MSC marc training mar103

Experimental Elastomer Analysis

MSC.Software Corporation

MA*V2008*Z*Z*Z*SM-MAR103-NT1

1

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Part Number:

MA*V2008*Z*Z*Z*SM-MAR103-NT1

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MA*V2008*Z*Z*Z*SM-MAR103-NT1

Experimental Elastomer Analysis

Table of Contents

Contents

Experimental Elastomer Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Table of Contents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

CHAPTER 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 Course Objective: FEA & Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Course Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 About MSC.Marc Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 About Axel Products, Inc.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Data Measurement and Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Typical Properties of Rubber Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Important Application Areas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

CHAPTER 2

The Macroscopic Behavior of Elastomers . . . . . . . . . . . . .

Microscopic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature Effects, Tg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time Effects, Viscoelasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Curing Effects (Vulcanization) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Damage, Early Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Damage, Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Damage, Chemical Causes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deformation States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 3

Material Models, Historical Perspective . . . . . . . . . . . . . . .

Engineering Materials and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neo-Hookean Material Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neo-Hookean Material Extension Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . Neo-Hookean Material Shear Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neo-Hookean Material Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Word About Simple Shear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-Constant Mooney Extensional Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Mooney-Rivlin Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ogden Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Foam Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Limitations and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscoelastic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Determining Model Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Experimental Elastomer Analysis

21 22 23 24 26 27 28 28 29 31 32 33 35 36 38 40 41 43 45 48 49 50 51

3

Contents

CHAPTER 4

Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Lab Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measuring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What about Shore Hardness? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Testing the Correct Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tensile Testing in the Lab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compression Testing in the Lab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equal Biaxial Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compression and Equal Biaxial Strain States . . . . . . . . . . . . . . . . . . . . . . . . . . . . Volumetric Compression Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Planar Tension Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscoelastic Stress Relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic Behavior – Testing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Friction Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Reduction in the Lab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Verification Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Testing at Non-ambient Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loading/Unloading Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test Specimen Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fatigue Crack Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental and Analysis Road Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 5

Material Test Data Fitting . . . . . . . . . . . . . . . . . . . . . . . . . .

Major Modes of Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Confined Compression Test (UniVolumetric). . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydrostatic Compression Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of All Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mooney, Ogden Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Visual Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Future Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adjusting Raw Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Consider All Modes of Deformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Three Basic Strain States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Curve Fitting with MSC.Marc Mentat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

53 54 55 57 58 59 60 61 63 65 66 67 68 70 71 73 74 76 78 79 80 81 82 83 84 87 88 89 90 91 92 93 94 95 98 99 100

Experimental Elastomer Analysis

Contents

CHAPTER 6

Workshop Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some MSC.Marc Mentat Hints and Shortcuts . . . . . . . . . . . . . . . . . . . . . . . . . . Model 1: Uniaxial Stress Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model 1: Uniaxial Curve Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model 1C: Tensile Specimen with Continuous Damage . . . . . . . . . . . . . . . . . . Model 1: Realistic Uniaxial Stress Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . Model 2: Equi-Biaxial Stress Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model 2: Equi-Biaxial Curve Fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model 2: Realistic Equal-Biaxial Stress Specimen. . . . . . . . . . . . . . . . . . . . . . . Model 3: Simple Compression, Button Comp. . . . . . . . . . . . . . . . . . . . . . . . . . . Model 4: Planar Shear Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model 4: Planar Shear Curve Fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model 4: Realistic Planar Shear Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model 5: Viscoelastic Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model 5: Viscoelastic Curve Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model 6: Volumetric Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 7

Contact Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Definition of Contact Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control of Rigid Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contact Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bias Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deformable-to-Deformable Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Potential Errors due to Piecewise Linear Description: . . . . . . . . . . . . . . . . . . . . Analytical Deformable Contact Bodies: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contact Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetry Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rigid with Heat Transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contact Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contact Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exclude Segments During Contact Detection. . . . . . . . . . . . . . . . . . . . . . . . . . . Effect Of Exclude Option:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contacting Nodes and Contacted Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . Friction Model Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coulomb ArcTangent Friction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coulomb Bilinear Friction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stick-Slip Friction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Glued Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Release Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interference Check / Interference Closure Amount . . . . . . . . . . . . . . . . . . . . . . Experimental Elastomer Analysis

107 108 109 113 133 145 149 153 165 168 176 180 195 198 200 213 217 218 220 221 222 223 224 224 225 226 227 229 231 232 233 234 235 236 237 238 239 241 241 5

Contents

Forces on Rigid Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

APPENDIX A The Mechanics of Elastomers. . . . . . . . . . . . . . . . . . . . . . Deformation States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Formulation of Elastomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Large Strain Viscoelasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Large Strain Viscoelasticity based on Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of Large Strain Viscoelastic Behavior . . . . . . . . . . . . . . . . . . . . . . .

APPENDIX B Elastomeric Damage Models . . . . . . . . . . . . . . . . . . . . . . Discontinuous Damage Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Damage Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

APPENDIX C Aspects of Rubber Foam Models . . . . . . . . . . . . . . . . . . . Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measuring Material Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

APPENDIX D Biaxial & Compression Testing . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overall Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Experimental Apparatus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analytical Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attachment A: Compression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

APPENDIX E Xmgr – a 2D Plotting Tool. . . . . . . . . . . . . . . . . . . . . . . . . Features of ACE/gr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Using ACE/gr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACE/gr Miscellaneous Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

APPENDIX F Notes and Course Critique . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Course Critique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

242 245 246 250 253 254 254 259 261 262 267 271 272 276 277 278 279 281 282 285 291 292 295 296 297 300 303 304 305 306 307

Experimental Elastomer Analysis

CHAPTER 1

Introduction

This course is to provide a fundamental understanding of how material testing and finite element analysis are combined to improve your design of rubber and elastomeric products. Most courses in elastomeric analysis stop with finite element modeling, and leave you searching for material data. This experimental elastomer analysis course combines performing the analysis and the material testing. It shows how the material testing has a critical effect upon the accuracy of the analysis.

Experimental Elastomer Analysis

7

Chapter 1: Introduction

Course Objective: FEA & Laboratory

Course Objective: FEA & Laboratory Left Brain

Right Brain

Computer Analytical Objective Logical

Laboratory Experimental Subjective Intuitive

W = NkT ( I 1 – 3 ) ⁄ 2 W = C1 ( I1 – 3 ) + C2 ( I2 – 3 ) N

W =

∑ n=1

μ α α α -----n- [ ( λ 1 n + λ 2 n + λ 3 n ) – 3 ] αn

1 1 2 2 W = G --- ( I 1 – 3 ) + ---------- ( I 1 – 3 ) + … 2 20N

8

Experimental Elastomer Analysis

Course Objective: FEA & Laboratory

Chapter 1: Introduction

Course Objective Discuss the TEST CURVE FIT ANALYSIS cycle specific to rubber and elastomers. Limit scope to material models such as Mooney-Rivlin and Ogden form strain energy models. Material Model (curve fit)

Test Material Specimen

Analyze Specimen ? n Correlatio ?

Analyze Part

Test Part

Experimental Elastomer Analysis

9

Chapter 1: Introduction

Course Objective: FEA & Laboratory

Course Objective (cont.)

Some important topics covered are: • What tests are preferred and why? • Why aren’t ASTM specs always the answer? • What should I do about pre-conditioning? • Why are multiple deformation mode tests important? • How can I judge the accuracy of different material models? • How do I double check my model against the test data?

10

Experimental Elastomer Analysis

Course Schedule

Chapter 1: Introduction

Course Schedule DAY 1 Begin 9:00

End 10:15

10:30 12:00 1:00 3:15

12:00 1:00 3:00 5:00

5:00

Topic Introduction, Macroscopic Behavior of Elastomers Laboratory Orientation Lunch Tensile Testing Tensile Test Data Fitting FEA of Tensile Test Specimen Adjourn

Chap. 1, 2, 3 4

5 6

DAY 2 - Chapter 6 + Lab Begin 9:00

End 10:30

10:45 12:00 1:00 3:15

12:00 1:00 3:00 5:00

5:00

Topic Equal Biaxial Testing, Compression, Volumetric Equi-Biaxial Test Data Fitting, Comp., Volumetric FEA of Biaxial Specimen, Comp., Volumetric Lunch Planar Shear Testing Planar Shear Test Data Fitting Data Fitting with All Test Modes FEA of Planar Test Specimen Adjourn

Experimental Elastomer Analysis

11

Chapter 1: Introduction

Course Schedule

Course Schedule (cont.) DAY 3 Begin 9:00

End 10:30

10:45 12:00 1:00

12:00 1:00 3:00

3:15 5:00

5:00

Topic

Chap.

Viscoelastic Testing Viscoelastic Data Fitting FEA of Viscoelastic Test Specimen Lunch Contact and Case Studies Specimen Test, FEA, Part Test Correlation Concluding Remarks Adjourn

6

7

•Keep Involved: Tell Me and I’ll Forget Show Me and I’ll Remember Involve Me and I’ll Understand

12

Experimental Elastomer Analysis

About MSC.Marc Products

Chapter 1: Introduction

About MSC.Marc Products MSC.Marc Products are in use at thousands of sites around the world to analyze and optimize designs in the aerospace, automotive, biomedical, chemical, consumer, construction, electronics, energy, and manufacturing industries. MSC.Marc Products offer automated nonlinear analysis of contact problems commonly found in rubber and metal forming and many other applications. For more information see: http://www.mscsoftware.com/products/products_detail.cfm?PI=1

Experimental Elastomer Analysis

13

Chapter 1: Introduction

About Axel Products, Inc.

About Axel Products, Inc. Axel Products is an independent testing laboratory, providing physical testing services for materials characterization of elastomers and plastics. See www.axelproducts.com.

14

Experimental Elastomer Analysis

Data Measurement and Analysis

Chapter 1: Introduction

Data Measurement and Analysis Experiment In 1927, Werner Heisenberg first noticed that the act of measurement introduces an uncertainty in the momentum of an electron, and that an electron cannot possess a definite position and momentum at any instant. This simply means that: Test Results depend upon the measurement Analysis Analysis of continuum mechanics using FEA techniques introduces certain assumptions and approximations that lead to uncertainties in the interpretation of the results. This simply means that: FEA Results depend upon the approximations Together This course combines performing the material testing and the analysis to understand how to eliminate uncertainties in the material testing and the finite element modeling to achieve superior product design.

Experimental Elastomer Analysis

15

Chapter 1: Introduction

Data Measurement and Analysis

Data Measurement and Analysis (cont.) Linear Material, How is Young’s modulus, E, measured? Tension/Compression P⁄A E = ⎛ -------------------⎞ ⎝ ( ΔL ) ⁄ L⎠

P, ΔL

Torsion Tc ⁄ J E = 2 ( 1 + υ ) ⎛ -------------⎞ ⎝ φ ⎠

T, φ

Bending 3

PL E = --------3δI

P, δ

Wave Speed 2

E = c ρ

Do you expect all of these E’s to be the same for the same material?

16

Experimental Elastomer Analysis

Typical Properties of Rubber Materials

Chapter 1: Introduction

Typical Properties of Rubber Materials Properties:

•It can undergo large deformations (possible strains up to 500%) yet remain elastic. •The load-extension behavior is markedly nonlinear. •Due to viscoelasticity, there are specific damping properties. •It is nearly incompressible. •It is very temperature dependent. Loading:

1. The stress strain function for the 1st time an elastomer is strained is never again repeated. It is a unique event. 2. The stress strain function does stabilize after between 3 and 20 repetitions for most elastomers. 3. The stress strain function will again change significantly if the material experiences strains greater than the previous stabilized level. In general, the stress strain function is sensitive to the maximum strain ever experienced. 4. The stress strain function of the material while increasing strain is different than the stress strain function of the material while decreasing strain. 5. After the initial straining, the material does not return to zero strain at zero stress. There is some degree of permanent deformation.

Experimental Elastomer Analysis

17

Chapter 1: Introduction

Typical Properties of Rubber Materials

Typical Loading of Rubber Materials (cont.) 6.0

Engineering Stress [MPa]

Experiment

4.0

Theory 2.0

0.0

0

1

2

3 4 5 Engineering Strain

6

7

Engineering Stress [MPa]

1.2

1.0

0.8

0.6

0.4

0.2

0.0 0.0

0.1

0.2

0.3

0.4

0.5

Engineering Strain

18

Experimental Elastomer Analysis

Important Application Areas

Chapter 1: Introduction

Important Application Areas – Car industry (tires, seals, belts, hoses, etc.) – Biomechanics (tubes, pumps, valves, implants, etc.) – Packaging industry – Sports and consumer industries

Experimental Elastomer Analysis

19

Chapter 1: Introduction

20

Important Application Areas

Experimental Elastomer Analysis

CHAPTER 2

The Macroscopic Behavior of Elastomers

Elastomers (natural & synthetic rubbers) are amorphous polymers, random orientations of long chain molecules. The macroscopic behavior of elastomers is rather complex and typically depends upon: – Time (strain-rate) – Temperature – Cure History (cross-link density) – Load History (damage & fatigue) – Deformation State

Experimental Elastomer Analysis

21

Chapter 2: The Macroscopic Behavior of Elastomers

Microscopic Structure

Microscopic Structure

• Long coiled molecules, with points of entanglement. Behaves like a viscous fluid. • Vulcanization creates chemical bonds (cross-links) at these entanglement points. Now behavior is that of a rubbery viscous solid. • Initial orientation of molecules is random. Behavior is initially isotropic. • Fillers, such as carbon black, change the behavior.

22

Experimental Elastomer Analysis

Temperature Effects, Tg

Chapter 2: The Macroscopic Behavior of Elastomers

Temperature Effects, Tg • All polymers have a spectrum of mechanical behavior, from brittle, or glassy, at low temperatures, to rubbery at high temperatures. • The properties change abruptly in the glass transition region.

• The center of this region is known as the Tg, the glass transition temperature. • Typical values of Tg (in oC) are: -70 for natural rubber, -55 for EPDM, and -130 for silicone rubber.

Experimental Elastomer Analysis

23

Chapter 2: The Macroscopic Behavior of Elastomers

Time Effects, Viscoelasticity

Time Effects, Viscoelasticity • Temp. & Time effects derive from long molecules sliding along and around each other during deformation. • A plot of shear modulus vs. test time:

• Material behavior related to molecule sliding (friction): •

24

strain-rate effects creep, stress-relaxation hysteresis damping

Experimental Elastomer Analysis

Time Effects, Viscoelasticity

Chapter 2: The Macroscopic Behavior of Elastomers

Time Effects, Viscoelasticity (cont.) • Different types of tests can be used to evaluate the short-time and long-time stress-strain behavior. • Our current favorite, the Stress-relaxation test:

• Gather data of strain, short-time stress, long-time stress.

Experimental Elastomer Analysis

25

Chapter 2: The Macroscopic Behavior of Elastomers

Curing Effects (Vulcanization)

Curing Effects (Vulcanization) • Curing creates chemical bonds – cross-linking. • Cross-link density directly affects the stiffness. • Cross-link density effect for Natural Rubber:

• Be careful that real parts and test specimens share the same curing history, thus same stiffness.

26

Experimental Elastomer Analysis

Damage, Early Time

Chapter 2: The Macroscopic Behavior of Elastomers

Damage, Early Time • Straining may break a fraction of the cross-links, reduces the overall stiffness and may cause plasticity. • Low cycle damage is very evident in filled elastomers, due to breakdown of filler structure and changes in the conformation of molecular networks. • Mullin’s Effect in carbon black filled NR: This is a textbook idealization. Real material behavior looks like: “Progressively Increasing Load History…” on page 60 (The loading curve and unloading curve are not coincident).

• Be careful that real parts and test specimens share the same load history, Preconditioning.

Experimental Elastomer Analysis

27

Chapter 2: The Macroscopic Behavior of Elastomers

Damage, Fatigue

Damage, Fatigue • Very early stages of understanding, see Gent’s Engineering with Rubber, Chapter 6, Mechanical Fatigue. http://www.amazon.com/exec/obidos/ASIN/1569902992/ref%3Ded%5Foe%5Fh/002-1221807-2520837

• Beyond scope of this course.

