)
With (1.22) 1 and (l.22h we deduce from (l.37) the important relations Eijk Eijk
EXAMPLE 1.1
Solution.
=6
.
Obtain the coordinate expression for the cross product w
( 1.38)
=u
x v.
Taking advantage of eqs. ( 1.20) and ( 1.35) we find that (l .39)
with the three components
( 1.40)
( l.41) ( 1.42) Consequently, (u x v) · ek equals
Eijk'uivi.
•
Now, using expressions ( 1.36), ( 1.39).1, the vector product u x v relative to { ek} may be written as
u xv= det
e:!
e:1
u1
U2
'll:1
'Vt
'V2
'lJ3
[ e1
l
= c ijk 'lli'lJ.i ek
( 1.43)
8
1 Introduction to Vectors and Tensors
The triple scalar (or box) product (u xv)· w represents the volume F of a parallelepiped 'spanned' by u, v, w forming a right-handed triad (see Figure 1.4).
uxv w
v
Volume= (u xv) ·w u
Figure 1.4 Triple scalar product.
By recalling definitions ( 1.29), ( 1.10) and ( 1.32), we have
l"
= (u x v) · w = (v x w) · u = (w x = ju x vi lwlcos8 = lullvlsinO lwlcosc5 .
u) · v ( 1.44)
~~
base area
height
Using index notation, then, from eqs. ( 1.20) and ( 1.35) we find with ( 1.2 I) the volume 1·'" to be
l"
= (u x v) · w = Eijk'lli'VfWk = (u2'V3 - 'U;fV2}'W1 + (u3'U1
-
'UJ V:1)W2
+ (u1 V2
-
'll2Vi)W:i
.
(I .45)
Hence, the triple scalar product ( 1.45):1 can be written in the convenient determinant form
(u x v) · w = det [
:~ :~~ ::~ ] 'U3
'V3
( 1.46)
'W3
Note that the vectors u, v, w are linearly dependent if and only if their triple scalar product vanishes (the parallelepiped has no volume). The product u x (v x w) is called the triple vector product and may be verified with ( 1.39).i, ( 1.38) 1, ( 1.22h and representations ( 1.20) and ( 1.24)4
9
1.2 Algebra of Tensors
U X
(v
X W}
= EijkUi(EmnjVmWn)ek = =
(8km8in -
= (u · w)v -
EkijEmn{UiVmWnek
c5kn8im)UiVmWnek
( 1.47)
(u · v)w ,
which is the so-called 'back-cab' rule well-known from vector algebra. Similarly,
(u x v) x w = (u · w)v - (v · w)u .
( 1.48)
The triple vector product is, in general, not associative, i.e. ( u x v) x w ¥= u x (v x w).
EXERCISES
1. Use the properties ( 1.5), ( 1.6), ( 1.8) and ( 1.9) to show that QO= 0
,
Ou= o ,
(-n)u = a(-u) .
2. By means of (l.30) and (1.28) derive the property
(au+ ,Bv) x w = a(u x w)
+ {3(v x
w) .
3. Prove the triple vector product ( 1.48) and show that the vector (u x v) x w lies in the plane spanned by the vectors u and v.
1.2 Algebra of Tensors A second-order tensor A, denoted by A, B, C ... (or in the literature sometimes written as A, B, C .. .), may be thought of as a linear operator that acts on a vector u generating a vector v. Thus, we may write
v=Au
( 1.49)
which defines a linear transformation that assigns a vector v to each vector u. Since A is linear we have
A(nu+v) = aAu+Av
( 1.50)
for all vectors u, v and all scalars a. Since most tensors used in this text are of order two, we shall often omit the adjective 'second-order'.
10
1 Introduction to Vectors and Tensors
If A and B are two (second-order) tensors, we can define the sum A + B, the difference A - B and the scalar multiplication a A by the rules (A± B}u = Au± Bu ,
( 1.51)
(o:A)u = cr(Au) ,
( 1.52)
where u denotes an arbitrary vector. The important second-order unit (or identity) tensor I and the second-order zero tensor 0 are defined, respectively, by the relations Io = ul = u and Ou = uO = o for all vectors u. Note that tensor 0 maps every u to the zero vector o. Tensor product. The tensor (or direct or matrix) product or the dyad of the vectors u and v is denoted by u ® v (some authors use the notation uv). It is a secondorder tensor which linearly tran~jorms a vector w into a vector with the direction of u following the rule ( 1.53) The dyad, not to be confused with the dot or cross product, has the linearity property (u ® v)(crw + x)
= a:(u ® v)w + (u 0
( 1.54)
v)x .
In addition, note the relations (nu
+ t'v) 0
w
u(v ® w) (u 0 v)(w ® x) A(u ® v)
=
a ( u 0 w)
+ f3 (v ® w)
= (u · v)w = w(u · v) , = (v · w)u ® x = u 0 x(v · w) =
(Au)® v .
( 1.55)
,
( 1.56) ,
( 1.57)
(l.58)
Generally, the dyad is not commutative, i.e. u 0 v # v 0 u and (u ® v)(w ® x) # (w ® x)(u ® v). A dyadic is a linear combination of dyads with scalar coefficients, for example, a(u ® v) + (J(w 0 x). Note further that no tensor may be expressed as a single tensor product, in general, A = u ® v + w ® x # y ® z. Any second-order tensor may be expressed as a dyadic. As an example, the secondorder tensor A may be represented by a linear combination of dyads formed by the (Cartesian) basis {ei}, i.e. ( 1.59) We call A, which is resolved along basis vectors that are orthonormal, a Cartesian tensor of order two (more general tensors will be considered in Section 1.6). The nine
1.2
11
Algebra of Tensors
Cartesian components of A with respect to { ei}, represented by Ai1, form the entries of the matrix [A], i.e.
(A]= [
~~:
( 1.60)
A:11
where by analogy with ( 1.22h
( 1.61) holds. Relation ( 1.60) is known as the matrix notation of tensor A.
EXAMPLE 1.2 Let A be a Cartesian tensor of order two. Show that the projection of A onto the orthonormal basis vectors ei is according to ( 1.62) where Aii are the nine Cartesian components of tensor A.
Solution. find that
Using representation ( 1.59) and properties ( 1.53 h, ( 1.2 I) and ( 1.61 ), we
Aei == Atk(el ® ek)ej = Atk(ek · ei)e,
==
AtkOkjel
= Atiet
.
(1.63)
On taking the dot product of ( 1.63 ),1 with ei, the nine Cartesian components Aii are completely determined, namely
ei · Aei
= ei · A11e1 = Atj(ei · e,) == Ali8il = AiJ = (A )iJ .
( 1.64)
In ( 1.64) we have used the notation (A)ii for characterizing the components of A.
•
If the relation v · Av > 0 holds for all vectors v # o, then A is said to be a positive semi-definite tensor. If the stronger condition v · Av > 0 holds for all nonzero vectors v, then A is said to be a positive definite tensor. Tensor A is called negative semidefinite if v · Av < 0 and negative definite if v · Av < 0 for all vectors v # o, respectively. The Cartesian components of the unit tensor I form the Kronecker delta symbol introduced in eq. ( 1.21 ). Thus, ( 1.65)
12
1 Introduction to Vectors and Tensors
We derive further the components of u ® v along an orthonormal basis { ei}. Using representation ( 1.62) and properties ( 1.53) and ( 1.23 )a, we find that
(u ® v) ii
= ei · (u ® v) ei = ei · u(v · ei) = (ei · u)vi ( 1.66)
where the coefficients uivi define the nine Cartesian components of u ® v. Writing eq. ( 1.66)4 in the convenient matrix notation we have U1V2
U1V3
U2V2
U2V:3
U3V2
U3V3
l
( 1.67)
Here, the 3 x 1 column matrix [ui] and the 1 x 3 row matrix [vi] represent the vectors u and v, while the 3 x 3 square matrix represents the second-order tensor u ® v. A scalar would be represented by a 1 x 1 matrix.
EXAMPLE 1.3 written as
Show that the linear transformation ( 1.49) that maps v to u may be
(l.68) where index .i is a dummy index.
Solution.
Using expressions ( 1.20), ( 1.59) and rules ( 1.53 ), ( 1.21) and ( 1.61 ), we find from ( 1.49) that viei
= Aiiuk(ei ®ej)ek = Aiiuk(eJ · ek)ei = Aiiuk8ikei = Aikukei .
(1.69)
A replacement of the dummy index k in eq. ( 1.69)4 by j gives the desired result (1.68). •
Dot product. The dot product of two second-order tensors A and B, denoted by AB, is again a second-order tensor. It follows the requirement {AB)u = A(Bu) for all vectors u.
( 1.70)
1.2
Algebra of Tensors
13
The summation, multiplication by scalars and dot products of tensors are governed mainly by properties known from ordinary arithmetic, for example,
A+B=B+A,
(1.71)
A+O=A,
(1.72)
A+ (-A)= 0
,
( I.73)
A+ (B + C) = (A+ B) + C , (a:A) = o:(A} = a:A , (AB}C = A{BC) = ABC ,
( 1.74) ( 1.75) ( 1.76)
A2 -AA '
( 1.77)
(A+B)C = AC+BC .
( 1.78)
Note that, in general, the dot product of second-order tensors is not commutative, i.e. AB # BA and also Au # uA. Moreover, relations AB = 0 and Au = o do not, in general, imply that A, B or u are zero. The components of the dot product AB along an orthonormal basis { ei }, as introduced in ( 1.18), read, by means of representation ( 1.62) and relations (I. 70), ( 1.63 ).i,
{AB)ii = ei · (AB}ei =
== ei · A(Bkiek) ==
ei ·
A{Bei)
= (ei · Aek)Bki
AikBki
( 1.79)
or equivalently ( 1.80)
For convenience, we adopt the convention that a repetition of only one index between a tensor and a vector (see, for example, eq. (1.68) with the dummy index j) or between two tensors (see, for example, eq. ( 1.80) 1 with the dummy index k) will not be indicated by a dot when symbolic notation is applied. Specifically that means for v = Au (vi = AifUj) and A == BC (Aii = BikCki) we do not write v = A· u and A = B · C, respectively.
Transpose of a tensor. The unique transpose of a second-order tensor A denoted by AT is governed by the identity v·ATu=u·Av=AV·U
( 1.81)
for all vectors u and v. The useful properties (l.82)
14
1
Introduction to Vectors and Tensors
(nA + ~Bfr {AB)'r
= aAT + (JBT = BTAT ,
( 1.83)
,
( 1.84)
(u®v)T = v®u
( 1.85)
hold. Hence, from identity ( 1.81) we obtain ei ·AT ei = ei · Aei, which gives, in regard to ( 1.62), the important index relation {AT)ii = Aii·
Trace and contraction. The trace of a tensor A is a scalar denoted by tr A. Taking, for example, the dyad u ® v and summing up the diagonal terms of the matrix form of that second-order tensor, we get the dot product u · v = 1Li'Vi, which is called the trace of u 0 v. We write tr( u ® v} = u · v = -urvi
( 1.86)
for all vectors u and v. Thus, with representation ( 1.59) and eqs. ( I.86)i, ( 1.21 ), ( 1.61) the trace of a tensor A with respect to the orthonormal basis {ei} is given by trA
= tr(Aiiei 0 ei) = Aiitr{ei ® ei) = Aii(ei · ei) = Aiioii ( 1.87)
= Aii ,
or equivalently
trA
= Aii = .A11 + A22 + A33
( 1.88)
.
We have the properties of the trace:
trAT
= trA
( 1.89)
,
tr{AB} = tr{BA) ,
( 1.90)
tr{A + B) = trA + trB ,
( 1.91)
tr{o:A)
= o:trA
(1.92)
.
In index notation a contraction means to identify two indices and to sum over them as dummy indices. In symbolic notation a contraction is characterized by a dot. A double contraction of two tensors A and B, characterized by two dots, yields a scalar. It is defined in terms of the trace by A :B
== tr(ATB) = tr{BTA)
= tr{ ABT) = tr{BAT) = B:A or AiiBii == BiiAii
.
( 1.93)
15
1.2 Algebra of Tensors
The useful properties of double contractions
I :A A : {BC)
= trA = A : I = {BTA) : C = {ACT)
( 1.94)
:B ,
A: (u ® v) = u ·Av= (u ® v) : A ,
( 1.95) ( 1.96)
( 1.97) = (u · w)(v · x) , ( 1.98) (ei ® ei) : (ek 0 el) = (ei · ek)(ei · e,) == 8ik8it Note that if we have the relation A : B = C : B, in general, we cannot conclude
(u ® v) : (w ® x)
hold. that A equals C. The norm of a tensor A is denoted by IAI (or llAll). It is a non-negative real number and is defined by the square root of A : A, i.e. ( l.99)
Determinant and inverse of a tensor. The determinant of a tensor A yields a scalar. It is defined by the determinant of the matrix [A] of components of the tensor, i.e. ( 1.100)
with properties
det(AB} detA T
= detA detB = detA .
,
(1.101) (I. I 02)
A tensor A is said to be singular if and only {f detA = 0. We assume that A is a nonsingular tensor, i.e. detA # 0. Then there exists a unique inverse A - t of A satisfying the reciprocal relation ( 1.103)
If tensors A and B are invertible, then the properties {AB)- 1
= B- 1A- 1
,
(A-1)-1 =A ' {a:A)- 1 = 1/o: A -I
( 1.105) ,
(A-l)T=(AT)-1, A-2 = A-1A-1
(1.104)
( 1.106) ( 1.107)
'
(I.] 08) ( 1.109)
16
1 Introduction to Vectors and Tensors
hold. Subsequently, in this text, we use the abbreviation (A-l}T
= A-T
( 1.110)
for notational convenience.
Orthogonal tensor.
An orthogonal tensor Q is a linear transformation satisfying
the condition Qu·Qv
= u·v
(l.111)
for all vectors u and v (see Figure 1.5). As can be seen, the dot product u · v is invariant during that transformation, which means that both the angle () between u and v and the lengths of the vectors, lnl, lvl, are preserved. Qu
u
Q
lul
v
Qv
Figure 1.5 Orthogonal tensor.
Hence, an orthogonal tensor has the property QTQ = QQT = I, which means that Q- = QT. Another important property is that det(QTQ) = (detQ) 2 with detQ = ± 1. If detQ = +1 ( -1), Q is said to be proper (improper) orthogonal corresponding to a rotation (reflection), respectively. 1
Symmetric and skew tensors. Any tensor A can always be uniquely decomposed into a symmetric tensor, here denoted by S, and a skew (or antisymmetric) tensor, here denoted by W. Hence, A
= S + W, while (l.112)
In this text, we also use the notation symA for Sand skew A for W. Tensors Sand W are governed by properties such as or
or
1.2
17
Algebra of Tensors
which in matrix notation reads
(SJ =
[ S11
812
S12
S22
S13 823
813
823
Saa
l
(W] = [
-l~'12
ll'12
-TV1a
0
W13 12a H
-H123
0
l
(l.114)
In accord with the notation introduced, the useful properties
s : B = s: BT= s: ~(B +BT)
w: B =
-W: BT=
w: ~(B -
(l.115)
' BT) '
....,
S:W= 0
(1.116) ( 1.117)
hold, where B denotes any second-order tensor. A skew tensor with property W = - WT has zero diagonal elements and only three independent scalar quantities as seen from eq. (1.114)2. Hence, every skew tensor W behaves like a vector with three components. Indeed, the relation holds:
Wu= w x u ,
(l.118)
where u is any vector and w characterizes the axial (or dual) vector of skew tensor W, with property lwl = {1/V2)1WI (the proof is omitted). The components of W follow from (1.62), with the help of (1.118), (1.20), (1.35), ( 1.21) and the properties of the permutation symbol
lVii =
ei
·Wei= ei · (w x ej) =
ei · (wkek
= ei · (wk€kjtel) = Wk€kjl8il = CkjiWk = -€ijkWk •
x ei) (1.119)
Therefore, with definition ( 1.33) we have
= -w3 = -€13kWk = W2
H112 =
ltf/13
-c12kwk
,
(1.120)
,
(1.121) (1.122)
where the components H112 , lV13 , H123 form the entries of the matrix [W] as characterized in ( 1.114)2. The inversion of ( 1. l l 9h follows with relations ( 1.38)2 and ( 1.22)2 after multiplication with the permutation symbol €ijp ( 1.123)
18
1 Introduction to Vectors and Tensors
which, after a change of the free index, gives finally Wk• --
-
~c.