Damage, Chemical Causes • Many other chemicals are known to damage elastomers and degrade the mechanical behavior: Ozone Oxidation Ultraviolet Radiation Oil, Gasoline

Brake Fluid Hydraulic Fluid

• Sometimes preconditioning of test specimens can be helpful in gauging these effects. • Typically, however, these are longer time effects.

28

Experimental Elastomer Analysis

Deformation States

Chapter 2: The Macroscopic Behavior of Elastomers

Deformation States • Shearing vs. Bulk Compression • Shearing Modulus, G , typical ~ 1 - 10 MPa p • Bulk Modulus, K = ----------------- , typical ~ 2 GPa ΔV ⁄ V 0

•

K hence ---- ∼ 10 3 → ∞

•

and υ → ---

G

1 2

• Ordinary solid (e.g., steel): K and G are the same order of magnitude. Whereas, in rubber the ratio of K to G is of the 3

order 10 ; hence the response to a stress is effectively determined solely by the shear modulus G when the material can shear. • We say rubber is (nearly) incompressible in those cases when it is not highly confined.

Experimental Elastomer Analysis

29

Chapter 2: The Macroscopic Behavior of Elastomers

Deformation States

Deformation States (cont.) • FEA Material Model calibration requires certain types of tests. • They require states of “pure” stress and strain, that is that the stress/strain state be homogeneous. •

homogeneous = uniform throughout (isotropic = identical in all directions)

• Or at least homogeneous throughout a large area/volume of the test specimen (minimize end effects). • It is good practice to model and analyze the test specimen itself to prove homogeneity. • The “button compression” test is notoriously bad from this perspective. • Keep in mind that many ASTM test standards are defined for characterization, or process control purposes. Many ASTM specs are NOT suitable for material model calibration.

30

Experimental Elastomer Analysis

CHAPTER 3

Material Models, Historical Perspective

It is useful to know the historical evolution of rubber material models. We will cover Neo-Hookean, Mooney, Mooney-Rivlin, and Ogden material models. Each model is based on the concept of strain energy functions, which guarantees elasticity.

Experimental Elastomer Analysis

31

Chapter 3: Material Models, Historical Perspective

Engineering Materials and Analysis

Engineering Materials and Analysis Clearly metals have been with us for a long time, unfortunately elastomers (natural and synthetic rubber) have just arrived relative to metals some 160 years ago. The study of elastomers has only spanned the last 60 years as shown in Table 1. If elastomers are to attain the position they seem to deserve in engineering applications, they must be studied comprehensively as have, for example, steel and other commonly used metals. TABLE 1. History of Metals, Elastomers, and Analysis Date

Metal

-4000

Copper, Gold

-3500

Bronze Casting

-1400

Iron Age

-1

Damascus Steel

Elastomer

1660

Hookean Materials

1800

Titanium

1840

Aluminum

1850 1879

3D Elasticity Vulcanization Parkesine

Rare earth metals

Colloids

1929

Aminoplastics

1933

Polyethylene

1933

PMMA

1939

Nylon

1940

Neo-Hookean

1940

PVC

1941

Polyurethanes

1943

PTFE

1949

Mooney-Rivlin

1950

Hill’s Plasticity

1955

Polyester

1965 1970

FEA Software Foams

1975

32

Analysis

Treloar

1980

> 200 Polymer compounds

1990

Recycle

Experimental Elastomer Analysis

Neo-Hookean Material Model

Chapter 3: Material Models, Historical Perspective

Neo-Hookean Material Model Definitions, Stretch ratios, Engineering Strain: L i + ΔL i λ i = -------------------- = 1 + ε i Li

eng. strain, ε i = ( ΔL i ⁄ L i ) t3

L1

t2

L3 L2

λ1 L1 λ3 L3

t1

λ2 L2

t1

t2 t3

Incompressibility: λ1 λ2 λ3 = 1

From Thermodynamics and statistical mechanics, First order approximation (neo-Hookean): 1 2 2 2 W = --- G ( λ 1 + λ 2 + λ 3 – 3 ) 2

Experimental Elastomer Analysis

33

Chapter 3: Material Models, Historical Perspective

Neo-Hookean Material Model

Neo-Hookean Material Model (cont.) Experimental Verification using Simple Extension λ1 = λ

λ2 = λ3 = 1 ⁄ λ

Hence:

NeoHookean Behavior Tension and Compression very Different

1 2 2 W = --- G ⎛ λ + --- – 3⎞ λ ⎠ 2 ⎝

Engineering Stress: 1 σ = dW ⁄ dλ = G ⎛ λ – ----2-⎞ = ⎝ ⎠ λ 1 = G ⎛ 1 + ε – ------------------2-⎞ ⎝ ⎠ (1 + ε)

Engineering Stress/(Shear Modulus)

5.0

5.0 Hookean (nu=.45) NeoHookean

15.0

25.0

0.8

0.4

0.0

0.4

0.8

Engineering Strain

σ 2 1 True Stress: t = ---------- = λσ = G ⎛⎝ λ – ---⎞⎠ λ 1⁄λ

Simple, one parameter material model Positive G guarantees material model stability

34

Experimental Elastomer Analysis

Neo-Hookean Material Extension Deformation

Chapter 3: Material Models, Historical Perspective

Neo-Hookean Material Extension Deformation Theory versus experiments:

6.0

Engineering Stress [MPa]

Experiment

4.0

Theory 2.0

0.0

0

Experimental Elastomer Analysis

1

2

3 4 5 Engineering Strain

6

7

35

Chapter 3: Material Models, Historical Perspective

Neo-Hookean Material Shear Deformation

Neo-Hookean Material Shear Deformation Experimental Verification using Simple Shear :

Y τ

atan γ

X 1 If λ 1 = λ , then λ 2 = --- and λ 3 = 1 λ

Equivalent shear strain γ : 1 γ = λ – --λ

Strain energy function: 1 1 2 1 2 W = --- G ⎛ λ + ----2- – 2⎞ = --- Gγ ⎠ 2 ⎝ 2 λ

Shear stress τ depends linearly on shear strain γ dW τ = -------- = Gγ dγ

36

Experimental Elastomer Analysis

Neo-Hookean Material Shear Deformation

Chapter 3: Material Models, Historical Perspective

Neo-Hookean Material Shear Deformation (cont.) Theory versus experiments:

1.6

Shear Stress [N/mm 2 ]

Theory 1.2

Experiment

0.8

0.4

0.0

0

Experimental Elastomer Analysis

1

2 3 Shear Strain

4

6

37

Chapter 3: Material Models, Historical Perspective

Neo-Hookean Material Summary

Neo-Hookean Material Summary TABLE 2. Basic Deformation Modes Mode

λ1

λ2

λ3

Biaxial

λ

λ

λ

–2

Planar Shear

λ

1

λ

–1

Uniaxial

λ

Simple Shear

2 2 γ γ 1 + ----- + γ 1 + ----2 4

λ

–1 ⁄ 2

λ

2 2 γ γ 1 + ----- – γ 1 + ----2 4

–1 ⁄ 2

1

Neo Hookean 1 2 2 2 W = --- G ( λ 1 + λ 2 + λ 3 – 3 ) 2 σ =

∂W = σ(ε) ∂λ

τ =

∂W = Gγ ∂γ

direct stresses

shear stress

Note: Shear Stress-Strain Relation is the same for Hookean and Neo Hookean.

38

Experimental Elastomer Analysis

Neo-Hookean Material Summary

Chapter 3: Material Models, Historical Perspective

Neo-Hookean Material Summary (cont.) TABLE 3. Hookean versus Neo Hookean Values of σ ⁄ G Hookean

Hookean as

Neo Hookean

σ⁄G=

υ→0

σ⁄G=

Biaxial

2( 1 – ν) -------------------- ε ( 1 – 2υ )

2ε

2{ 1 + ε – (1 + ε) }

Planar Shear

2(1 – ν – ν ) ------------------------------- ε ( 1 – 2υ )

2ε

{1 + ε – (1 + ε) }

Uniaxial

2 ( 1 + υ )ε

2ε

{1 + ε – (1 + ε) }

Mode

2

–5

–3

–2

Hookean and Neo Hookean Material Models Poisson’ Ratio = 0.45

Engineering Stress/Shear Modulus

10.0

5.0

Hookean Biaxial Hookean Planar Shear Hookean Uniaxial New Hookean Biaxial Neo Hookean Planar Shear Neo Hookean Uniaxial

0.0

-5.0

-10.0 -1.0

-0.5

0.0

0.5

1.0

Engineering Strain

Experimental Elastomer Analysis

39

Chapter 3: Material Models, Historical Perspective

A Word About Simple Shear

A Word About Simple Shear The simple shear mode of deformation is called simple shear because of two reasons: first it renders the stress strain relation linear for a NeoHookean material; secondly it is simple to draw. Linear Stress Strain Relation comes from substituting the simple shear deformations modes of: ⎛ λ2 = ⎝ 1

2 2 γ γ ⎞ ------1+ +γ 1+ 2 4⎠

⎛ λ2 = ⎝ 2

2 2 γ γ ⎞ ------1+ –γ 1+ 2 4⎠

2

(λ 3 = 1)

into 1 2 1 2 2 2 W = --- G ( λ 1 + λ 2 + λ 3 – 3 ) = --- Gγ 2 2

and then τ =

∂W = Gγ ∂γ

Secondly the mode is simple to draw. τ

atan γ

40

Experimental Elastomer Analysis

2-Constant Mooney Extensional Deformation

Chapter 3: Material Models, Historical Perspective

2-Constant Mooney Extensional Deformation Basic assumptions: (1) The rubber is incompressible and isotropic (2) Hooke’s law is obeyed in simple shear Strain energy function with two constants: W =

2 C1 ( λ1

+

2 λ2

+

2 λ3

⎛1 ⎞ 1 1 ----------– 3 ) + C 2 ⎜ 2 + 2 + 2 – 3⎟ ⎝ λ1 λ2 λ3 ⎠

Simple shear: ⎛ 2 1 ⎞ 2 W = ( C 1 + C 2 ) ⎜ λ 1 + ----2- – 2⎟ = ( C 1 + C 2 )γ ⎝ λ1 ⎠ τ = dW ⁄ dγ = 2 ( C 1 + C 2 )γ

Hence G = 2 ( C 1 + C 2 ) C 2⎞ C2 1 ⎞⎛ σ ⎛ ---------------------------------------σ = 2 λ – 2 C1 + or = C1 + 2 ⎝ ⎠⎝ ⎠ λ λ λ 2(λ – 1 ⁄ λ )

Experimental Elastomer Analysis

41

Chapter 3: Material Models, Historical Perspective

2-Constant Mooney Extensional Deformation

2-Constant Mooney Extensional Deformation (cont)

Theory versus experiments 0.4

G

0.3

E D

2

2

σ/2(λ–1/λ (N/mm2) ) σ/2(λ−1/λ2) (N/mm

F

C A 0.2

0.1 0.5

B

0.6

0.7

0.8

0.9

1.0

1/λ 1/λ

42

Experimental Elastomer Analysis

Other Mooney-Rivlin Models

Chapter 3: Material Models, Historical Perspective

Other Mooney-Rivlin Models Basic assumptions: (1) The rubber is incompressible and isotropic in the unstrained state (2) The strain energy function must depend on even powers of λ i The three simplest possible even-powered functions (invariants): 2

2

2

I1 = λ1 + λ2 + λ3 2 2

2 2

2 2

I2 = λ1 λ2 + λ2 λ3 + λ3 λ1 2 2 2

I3 = λ1 λ2 λ3

Incompressibility implies that I 3 = 1, so that: W = W ( I 1, I 2 )

Mooney material in terms of invariants: W = C1 ( I1 – 3 ) + C2 ( I2 – 3 )

(Mooney’s original notation)

W = C 10 ( I 1 – 3 ) + C 01 ( I 2 – 3 ) (Mooney-Rivlin notation)

Experimental Elastomer Analysis

43

Chapter 3: Material Models, Historical Perspective

Other Mooney-Rivlin Models

Other Mooney-Rivlin Models (cont) Some other proposed energy functions: The Signiorini form: W = C 10 ( I 1 – 3 ) + C 01 ( I 2 – 3 ) + C 20 ( I 1 – 3 )

2

The Yeoh form: 2

W = C 10 ( I 1 – 3 ) + C 20 ( I 1 – 3 ) + C 30 ( I 1 – 3 )

3

Third order Deformation Form (James, Green, and Simpson): W = C 10 ( I 1 – 3 ) + C 01 ( I 2 – 3 ) + C 11 ( I 1 – 3 ) ( I 2 – 3 ) + 2

C 20 ( I 1 – 3 ) + C 30 ( I 1 – 3 )

44

3

Experimental Elastomer Analysis

Ogden Models

Chapter 3: Material Models, Historical Perspective

Ogden Models Slightly compressible rubber: N

W =

∑ n=1

– αn

1

μ n -------⎛ --3- ⎞ αn αn αn 3 ------ J ( λ 1 + λ 2 + λ 3 ) – 3 + 4.5K ⎜ J – 1⎟ αn ⎝ ⎠

2

μ n and α n are material constants, K is the initial bulk modulus, and J is the volumetric ratio, defined by J = λ1 λ2 λ3

The order of magnitude of the volumetric changes per unit volume should be 0.01 Usually, the number of terms taken into account in the Ogden models is N = 2 or N = 3 . The initial bulk modulus is usually estimated instead of being measured in a volumetric test.

Experimental Elastomer Analysis

45

Chapter 3: Material Models, Historical Perspective

Ogden Models

Ogden Models Let’s suppose we want to fit a 1-term Ogden for tension. 1.) Assume incompressible (J=1) then μ α α α W = --- [ ( λ 1 + λ 2 + λ 3 ) – 3 ] α

2.) Strain mode is tension, thus λ 1 = λ

λ 2 = λ 3 = 1 ⁄ λ and

α

– --⎞ μ⎛ α 2 W = --- ⎜ λ + 2λ – 3⎟ α⎝ ⎠ α ⎛ – ⎛ --- + 1⎞ ⎞ ⎝ ⎠ 2 ⎟, 3.) Compute engineering stress, σ = dW ⁄ dλ = μ ⎜⎜ λ α – 1 – λ ⎟ ⎝ ⎠ α ⎛ – ⎛⎝ --- + 1⎞⎠ ⎞ 2 α–1 ⎟ = σ ( μ , α, ε ) – (1 + ε) or σ = dW ⁄ dλ = μ ⎜⎜ ( 1 + ε ) ⎟ ⎝ ⎠

4.) Fit data, say to st_18.data that has 60 stress-strain points. Find i = 1, 60 , has the “best fit.” μ, and α such that σ i = σ ( μ, α, ε i ) 5.) Panic σ i = σ ( μ, α, ε i ) is nonlinear. Ok, use program and μ = 25.78 α = 0.05298

....but other values are possible and perhaps unstable...visualize...