= dcI>(t}/dt.
In general, the n-th derivative of u and A (for any desired n) denoted by d 11 u/dt 1i and dn A/dtn, is a vector-valued and tensor-valued function whose components are d 11 [ui(t)]/dtn and d 11 [..4ij(t)]/dtn, respectively. Then-th derivative of is simply d 11 cI>(t)/dtn. Note that if , u, A are constant quantities, then their derivatives are equal to zero. By applying the usual rules of differentiation, we obtain the identities:
u±v=ti±v'
Cilii = ci> u + cI> ti ' u0v = ti©v+u0v , A±B=A±B
T
·T
trA
= trA
( 1.232) (1.233) ( 1.234) (I .235)
A =A
.
c[231)
.
( 1.236)
Here and elsewhere the overbars cover quantities to which the dot operations are applied.
41
1.7 Scalar, Vector, Tensor Functions
EXAMPLE 1.9
Show the identity
( 1.237)
Solution. Starting from A - l A = I we find by means of the chain rule that A- 1A+ A- 1A = 0. Finally, by multiplying the relation with A - l from the right-hand side we obtain the desired result (I .237). •
Gradient of a scalar-valued (tensor) function. We consider a nonlinear and smooth scalar-valued (tensor) function {A) of one second-order tensor variable A. Thus, for each A, (A) results in a scalar. The aim is to approximate the nonlinear function at A by a linear function. Firstorder (Taylor's) expansion for at A yields, using property ( l.93)i,
cI>(A + dA) = {A} d =
+ d + o(dA)
a~~A) :dA =tr [ (a~~))
T
dA]
,
( 1.238) ( 1.239)
where d denotes the total differential. The remainder is characterized by o( dA), where o( •) denotes the Landau order symbol. It is a small error that tends to zero faster than dA ---+ 0, i.e.
. o(dA} hm jdAI =0 . dA-tO
( 1.240)
The term D(A)/DA (or gradcI>(A)) denotes the gradient (or derivative) of function cl> at A, characterizing a second-order tensor. In order to make the derivation clear, the tensor variable to be derived is sometimes indicated in the literature by a corresponding index, for example, gradA {A). Note that the first variation of cI> is developed in the same way as its total differential. In Chapter 8 the concept of linearization is treated in more detail.
EXAMPLE 1.10 relation
If the second-order tensor A is invertible, establish the important
BdetA _ d A A-T BA - et .
( 1.241)
42
Solution.
1 Introduction to Vectors and Tensors With ( 1.10 I) we find that det(A + dA) = det[A(I + A- 1dA)] = detAdet(I + A- 1dA) .
( 1.242)
By means of eq. ( 1.168), we conclude with A replaced by A - I dA and with ,.\ that
= -1
det(A- dA +I) = 1 + /A-1dA + IIA-1c1A + IIIA-1dA 1
= 1 + tr{A- 1dA) + o{dA)
( 1.243)
.
In eq. (1.243) we have used the fact that in regard to (l.171) and (l.172) the second invariant is quadratic in dA and the third invariant is cubic in dA. Therefore, both are suitably small and summarized by the remainder of order o( dA). Replacing cI> (A) in relations ( 1.23 8) and ( 1.239) by the scalar-valued function detA, we deduce that
det(A + dA)
= detA +tr [ (
DdetA f)A
'l'
)
l
dA + o(dA) .
(1.244)
On the other hand we find by means of (l.242h, (l.243h and property (1.92) that det(A + dA)
= detA(l + tr(A- 1dA)] + o{dA) = detA + tr(detAA- 1dA) + o{dA)
.
(1.245)
Using rule (l.93) 1 and property (1.83) we find by comparing (1.244) with (l.245h that EJdetA
aA : dA = {detAA - 1)T
: dA
= detA A - T
which proves finally eq. ( 1.24 I}, since this holds for all dA.
: dA ,
(1.246)
•
Gradient of a tensor-valued (tensor) function.
We consider a nonlinear hnd smooth tensor-valued (tensor) function A {B) of one second-order tensor variable B. Thus, for each B, A(B) gives the value of a second-order tensor. In order to compute the total differential dA we consider first-order (Taylor's) expansion for A at B. We obtain A{B +dB)= A(B) + dA + o(dB) ,
cIA
= 8A(B) . dB
8B
.
.
( 1.247) ( 1.248)
The term 8A(B)/ EJB (or grad 8 A(B)) denotes the gradient (or derivative) of function A at B, characterizing afourth-order tensor. Note that BA/ DA produces the fourthorder unit tensor Il, as given in eq. (I. I 6 I).
1.7
43
Scalar, Vector, Tensor Functions
EXAMPLE 1.11 Assume that A is an invertible smooth tensor-valued function with property A = AT. Show that
aA-
1
aA
(
)
1
.. = - ?(AU,1 Aijt +All' Ak}) . 11kl
( 1.249)
-
Solution. Starting from the identity that 8{A- 1A)/8A = ((])gives the fourth-order zero tensor 0, we continue in index notation and find by means of the product rule
8Ahr!
,, _ 8Ami
4mj + "..:.iim1 {) 4 = 0 ./ aA kl kl
(1.250)
.6
and finally, after multiplication with (A - l )jn from the right-hand side, and using property ( 1.61 ),
8Aj;,: A
a
,t -'~kl
4-1 -
mj-- jn -
aaAkl A~,!
r Umn
8Ai;ll a"4kl
-
A-1 8Ami A-1 im {)A
kl
jn
'
1 ( A-1 r r 4-1 4-1 r r 4- I ) =- 2 imUmkUjt/ jn +" imUm/Ujk-'I jn
A-I ,t-lA-1) = - -1 (A-1 ik In + _.""iil kn I
2
'
•
( 1.251)
A replacement of the free index n by .i gives ( 1.249). Looking at the indices in ( 1.251) it is important to note that aA- 1 /BA# -A- 1 ® A- 1• •
EXERCISES
1. Suppose that Q = Q(t) is a given orthogonal tensor-valued function of a scalar, such as time t. Show that QQT is a skew tensor with the property QQT = -(QQT)T. 2. If the tensor A(t) is invertible, prove the important relation
detA(t) = detA(t)tr(A - l (t)A{t)] = detA(t)A(t)-T : A(t) and check the result with the matrix representation of the tensor A(t) given by
where a denotes a scalar.
44
1 Introduction to Vectors and Tensors 3. Consider two smooth tensor-valued functions A and B of a tensor argument. (a) Show the properties
DtrA =I
DA
Dtr(AJ) 8A
'
= 3(AT)2
(1 252)
'
.
(n is a positive integer) .
(b) If A is an invertible smooth tensor-valued function, show that (l.253) ( 1.254) 4. Let be a smooth scalar-valued function and A, B smooth tensor-valued functions of a tensor variable C. Obtain the properties
= A . DB B . 8A ac · ac + · ac ' 8( A) = A D aA 0 ac ac + ac ·
8(A : B)
(l.255) (1.256)
5. Alternatively to ( 1.241), show that
DdetA 8A
= {AT)2 -
I AT i
+
I I 2
r .
(1.257)
Hint: By means of property ( 1.92) and scalar invariants ( 1.170)-( I. I 72) take the trace of the Cayley-Hamilton equation ( 1.174) and derive it with respect to tensor A, use result ( 1.252).
1.8 Gradients and Related Operators In this section we consider scalar, vector and tensor-valued functions that assign a scalar, vector and tensor to each material point x varying over some region of definition.