46

Experimental Elastomer Analysis

Ogden Models

Chapter 3: Material Models, Historical Perspective

Ogden Models 0.05298 ⎛ – ⎛ ------------------- + 1⎞ ⎞ ⎝ ⎠ 2 0.05298 – 1 ⎜ ⎟. 6.) Plot σ = 25.78 ⎜ ( 1 + ε ) – (1 + ε) ⎟ ⎝ ⎠

7.) Repeat plot of engineering stress versus engineering strain for biaxial and planar shear where: TABLE 4. Basic Deformation Modes Mode

λ1

λ2

λ3

Biaxial

λ

λ

λ

–2

Planar Shear

λ

1

λ

–1

1.357

0 0

8.894 (x.1)

uniaxial/experiment biaxial/ogden

uniaxial/ogden planar_shear/ogden

1

8. Estimate K = 2500(25.78)0.05298 = 3414. Experimental Elastomer Analysis

47

Chapter 3: Material Models, Historical Perspective

Foam Models

Foam Models Elastomer foams: N

W =

∑ n=1

μ α α α -----n- [ λ 1 n + λ 2 n + λ 3 n – 3 ] + αn

N

∑ n=1

μn β ----- ( 1 – J n ) βn

μ n , α n and β n are material constants

48

Experimental Elastomer Analysis

Model Limitations and Assumptions

Chapter 3: Material Models, Historical Perspective

Model Limitations and Assumptions This material model assumes that the rate of relaxation is independent of the load magnitude. For instance, for relaxation tests at 20%, 50%, and 100% strain, the percent reduction in stress at any time point should be the same. The relaxation is purely deviatoric, there is no relaxation associated with the dilatational (bulk) behavior. When used with a Mooney-Rivlin form model, the material is assumed to be incompressible. In MSC.Marc some small compressibility is introduced for better numerical behavior, namely if no bulk modulus is specified, then MSC.Marc computes the following for the bulk modulus: K = 10000 ( C 10 + C 01 )

When used with an Ogden model, the material may be slightly compressible, and if a bulk modulus is not supplied, it is estimated as: N

K = 2500

∑

μn αn

n=1

Experimental Elastomer Analysis

49

Chapter 3: Material Models, Historical Perspective

Viscoelastic Models

Viscoelastic Models MSC.Marc has the capability to perform both small strain and large strain viscoelastic analysis. The focus here will be on the use of the large strain viscoelastic material model. MSC.Marc’s large strain viscoelastic material model is based on a multiplicative decomposition of the strain energy function W ( E ij, t ) = W ( E ij ) × R ( t )

where W ( E ij ) is a standard Mooney-Rivlin or Ogden form strain energy function for the instantaneous deformation. And R ( t ) is a relaxation function in Prony series form: N

R(t) = 1 –

∑

n

n

δ ( 1 – exp ( – t ⁄ λ ) )

n=1

where δ n is a nondimensional multiplier and λ n is the associated time constant.

50

Experimental Elastomer Analysis

Determining Model Coefficients

Chapter 3: Material Models, Historical Perspective

Determining Model Coefficients This material model requires two different types of tests be conducted and two separate curve fits be performed. The time-independent function, W ( E ij ) , is determined from standard uniaxial, biaxial, etc., stress-strain tests. These tests are covered in more detail in Chapter 5 and demonstrated in Chapter 6. The time-dependent function, R ( t ) , is determined from one or more stress relaxation tests. This is a test at constant strain, where one measures the stress over a period of time. For example, R ( t ) is determined in “Model 5: Viscoelastic Curve Fit” on page 200.

Experimental Elastomer Analysis

51

Chapter 3: Material Models, Historical Perspective

52

Determining Model Coefficients

Experimental Elastomer Analysis

CHAPTER 4

Laboratory

Need to know: What are the actual tests used to measure elastomeric properties. The limitations of common laboratory tests. How to specify a laboratory experiment as required by your product requirements. Let’s understand the specimen testing better to achieve better correlation and confidence in our component analysis.

Experimental Elastomer Analysis

53

Chapter 4: Laboratory

Lab Orientation

Lab Orientation

Safety Tour of Lab

Laboratory Dangers High Pressure Hydraulics Class II Lasers Instrument Crushing

Wear Safety Glasses Don’t Look Into Lasers Don’t Touch Specimens or Fixtures When Testing

54

Experimental Elastomer Analysis

Basic Instrumentation

Chapter 4: Laboratory

Basic Instrumentation Electromechanical Tensile Testers

Servo-hydraulic Testers

Experimental Elastomer Analysis

55

Chapter 4: Laboratory

Basic Instrumentation

Basic Instrumentation (cont.) Wave Propagation Instrument

Automated Crack Growth System

Aging Instrumentation

56

Experimental Elastomer Analysis

Measuring

Chapter 4: Laboratory

Measuring Force Strain Gage Load Cells

Position Encoders and LVDT’s

Strain Clip-on Strain Gages Video Extensometers Laser Extensometers

Temperature Thermocouples

Experimental Elastomer Analysis

57

Chapter 4: Laboratory

Measurements

Measurements Force, Position, Strain, Time, Temperature Testing Instrument Transducers Load Cell (0.5% - 1% of Reading Accuracy in Range) Position Encoder (Approximately +/- 0.02 mm at the Device) Position LVDT (Between +/- 0.5 to +/- 1.0% of Full Scale) Video Extensiometer (Function of the FOV) Laser Extensiometer (+/- 001 mm) Time (Measured in the Instrument or at the Computer) Thermocouple

58

Experimental Elastomer Analysis

What about Shore Hardness?

Chapter 4: Laboratory

What about Shore Hardness? Perhaps the Most Common Rubber Test Useful in General Easy to Perform at the Plant Generally Useless for Analysis!

“The Shore Round Style Durometer was introduced in 1944. It is a general purpose device that is considered the most widely used instrument throughout the world for the hardness testing of cellular, soft and hard rubber, and plastic material.” http://www.instron.com

Experimental Elastomer Analysis

59

Chapter 4: Laboratory

Testing the Correct Material

Testing the Correct Material Consistent within The Experimental Data Set

Cut Specimens from Same Material 150mm x 150mm x 2mm Sheet

Cut All Specimens from the Same Slab Verify that The Tested Material is the Same as the Part Processing Color Cure Progressively Increasing Load History… All Are Same Compound

60

Experimental Elastomer Analysis

Tensile Testing in the Lab

Chapter 4: Laboratory

Tensile Testing in the Lab What is Simple Tension? Uniaxial Loading Free of Lateral Constraint Gage Section: Length: Width >10:1 Measure Strain only in the Region where a Uniform State of Strain Exists No Contact

2

3

1

Cut Specimens from Same Material 150mm x 150mm x 2mm Sheet

Experimental Elastomer Analysis

61

Chapter 4: Laboratory

Tensile Testing in the Lab

Tensile Testing in the Lab (cont.) Some Common Elastomers Exhibit Dramatic Strain Amplitude and Cycling Effects at Moderate Strain Levels. Conclusions: Test to Realistic Strain Levels Use Application Specific Loadings to Generate Material Data

62

Experimental Elastomer Analysis

Compression Testing in the Lab

Chapter 4: Laboratory

Compression Testing in the Lab It is Experimentally Difficult to Minimize Lateral Constraint due to Friction at the Specimen Loading Platen Interface Friction Effects Alter the Stressstrain Curves

2

3

1

The Friction is Not Known and Cannot be Accurately Corrected Even Very Small Friction Levels have a Significant Effect at Very Small Strains

Experimental Elastomer Analysis

63

Chapter 4: Laboratory

Compression Testing in the Lab

Compression Testing in the Lab (cont.) Friction Effects on Compression Data

Analysis by Jim Day, GM Powertrain

64

Experimental Elastomer Analysis

Equal Biaxial Testing

Chapter 4: Laboratory

Equal Biaxial Testing Why? Same Strain State as Compression Cannot Do Pure Compression Can Do Pure Biaxial Analysis of the Specimen justifies Geometry

2

3

Experimental Elastomer Analysis

1

65

Chapter 4: Laboratory

Compression and Equal Biaxial Strain States

Compression and Equal Biaxial Strain States There is also no ASTM Specification for equal biaxial strain tests. None the less, in common practice either square or circular frames shown below are used. The equal biaxial strain state is identical to the compression button’s strain state, simply substitute Λ = λ –2 .

Λ = λ

–2

λ2 = Λ

λ3 = Λ

–1 ⁄ 2

λ1 = Λ

–1 ⁄ 2

λ1 = λ

λ3 = λ

66

–2

λ2 = λ

Experimental Elastomer Analysis

Volumetric Compression Test

Chapter 4: Laboratory

Volumetric Compression Test Direct Measure of the Stress Required to Change the Volume of an Elastomer Requires Resolute Displacement Measurement at the Fixture Initial Slope = Bulk Modulus Typically, only highly constrained applications require an accurate measure of the entire PressureVolume relationship. Base Data Set 300 VOLCOMP_B

Pressure (MPa)

250 200

2

150

100

Bulk Modulus = 2.1 GPa

50

3

1

0 0.00

0.02

0.04

0.06

0.08

0.10

Volumetric Strain

Experimental Elastomer Analysis

67

Chapter 4: Laboratory

Planar Tension Test

Planar Tension Test Uniaxial Loading Perfect Lateral Constraint All Thinning Occurs in One Direction Strain Measurement is Particularly Critical Some Material Flows from the Grips The Effective Height is Smaller than Starting Height so >10:1 Width:Height is Needed

2

Similar Stress-strain Shape to Simple Tension and Biaxial Extension Match Loadings between Strain States

3

1

Base Data Set 0.6

Engineering Stress (MPa)

PT23C_B

0.5 Planar Tension

0.4 0.3 0.2 0.1 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Engineering Strain

68

Experimental Elastomer Analysis

Planar Tension Test

Chapter 4: Laboratory

Planar Tension Test (cont.) A Small but Significant amount of Material will Flow From the Planar Tension Grips.

Experimental Elastomer Analysis

69

Chapter 4: Laboratory

Viscoelastic Stress Relaxation

Viscoelastic Stress Relaxation Viscoelastic Behavior Can be Assumed to Reasonably Follow Linear Viscoelastic Behavior in Many Cases Is not the same as aging! Describes the short term reversible behavior of elastomers. Tensile, shear and biax have similar viscoelastic properties!

Stress

A totally “relaxed” Stress-strain Curve can be Constructed. Decades of data in time are equally valuable for fitting purposes.

7

0.8 0.7

Strain

Strain = 50 %

0.6

5

Stress (MPa)

Stress (MPa)

6

4 3

0.5 0.4

Strain = 30 %

0.3 0.2 0.1

2 0.0

0

2000

4000

6000

8000

Time (s)

1 0 0

500

1000

1500

2000

Time (Seconds)

70

Experimental Elastomer Analysis

Dynamic Behavior – Testing

Chapter 4: Laboratory

Dynamic Behavior – Testing Types of Dynamic Behavior Large strains at high velocity Small sinusoidal strains superimposed on large mean strains

Experimental Elastomer Analysis

71

Chapter 4: Laboratory

Dynamic Behavior – Testing

Dynamic Behavior – Testing (cont.) Mean Strain and Amplitude Effects are Significant

72

Experimental Elastomer Analysis

Friction Test

Chapter 4: Laboratory

Friction Test Friction is the force that resists the sliding of two materials relative to each other. The friction force is:

Coulomb performed many experiments on friction and pointed out the difference between static and dynamic friction. This type of friction is referred to as Coulomb friction today.

Increasing Normal Force

These two laws of friction were discovered experimentally by Leonardo da Vinci in the 13th century, and latter refined by Charles Coulomb in the 16th century.

Friction Test

Friction Force

(1) approximately independent of the area of contact over a wide limit and (2) is proportional to the normal force between the two materials.

Position

In order to model friction in finite element analysis, one needs to measure the aforementioned proportionally factor or coefficient of friction, μ . The measurement of μ is depicted here where a sled with a rubber bottom is pulled along a glass surface. The normal force is known and the friction force is measured. Various lubricants are placed between the two surfaces which greatly influence the friction forces measured.

Experimental Elastomer Analysis

73

Chapter 4: Laboratory

Data Reduction in the Lab

Data Reduction in the Lab The stress strain response of a typical test are shown at the right as taken from the laboratory equipment. In its raw form, it is not ready to be fit to a hyperelastic material model. It needs to be adjusted. The raw data is adjusted as shown below by taking a stable upload cycle. In doing this, Mullins effect and hysteresis are ignored. This upload cycle then needs to be shifted such that the curve passes through the origin. Remember hyperelastic models must be elastic and have their stress vanish to zero when the strain is zero.This shift changes the apparent gauge length and original cross sectional area. There is nothing special about using the upload curve, the entire stable cycle can be entered for the curve fit once shifted to zero stress for zero strain. Fitting a single cycle gives an average hyperelastic behavior to the hysteresis in that cycle. Also one may enter more data points in important strain regions than other regions. The curve fit will give a closer fit were there are more points.

74

Raw Data

Adjusted Data

Fit for Arruda-Boyce

Experimental Elastomer Analysis

Data Reduction in the Lab

Chapter 4: Laboratory

Data Reduction in the Lab (cont.) Data Reduction Considerations for Data Generated using Cyclic Loading 1. Slice out the selected loading path. 2. Subtract and note the offset strain. 3. Divide all strain values by (1 + Offset Strain) to account for the “new” larger stabilized gage length. 4. Multiply all stress values by (1+ Offset Strain) to account for “new” smaller stabilized cross sectional area. 5. The first stress value should be very near zero but shift the stress values this small amount so that zero strain has exactly zero stress. 6. Decimate the file by evenly eliminating points so that the total file size is manageable by the particular curve fitting software.

Experimental Elastomer Analysis

75

Chapter 4: Laboratory

Model Verification Experiments

Model Verification Experiments Attributes of a Good Model Verification Experiment The geometry is realistic. All Relevant Constraints are Measurable. The Analytical Model is Well Understood

76

Experimental Elastomer Analysis

Model Verification Experiments

Chapter 4: Laboratory

Model Verification Experiments (cont.) The Contribution of the Flashing on the Part was Unexpected, Initially Not Modeled, But Very Significant to the Actual Load Deflection.

Experimental Elastomer Analysis

77

Chapter 4: Laboratory

Testing at Non-ambient Temperatures

Testing at Non-ambient Temperatures Testing at the Application Temperature Measure Strain at the Right Location Perform Realistic Loadings Elastomers Properties Can Change by Orders of Magnitude in the Application Temperature Range.

78

Experimental Elastomer Analysis

Loading/Unloading Comparison

Chapter 4: Laboratory

Loading/Unloading Comparison

Experimental Elastomer Analysis

79

Chapter 4: Laboratory

Test Specimen Requirements

Test Specimen Requirements Where do these specimen shapes come from? 1. The states of strain imposed have an analytical solution. 2. A significantly large known strain condition exists free of gradients such that strain can be measured. 3. The state of strain is homogeneous for homogeneous materials. 4. The specimen shapes are such that different states of strain can be measured under similar loading conditions. 5. The specimen shapes are such that different states of strain can be measured with the same material.

80

Experimental Elastomer Analysis

Fatigue Crack Growth

Chapter 4: Laboratory

Fatigue Crack Growth Provides Great Potential. Not well understood.

Experimental Elastomer Analysis

81

Chapter 4: Laboratory

Experimental and Analysis Road Map

Experimental and Analysis Road Map TABLE 5. Experimental Tests Test

Description

Notes

1

Uniaxial

1a

Uniaxial - Rate Effects

2

Biaxial

2a

Biaxial - Temperature Effects

3

Planar Shear

4

Compression Button

5

Viscoelastic

6

Volumetric Compression

7

Friction Sled

8

Viscoelastic Damper

Planned

9

Foam

Planned

TABLE 6. Analysis Workshop Models Model

82

Description

Notes

1

Uniaxial

2

Biaxial

3

Planar Shear

4

Compression Button

5

Viscoelastic

6

Volumetric Compression

7

Friction Sled

Planned

8

Viscoelastic Damper

Planned

9

Foam

Planned

10

Damage

Planned

Experimental Elastomer Analysis

CHAPTER 5

Material Test Data Fitting

The experimental determination of elastomeric material constants depends greatly on the deformation state, specimen geometry, and what is measured.