1.8
45
Gradients and Related Operators
We call a tensor A and a vector u whose components Aii and ui depend on three (Cartesian) coordinates Xi a tensor field A{x) and a vector field u(x) in space at a fixed time, respectively. A scalar field cI>(x) of a body is a function that assigns a scalar cl> to each material point x. Subsequently, we consider a smooth scalar field cI>(x) (for example, a temperature, density or mass field). First-order (Taylor's) expansion for at x yields by analogy with ( 1.238) and ( 1.239)
Gradient of a scalar field.
(x + dx) == (x) d
+ dcI> + o( dx)
a == - · dx
ax
,
( 1.258) ( 1.259)
'
denoting the total d(fferential of . The remainder o( dx) tends to zero faster than dx -t o. The total differential, characterized as the difference between the scalars at x + dx and x, may be written in index notation as ( 1.260) In the literature the partial derivatives of with respect to Xi is frequently denoted by the subscript comma, i.e. a multiplied by the appropriate basis vectors is a vectorjield (8cI>/8:1:i)ei, called the gradient (or derivative) of the scalar field cI>(x). By analogy with ( 1.261 ), the dot product, cross product and tensor product of the vector operator V with a smooth vector or tensor field ( •) is governed by the rules
V®(•)
= ~:;/ ®ei
These relations turn out to be very useful for subsequent derivations.
. (1.264)
1 Introduction to Vectors and Tensors
46
Concept of directional derivative.
We consider cI> (x) describing a scalar field that varies throughout a three-dimensional space. We recall that (x) = (xi, .1: 2 , x:1) = const denotes a so-called level surface characterized by all points x resulting in the same value cl>. All neighboring points x + clx on a level surface in space are therefore governed by the relation d = 0. A normal to the surface is defined by three values, namely 8cI> I I~ acI> I a:r2' 8. We call the dot product gradcI> · u a directional derivative (or a Gateaux derivative) of at x in the direction of the vector u. Note that for the definition of a directional derivative the normalized vector u is often used. As we let u vary, the directional derivative gradcI> · u takes its maximum (fora fixed point x) when u approaches the direction of grad (compare with the expression for the dot product, i.e. ( 1.10) ). For this particular situation u points in the direction of the normal vector n and cos(} = 1. Similarly, gradcI> · u takes its minimum value (for a fixed point x) when u approaches the opposite direction of the gradient (i.e. when cos(}= -1).
The directional derivative of in the direction of n is called the normal derivative; we write
gradcI> · n
= jgradI
.
( 1.265)
For the case in which u is a unit vector the normal derivative represents the maximum of all the directional derivatives of at x.
1.8
47
Gradients and Related Operators
For further consideration we briefly note the equivalent representation for the directional derivative of at x, namely
(l.266) characterizing the rate of change of along the (straight) line through x in the direction u (for a more detailed account see, for example, ABRAHAM et al. [1988] and MARSDEN and HUGHES [1994]). Usually D( •)is called the Gateaux operator and c is a scalar parameter. By means of the chain rule we immediately find from definition (l.266) that D 0 (x) = gradcI> · u. Note that the directional derivative satisfies the usual rules of differentiation, i.e. the chain rule, product rule and so forth. Later in the text we will see that the Lie time derivative, Section 2.8, and the variation and linearization of fields, Sections 8.1 and 8.4, are equivalent operations which are based on the powerful concept of the directional derivative.
EXAMPLE 1.12
Suppose that the scalar field
cI>(x)
= :z:i + 3x2xa
(1.267)
describes some physical quantity in space. Compute the directional derivative of in the direction of the vector u = 1/ J3( e 1 + e 2 + e3 ) at the point x with coordinates (2, -1, 0) and interpret the result. The parametric expression x +ell represents a line through the given point x with direction u. We find the components of x + en (for arbitrary, but fixed e) as
Solution.
{2 + c/V3, -1 + c:/J3, c/J3). Hence, according to definition ( 1.266) and given expression ( 1.267), D 0 (2, -1, 0) is the derivative of 2
(x +cu)= ( 2 +
c )
J3
+3(
-1
e ) e 4 ') c + J3 J3 = 3 c- + J3 + 4
(1.268)
with respect to the one variable e, evaluated at zero. Therefore,
Duib (2 -1 0) = -d '
'
de
(4-e + -J3e +4) 2
3
1
=- . e=O J3
(1.269)
As noted above the directional derivative may also be computed by the dot product grad · u. In matrix notation we equivalently find from the given expression ( 1.267) that 1 [ 3x3 2:i: 1 D0 cJ>(x) = [grad][u] = J3 3:r:2
l[ l 1 1 1
( 1.270)
1 Introduction to Vectors and Tensors
48
which, evaluated at :c 1 = 2, x 2 = -1, x:l = 0, gives the desired result of 1/ J3. Hence, if we are located at point (2, -1, 0) and walk away in the given direction u, we recognize an increase in some physical quantity, at a rate of 1/ J3 per distance, which is here
lnl =
1.
•
Divergence of a vector field. The introduced vector operator V dotted into any smooth vectorfield u(x) (for example, aforce, velocity or acceleration.field) is called the divergence of u(x). It is a scalar.field denoted by divu (or V · u). With ( 1.264) 1 and eqs. ( 1.20), ( 1.21) and ( 1.22)2 we find the important rule
au.·
Ou ·
8-ui
divu = V · u = - 1 e·1 · ei = - 1 8·i = 8:ri 8:i-i J 8:i;i Du1 8u.2 8u3
=8x1 - +a.1;2 - +8:1:3 -.
If divu
( 1.271)
= 0, then the vector field u(x) is said to be solenoidal (or divergence-free).
Curl of a vector field. The cross product of the vector operator V with a smooth vector field u(x) is called the curl (or rotation) of u. It is a vector field denoted by curio (or V x u, or rotu). With (I .264h and eqs. ( l .20) and ( 1.35) we find that
curio= V
D-u.
X U
= D J ei
X ej
=
Xi
a'u.
Eijk EJ J ek
( 1.272)
Xi
and with definition ( 1.33)
( l.273)
If curio = o, then the vector field u(x) is said to be irrotational (or conservative, A or curl-free). We can show further that curl grad cI> = o , div curio
=0
.
( 1.274) ( 1.275)
We now consider any vector field u of the form u = grad cI>, where is some scalar field, called the potential of u. According to ( 1.274) it readily follows that the vector field u is automatically irrotationa/, since curio = o.
EXAMPLE 1.13 Show that curio is determined to be twice the axial vector w of the skew part of tensor gradu.
1.8
49
Gradients and Related Operators
Solution. We denote the skew part of gradu as W, with components H'ii = 8ui/axi. With the index relation H'ii = -lVii for skew tensors and (l.124)2, we deduce from (l.272h that
•
(1.276)
Gradient of a vector field. The gradient (or derivative) of a smooth vector.field u(x) is defined to be a (second-order) tensorjield. It is denoted by gradu (or Vu, or au/ax or sometimes more explicitly by V ® u) and according to (1.264)3 is given as gradu
=V
®u
with Cartesian components (gradu)ij
[gradu]
(1.277)
,
= 8uif 8xi. In matrix notation this reads as 8u1
a'U1
8u1
Bx1
ax2 a'U2
8xa
8u2
=
8-ui 8 J
= - x· ei ® ei
8u2
-
8:1:1
8x2
8u3
GU3
au3
a:r1
8x2
8xa
( 1.278)
8:c 3
It is easily seen from eq. ( 1.278) that the sum tr(gradu) of the diagonal elements of the matrix is equivalent to the divergence of u characterized in eq. ( 1.271 ). Thus, tr{gradu)
= gradu : I = divu
(1.279)
.
In addition, the transposed gradient of a smooth vectorfield u(x) of position xis denoted by gradTu (or uV, or (8u/8x)T or sometimes more explicitly by u ® V). It 1s given as gradT U
=U 0
V
= ei 0
with Cartesian components (gradTu)ij
au = Bu· a:ti. ei ® ej 8 1
-
,
(1.280)
Xi
= 8ui/8xi.