Experimental Elastomer Analysis

83

Chapter 5: Material Test Data Fitting

Major Modes of Deformation

Major Modes of Deformation Uniaxial Tension λ1 = λ2 = λ

λ2 = λ3 = 1 ⁄ λ

2

2 3

1

Biaxial Tension (equivalent strain as uniaxial compression) λ1 = λ2 = λ

λ3 = 1 ⁄ λ

2

3 1

84

2

Experimental Elastomer Analysis

Major Modes of Deformation

Chapter 5: Material Test Data Fitting

Major Modes of Deformation (cont.) Planar Tension, Planar Shear, Pure Shear λ1 = λ

λ2 = 1

λ3 = 1 ⁄ λ

2 1

3

Simple Shear

Experimental Elastomer Analysis

85

Chapter 5: Material Test Data Fitting

Major Modes of Deformation

Major Modes of Deformation (cont.) Volumetric (aka Hydrostatic, Bulk Compression)

86

Confined Compression

Hydrostatic Compression

F

F

Experimental Elastomer Analysis

Confined Compression Test (UniVolumetric)

Chapter 5: Material Test Data Fitting

Confined Compression Test (UniVolumetric) F, L

Strain State: λ1 = 1

λ2 = 1

λ3 = L ⁄ L0

Stress State: σ1 = σ2 = σ 3 = – F ⁄ Ao = p

For this deformation state we have λ1 λ2 λ3 = V ⁄ V0 = L ⁄ L0 ,

and the uniaxial strain is equal to the volumetric strain or ΔL ⁄ L 0 = ΔV ⁄ V 0 .

Volumetric Data For Mentat Curve Fitting 400.0

The bulk modulus becomes p p K = ----------------- = ----------------ΔV ⁄ V 0 ΔL ⁄ L 0

Pressure [Mpa]

p

300.0

200.0

MSC.Marc Mentat uses the pressure, ⎛ 1---⎞ ΔV ⁄ V ⎝ 3⎠ 0 p , versus a “uniaxial equivalent” of the volumetric strain namely, ⎛1 ---⎞ ΔV ⁄ V , to determine the bulk 0 ⎝ 3⎠ modulus as shown on the right. Take care to divide the volumetric strain by 3, because ΔV ⁄ V 0 = ΔL ⁄ L 0 you may forget. 100.0

0.0 0.000

Experimental Elastomer Analysis

0.010

0.020 Equivalent Uniaxial Strain [1]

0.030

0.040

87

Chapter 5: Material Test Data Fitting

Hydrostatic Compression Test

Hydrostatic Compression Test Strain State:

F, L

λ1 = λ2 = λ3 = λ = ( V ⁄ V0 )

1⁄3

Stress State: σ1 = σ2 = σ 3 = – F ⁄ Ao = p

For this strain state we have λ = ( 1 + ΔV ⁄ V 0 )

1⁄3

≅ 1 + ⎛⎝ 1--3-⎞⎠ ΔV ⁄ V 0

and since λ = 1 + ΔL ⁄ L0

the uniaxial strain becomes one third the volumetric strain or ΔL ⁄ L 0 =

⎛1 ---⎞ ΔV ⎝ 3⎠

⁄ V0 .

The bulk modulus becomes p p K = ----------------- = -------------------------ΔV ⁄ V 0 3 ( ΔL ⁄ L 0 )

Again MSC.Marc Mentat uses the pressure, p , versus a “uniaxial equivalent” of the volumetric strain namely, ⎛⎝ 1--3-⎞⎠ ΔV ⁄ V 0 , to determine the bulk modulus.

88

Experimental Elastomer Analysis

Summary of All Modes

Chapter 5: Material Test Data Fitting

Summary of All Modes Mode: Maping

Uniaxial

Planar

λX 1

λX 1

λX 2 X3 -----2λ

X2 -----λ X3

λ 0 0 0 λ 0 1 0 0 ----2λ

λ 0 0 1 0 --- 0 λ 0 0 1

λX 1 x1

X2 ------λ X3 ------λ

X = x2 x3

Deformation Gradient F =

Figer Tensor

2

Principal Stretch Ratios λ i , i = 1, 2, 3 2 λi1

λ 0 0 1 0 ------- 0 λ 1 0 0 ------λ λ 0 0 1 0 --- 0 λ 1 0 0 --λ

b = F FT

b–

Biaxial

= 0

λ

2

0 0

0 λ

2

0 1 0 0 ----4λ

λ

Simple Shear

UniVolumetric

Volumetric

X1 + γ X2

X1

λX 1

X2

X2

λX 2

X3

λX 3

λX 3

1γ 0 0 1 0 0 0 1

1 0 0 0 1 0 0 0 λ

λ 0 0 0 λ 0 0 0 λ

2

0 0 1 0 ----2- 0 λ 0 0 1

1+γ γ 0

2

γ 0 1 0 01

1/ λ

λ λ

1/ λ

1 /λ

2

λ 1/λ 1

0 1 0

0 λ

2

0 0 2

0

0 0 λ

2

2

γ γ 1 + ---- + γ 1 + ---2 4 2

2

γ γ 1 + ---- – γ 1 + ---2 4 1

γ

λ

0 0 λ

2

λ

2

1 0 0

1 1 λ

λ λ λ

τ

Shape

Experimental Elastomer Analysis

89

Chapter 5: Material Test Data Fitting

General Guidelines

General Guidelines Its just curve fitting! No Polymer physics as basis Don’t use too high order fit Remember polynomial fit lessons (high school?)

Number of Data Points Don’t use too many Regularize if needed Add/Subtract points if needed Weighting of Points

Range and Scope of Data Check fit outside range of data Check fit in other modes of deformation – scope

90

Experimental Elastomer Analysis

Mooney, Ogden Limitations

Chapter 5: Material Test Data Fitting

Mooney, Ogden Limitations Phenomenological models – not material “law” These models are mathematical forms, nothing more

Summary of phenomenological models given by Yeoh (1995) “Rivlin and Saunders (1951) have pointed out that the agreement between experimental tensile data and the Mooney-Rivlin equation is somewhat fortuitous. The Mooney-Rivlin model obtained by fitting tensile data is quite inadequate in other modes of deformation, especially compression.”

Using only uniaxial tension data is dangerous! Mooney model in MSC.Marc allows no compressibility Ogden model does allow compressibility

Experimental Elastomer Analysis

91

Chapter 5: Material Test Data Fitting

Visual Checks

Visual Checks Extrapolations can be dangerous Always visually check your model’s predicted response Check it outside of the data’s range (see below) Check it outside the test’s scope dσ • dε > 0 σ

Real Material

Predicted Response Predicted Response

dσ • dε < 0 DATA

ε

Real Material

92

Experimental Elastomer Analysis

Material Stability

Chapter 5: Material Test Data Fitting

Material Stability Unstable material model -> numerical difficulties in FEA Druckers stability postulate, dσ • dε > 0 Graphically: σ

dσ 11 • dε 11 > 0

dσ 11 • dε 11 < 0

ε

Remember effects of Newton-Raphson and strain range

Experimental Elastomer Analysis

93

Chapter 5: Material Test Data Fitting

Future Trends

Future Trends Statistical Mechanics Models Based on single-chain polymer chain physics Build up to network level using non-gaussian statistics

8 Chain model by Arruda-Boyce (1993) 2 parameter model, can be expressed in terms of I1 Paper: “A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials”, J. Mech. Phys. Solids, V41 N2, pp 389-412.

Also similar is the Gent model (1996) Paper: “A new Constitutive Relation for Rubber”, Rubber Chem. and Technology, v. 69, pp 59-61.

Claim: alleviates need to gather test data from multiple modes

94

Experimental Elastomer Analysis

Adjusting Raw Data

Chapter 5: Material Test Data Fitting

Adjusting Raw Data The stress strain response of the three modes of deformation are shown below as taken from the laboratory equipment. In its raw form

The Raw Data (4 points/sec) 2.0

Engineering Stress [Mpa]

Equal Biaxial Pure Shear

1.5

Tension 1.0

0.5

0.0 0.0

0.2

0.4 0.6 0.8 Engineering Strain [1]

1.0

it is not ready to be fit to a hyperelastic material model. It needs to be adjusted.

Experimental Elastomer Analysis

95

Chapter 5: Material Test Data Fitting

Adjusting Raw Data

Adjusting Raw Data (cont.) The raw data is adjusted as shown below by taking the 18th upload cycle. In doing this Mullins effect is ignored. This 18th upload cycle Adjusting The Raw Data Shift to the Origin 2.0

Engineering Stress [Mpa]

1.5

Equal Biaxial Shifted Equal Biaxial Pure Shear Shifted Pure Shear Tension Shifted Tension ε = (ε ' – εp) ⁄ ( 1 + ε ) p

σ = σ ' ( 1 + εp)

1.0

0.5

0.0 0.0

0.2 εp

0.4 0.6 Engineering Strain [1]

0.8

1.0

then needs to be shifted such that the curve passes through the origin. Remember hyperelastic models must be elastic and have their stress vanish to zero when the strain is zero. This shift changes the apparent gauge length and original cross sectional area.

96

Experimental Elastomer Analysis

Adjusting Raw Data

Chapter 5: Material Test Data Fitting

Adjusting Raw Data (cont.) There is nothing special about taking the upload cycle, for instance the curve fitting may be done on the download path or both upload and download paths as shown below. The intended application can help you 1

Engineering Stress [Mpa]

Fit to upload

Fit to upload & download

0 0

uniaxial/experiment

Engineering Strain [1]

1

uniaxial/neo_hookean 1

decide upon the most appropriate way to adjust the data prior to fitting the hyperelastic material models.

Experimental Elastomer Analysis

97

Chapter 5: Material Test Data Fitting

Consider All Modes of Deformation

Consider All Modes of Deformation The plot below illustrates the danger in curve fitting only the tensile data, namely the other modes may become too stiff. This is why MSC.Marc Mentat always draws the other modes even when no experimental data is present. Below, a 3-term Ogden provides a great fit to the tensile data, but spoils the other modes. This can be avoided by looking for a balance between the various deformation modes.

98

Experimental Elastomer Analysis

The Three Basic Strain States

Chapter 5: Material Test Data Fitting

The Three Basic Strain States After shifting each mode to pass through the origin, the final curves are shown below. Very many elastomeric materials have this basic shape of the three modes, with uniaxial, shear, and biaxial having The Three Basic Strain States General Elastomer Trends

Engineering Stress [Mpa]

2.0

Equal Biaxial Pure Shear Tension

1.5

1.0

0.5

0.0 0.0

0.2

0.4 0.6 Engineering Strain [1]

0.8

1.0

increasing stress for the same strain, respectively. Knowledge of this and the actual shape above where say at a strain of 80%, the ratio of equal biaxial to uniaxial stress is about 2 (i.e., 1.3/0.75 = 1.73) will become very important as we fit this data with hyperelastic material models. Furthermore, this fit reduces the 10,000 data points taken from the laboratory to just a few constants.

Experimental Elastomer Analysis

99

Chapter 5: Material Test Data Fitting

Curve Fitting with MSC.Marc Mentat

Curve Fitting with MSC.Marc Mentat Objective: Fit experimental data of Mooney or Ogden materials with MSC.Marc Mentat. Begin at the main menu. MATERIAL PROPERTIES TABLES READ RAW (name of file) TABLE TYPE experimental_data OK RETURN EXPERIMENTAL DATA FITTING UNIAXIAL (pick table1) OK ELASTOMERS NEO-HOOKEAN UNIAXIAL COMPUTE APPLY OK SCALE AXES

100

Experimental Elastomer Analysis

Curve Fitting with MSC.Marc Mentat

Chapter 5: Material Test Data Fitting

Curve Fitting with MSC.Marc Mentat (cont) The resulting display of the material model is similar to this. The numerical coefficients for the model are shown in the popup menu. Use the APPLY button to copy these coefficients to your material model.

Notice that the uniaxial, biaxial, planar shear and simple shear modes are shown, where the uniaxial mode matches the material input. To turn some modes off, or make other display modifications go to PLOT OPTIONS. PLOT OPTIONS SIMPLE SHEAR PLANAR SHEAR RETURN SCALE AXES

Experimental Elastomer Analysis

(this toggles it off) (this toggles it off)

101

Chapter 5: Material Test Data Fitting

Curve Fitting with MSC.Marc Mentat

Curve Fitting with MSC.Marc Mentat (cont) Objective: Fit experimental data of Viscoelastic materials with MSC.Marc Mentat. Begin at the main menu. MATERIAL PROPERTIES TABLES READ RAW (name of file) TABLE TYPE experimental_data OK RETURN EXPERIMENTAL DATA FITTING ENERGY RELAX (pick table1),OK ELASTOMERS ENERGY RELAX RELAXATION # OF TERMS 3 COMPUTE APPLY, OK SCALE AXES

102

Experimental Elastomer Analysis

Curve Fitting with MSC.Marc Mentat

Chapter 5: Material Test Data Fitting

Curve Fitting with MSC.Marc Mentat (cont) Mooney-Rivlin fitting is linear, uses least squares fitting The least squares error is given by either: Ndata

error

R

∑

=

i

i

σ calc ⎞ ⎛ -⎟ ⎜ 1 – ----------------------i ⎝ σ measured⎠

2

or

Ndata A

error =

∑

i

i

( σ measured – σ calc )

2

i

The error R and error A are relative or absolute respectively Ndata is the total number of data points i

σ calc is the calculated stress i

σ measured is the measured engineering stress

Relative error is the default Engineering judgement is best to determine the best fit based upon physical not mathematical reasons.

Experimental Elastomer Analysis

103

Chapter 5: Material Test Data Fitting

Curve Fitting with MSC.Marc Mentat

Curve Fitting with MSC.Marc Mentat (cont) Ogden fitting is nonlinear, uses downhill-simplex method Downhill-simplex method is an iterative method Uses a number of start points abs ( error max – error min ) tol --------------------------------------------------------------- < ------Continues until: abs ( error max + error min ) 2 tol is set using CONVERGENCE TOLERANCE error min can be set with the ERROR LIMIT button

104

Experimental Elastomer Analysis

Curve Fitting with MSC.Marc Mentat

Chapter 5: Material Test Data Fitting

Curve Fitting with MSC.Marc Mentat (cont) Viscoelastic fitting is linear, uses least squares fitting A Prony series (exponential decay) is fit to the test data The least squares error is given by: Ndata

error

R

=

∑ i

i

σ calc ⎞ ⎛ -⎟ ⎜ 1 – ----------------------i ⎝ σ measured⎠

2

For a good fit, the number of Prony series terms should equal the number of time decades in the test data

Experimental Elastomer Analysis

105

Chapter 5: Material Test Data Fitting

106

Curve Fitting with MSC.Marc Mentat

Experimental Elastomer Analysis

CHAPTER 6

Workshop Problems

These problems are to provide self paced examples to develop skills in performing elastomer material curve fitting and finite element analysis using MSC.Marc and MSC.Marc Mentat. Workshop data files are in the product directory ..mentat2008r1/examples/training/mar103 and usually coppied to your working directory eea/ wkshops_A/ or eea/wkshops_B/. Subfolders are: uniaxial biaxial planar comp visco volume test_data (raw data)

Experimental Elastomer Analysis

107

Chapter 6: Workshop Problems

Some MSC.Marc Mentat Hints and Shortcuts

Some MSC.Marc Mentat Hints and Shortcuts 1. Enter MSC.Marc Mentat to begin, Quit to stop 2. Mouse in Graphics: Left to pick, Right to accept pick 3. Mouse in Menu: Left to pick another menu or function, Middle for help, Right to return to previous menu. means keyboard return. 4. Save your work frequently. Go to FILES and select SAVE AS and specify a file name. Use SAVE from then on. This will save the current MENTAT database to disk. 5. Dialog region at the lower left of screen displays current activity and prompts for input. Check this region frequently to see if input is required. 6. Dynamic Viewing can be used to position the model in the graphics area. When activated, the mouse buttons: Left – translates the model Right – zooms in/out Middle – rotates in 3D Use RESET VIEW and FILL to return to original view. Be sure to turn off DYNAMIC VIEW before picking in the graphics area. 7. CTRL P/N recall Previous/Next commands entered.