Divergence and gradient of a (second-order) tensor field. The vector operator V dotted into any smooth (second-order) tensor.field A(x) is a vector.field denoted by div A (or V ·A), which is the divergence of A(x). With (l.264)i, representation (1.59) and properties ( 1.53 }, ( 1.21) and ( 1.61) we have
( 1.281)
1 Introduction to Vectors and Tensors
50
Note that, especially in the mathematical community, the alternative definition div A = 8Aii/ 8xiei of the divergence of a second-order tensor field A is frequently used. The gradient (or derivative) of a smooth (second-order) tensor field A(x) is defined to be a (third-order) tensor.field. It is denoted by gradA (or VA, or DA/Dx, or more explicitly by V ®A). Using ( 1.264)3 and representation ( 1.59), we may write
gradA
=V
®A
=
DA·· 11
aXk. ei ® eJ ® eh
( 1.282)
.
Laplacian and Hessian. The vector operator V dotted into itself gives the Laplacian (or the Laplacian operator) \7 2 (or V · V, or in the literature sometimes denoted by
~).
With the rules ( 1.261 ) 1, ( 1.264) 1 and properties ( 1.2 I), ( l.22h we find that
" . v-(•) =
V · V(•) == V
2 a (.) a (.) · -ei =
D:ri
= a~(·~
oij
8xi83-j
Dxia:c:i
= a2(~)
ei · e·3
.
( 1.283)
8.'ti
The Laplacian V 2 operated upon a scalar field cI> yields another scalar field. One example is Laplace's equation \7 2 cI> = 0. Another example is the inhomogeneous Laplace's equation often referred to as Poisson's equation, i.e. \7 2 = w. The differential equations of Laplace and Poisson occur in many fields of physics and engineering, for example, in electrostatics, in elasticity theory, in heat conduction problems of solids and so on. The operator VV (or more explicitly denoted by V ® V) is called the Hessian. With the rules ( 1.261 )i and ( 1.264h we obtain
a(.)
VV(•)=V0V(•)=V®-a· ei= :Ci
a2 (.)
aXi a·Xj ei®ei
( 1.284)
for the Hessian. Note the properties
curlcurlu =grad (divu) - \7 2 u , 2
\7 (u · v)
? ? = v-u · v + 2gradu · gradv + u · v-v .
( 1.285) ( 1.286)
If a vector field u is both solenoidal (divu = O} and irrotational (curio = o ), then \7 u = o (see eq. ( 1.285)). For this case the vector field u(x) is said to be harmonic. If a scalar field cI> satisfies Laplace's equation \7 2 cI> = 0, then cI> is said to be harmonic. 2
EXERCISES
1. Consider a smooth scalar field = x 1:z:2 :r3 - :r: 1 • Find a vector n of unit length normal to the level surface cl>= canst passing through {2, 0, 3).
1.8
Gradients and Related Operators
51
2. Assume a force with magnitude F acting in the direction radially away from the origin at point (2a, 3a, 2v'3c) on the surface of a hyperboloid cI>(x) == cI>(1: 1, 1: 2 , 'JI 2 ., I ., 2 I " . :1:3 ) = :r:1 a + 1:2 er - :z::1 c- = 1. Compute the components of the force located in the tangent plane to the surface at the point (2a, 3a, 2v'3c). 3. Consider a scalar field {:r 1 , :v2} = e:ra cos(3:i: 1 - 2:r 2 }. Compute the directional derivative of cI> in the direction of the line :1: 1 = :r2 /3 - 1 at the point with coordinates (0, 1), for increasing values of:1: 1• 4. If U = u(x) == :r1:1:2J:ae1 + :r1:r2C2 + :r1 e3, determine divu, curio, graclu, V7 2u, respectively. Verify that ( 1.285) is satisfied. 5. Apply the operator V to products of smooth scalar fields cl>, \JI, vector fields u, v and tensor fields A. Establish the important identities div( cI>u) = divu + u · grad ,
( 1.287)
div{cpA) = cI>divA + Agrad ,
( 1.288)
div(ATu) = div A· u +A : gradu ,
( 1.289)
div(u x v) = v ·curio - u · curlv , div(u®v) = (gradu)v+udivv , grad ( \JI)
( 1.290)
= (grad cp) W + grad \JI
,
gTad(u) = u ® grad + gradu , grad{u · v) = {gracfru) v + (gradTv) u , curl{ cI>u)
= grad x u + curlu
( 1.291)
,
curl(uxv) =udivv-vdivu+(gradu)v-(gradv)u.
(1.292)
6. Consider the inverse square law defined by the particular vector field x u =a lxFI '
with the positive constant a and x = :ciei. The region of u is the three-dimensional space, excluding the origin whose gradient is u. (
== -a/lxl>
52
1 Introduction to Vectors and Tensors
1.9 Integral Theorems In the following we summarize some results of the integral theorems of Gauss and Stokes which are of essential importance in the field of continuum mechanics.
Divergence theorem. Suppose u(x) and A(x) are any smooth vector and tensor fields defined on some convex three-dimensional region in physical space with volume v, and on a closed surfaces bounding this volume (see Figure 1.10). n
Surface element ds "
Closed surface s
Volume v
Figure 1.10 Volume v and bounded closed surface s (with infinitesimal surface element ds and associated unit normal vector n).
Then, for u and A we have - without proof
/ u · nds = / divudv
or
v
s
J
Ands
=
s
{
.
'Uinids
=
divAdv
or
I
1
axi
v
s
J
J8u·-
Ai1·n1·ds
=
v
s
ti
I
dv
'
--dv aA;j
8x·J
(1.293)
(1.294)
(given here in both symbolic and index notation), where n is the outward uni1' normal field acting along the surface s, dv and ds are infinitesimal volume and surface elements at x, respectively. The important transformation of a surface integral into a volume integral (i.e. (1.293), (1.294)) is known as the divergence theorem (or Gauss' divergence theorem). Similar results hold for higher-order tensors. The surface integral J'l u · n, u, and A be smooth scalar, vector and tensor fields defined in v and on s
1 Introduction to Vectors and Tensors
54
and let n be the outward unit normal field acting along the surface s. Show that
div(u)dv ,
( 1.297)
/ Ands = / div(cI>A)d-u ,
( 1.298)
/ cI>u · nds s
= / v
v
/ n x uds = / curludv , v
/ u 0 nds = / gradudv , ti
/ u ·Ands= / div(ATu)dv .
(1.299)
v
s
2. Let u and A be smooth vector and tensor fields defined in v and on .s and let n be the outward normal to s. Use index notation to show that
/ u x Ands=/ (u x div A+£ : AT)dv ,
(1.300)
v
where
e is the third-order permutation tensor as expressed in eq. ( 1.143 ).
3. Let cI> and u be smooth scalar and vector fields defined on s and c and let n act on s. Show that
.f
f c
c
dx = / n x gradcI>ds , s
u x dx = /[(divu)n - (gradTu)n]ds s
2
Kinematics
In the real world all physical objects are composed of molecules which are formed by atomic and subatomic particles. A microscopic system is studied by means of magnifying instruments such as a microscope. Microscopic studies are effective at the atomic level and very important in the exploration of a variety of physical phenomena. The atomistic point of view, however, is not a useful and adequate approach for common engineering applications. The microscopic approach is used in this text only briefly within Chapter 7. We use the method of continuum mechanics as a powerful and effective tool to explain various physical phenomena successfully without detailed knowledge of the complexity of their internal (micro)structures. For example, think about water which is made up of billions of molecules. A good approximation is to treat water as a continuous medium characterized by certain field quantities which are associated with the internal structure, such as density, temperature and velocity. From the physical point of view this is an approximation in which the very large numbers of particles are replaced by a few quantities; we consider a macroscopic system. Hence, our primary interface with nature is through these quantities which represent averages over dimensions that are small enough to capture high gradients and to reflect some microstructural effects. Of course the predictions based on macroscopic studies are not exact but good enough for the design of machine elements in engineering. The subject of continuum mechanics roughly comprises the following basic ingredients: (i) the study of motion and defonnation (kinematics), (ii) the study of stress in a continuum (the concept of stress), and
(iii) the mathematical description of the fundamental laws of physics governing the motion of a continuum (balance principles). The aim of the fallowing three Chapters 2-4 is to derive the essential equations within the basic fields (i)-(iii). Note that the provided results are applicable to all
55
2
Kinematics
classes of materials, regard.less of their internal physical structure. In -order to explain the macroscopic behavior of physical objects, :first of all we must understand the motion and deformation that cause stresses in a material (or are caused by stress.es) arising from forces and moments. Thus, to study the motion and (finite) d.eformation of a continuum, i.e. kinematics, is essential and mainly the aim -of the following chapter. The reader who requires additional information on the subject of this -chapte.r i-s referred to the monographs by, for example, MALVERN [ l 969], WANG and TRUESDELL .(1973], CHADWICK [1976], TRUESDELL [1977], GURTIN [1981.a], CIARLBT :[1988], TRUESDELL and NOLL [1992], MARSDEN and HUGHES [.1994) or OGDEN [1997].