108

Experimental Elastomer Analysis

Model 1: Uniaxial Stress Specimen

Chapter 6: Workshop Problems

Model 1: Uniaxial Stress Specimen Objective: To model an elastomeric material under a uniaxial stress deformation mode. To focus on curve fitting elastomeric test data, a fully runnable procedure file is provided that will build and (and run) an initial model. However, the model contains only a trivial neo-Hookean material model with C10 = 0.5. It will be your job to modify the model by reading in the test data and curve fitting it using various material models. In a terminal window, use the cd command to move to the wkshops_A/uniaxial or the wkshops_B/uniaxial directory. Type “mentat” to start the MSC.Marc Mentat program, then starting from the main menu proceed as follows: UTILS PROCEDURES EXECUTE pick the file named uni_neo05.proc OK OK

This will produce and run a uniaxial stress model. Please familiarize yourself with this model. Look at the BC’s, the material specification, the contact bodies and contact table, and the loadcase.

Experimental Elastomer Analysis

109

Chapter 6: Workshop Problems

Model 1: Uniaxial Stress Specimen

After the procedure file is finished the final picture on your screen will look like this. Here is a brief summary of the uniaxial model we have created: • A single brick element, full integration, Herrmann. • Boundary conditions on x=0 & y=0 faces to prevent free translation in space. • Material model is neo-Hookean with C10 = 0.5 • Rigid contact surfaces are used to impose deformation. lower rigid body, cbody2, is stationary. upper rigid body, cbody3, is moved so as to first push, then pull, the brick element. • Loading is performed in 40 equal time increments. Increment 10 is full compression of 50%, increment 30 is full extension of 200%, increment 40 returns the brick to it’s original configuration. Now let’s look at the results of this analysis before curve fitting our uniaxial test data.

110

Experimental Elastomer Analysis

Model 1: Uniaxial Stress Specimen

Chapter 6: Workshop Problems

All of the postprocessing functions are accessed from RESULTS, which is located on the topmost MAIN menu. We are especially interested in deformed shape plots and XY plots of stress vs. strain. MAIN RESULTS OPEN DEFAULT DEF & ORIG SKIP TO INC 10 PLOT SURFACES WIREFRAME REGEN RETURN CONTOUR BAND SCALAR Displacement Z, OK SCALAR PLOT SETTINGS #LEVELS 5 , RETURN SKIP TO INC 30 FILL REWIND MONITOR

Experimental Elastomer Analysis

111

Chapter 6: Workshop Problems

Model 1: Uniaxial Stress Specimen

Now let’s generate the stress-strain plot that the MSC.Marc analysis has calculated. When we curve fit the actual test data, this analysis stressstrain curve should match the curve fit response exactly. HISTORY PLOT COLLECT GLOBAL DATA NODE/VARIABLES ADD GLOBAL CRV. Pos Z cbody3 Force Z cbody2 FIT, RETURN

Since the original area is one, and since the original length in the z-direction is one, the above plot is the engineering stress versus the engineering strain for a uniaxial stress specimen with neo-Hookean behavior. We use the Body 2 force just to get the sign correct. Another way of getting engineering stress-strain output is to use the user subroutine PRINCA.F. This is a plotv routine that calculates principal values of engineering stress & strain as well as principal stretch ratio. If available try re-running this analysis with the princa.f routine. Q: Why is it ok to use a one element model for this problem? A: ____________________________________________________ RETURN, CLOSE, SHORTCUTS SHOW MODEL OK, MAIN

112

Experimental Elastomer Analysis

Model 1: Uniaxial Curve Fit

Chapter 6: Workshop Problems

Model 1: Uniaxial Curve Fit Using this model file, go to the material definition stage and redefine the material by reading the uniaxial data, filename st_18.data, and proceed to re-run the problem using neo-Hookean, Mooney 2-term, Mooney 3-term, and Ogden 2-term fits. MATERIAL PROPERTIES EXPERIMENTAL DATA FITTING TABLES READ RAW FILTER: type *.data pick file st_18.data, OK

Experimental Elastomer Analysis

113

Chapter 6: Workshop Problems

Model 1: Uniaxial Curve Fit

Make the table type experimental_data, and associate this data with the uniaxial button. Your screen should look similar to the one below, and we are ready to start curve fitting the data. TABLE TYPE experimental_data, OK, RETURN UNIAXIAL table2

114

Experimental Elastomer Analysis

Model 1: Uniaxial Curve Fit

Chapter 6: Workshop Problems

Choose the neo-Hookean curve fitting routine and base the curve fit on just uniaxial data. The compute button will compute the model coefficients. By default, responses for many modes are plotted. The single neo-Hookean coefficient, C10, is 0.265. ELASTOMERS NEO-HOOKEAN UNIAXIAL COMPUTE, OK SCALE AXES PLOT OPTIONS SIMPLE SHEAR, RETURN (this turns off simple shear)

Experimental Elastomer Analysis

115

Chapter 6: Workshop Problems

Model 1: Uniaxial Curve Fit

Comments: We have just fit a neo-Hookean model using only uniaxial data. MSC.Marc Mentat by default shows the model’s response in all major modes of deformation. This is very important. You should always know your model’s response to each mode of deformation. Look again at the previous stress-strain plot. Notice the relative magnitude of the responses. Uniaxial is the lowest magnitude, the planar shear is higher, and the biaxial response is the highest. This is typical of most elastomers. See, for example, the stress-strain plot on the front cover of these notes. Always start fitting with simple models first. If a simple model captures the curvature of the test data, use it! Proceed to higher order and more complex models only as needed. Go back and use the EXTRAPOLATION feature and replot the neoHookean results from -0.5 to 2.0 strain. It is very important to look at the model’s response over a wide range of strain, including both tension and compression. We are looking for stability limits (maxima in the stressstrain curve). Mooney form models with all positive coefficients guarantee stability in all modes, for all strain. The simpler the material model, the higher probability it will be stable over a wider strain range. Later, after curve fitting several choices of models and selecting the best one, we will re-run our simple analysis.

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Here’s how to use the extrapolation feature to extend the strain range over which we plot the model’s response. We see that our neo-Hookean model is stable for all deformation modes. NEO-HOOKEAN EXTRAPOLATION EXTRAPOLATE LEFT BOUND, enter -0.5, RIGHT BOUND, enter 2.0, , OK COMPUTE, OK SCALE AXES

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Now fit a Mooney 2-term material model. Turn the extrapolation feature off for now. The Mooney coefficients are C10 = 0.074 and C01 = 0.280. Positive coefficients guarantee stability. Notice the relative magnitudes now – the biaxial stiffness is about 4 times the earlier material model. Of course, the fit to the uniaxial data is better, with more terms this model can capture a higher curvature in the stress-strain data. MOONEY(2) EXTRAPOLATION EXTRAPOLATE, OK COMPUTE, OK SCALE AXES

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(we want to turn it off)

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Now fit a Mooney 3-term material model. The Mooney coefficients are C10 = -0.735, C01 = 1.21, and C11 = 0.194. The uniaxial response is fantastic! The presence of a negative coefficient means that the material model might be unstable. We need to visually determine the stability range of the model. Note that the peak stress for the biaxial response has gone from 1.0 (neo-Hookean), to 4.5 (Mooney 2-term), to 36 (Mooney 3term). Which one is correct? MOONEY(3) COMPUTE, OK SCALE AXES

MOONEY(3), EXTRAPOLATION EXTRAPOLATE, OK COMPUTE, OK SCALE AXES (after viewing this turn extrapolate back off)

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Comments: Which biaxial fit is correct? Well, we don’t know because we haven’t (yet) performed a biaxial test. This is the great difficulty with the Mooney form and Ogden form material models – they are just curve fits. There is no “rubber physics” embedded in these equations. As we see here, a curve fit to uniaxial data will have a good response for that mode of deformation. But the responses for the other modes of deformation are all over the map. A rule of thumb based on observations of natural rubber and some other elastomers is that the tensile equi-biaxial response should be about 1.5 to 2.5 times the uniaxial tension response. We have seen many instances of higher order Mooney and Ogden models (using only uniaxial data) returning biaxial responses that are far too high. These are clearly bad material models. Try playing with the POSITIVE COEFFICIENTS option to see how much the responses change. For the curve fitting examples, you may need to toggle certain things on & off to better view and understand the computed fit. Keep these features in mind throughout all of these exercises: • EXTRAPOLATION on/off • PLOT OPTIONS, PREDICTED MODES (select subsets of UNIAXIAL, BIAXIAL, PLANAR SHEAR) • PLOT OPTIONS, LIMITS, YMAX, etc. (you may need to set plot limits by hand for better viewing)

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Now fit an Ogden 2-term material model. The uniaxial response is very good, but the biaxial response is now even higher than the Mooney 3-term. Ogden coefficients come in pairs, the moduli are μi and the exponents are α i . If each μ i and α i have the same sign then stability is guaranteed. If a μ i is positive and its corresponding α i is negative (or vice versa) then the material model might be unstable. Thus we may need to visually determine the stability range of the model. OGDEN COMPUTE, OK

This plot is to the same scale (ymax) as the Mooney 2-term plot.

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Comments: We are now finished with the curve fitting portion of this uniaxial exercise. We see that the Mooney 3-term and Ogden 2-term fit the uniaxial test data very well. However, we are concerned (or should be!) that the equi-biaxial response for some models (M 3-term, O 2-term) are too high and could make the material model overly stiff if that mode of deformation exists in our analysis. We need equi-biaxial test data to get a better fit to that mode. Let’s run this uniaxial analysis with the Ogden 3-term model. We select the curve fit model by pressing the APPLY button. Now go back and view the material model. Submit the analysis, then we will postprocess and show the analysis calculated stress-strain curve. OGDEN # OF TERMS = 3, OK COMPUTE, APPLY, OK PLOT OPTIONS (turn off all – leave uniaxial only) COPY TO GEN. XY PLOTTER RETURN (thrice) MECH. MATERIALS TYPE, MORE OGDEN (look at the material properties) OK FILES SAVE AS ogden3, OK MAIN JOBS RUN SUBMIT1 MONITOR 122

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Now go to postprocessing and generate the engineering stress-strain curve (we did this earlier with the original model). We will also save the analysis generated stress-strain curve to an external file for comparison to the test data. MAIN RESULTS OPEN DEFAULT HISTORY PLOT COLLECT DATA 14 30 1 (this collects just the tensile part) NODE/VARIABLES ADD GLOBAL VAR. Pos Z cbody3 Force Z cbody2 FIT, RETURN COPY TO GEN. XY PLOTTER SAVE type ogden3.tab

This last command saves the table to an external file named ogden3.tab (.tab is just to remind us that it is table data).

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To compare the two stress-strain curves, we will use MSC.Marc Mentat’s generalized plotter feature. UTILS GENERALIZED XY PLOT FIT SHOW IDS = 0

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Zoom in and tilt the plot and you will notice three curves: the data, the fit, and the response of our model.

O gd ns e en (3 )f it

es po

D

at a

R

tress Engineering S

Note that the model must follow the hyperelastic material model (Ogden(3)) exactly.

Engin

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tra S g n i eer

in

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One may also use xmgr to read the file ogden3.tab that was generated in MSC.Marc Mentat. From a terminal window type: xmgr st_18.data ogden3.tab A graphics screen will appear in which the experimental data is shown in black and the analysis generated stress-strain curve is shown in red. Of course, the test data only extends to about 100% strain whereas we performed our analysis out to 200% strain.

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Stress and Strain Measures In order to plot the engineering stress and strain measures in this example, we plotted “Pos Z cbody3” versus “Force Z cbody2” and because the original length and cross-sectional area are unity, “Pos Z cbody3” versus “Force Z cbody2” is the engineering strain versus the engineering stress. Since a total Lagrangian formulation is being used, the stress and strain measures (or Lagrangian measures) on the post file are Cauchy stress and Green-Lagrange strain which are different than the engineering measures. In this section, we shall convert the Lagrangian measures to engineering measures using the copy to clipboard feature available on the PC version of Mentat. MAIN RESULTS OPEN DEFAULT HISTORY PLOT SET NODES 8 # (pick node 8) COLLECT GLOBAL DATA (this collects all the data) NODE/VARIABLES ADD GLOBAL VAR. Pos Z cbody3 Force Z cbody2 FIT, RETURN COPY TO CLIPBOARD

With the plotted values stored in the clipboard, open Excel and paste the clipboard into the worksheet.

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Stress and Strain Measures Now we shall repeat the above for the stress and strain values on the post file (Lagrangian measures). NODE/VARIABLES CLEAR CURVES ADD VARIABLE Comp 33 of Total Strain Comp 33 of Cauchy Stress FIT, RETURN COPY TO CLIPBOARD

With the plotted values stored in the clipboard, paste the clipboard into the worksheet starting in column, the top of the worksheet should look like: Pos Z cbody3 0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35 -0.4 -0.45 -0.5 -0.375 -0.25

Force Z cbody2 0 -0.158024 -0.334548 -0.534053 -0.762458 -1.02772 -1.34074 -1.71677 -2.17766 -2.75563 -3.4998 -1.9349 -1.02772

Comp 33 of Total Strain Node 8 0 -0.04875 -0.095 -0.13875 -0.18 -0.21875 -0.255 -0.28875 -0.32 -0.34875 -0.375 -0.304688 -0.21875

Comp 33 of Cauchy Stress Node 8 0 -0.150124 -0.3011 -0.453959 -0.609991 -0.770832 -0.938581 -1.11599 -1.30671 -1.51575 -1.75011 -1.20941 -0.770831

Save this Excel file as neohookean05_job1.t16.xls.

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Stress and Strain Measures Now we can plot the two different strain and stress measures in Excel as: Strain and Stress Measures 10

Engineering Strain Versus Engineering Stress [MPa]

8

Green Lagrange Strain Versus Cauchy Stress [MPa]

6

Stress

4

2

0 -1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

-2

-4

-6 Strain

This plot allows us to clearly see the difference between the two measures and notice that for small values of strain, the difference becomes very small.