2.1
Configurations, and Motions of Continuum Bodies
Under an electron microscope we see the discontinuous atomic structure of matter. The molecules may be crystalline or .randomly oriented. Between the particles there are large gaps. Theories considering the discrete structure of matter are ·molecular and atomistic theories. They are based on a discrete particle approach. For macroscopic systems such theories tend .to become too complicated to yield the desired results and therefore do not.meet our needs. However, under certain circumstances the microscopic approach is indispensable for the study of physical phenomena. A review of atomistic models is :presented in the work of ORTIZ f1999, and references the.rein_]. Notation of a particle nod a continuum body. Macroscopic systems usually can be described successfully with a continuum a_pproach (macroscopic .approach). Such an approach leads to the continuum theory. The continuum theory has been developed .independently of the molecular and atomistic theory and is meeting our needs. A fundamental assu-mp.tion therein states that a body, denoted by B, may be viewed as having a continuous (or at least a piecewise continuous) distribution of matte.r .in space and time. The body is imagined as being a composition of a (continuous) set of parti .. cles (or continuum particles or material points), represented by P E B as indicated . p·1gure 'J-· I . m. lt is important to note that the notion 'particle' (or 'continuum .particle' or 'material point') refers to a part of a body and does not imply any assoc.iation with the point mass -of Newtonian mechanics or the discrete particle of the atomistic theory as .mentioned above. A typical continuum particle is an accumulation of a large number of molecules, yet is small enough to be considered as a particle. The behavior of a continuum partic.le is a consequence of the collective behavior of all the molecules constituting that particle. Hence, in a macroscopic study we are concerned with the mec.hanics of a body in which both mass and volume are continuous (o.r at least piecewise continuous) .func-
2.. 1 Configurations, and Motions of Continuum Bodies Current configuration
Reference configuration .
.·.
.
. .
,; :)ii:::::::::
.
57
:· .
x
:~i,: >: ; :
:. ·:... ::. :·' ·' ._: .·. _· ... ·. The motion xis assumed to be uniquely invertible. Consider (x 1 t), the position of X, which is associated with the place x at time ·t, is specified uniquely by (2 ..I) as
:.x
x
\ve
·.point .......... ... ;
X
= x·~
1
(x,t)
,
(22)
\vith the inverse -motion, denoted by x-l., i.e. the .inverse of the mapping x~ as defined -~~. ~q. (2.1). For a given time t, the inverse motion (2.2) carries points located at .n to ·.p_o_i_nts in. the reference configuration n0 • .In (2.1) and (2 . 2) respectively the pairs (X, t) .... . :and (x, t) denote independent variables . _:· ... ~ A motion x of a body will generally change its shape~ position and orientation. ·Acontinuum body which is able to change its shape is said to be deformable. By a ~~formation x (or inverse deformation x- 1) of a body we mean a motion (or inverse :motion) of a body that is independent of time. .
·.'·::'-:=.==.:··~·~···:··."·."·'·:·····:·:·, ................... , .............,., ..... :~, ....... ,.,_... , ....................
··········· ······················:·:·:······."·······:·····:·.·.·.... : ....... ,...................................... ············ ················:·:·.··:'•." . ••••'··'·····.·:···'·."·:·.'."•."•"."•.'."·.··.··.·,.: ..·····."·.··,.·,··· . ·····,:•.•."•:.··
EXAMPLE 2.1 Let a deformed configuration of an initially .rectangular region with lengths L 1 and L 2 be given as shown in Figure 2 . 2. The time-dependent angle fJ is :given by O(t) = wl:, where w denotes the angular velocity. :_'.="·:·:.-:Determine the motion of a .particle given by the position vector XE .n 0 and time t In: particular, determine the -motion x = x(X, .t) at (L 1 /2, .£ 2 /2, 1) and at time t = "/4, ·with w = 1~
·60
2 Kinematics
:t =0 . f!o
Figure 2.2 Deformed configuration of an initially rectangular region.
Solution. The reference configuration of the region 0 0 of interest is characterized by (Xt, ""\2 , ..X3_) .at t = 0, with 0 < .X1 < L1 , 0 < .X2 < L2 and 0 < .1Ya < 1. The motion results in accord with (2.1)
= X2(..(Y1, .1Y2> ..t\3, t) = .X2 = L2/2 x3 = x3(.:Y1, . .Y2,..1Y3,t) = .x·3=1 . ,
:i:2
where c(t)
= tanO(t) is a positive scalar.
,
(2.3)
•
Material .and spatial descri.ptions.. The so-called material (or referential) description is a characterization o.f the motion (or any other quantity) with respect to the material coordinates {.X1 , ~~2 , ~Y3 ) and time t, given by eq. (2.1). In the material description attention is. paid to a particle, and we observe what happens to the particle as it moves. Traditionally the material description .is often referred to as the Lagrangian description (or :Lagrangian form). Note .that at t 0 we have the consistency condition X = x., .x·.:, = Xa. The so-called Eule-rian (or spatial) description (or Eulerian form)~ is .a charac ... terization of -the -motion (or any other quantity) with .respect to the spatial coordinates (:t.t, x 2 , x 3 ) and time t, given by eq. (2.2). In the spatial description attention is paid to a point in space, and we study what ·happens at the point as time changes. In :fluid mechanics we quite often work in the Eulerian description in which we refer all relevant quantities to the position in space at time t. It is not useful to refer the quantities to the material coordinates ..iY1i, A = 1, 2, 3, at t = 0, which are, in general, not known -in fluid mechanics. However, in solid mechanics we use both types
=
2.2
Displacement, Velocity, Acceleration Fields
61
of description. Due to the fact that the constitutive behavior of solids is often given in terms of material coordinates we often prefer the Lagrangian description.
EXERCISE
:I. Consider the two motions defined by the following sets of ·equations:
= o:t:i e + X
x
:1:2
= -fJ -
,
(8 +wt)
1e-111"cos:
.~,
where .n, /3, ·/, c5 and w are scalar constants and e denotes a fixed unit vector. Show that the path lines are straight lines and circles, respectively.
2.2 Displacement, Velocity, A-cceleration Fields Displacement field.
The vector field U(X, 't) = x(X, t) - X
(2.4)
represents the displacement fie=ld of a typical particle and relates its position X in the undeformed configuration to its position x in the de.formed configuration at time t (see Figure 2.3). Relation (2.4) holds for all particles. The displacement fie.Id U is a function of the referential position X and ti.me t, which characterizes the material description (Lagrangian form) of the displacement field. The displacement ·field in the spatial description {Eulerian form), denoted u, .is a function of the current position x and time t. We write
u(x, t)
= x - X(x, t)
.
(2.5)
Representation (2.5) specifies the current position x of a particle at time t which results from its referential position X(x, t) plus its displacement u(x, t) from that position. Representation (2.4) -expresses the displacement of a particle at time t in terms of its ·referential position X. The two descriptions are related by means of the motion x = x(X, t), name-Iy
U(XJ)
= U[x-t.{x, t), t] =
u(x, t) .
(2.6)
62
2
Kinematics
U(X, .t)
lime t = 0
= u(x, t)
x
x
lime t
Figure 2.3 Displacement fi.eld U of a typical particle.