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Stress and Strain Measures All of these measures are related, and we now will show how to convert from the Green-Lagrange strain and Cauchy stress to engineering values for this particular problem. The uniaxial direction in the model is in the “z” or direction so we will use the 33 component of stress and strain. Letting E 33 and t 33 be the 33 component of Green-Lagrange strain and Cauchy stress ε 33 and σ 33 be the engineering measures, respectively, we have for this deformation mode the following relations: ε 33 =

2E33 + 1 – 1

and t 33 --------------------σ 33 = ( 1 + ε 33 )

The above formulas come from the definition of Green-Lagrange strain, T 1 E ij = --- ( [ F ] [ F ] – δ ij ) (see Appendix A on page 250) where [ F ] is the 2

deformation gradient that is determined from the stretch ratios (see “Summary of All Modes” on page 89) From incompressibility we have

A 0 L 0 = AL

and then

F 33 F 33 L t 33 t 33 ⎛ ⎞ -------------------------------------------σ 33 = = = = ⎝ A ⎠L A0 λ ( 1 + ε 0 33 )

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Stress and Strain Measures The Excel file can be used to verify the conversion as: A Pos Z cbody3 0.00 -0.05 -0.10 -0.15 -0.20 -0.25 -0.30 -0.35 -0.40 -0.45 -0.50 -0.38 -0.25 -0.13 0.00 0.13 0.25 0.38 0.50 0.63 0.75 0.88 1.00 1.13 1.25 1.38 1.50 1.63 1.75 1.88 2.00 1.80 1.60 1.40 1.20 1.00 0.80 0.60 0.40 0.20 0.00

B Force Z cbody2 0.00 -0.16 -0.33 -0.53 -0.76 -1.03 -1.34 -1.72 -2.18 -2.76 -3.50 -1.93 -1.03 -0.43 0.00 0.33 0.61 0.85 1.06 1.25 1.42 1.59 1.75 1.90 2.05 2.20 2.34 2.48 2.62 2.75 2.89 2.67 2.45 2.23 1.99 1.75 1.49 1.21 0.89 0.51 0.00

C D =SQRT(2*C3+1) - 1 Comp 33 of Total Strain NComp 33 of Cauchy Stres Convert E33 to ε33 0.00 0.00 0.00 -0.05 -0.15 -0.05 -0.10 -0.30 -0.10 -0.14 -0.45 -0.15 -0.18 -0.61 -0.20 -0.22 -0.77 -0.25 -0.26 -0.94 -0.30 -0.29 -1.12 -0.35 -0.32 -1.31 -0.40 -0.35 -1.52 -0.45 -0.38 -1.75 -0.50 -0.30 -1.21 -0.38 -0.22 -0.77 -0.25 -0.12 -0.38 -0.13 0.00 0.00 0.00 0.13 0.38 0.13 0.28 0.76 0.25 0.45 1.16 0.38 0.63 1.58 0.50 0.82 2.02 0.63 1.03 2.49 0.75 1.26 2.98 0.87 1.50 3.50 1.00 1.76 4.04 1.12 2.03 4.62 1.25 2.32 5.22 1.37 2.63 5.85 1.50 2.95 6.50 1.62 3.28 7.19 1.75 3.63 7.91 1.87 4.00 8.66 2.00 3.42 7.48 1.80 2.88 6.37 1.60 2.38 5.34 1.40 1.92 4.38 1.20 1.50 3.50 1.00 1.12 2.68 0.80 0.78 1.93 0.60 0.48 1.25 0.40 0.22 0.61 0.20 0.00 0.00 0.00

=D3/(1+E3) Convert t33 to σ33 0.00 -0.16 -0.33 -0.53 -0.76 -1.03 -1.34 -1.72 -2.18 -2.76 -3.50 -1.94 -1.03 -0.43 0.00 0.33 0.61 0.85 1.06 1.25 1.42 1.59 1.75 1.90 2.05 2.20 2.34 2.48 2.62 2.75 2.89 2.67 2.45 2.22 1.99 1.75 1.49 1.21 0.89 0.51 0.00

Where columns E and F show the formulas to convert from the total Lagrangian to engineering measures of stress and strain, and columns E and F are identical to columns A and B, respectively. This file, neohookean05_job1.t16.xls, is also available in the uniaxial directory. Finally, although all of the examples in this workshop are in a total Lagrange framework, the stress and strain measures for the updated Lagrange framework are Cauchy stress and Logarithmic strain, E ij , where E 33 = ln ( 1 + ε 33 ) . Experimental Elastomer Analysis

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Stress and Strain Measures Summarizing the various stress and strain measures used we have: Stress and Strain Measures Stress Measure

Strain Measure

Engineering

Engineering

Total Lagrange

Cauchy

Green-Lagrange

Updated Lagrange

Cauchy

Logarithmic

Curve Fitting Analysis

In our uniaxial example, these measures are related as: Stress and Strain Measures Stress Measure

Strain Measure

σ 33

ε 33

Total Lagrange

t 33 = ( 1 + ε 33 )σ 33

E 33 = ( 1 + ε 33 ) – 1

Updated Lagrange

t 33 = ( 1 + ε 33 )σ 33

E 33 = ln ( 1 + ε 33 )

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Model 1C: Tensile Specimen with Continuous Damage Objective: To model an elastomeric material under a cyclical uniaxial deformation mode subjected to damage accumulated from continuously varying strain cycles. For instance, looking at the test data below, we notice that upon repeated cycling the peak stress decays. Tensile Data Continuous Damage

Engineering Stress [Mpa]

1.0

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4 0.6 0.8 Engineering Strain [1]

1.0

This damage can be due to polymer chain breakage, multi-chain damage, and detachment of filler particles from the network entanglement.

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Model 1C: Tensile Specimen with Continuous Damage

In this workshop problem, we will simulate this behavior using the continuous damage model discussed in Appendix B. To clarify the behavior let’s plot the peak stress versus the cycle number as shown below. Tensile Data Continuous Damage for Engineering Strain = 1.00

Engineering Stress [Mpa]

1.10

1.05

1.00

0.95

0.90 0.0

2.0

4.0

6.0

8.0

10.0

Cycle Number

If our application experiences, this kind of behavior then we may wish to simulate this continuous damage. We would start by doing any normal hyperelastic curve fit. However, we would use the 1st cycle of the stress strain curve, not the steady state behavior in the file st_18.data which was for the 10th cycle shown above. We are now ready to begin modeling this continuous damage. In a terminal window, use the cd command to move to the wkshops_A/uniaxial or the wkshops_B/uniaxial directory.

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From NT (Windows 2000) just click on the uni_neo05.proc file or from unix Type “mentat” to start the MSC.Marc Mentat program, then starting from the main menu proceed as follows: UTILS PROCEDURES EXECUTE pick the file named uni_neo05.proc OK MAIN

This will produce and run a uniaxial stress model. Using this model file, we will go to the material definition stage and redefine the material by reading the uniaxial data, filename st_1st.tab, damage data, st_cont.tab, loading data st_load.tab and proceed to re-run the problem using an Ogden 1-term fit with continuous damage. MATERIAL PROPERTIES EXPERIMENTAL DATA FITTING TABLES READ NORMAL FILTER: type st* pick file st_1st.tab, OK (different data from st_18.data)

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Your screen should look similar to the one below

While we are here let’s read some more tables. READ NORMAL FILTER: type st* pick file st_cont.tab pick file st_load.tab RETURN

Now we are ready to start curve fitting the data. UNIAXIAL table2 CONSTANT pick st_const table 136

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ELASTOMERS MORE CONTINUOUS DAMAGE CONSTANT NUMBER OF TERMS = 2 FREE ENERGY = 1.07 (this is just the 1st peak stress) COMPUTE APPLY, OK, RETURN

OGDEN UNIAXIAL NUMBER OF TERMS = 1 COMPUTE, APPLY, OK SCALE AXES PLOT OPTIONS SIMPLE SHEAR (this turns off simple shear) Experimental Elastomer Analysis

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Model 1C: Tensile Specimen with Continuous Damage

RETURN (twice)

Let’s review the material properties to check that the curve fit has been properly applied to the selected material. MAIN MATERIAL PROPERTIES MORE OGDEN, DAMAGE EFFECTS - RUBBER, OK OK

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Now we can complete the model and run the analysis. The remaining item to finish is to attach a table to the contact body to cycle the loading several times from a strain of 0 to a strain of 1. MAIN CONTACT CONTACT BODIES EDIT (pick cbody3) RIGID POSITION (Z) TABLE (pick table st_load) OK (twice) MAIN LOADCASE MECHANICAL

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Model 1C: Tensile Specimen with Continuous Damage

STATIC TOTAL LOADCASE TIME = 940 # STEPS = 20 OK MAIN FILES SAVE AS ogden_damage OK MAIN JOBS RUN, SUBMIT1, MONITOR, OK MAIN RESULTS OPEN DEFAULT HISTORY PLOT COLLECT DATA 1 19 2 NODE/VARIABLES ADD GLOBAL VAR. Time Force Z cbody2 FIT, RETURN COPY TO GEN. XY PLOTTER SAVE type ogden_damage.tab

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Here we see the peak engineering stress drop upon subsequent applications of the prescribed displacements. Let’s run this same example but increase the number of load cycles by using the BEGIN/END SEQUENCE feature of MSC.Marc. This can be done by closing the post file, going to jobs, editing the input file to MSC.Marc then executing the edited input file. MAIN RESULTS CLOSE MAIN JOBS RUN ADVANCED JOB SUBMISSION EDIT INPUT

Here we need to locate the first occurrence of the “auto load” keyword. Experimental Elastomer Analysis

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Before this keyword, we need to enter the following: begin sequence, 100,

Now locate the second occurrence of the keyword continue and insert after it the following: end sequence

Now delete all input records after the end sequence record inserted. The tail end of the input data set will look like: begin sequence,100, auto load 1 0 time step 4.700000000000000+1 motion change 2 2 0 0.000000000000000+0 3 -1 0.000000000000000+0 continue auto load 1 0 time step 4.700000000000000+1 motion change 2 2 0 0.000000000000000+0 3 -1 0.000000000000000+0 continue end sequence

10

0.000000000000000+0 0.000000000000000+0 0.000000000000000+0 0.000000000000000+0 1.000000000000000+0 0.000000000000000+0

10

0.000000000000000+0 0.000000000000000+0 0.000000000000000+0 0.000000000000000+0 0.000000000000000+0 0.000000000000000+0

This change to the input file will run with 100 repetitions of the load sequence above.

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Save the input file and run the job by selecting the execute button, namely: OK RUN, EXECUTE1, MONITOR, OK MAIN RESULTS OPEN DEFAULT HISTORY PLOT COLLECT DATA 1 1999 2 NODE/VARIABLES ADD GLOBAL VAR. Time Force Z cbody2 FIT, RETURN

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Model 1C: Tensile Specimen with Continuous Damage

We now see that the engineering stress asymptotically approaches a value of 0.972 [Mpa] from its initial value of 1.114 [Mpa]. As shown below, the peak stress drops by about 13% from the initial load to an infinite number of repeated loadings. Although this drop may not appear to be large, other materials may demonstrate larger drops in peak stress upon repeated loadings and be more worthy of damage modeling. Tensile Simulation - Continuous Damage 1-Term Ogden and Original Data

Engineering Stress [Mpa]

1.0

1-Term Ogden Original Data

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Engineering Strain [1]

Should one desire to use a Mooney material model, the model would have to be converted to an updated Lagrangian formulation, by changing to element type 7, and choosing the “LARGE STRAIN-UPDATED LAGRANGE” rubber elasticity procedure. Finally, the hyperelastic fit above can be made better by simultaneously using other deformation modes as we shall see in subsequent exercises.

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Model 1: Realistic Uniaxial Stress Specimen Extra Credit Problem Statement:

This problem is in the subdirectory named ./uniaxial/big. Geometry is 45.5 L x 4 H x 2 W (mm) between grips, 10 mm length under grip. The x and z planes of symmetry are used. Read model from file uniaxial_specimen.mud. Grips are modeled as discrete rigid surfaces that squeeze then pull. The friction of the grip pulls the specimen. Run analysis with Mooney 1-term model (C = 0.265) and plot engineering stress-strain, compare with original test data. Use princa.f usersub if possible.

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Results: uniaxial_specimen.mud

Y X Z

4

Results from uniaxial_specimen.mud 1.0

Engineering Stress [MPa]

0.8 0.6

Curve Fit

0.4

Model

0.2 Engineering Strain -0.2

0.2 -0.2

146

0.4 0.6 0.8 1.0 Grip Squeeze Causes Specimen Buckling (Compression)

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Results: uniaxial_specimen.mud The engineering stress and strain are extracted from the post file from a history plot of Displacement X Node 628 vs. Force X Force X left_grip. The data is copied into the clip board and placed in Excel. Node 628 is on the far side of one of the elements attached to the symmetry plane called left_grip. The length of the element is 1, so the engineering strain is just the displacement of this node. Since the cross-sectional area of the model is 4.0, the engineering stress is simply the x component of force on the wall (Force X Force X left_grip) divided by 4. This is plotted over all of the increments and compared to the curve fit which is 1 1 σ = dW ⁄ dλ = G ⎛ λ – ------⎞ = 2 ( 0.265 ) ⎛ ( 1 + ε ) – -------------------⎞ ⎝ ⎝ 2⎠ 2⎠ (1 + ε) λ

Of course the curves agree identically as they should. The more important issue is with the grips. As the grips are squeezed onto the specimen by displacement control, the material flows out of the grip and puts the specimen in compression. Although this is not too noticeable on the tension specimen it is very noticeable on the planar specimen; care must be taken not to prestress the specimen before the testing begins by expanding the distance between the grips to account for the longer specimen. This will be seen as you perform the planar tension test later.

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Model 1: Realistic Uniaxial Stress Specimen (cont.) What happened to our specimen model with 600 elements? ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ How does the specimen model compare to the one element test case? ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ How would we convert measured force versus stroke to engineering stress versus engineering strain? How realistic is this? ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ Where is the actual gauge length in the specimen model? ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________

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Model 2: Equi-Biaxial Stress Specimen Objective: To model an elastomeric material under a equi-biaxial stress deformation mode. To focus on curve fitting elastomeric test data, a fully runnable procedure file is provided that will build and (and run) an initial model. However, the model contains only a trivial neo-Hookean material model with C10 = 0.5. It will be your job to modify the model by reading in the test data and curve fitting it using various material models. In a terminal window, use the cd command to move to the wkshops_A/biaxial or the wkshops_B/biaxial directory. Type “mentat” to start the MSC.Marc Mentat program, then starting from the main menu proceed as follows: UTILS PROCEDURES EXECUTE pick the file named eb_neo05.proc OK OK

This will produce and run a biaxial stress model. Please familiarize yourself with this model. Look at the BC’s, the material specification, the contact bodies and contact table, and the loadcase.

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Model 2: Equi-Biaxial Stress Specimen

After the procedure file is finished the final picture on your screen will look like this. Here is a brief summary of the biaxial model we have created: • A single brick element, full integration, Herrmann. • Boundary conditions on y=0 face to prevent free translation in space. • Material model is neo-Hookean with C10 = 0.5 • Rigid contact surfaces are used to impose deformation. cbody2 & cbody5 are stationary. cbody3 & cbody4 are moved so as to impose displacements in the Z & X directions respectively. • Loading is performed in 30 equal time increments. Increment 10 is biaxial compression of 50% (compression in X & Z), increment 30 is biaxial extension of 200%(extension in X & Z). Now let’s look at the results of this analysis before curve fitting our biaxial test data.

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All of the postprocessing functions are accessed from RESULTS, which is located on the topmost MAIN menu. We are especially interested in deformed shape plots and XY plots of stress vs. strain. MAIN RESULTS OPEN DEFAULT DEF & ORIG SKIP TO INC 10 CONTOUR BAND SCALAR Displacement Z, OK SCALAR PLOT SETTINGS #LEVELS 5 , RETURN SKIP TO INC 30 REWIND MONITOR

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Model 2: Equi-Biaxial Stress Specimen

Now let’s generate the stress-strain plot that the MSC.Marc analysis has calculated. When we curve fit the actual test data, this analysis stressstrain curve should match the curve fit response exactly. HISTORY PLOT SET NODES (pick node 8 shown) END LIST COLLECT DATA 0 30 1 NODE/VARIABLES ADD VARIABLE Displacement Z Force Z cbody2 FIT, RETURN RETURN CLOSE, MAIN

Pick

Since the original area is one, and since the original length in the zdirection is one, this plot is the engineering stress versus the engineering strain. We use the Body 2 force just to get the sign correct. Notice how much different compression is for biaxial than uniaxial behavior. Of course, biaxial compression is very hard to simulate with a physical test, and only tension is usually done.

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Model 2: Equi-Biaxial Curve Fit Now we will read in both the uniaxial and biaxial test data and simply repeat fitting the four material models. The difference is that we will now use both sets of data. Start from the MAIN menu. MATERIAL PROPERTIES EXPERIMENTAL DATA FITTING TABLES READ RAW FILTER: type *.data pick file st_18.data, OK

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Model 2: Equi-Biaxial Curve Fit

Since we will be reading more than one set of test data, let’s name the datasets. Then make the table type experimental_data, and associate this data with the uniaxial button. NAME uniaxial TABLE TYPE experimental_data, OK, RETURN UNIAXIAL uniaxial

Repeat the above sequence to read in the file eb_18.data and name this dataset biaxial. Associate this dataset with the biaxial button. Your screen should look similar to the one below and we are ready to start curve fitting the data.

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Choose the neo-Hookean curve fitting routine and base the curve fit on all the data. The compute button will compute the model coefficients. By default, responses for many modes are plotted. Turn off the plotting of simple shear. ELASTOMERS NEO-HOOKEAN USE ALL DATA COMPUTE, OK SCALE AXES PLOT OPTIONS SIMPLE SHEAR, RETURN (this turns off simple shear)

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Model 2: Equi-Biaxial Curve Fit

Comments: We have just fit a neo-Hookean model using both uniaxial and biaxial data. MSC.Marc Mentat by default shows the model’s response in all major modes of deformation. This is very important. You should always know your model’s response to each mode of deformation. Compare this plot with the uniaxial only stress-strain plot on (page 115). Both plots are very similar. The uniaxial only C10 was 0.265, while the new material model based on both uniaxial and biaxial data gives C10 = 0.280. These neo-Hookean coefficients are quite close, telling us that the earlier model was pretty good. We would prefer to use the latest model since it is based on more information and gives a better fit to the biaxial test data. If you can accept the differences between the test data and fitted response, this material model is quite adequate (and stability is guaranteed because the coefficient is positive). For scoping analysis and the initial stage of an analysis, this model is sufficient.

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Now fit a Mooney 2-term material model. Make sure extrapolation is off. The Mooney coefficients are C10 = 0.247 and C01 = 0.0270. Notice the relative magnitudes now – the biaxial response is much different than before (page 118) and the coefficients are much different as well. (Uniaxial coeff’s were C10 = 0.074 and C01 = 0.280). This confirms our suspicion that the earlier Mooney 2-term model based on only uniaxial data misrepresented the biaxial behavior. MOONEY(2) COMPUTE, OK SCALE AXES

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Model 2: Equi-Biaxial Curve Fit

To see the old (uniaxial data only) fit response plotted along with the new data, use the EVALUATE feature. MOONEY(2) EVALUATE type in the old coeff’s as prompted at the command line ENTER C10: 0.074 ENTER C01: 0.280 All coefficients entered. Continue? y

So this is the uniaxial only model response. Notice how overly stiff the biaxial model response (yellow/light grey line) is compared to the actual biaxial test data (yellow/light grey line with squares).

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Now fit a Mooney 3-term material model. The Mooney coefficients are C10 = 0.246, C01 = 0.029, and C11 = -0.0004. This is essentially the same as the Mooney 2-term material model from the previous page. The biaxial data is adding additional constraint to the fit. The third term is almost zero, thus the fit has not changed. One would not choose this model over the Mooney 2-term fit. MOONEY(3) COMPUTE, OK SCALE AXES

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Model 2: Equi-Biaxial Curve Fit

Now fit an Ogden 2-term material model. The uniaxial and biaxial model responses are slightly better than the Mooney models. However, the first pair of coefficients (modulus term of -2.55E-6 and exponent of -10.5) only contribute to the response at high strains. Set the NUMBER OF TERMS to 1 and re-fit the data. OGDEN COMPUTE, OK

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Comments: We are now finished with the curve fitting portion of this uniaxial and biaxial data exercise. As you saw, the addition of biaxial information was very valuable. The earlier Mooney and Ogden uniaxial only fits were way off base! However, it is interesting to note that the earlier neo-Hookean fit was pretty decent. This gives more merit to keeping the material as simple as possible. Let’s run this biaxial analysis with the Mooney 2-term model. Go back to MOONEY(2) and fit it again, press the APPLY button. Submit the analysis, then we will postprocess and show the analysis calculated stress-strain curve. MOONEY(2) COMPUTE APPLY, OK PLOT OPTIONS COPY TO GEN. XY PLOTTER, RETURN RETURN (twice) MECHANICAL MATERIALS TYPE, MORE MOONEY look at the material properties OK FILES SAVE AS moon2, OK RETURN (twice) JOBS RUN SUBMIT1 MONITOR Experimental Elastomer Analysis

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Model 2: Equi-Biaxial Curve Fit

Now go to postprocessing and generate the engineering stress-strain curve (we did this earlier with the original model). We will also save the analysis generated stress-strain curve to an external file for comparison to the test data. MAIN RESULTS OPEN DEFAULT HISTORY PLOT SET NODES (pick node 8 shown) END LIST COLLECT DATA 14 30 1 NODE/VARIABLES ADD VARIABLE Displacement Z Force Z cbody2 FIT, RETURN COPY TO GEN. XY PLOTTER SAVE type moon2.tab

Pick

This last command saves the table to an external file named moon2.tab (.tab is just to remind us that it is table data).

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To compare the two stress-strain curves we will use MSC.Marc Mentat’s generalized plotter feature. UTILS GENERALIZED XY PLOT FIT SHOW IDS = 0

Engineering Stress [Mpa]

Biaxial Response

Biaxial Data Uniaxial Data Biaxial Fit

Uniaxial Fit

Engineering Strain

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Model 2: Equi-Biaxial Curve Fit

To compare the two stress-strain curves we will use XMGR. From a terminal window type: xmgr eb_18.data moon2.tab A graphics screen will appear in which the experimental data is shown in black and the analysis generated stress-strain curve is shown in red. Of course, the test data only extends to about 100% strain whereas we performed our analysis out to 200% strain.

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Model 2: Realistic Equal-Biaxial Stress Specimen Extra Credit Problem Statement:

This problem is in the subdirectory named ./biaxial/big. Geometry is 86 Dia x 2 Thick (mm), 16 Grips around full circumference (22.5 deg). Mesh uses symmetry at X=0, Y=0, and Z=0. Read model from file bi_glue.mud. Grips are modeled as discrete rigid surface, Grips are 10 mm in dia., placed on a 71 mm dia., friction coefficient is infinite. Run analysis with Ogden 3-term model and plot engineering stress-strain, compare with original test data. Use princa.f usersub if possible.

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Results: bi_glue.mud

Model 2 Equal-Biaxial Specimen Model versus Data

2.0

Specimen Data Specimen Model Engineering Stress [Mpa]

1.5

1.0

0.5

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Engineering Strain

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Model 2: Realistic Equal-Biaxial Stress Specimen (cont.) What happened to our specimen model with 1128 elements? ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ How does the specimen model compare to the test data? ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ How would we convert measured force versus stroke to engineering stress versus engineering strain? How realistic is this? ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ Where is the actual gauge length in the specimen model? ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________

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Model 3: Simple Compression, Button Comp.

Model 3: Simple Compression, Button Comp. Objective: To model a neo-Hookean elastomeric material under a compressive deformation mode with and w/o friction. In a terminal window use the cd command to move to the wkshops_A/comp or the wkshops_B/comp directory. In MSC.Marc Mentat, go to FILES and read in the comp_start.mud file. This file contains two separate models. We will call the top model the “uniaxial” model, meaning that its end conditions are free of friction and the specimen will not barrel. The bottom model (lower in Z) we will call the “button” compression model, meaning that its ends are glued to the platens simulating a high friction condition, or actual bonding. Both models already have boundary conditions and material properties assigned. OPEN choose file comp_start, OK SAVE AS type in comp, OK PLOT ELEMENTS SOLID REGEN, RETURN VIEW LOAD VIEW (select file OBL.VIEW from list), OK RETURN MAIN

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CONTACT CONTACT BODIES DEFORMABLE, OK ELEMENTS ADD (pick top elements) NAME, uniaxial NEW DEFORMABLE, OK ELEMENTS ADD (pick bottom elems) NAME, button NEW RIGID DISCRETE, OK SURFACES ADD (pick z=30 surface) NAME, uni_bot NEW RIGID DISCRETE, OK SURFACES ADD (pick z=43 surface) NAME, uni_top ID BACKFACES

Chapter 6: Workshop Problems

z=43 z=30

z=13 z=0

(Make sure gold side of surfaces touch the deformable brick. If not flip surfaces until this happens, otherwise, continue.) SAVE

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CONTACT (cont’d) NEW RIGID DISCRETE, OK SURFACES ADD (pick z=0 surface) NAME, but_bot NEW RIGID DISCRETE, OK SURFACES ADD (pick z=13 surface) NAME, but_top

(Make sure gold side of surfaces touch the deformable brick. If not flip surfaces until this happens, otherwise, continue.) EDIT uni_top, OK RIGID VELOCITY PARAMETERS VELOCITY Z=-6 OK (twice)

(repeat the above sequence for the but_top contact surface) SAVE, RETURN

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Construct your contact table to look like the one below. Notice that the elements “uniaxial” touch uni_bot and uni_top, while elements “button” are glued to but_bot and but_top. All separation forces are zero. Return to the MAIN menu.

CONTACT TABLE NEW PROPERTIES

Make elements “uniaxial” touch uni_bot and uni_top, while elements “button” are glued to but_bot and but_top. OK, MAIN

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LOADCASES MECHANICAL STATIC STEPPING PROCEDURE FIXED PARAMETNERS # OF STEPS=12 , OK (twice), MAIN JOBS MECHANICAL lcase1 ANALYSIS OPTIONS LARGE DISPLACEMENT, OK JOB RESULTS CAUCHY STRESS TOTAL STRAIN, OK OK INITIAL LOADS xsym ysym CONTACT CONTROL INITIAL CONTACT CONTACT TABLE ctable1 OK (3 times) JOBS SAVE RUN SUBMIT1 MONITOR OK, MAIN 172

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RESULTS OPEN DEFAULT DEF & ORIG SKIP TO INC 12 PLOT SURFACES WIREFRAME REGEN RETURN

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POST PROCESSING HISTORY PLOT

Construct time history of Pos Z uni_top vs. Force Z uni_top. This is the true uniaxial response. Construct the same for Pos Z but_top vs. Force Z but_top. This is response that mixes shearing and bulk compression (remember bulk, or hydrostatic, compressive stiffness is many times higher than the shear stiffness)

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POST PROCESSING HISTORY PLOT COLLECT GLOBAL DATA NODES/VARIABLES ADD GLOBAL CURVE POS Z UNI_TOP FORCE Z UNI_TOP ADD GLOBAL CURVE POS Z BUT_TOP FORCE Z BUT_TOP

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Model 4: Planar Shear Specimen

Model 4: Planar Shear Specimen also known as Planar Tension, Pure Shear

Objective: To model an elastomeric material under a planar shear stress deformation mode. To focus on curve fitting elastomeric test data, a fully runnable procedure file is provided that will build and (and run) an initial model. However, the model contains only a trivial neo-Hookean material model with C10 = 0.5. It will be your job to modify the model by reading in the test data, and curve fitting it using various material models. In a terminal window, use the cd command to move to the wkshops_A/planar or the wkshops_B/planar directory. Type “mentat” to start the MSC.Marc Mentat program, then starting from the main menu proceed as follows: UTILS PROCEDURES EXECUTE pick the file named ps_neo05.proc OK OK

This will produce and run a planar shear stress model. Please familiarize yourself with this model. Look at the BC’s, the material specification, the contact bodies and contact table, and the loadcase.

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After the procedure file is finished, the final picture on your screen will look like this. Here is a brief summary of the planar shear model we have created: • A single brick element, full integration, Herrmann. • Boundary conditions on y=0 face to prevent free translation in space. • Material model is neo-Hookean with C10 = 0.5 • Rigid contact surfaces are used to impose deformation. cbody2, cbody4 & cbody5 are stationary. cbody3 is moved so as to impose displacement in the Z direction. • Loading is performed in 30 equal time increments. Increment 10 is compression of 50% (compression in Z), increment 30 is extension of 200% (extension in Z). Now let’s look at the results of this analysis before curve fitting our planar shear test data.

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Model 4: Planar Shear Specimen

All of the postprocessing functions are accessed from RESULTS, which is located on the topmost MAIN menu. We are especially interested in deformed shape plots and XY plots of stress vs. strain. MAIN RESULTS OPEN DEFAULT DEF & ORIG SKIP TO INC 10 CONTOUR BAND SCALAR Displacement Z, OK SETTINGS #LEVELS 5 , RETURN SKIP TO INC 30 REWIND MONITOR

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Now let’s generate the stress-strain plot that the MSC.Marc analysis has calculated. When we curve fit the actual test data, this analysis stressstrain curve should match the curve fit response exactly. HISTORY PLOT SET NODES (pick node 8 shown) END LIST COLLECT DATA 0 30 1 NODE/VARIABLES ADD VARIABLE Displacement Z Force Z cbody2 FIT, RETURN RETURN

Pick

Since the original area is one, and since the original length in the zdirection is one, this plot is the engineering stress versus the engineering strain. We use the Body 2 force just to get the sign correct. You will usually see this test performed only in tension, but some labs will perform a plane strain compression test. CLOSE, MAIN

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Model 4: Planar Shear Curve Fit

Model 4: Planar Shear Curve Fit Now we will read in both the uniaxial, biaxial, and planar shear test data and repeat fitting the four material models. The difference is that we will now use all sets of data. Start from the MAIN menu. MATERIAL PROPERTIES EXPERIMENTAL DATA FITTING TABLES READ RAW FILTER: type *.data pick file st_18.data, OK

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Since we will be reading more than one set of test data, let’s name the datasets. Then make the table type experimental_data, and associate this data with the uniaxial button. NAME uniaxial TABLE TYPE experimental_data, OK, RETURN UNIAXIAL uniaxial

Repeat the above sequence to read in the file eb_18.data and name this dataset biaxial. Associate this dataset with the biaxial button. Repeat again to read in the file ps_18.data and name this dataset planar. Associate this dataset with the planar shear button.

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Model 4: Planar Shear Curve Fit

Choose the neo-Hookean curve fitting routine and base the curve fit on all the data. The compute button will compute the model coefficients. By default, responses for many modes are plotted. Turn off the plotting of simple shear. ELASTOMERS NEO-HOOKEAN USE ALL DATA COMPUTE, OK SCALE AXES PLOT OPTIONS SIMPLE SHEAR, RETURN (this turns off simple shear)

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Summary of neo-Hookean fits: We have just fit a neo-Hookean model using three sets of data, uniaxial, biaxial, and planar shear. MSC.Marc Mentat by default shows the model’s response in all major modes of deformation. This is very important. You should always know your model’s response to each mode of deformation. Compare this plot with the uniaxial only stress-strain plot on (page 115), and the uniaxial+biaxial fit on (page 155). All the plots are very similar. The uniaxial only C10 was 0.265, the uniaxial and biaxial data gives C10 = 0.280, and the fit of all three sets of data simultaneously gives C10 = 0.276. These neo-Hookean coefficients are quite close, telling us that all of the neo-Hookean models are pretty good. We would prefer to use the latest model since it is based on more information and gives a better fit to all the test data. If you can accept the differences between the test data and fitted response, this material model is quite adequate (and stability is guaranteed because the coefficient is positive). For scoping analysis and the initial stage of an analysis, this model is sufficient.

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Model 4: Planar Shear Curve Fit

Now fit a Mooney 2-term material model. Make sure extrapolation is off. The Mooney coefficients are C10 = 0.244 and C01 = 0.0270. Compare these results to those of the uniaxial+biaxial fit on page 157. There is very little difference in the fit and the coefficients have changed only slightly. MOONEY(2) COMPUTE, OK SCALE AXES

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Summary of Mooney 2-term fits: We have now completed a series of Mooney 2-term fits that used progressively more information as the basis for the curve fitting. The table below summarizes the coefficients calculated in each case. The conclusion is that adding biaxial data had a big influence on the quality of the fit and changed the coefficients greatly. Adding the planar shear data did not cause further big changes. Mooney 2-term Fitting Summary Uniaxial Uniaxial + Biaxial Uniaxial+Biaxial+Planar Data Data Shear Data C10

0.074

0.247

0.244

C01

0.280

0.027

0.027

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Model 4: Planar Shear Curve Fit

Now fit a Mooney 3-term material model. The Mooney coefficients are C10 = 0.239, C01 = 0.035, and C11 = -0.0015. This is essentially the same as the Mooney 2-term material model from the previous page. The third term is almost zero, thus the fit has not changed. One would not choose this model over the Mooney 2-term fit. MOONEY(3) COMPUTE, OK SCALE AXES

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Summary of Mooney 3-term fits: We have now completed a series of Mooney 3-term fits that used progressively more information as the basis for the curve fitting. The table below summarizes the coefficients calculated in each case. The conclusion is that adding biaxial data had a big influence on the quality of the fit and changed the coefficients greatly. Adding the planar shear data did not cause further big changes. Mooney 3-term Fitting Summary Uniaxial Data

Uniaxial + Biaxial Uniaxial+Biaxial+Planar Data Shear Data

C10

-0.735

0.246

0.239

C01

1.21

0.029

0.035

C11

0.194

-0.0004

-0.0015

Mooney 2-term Fitting Summary Uniaxial Data

Uniaxial + Biaxial Uniaxial+Biaxial+Planar Data Shear Data

C10

0.074

0.247

0.244

C01

0.280

0.027

0.027

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Model 4: Planar Shear Curve Fit

Now fit an Ogden 2-term material model. The fit is similar to the earlier one based on just uniaxial and biaxial data. Indeed, adding the planar shear data has caused the biaxial fit to be worse. OGDEN COMPUTE, OK

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Just for fun, try fitting an Ogden 3-term material model to just the uniaxial and planar shear data. You will have to clear the table associated with the biaxial button to do this. The results should look like the figure below. Removing the biaxial data is like removing a constraint. The uniaxial and planar shear response improve quite a bit. However, the biaxial fit response is very bad, with a stability point at about 30% strain.