Note that U and u have the same values. They represent functions of different arguments. The vectors U and u are referred to the material coordinates UA and the spatial coordinates ··11.a, respectively,. Since we agreed that the initial configuration coincides with the reference configuration, the displacements vanish in .the reference configuration. For a so-called rigid-body translation any particle moves an identical distance, with the same magnitude and the same direction at time t. For this case the displacement field is independent of X (it can be a function of time t).
Velocity and acceleration field.
.In solid mechanics the motion and the deformation of a continuum body are, in general, described in terms of the displacement ·field. However, the primary field ·quantities in fluid mechanics describing the :fundamental kinematic properties are the velocity fie.Id .and the acceleration field. The "first and second derivatives of the motion x with respect to time t are .perform.ed by holding X fixed. We obtain
V(X, t)
= Dx.~X, t) t
'
A(x ,t. )
= 8V(X, t) = D2 x(X, 't_) '" a·t a')t'
where V(X, t) and A(X, t) denote the material description of the veloc:ity field and .the acceleration field, respectively. They are functions of the material coordinates (.X 1 , ..Y2 , ~\:1 ) and time t representing the time rate of change of position and velocity .of a particle X at time t.
2.2
·63
Displacement, Velocity, Acceleration Fields
Quite often we need a formuJation in terms of the spatial coordinates (:1:J, :l:2 , :i::i) chm·acterizing a fixed location in spac-e at time t. As mentioned above, such a description is known as the Eulerian or spatial description in which now we like to define the velocity field and the acceleration field. :.":". . . The two -descriptions are transformed into -each other by using the motion X· In · · the spatial description, X may be expressed in tenns of x and t Hence, from (2. 7) we conclude by analogy with eq. (2.6) that by change of variables
V(X, f)
= V[x- 1 (x, t), tJ =
A(X, t)
= A[x- 1 (x, t), t;] = a(x, t) .,
v{x~ t)
(2.8) (2 ..9)
:·\< . where v(x, t) and a(x, t) denote the spatial description of the velocity ·field and the : 0. Find the velocity and acceleration components in terms of the material ··_and spatial coordinates and time, i.e. '1~1 = V4( ..Y..:t, t), AA = AA( . .~A, t) and 'Va = ·.Va { Xa, t)" CLa = a,, (:i;,1 , t), respectively.
:_._ ~olution. For the given motion ~l;,1 = Xc1(.X 1, ..:Y2 , ..Y3 , t), the velocity and accelera:_.·:_.:Hon components in tenns of the material coordinates .XA •. A = l_, 2, 3, and time t foHow :::·._:._from eqs. (.2.7) and have the forms T /•
,.,. 1
4
== e t ...-·i."\... J + c -t. . .v· -·\ 2
\·... 4 1 = e·t . "'. it
·-t "\.·"' ... ·\.2
- e
'
'
T...
•/2
'\.·.. = e ·__..\·v· 1 ·- .e -·t ..:\.2
,
+ (::."-t ..-·\v·2
_,
{,
4 _ £,) ....·v·\ I
" 2 -
(2.11) A.:1
=0
.
(2.12)
... ·. ··:. . . In order to find the velocity and acceleration components in terms of the spatial \·:.:.:_::·coordinates ~r.," u = 1, 2, 3, .and Lime t, we substitute simply the given motion into '.:./.: :· .:·~e_rived relations (2.l l) and (2 ..12). Thus, according to eqs. (2~.8) and (2.9) we find that .
..
.
V1 a1
== :1:1
=
= V2
~r2
'
,
'l.13
(2.] 3)
·== 0 ' ll:J
= -0
.
Iii
(2 ..14)
64
2
Kinematics.
EXERCISES
I. Consider the motion of a continuum body given by the equations
:r1
= . .Y.1 (1 + nt:i)
,. ,. .{.··, - "A - ... '
,
(2.15)
~
')
where a is a constant. Determine the displacement, velocity and acceleration fields in each of the material and spatial descriptions.
2. A motion of a continuum body is defined by the velocity -components 3;1:1
'll1
=- ' l+t
V2
=l
:r:'>
~t
(2.16)
1
Assume that the reference configuration of the continuum body is at t the consistency -condition X
= O, be a smooth spatial field which assigns a scalar (x, t) to each point x at t. By the chain rule we find from (2.23) that
(x, t)
=
DcI~(x
~
or
' t
t)
=
.
8cI>(x, t) [) ·t
D-
.
+ grad(x, t) · v(x, t) , D, which describes the changing position of a particle.
2.3
Material_, Spatial Der.ivativcs
67
By analogy with the above, the ·material tim-e derivative of a smooth spatial field v(x, J;), which is vector-valued, is given by
··( ) v x,t
= Dv(x,t) Dt = Dv(x,t) at . vo.
or
Dva.
(
) .
+gradv x,t v(x,t
Dva
) (2.26)
Dva
= -D -a Vti . t = -D ·t; + ··=1:b
where v = v{x, -tL (vu, a = 1, 2, 3), and its material ti.me .derivative v = v(x, :I:), (t}ll., a = 1, 2, 3), may also be viewed as the spatial description of a velocity and acceleration field, referred to brie-fly as the spatial velocity .field nnd the spatial acceleration .field, respectively. The spatial acceleration field we often denote by a = v, (c1u = 'Vu, n = l_, 2, 3). Hence, we may write eq. (2.26h as a
= -Dv + (gradv)v ' 0. t
(2.27)
or
(the arguments of the functions have been omitted for simplicity). The first term on the right-hand side, name.1y av I at, describes the ·local acceleration (local rate of change of the velocity field). The second term is quadratically nonlinear in the velocity field and describes the convective acceleration field (convective rate of change of the velocity field). Relation (2.27) turns out to be very useful, because the spatial acceleration field a(x, t) m~y be determined from v(x, t) without knowing the motion explicitly, which is an important question in applied :fluid mechanics, By virtue of (2.7) 1 and (2.8), the spatial velocity field v(x, t) may be expresseµ as the material time derivative of the motion x = x(X, t), i.e.
. ax
V=X=----
8t
or
.
Vu
iJ:c a
= :i;n = -at .,
(2.28)
and from (2.5) we find that v == ii. The spatial acceleration field re.ads by analogy with relation (2 .19) as
a=v=u .
(2.29)
A very useful short-hand notation of lhe ope.rator D( •) /Dt which acts on spatial fields reads, in symbolic and index notation,
D(•) Dt
= D(•) + grad(•)·v Dt
-
or
D(~)
D(•)
D(•)
- - = - - +-.-'Vo , .Dt D:r(L
ut
(2~30)
which should be compared with eqs. (.2.25) and (2.26). Relation (2.30) shows clearly the relationship between the ·material and spatial t\me derivatives.
68
--~---=--~---,~~""""":'''~-,,~,~-~,-~-.-~
2 Kinematics
..... ,,-.,:·---· - - · - - - ___ ,
..
-·--------------·-·-"-·'"·~,,.,.~~:,,, _,.,.,,:·~~~:~~-,.~
...
..,,,.~,,,,,, ~,,,,,,,,,~.,,.,,~
...
..,,,,~,,,,. ,_.,,,.,,,~~-,
....... ,.
,~,.,,_,~.,,.~
... -.,"':"""""-.-...... ,.,_,.,. ___ ,
~---,·-~·-.-- ~-·
.. -· -.-:·
EXAMPLE 2.4 In the current configuration of a continuum body a certain physical quantity is given in space and time according to t-1.
(x, t}
= - TXT ,
(2.31)
with :place x = :1:iei. The region of the spatial (scalar) field · = , i . e. gradqi(x, t) =
8 (x, t:)
ax
-1
x
= t lxl3 .
(2.34)
Note that the spatial vector field gTadcI> is harmonic and describes a point source at .a certain time t (compare with Exercise ·6 on p. 51 ). By virtue of the derived solutions (2.33), (2.34) and the given velocity field (2.32) we obtain finally (2.35)
which gives the desired solution {2.3 .l ), the material time derivative of at x and at • time t. , ·' .... ~··,, ... , •.... : ··'····· '··· ~, ... ~·· ,, '.·: ~'·'···." ... , .. : .. ,.,, ~ .. ,., •. , , ~···. ~·, .... ,, ~-, ~,, ... ~'·' ~'·'····: .,'·····~··.··'·-''' ~ .. , ........ ·,··: ,, .. ·.·~···: ,,.,.,,.,,._ .... ,·.·~·~''' ~- '·' ., .. : ... c ~··,: ...... : , ~···: '···~·······, ~, ... , '.·:: '·' ~'·'····.. ·......... : ~···· ,_., ... '···, ... '···." .... ·: '···."····: ·'···." '·~'··'·"~', ~, '."·"·'·"·"·'·"•"·'·~·,,: .······: ,. : ,.~···': ..... :: ,., : ... -~·:: ~.
at
Types of motions. .If the velocity field is independent of time~ i.e. av I = 0 and v = v(x), the associated motion is said to be steady and (2.27) reduces to a = (gradv)v. On the other hand, if the velocity field at each instant :is independent of position x, i.e. gradv = 0 and v = v{t), the associated motion is said to be uniform and (2.27) reduces to a = av I If the velocity .field has components of the form
at.
2.3
=
v-i ( :1;.1> :1;2 , t), v2
=
69
l\llaterial, Spatial Derivatives
v 2 ( x 1 , :r 2 , t ).,-.v:1
=
0 the associated mot.ion is caned plane. A motion is potential if there exists a spatial velocity field v = grad lJ>, where the spatial scalar fie]d is called the veloc-ity potential. A motion satisfying c;url v = o is irrotational. By recalling relation (1.274) we conclude that if a motion is potential then it is automatic.ally an .irrotational motion.
v1
EXERCISES
:1. For a one-dimens.ional problem the displacement field U the equation
= U (.X, t)
is given by
U =ct)( ,
with c denoting a constant. The relation between the spatial and the material coordinates is given by :r = (1 + ct) .X. Derive the first and second derivatives of U with respect to time t in both the material and spatial descriptions. 2. Recall Example 2.2 on p. 63. The spatial components a. 1 = ;r 1 , a 2 = J: 2 , a.~i = 0, i.e. eq. (2.14), are derived directly from Lhe acceleration in the material description, Le. eq. {2.12)~ by means of given motion (2.10).
Show that the spatial components a.a, a = 1., 2, 3, may also be derived from expression {2 ..27) using eq. (2.13 ), i.e~ ·v 1 = :1: 2 , v2 = :r 1, v:i = 0. 3. ln a certain region the spatial veloc.ity components of v = v(x, t) are given as
.·,l . l -_ _ (l;(··.a .I..
; .2·) (:,_l-/Jt + ,.J,.. t •l'2
·'
'l 1'•2
=
(\i ( -4;
'1~2.,r'' 1ol.I 2
• '
+ •.,.a) •2
Ll
L-
-{Jl
.'7
where n, /3 > 0 are given constants. Find the components of the spatial acceleration field a ;::: a(x.~ t) at point (1, 0, 0) and time t = 0.
4. Consider a plane motion defined by the velocity components V··1
...
=
{J (T) ( •-r•. 'l"• ·f··) ± ~· I. ~ ' ' 2 ' . I
[h:1
,
where cl> is a harmonic scalar field. Show that the plane motion is .irro.tational and find the spatial acceleration field.
70
2
Kinematics
2.4 Deformation Gradient One goal of th.is section is to study the deformation (i.e. the changes of size and shape) of a continuum body occurring when moved from the reference configuration Q0 to some current configuration n. Fo.r simplicity, we often omit the arguments of the tensor quantities in subsequent considerations. Deformation gradient. As we know from Section 2.1, a typical point X E 0 0 identified by the position vector X maps into the point x E n with position vector x . Now we want to know how curves and tangent vectors deform. Consider a material (or undeformed) curve x = r(~) c no, (. YA = rA(~)), where f, denotes a parametrization (see Figure 2.4_). The material curve is associated
with the reference -configuration n0 of the continuum body. Hence, the material curve is not .a function of time. During a certain motion x the material curve deforms into a spatial (or deformed) curve x = 1(~, t) c n, (:z;a = 1'n(~, t) ), at ti.me'/:.
r.~· ..
..:.·•. •..
..i'(~, t)
·.. .
.
. . . . . . ·.. ~ ..
...
.n
~'
i time. t" = 0 J.;
[
3
0
"Figure 2.4 Defonnation of a material curve r
... .... .
••• .
]
time t
··-.i
c no into a spatial curve I
C
n.
The spatial curve at a fixed time t is then defined by the parametric equation x = 1'(C t)
= x(r(€)J)
or
We denote the spatial tangent vector to the spatial curve as dx and the material tangent vector to the material curve as dX. They are de-fined by dx ==
,../(~ I ~'
·t)d ~c
'
dX
= I''(c)d.C ~ ~
'
(2.37)
with the abbreviation ( e )' = D( •) / D~. In the literature the tangent vectors dx and dX, vihich are infinitesimal vector elements in the current and reference configuration (see
2.4
71
Deformation Gradient
:. figure 2.4), are often referred to as the spatial (or deformed) line element and the :,·material (or undeformed) line element, respectively. ~.{~·:=.:·::·By using (2.36) and the chain rule we find that 1'(~, t;) = (Bx(X, t)/DX)r'(~) . . Hence, from eq. (2.37_) we deduce the fundamental relation .. ..
:
··:·: .=
.
dx
..
= F(X, t)dX
or
F (X, t_) =
·Bx (X, t) · DX
(2.38)
\vhere the definition ....
"f: . . ·· .
r:t
or
r aA
= Grad.x (X. t ) 1
G:rl"afI..\"~ 1, ,.\ :;:::; 1 or A < L. respectively. With definitions (l..15h, (2.·60) and property (1..8.1), the square of,.\ .is computed according to ,\
2
= Ano · A110 == Fao · Fao
rr = a 0 · F Fao = a 0 · Ca0
(2.62)
,
(2.63)
or
where we have introduced the right Cauchy-Green tensor C as an important strain measure in material coordinates (F is on the right). Frequently in the literature C is referred to as the Green deformation tensor. From (2.63) we learn that to determine the stretch of a fiber one only needs the direction a 0 at a point X E 0 0 and the se-condorder tensor C. Note that C is symmetric and positive definite at each X E f2 0 . Thus, and
u · Cu > 0
for all
u =!= o . (2.64)
2.5 Strain Tensors
79
·.: · Consequently, given the nine components Fa.4 , it is easy to compute the six components ·. CAB = Cn.-t via (2.63), but given CAH it is impossible to compute the nine components ·: ~1 .AA With definition (2.63) and eqs . (l. l.01.) and (2.51.) we find that
detC
= (detF) 2 .= J 2 > O
.
(2.65)
··:·The so-called Piola deformation tensor, denoted by B, is defined by the inverse of the ·..dghtCauchy~Oreen tensor, i.e. "B = c- 1, with c- 1 = (FTF)- 1 == F- 1 _F~T . As a further strain measure we define the change in the squared lengths, i.e.. (/\de )22 ·. ·clc • With (2.62)a, the use of the unit tensor I and eq. (2.57)2 we have (2.66)
E
1 T = -(F· F- I) 2
E~w = ~(Fa,i.Fi,11 -
or
(2.·67)
i5A11) ,
-
.·.· where the introduced normalization factor 1/2 wiH be evident within the linear theory. ·. · : · This expression describes a strain measure .in the direction of a 0 at point X E .n 0 . In _:_.· (2.67) we have introduced the commonly used strain tensor E, which is known as the · ~reen-Lagrange strain .tensor. Since I and C ~re sym.metric we deduce from (2.67) . . that E = ET also. So far the introduced strain tensors operate so]ely on )he material vectors a 0 , X. ·.· Thus, C, .E and their inverse are also referred to as material strain tensors. :·. · $pati.al strain tensors. In order to relate strain measures to quantities which are · ·:_·.:.associated with the current configuration we continue with arguments entirely similar ._:. · ~