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Model 4: Planar Shear Curve Fit

Summary of Ogden 2-term fits: We have now completed a series of Ogden 2-term fits that used progressively more information as the basis for the curve fitting. The table below summarizes the coefficients calculated in each case. We know the uniaxial only data fit had too little information as its basis, and it’s biaxial response was very bad. The last two fits, however, were relatively similar and yet their coefficients are markedly different. We see this in many Ogden fits and it is attributed to the many local minima that exist in the Ogden equation set. Ogden 2-term Fitting Summary

190

Uniaxial Data

Uniaxial + Biaxial Data

Uniaxial+Biaxial+Planar Shear Data

μ1

-3.01

-2.55E-6

-0.353

α1

0.733

-10.5

-.582

μ2

-0.861

1.00

0.592

α2

-4.91

1.18

1.60

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Comments: We are now finished with the curve fitting portion of this exercise. The further addition of planar shear data did not change the material models very much. Let’s run this planar shear analysis with the Mooney 2-term model. Go back to MOONEY(2) and fit it again, press the APPLY button. Submit the analysis, then we will postprocess and show the analysis calculated stress-strain curve. MOONEY(2) COMPUTE APPLY, OK RETURN (twice) MECHANICAL MATERIALS TYPE, MORE MOONEY (look at the material properties) OK FILES SAVE AS moon2, OK RETURN (twice) JOBS RUN SUBMIT1 MONITOR, OK

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Model 4: Planar Shear Curve Fit

Now go to postprocessing and generate the engineering stress-strain curve (we did this earlier with the original model). We will also save the analysis generated stress-strain curve to an external file for comparison to the planar shear test data. MAIN RESULTS OPEN DEFAULT HISTORY PLOT SET NODES (pick node 8 shown) END LIST COLLECT DATA 14 30 1 NODE/VARIABLES ADD VARIABLE Displacement Z Force Z cbody2 FIT, RETURN COPY TO GEN. XY PLOTTER SAVE type moon2.tab

Pick

This last command saves the table to an external file named moon2.tab (.tab is just to remind us that it is table data).

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For the moment, we shall use the generalized xy plotter to compare the response of the model to the curve fit. MAIN RESULTS CLOSE, RETURN UTILS GENERALIZED XY PLOT DATA FIT FIT, FILL

Planar Shear Response

Planar Shear Data

Planar Shear Curve Fit

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Model 4: Planar Shear Curve Fit

To compare the two stress-strain curves we will use XMGR. From a terminal window type: xmgr ps_18.data moon2.tab A graphics screen will appear in which the experimental data is shown in black and the analysis generated stress-strain curve is shown in red. Of course, the test data only extends to about 100% strain whereas, we performed our analysis out to 200% strain.

Planar Shear Response

Planar Shear Data

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Model 4: Realistic Planar Shear Specimen Extra Credit Problem Statement:

This problem is in the subdirectory named ./planar/big. Geometry is 75 L x 12 H x 2 W (mm) between grips. Read model from file pt_45.mud. Grips are modeled as discrete rigid surfaces, with glue. Run analysis with Mooney 1-term model and plot engineering stressstrain, compare with original test data. Use princa.f usersub if possible.

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Results: pt_45.mud

Model 4: Planar Shear (pt_45.mud) Neo Hookean: G = 2(2.71964-1)

1.5

Engineering Stress [Mpa]

G(1+x-(1+x)^-3) max princ engg. stress l [124] max princ engg. stress l [614] max princ engg. stress l [615]

1.0

0.5

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Engineering Strain

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Model 4: Realistic Planar Stress Specimen (cont.) What happened to our specimen model with 612 elements? ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ How does the specimen model compare to the test data? ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ How would we convert measured force versus stroke to engineering stress versus engineering strain? How realistic is this? ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ Where is the actual gauge length in the specimen model? ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________

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Model 5: Viscoelastic Specimen

Model 5: Viscoelastic Specimen Objective: To model a viscoelastic neo-Hookean elastomeric material under a uniaxial stress deformation mode with a load to 50% strain and hold for 7200 seconds. Begin at the main menu. To focus on curve fitting elastomeric test data, a fully runnable procedure file is provided that will build an initial model. However, the model contains only a trivial neo-Hookean material model with C10 = 0.5. It will be your job to modify the model by reading in the test data, and curve fitting it to add viscoelastic effects. In a terminal window, use the cd command to move to the wkshops_A/visco or the wkshops_B/visco directory. Type “mentat” to start the MSC.Marc Mentat program, then starting from the main menu proceed as follows: UTILS PROCEDURES EXECUTE pick the file named visco.proc OK OK

This will produce a uniaxial stress model. Please familiarize yourself with this model. Look at the BC’s, the material specification, the contact bodies and contact table, and the loadcase.

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After the procedure file is finished the final picture on your screen will look like this. Here is a brief summary of the uniaxial model we have created: • A single brick element, full integration, Herrmann. • Boundary conditions on x=0 & y=0 faces to prevent free translation in space. • Material model is neo-Hookean with C10 = 0.5, no viscoelastic properties are included. • Rigid contact surfaces are used to impose deformation. lower rigid body, cbody2, is stationary. upper rigid body, cbody3, is position controlled and moves +0.5 in the Z direction at time zero to achieve 50% strain. • Seven loadcases are used to mirror the test data sampling times. This problem is not run in this trivial form since no viscoelastic properties have been added yet. We will now read in the material data and perform the curve fit(s).

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Chapter 6: Workshop Problems

Model 5: Viscoelastic Curve Fit

Model 5: Viscoelastic Curve Fit For curve fitting, we need two different types of test data. First we need to create a table of instantaneous strain, stress data to fit a standard Mooney or Ogden model. Then we need to read a file of time, stress information that will be used to curve fit a relaxation function. We will create the instantaneous table from our viscoelastic test data. For this exercise, we have 30% strain and 50% strain visco tests. Look at the first line from each data file – named 30percent.data and 50percent.data. We will take the first stress point from each file as the instantaneous stress. These first stress points are 0.7524 and 1.1695 respectively. Go to the material definition stage and create the following table of instantaneous strain, stress data. MAIN MATERIAL PROPERTIES EXPERIMENTAL DATA FITTING TABLES NEW 1 INDEPENDENT VARIABLE ADD POINT 0, 0 0.30, 0.7524 0.50, 1.1695 SAVE

200

Experimental Elastomer Analysis

Model 5: Viscoelastic Curve Fit

Chapter 6: Workshop Problems

Make the table type experimental_data, and associate this data with the uniaxial button. Your screen should look similar to the one below, and we are ready to start curve fitting the data. TABLE TYPE experimental_data, OK, RETURN UNIAXIAL table2

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Chapter 6: Workshop Problems

Model 5: Viscoelastic Curve Fit

Choose the neo-Hookean curve fitting routine and base the curve fit on just uniaxial data. The compute button will compute the model coefficients. By default, responses for many modes are plotted. The single neo-Hookean coefficient, C10, is 0.542. Notice that the model’s uniaxial response does not exactly match the data. ELASTOMERS NEO-HOOKEAN UNIAXIAL COMPUTE, APPLY, OK SCALE AXES

202

Experimental Elastomer Analysis

Model 5: Viscoelastic Curve Fit

Chapter 6: Workshop Problems

Comments: For simplicity, we have fit a neo-Hookean model using only uniaxial data. All of the previously discussed issues regarding using only one mode of deformation still apply here! We are simply ignoring them for purposes of this exercise. We have used the first data point from the stress relaxation test to define our “instantaneous” or short time behavior. We could have used data from a separate simple tension test (non-relaxation), but this would add to our uncertainty. Test sample differences (cure, preconditioning, etc.), test strain-rate differences, and other such influences may cause correlation difficulties. We have based our neo-Hookean model on both 30% and 50% strain data. If we wanted near perfect correlation between one test and one analysis, we could have based the neo-Hookean model on just the 50% strain test. Now we are ready to read in one set of relaxation test data, curve fit, and run our uniaxial stress relaxation analysis.

Experimental Elastomer Analysis

203

Chapter 6: Workshop Problems

Model 5: Viscoelastic Curve Fit

For the 50 durometer silicone rubber we have been using in this class, we will perform 2 stress relaxation tests – one at 30% strain and at 50% strain. For completeness, we show these two sets of data below.

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Experimental Elastomer Analysis

Model 5: Viscoelastic Curve Fit

Chapter 6: Workshop Problems

Remember, that a key limitation of this large strain viscoelastic material model is that it assumes the relaxation rate (and thus overall stress relaxation at any time) is independent of the imposed strain. It would be reasonable to check our test data to see if this material satisfies this assumption. We do so by normalizing each dataset (the 30% and 50% strain stress relaxation datasets) and plotting both. This has been done and is shown below. Our 50 durometer silicone rubber satifies this assumption nicely within this range of strain. Q: What to do if your material shows markedly different relaxation rates at different strain levels?

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Chapter 6: Workshop Problems

Model 5: Viscoelastic Curve Fit

Continuing from our previous location in the menu hierarchy, we will now read in one set of stress relaxation data. Choose the 50% strain data (we have set up the analysis for 50% strain loading). RETURN TABLES READ RAW FILTER: type *.data pick file 50percent.data COPY TO GEN. XY PLOTTER, RETURN (twice) TABLE TYPE experimental_data, OK, RETURN ENERGY RELAX. table3 ELASTOMERS ENERGY RELAXATION RELAXATION (on) COMPUTE, OK SCALE AXES

We have done this initial fit with the default of two terms in the prony series. This is a pretty crude fit. A rule of thumb is to use as many terms as there are time decades of data. We have 5 decades of data. Re-fit the data using 3, then 4, then 5 terms and watch especially the relaxation time values. Notice that finally you will have a relaxation time value in each decade.

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Experimental Elastomer Analysis

Model 5: Viscoelastic Curve Fit

Chapter 6: Workshop Problems

The final 5 term prony series fit will look like this. Note the coefficients in the upper right portion of the screen. We are happy with this fit and are ready to APPLY it to the current material definition. From the menu shown below, do the following: APPLY, OK

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Chapter 6: Workshop Problems

Model 5: Viscoelastic Curve Fit

Comments: We are now finished with the curve fitting portion of this viscoelastic exercise. Let’s save our changes to the model and run the analysis. SAVE MAIN JOBS RUN SUBMIT1 MONITOR OK (when finished)

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Experimental Elastomer Analysis

Model 5: Viscoelastic Curve Fit

Chapter 6: Workshop Problems

Now go to postprocessing and generate the engineering stress-time relaxation curve. We will also save the analysis generated stress-time curve to an external file for comparison to the test data. MAIN RESULTS OPEN DEFAULT HISTORY PLOT COLLECT DATA 1 60 1 NODE/VARIABLES ADD GLOBAL CRV Time Force Z cbody2 FIT, RETURN COPY TO GEN. XY PLOTTER

RETURN SAVE type visco50.tab

This last command saves the table to an external file named visco50.tab (.tab is just to remind us that it is table data).

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Chapter 6: Workshop Problems

Model 5: Viscoelastic Curve Fit

We can now use the GENERALIZED XY PLOTTER to compare the response with the data. CLOSE UTILS GENERALIZED XY PLOT DATA FIT (this get the data fit curves)

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Experimental Elastomer Analysis

Model 5: Viscoelastic Curve Fit

Chapter 6: Workshop Problems

Instead of using the GENERALIZED XY PLOTTER, the two stressstrain curves can be compared by using XMGR. From a terminal window type: xmgr 50percent.data visco50.tab A graphics screen will appear in which the experimental data is shown in black and the analysis generated stress-time curve is shown in red (and dashed). You will not see all the text labels. Q: Why is there a difference between the two lines?

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Chapter 6: Workshop Problems

Model 5: Viscoelastic Curve Fit

Q: Why is there a difference between the two lines? A: Recall from (page 207) that the 5 term prony series fit the data extremely well, the fit and data lines were virtually indistinguishable. So why does the MSC.Marc result also not lie directly on top of the test data? The difference is caused by the error in the instantaneous neoHookean model. Remember (page 202) that the neo-Hookean model with C10 = 0.542 did not pass exactly through the 50% strain point. This error causes all the difference in the stress-time plot shown on the previous page. To achieve a better correlation of MSC.Marc result to the 50% strain test data, base the neo-Hookean fit on just the 50% strain data. Doing so gives a C10 = 0.554 and the MSC.Marc results will now match the relaxation test data very closely.

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Experimental Elastomer Analysis

Model 6: Volumetric Fit

Chapter 6: Workshop Problems

Model 6: Volumetric Fit Although the curve fit of this deformation mode ultimately establishes the bulk modulus, it is the least significant test regarding the material behavior since the estimated bulk modulus in Ogden fits is quite adequate. The value of this workshop exercise is just knowing how to perform the fit if required. For historical reasons, the curve fitting in Mentat is based upon the hydrostatic compression test (Chapter 5 on page 88) yet the actual test performed is the confined compression test or univolumetric test (Chapter 5 on page 87). Hence we will find that we need to adjust the data to accommodate for this. The balance of the fitting is similar to all of the other modes we have already performed; so let’s begin. FILES volumetric.mfd OK MAIN MATERIAL PROPERTIES EXPERIMENTAL DATA UNIAXIAL st (pick table st) BIAXIAL eb (pick table eb) PLANAR SHEAR ps (pick table ps) ELASTOMERS OGDEN USE ALL DATA COMPUTE, OK Experimental Elastomer Analysis

213

Chapter 6: Workshop Problems

Model 6: Volumetric Fit

Model 6: Volumetric Fit SCALE AXES DATA FIT PLOT COPY TO GENERALIZED XY PLOTTER RETURN (thrice) 1.571

0 0

9.1 (x.1)

uniaxial/experiment biaxial/experimen t planar_shear/experiment simple_shear/ogden

uniaxial/ogden biaxial/ogden planar_shear/ogden

So far we have just fit the first three modes (uniaxial, biaxial and planar shear) and now we will add the univolumetric data and fit. For your convenience, the data tables are already in the model file and the univolumetric data is contained in volume. Furthermore the “x axis” of this table has been scaled by 1/3 and is in table eq_uni_volume. It is this last table that we will use for the volumetric fit. 214

Experimental Elastomer Analysis

Model 6: Volumetric Fit

Chapter 6: Workshop Problems

Model 6: Volumetric Fit MATERIAL PROPERTIES EXPERIMENTAL DATA VOLUMETRIC eq_uni_volume (pick table eq_uni_volume) ELASTOMERS OGDEN USE ALL DATA COMPUTE, OK

At this point it is worth mentioning that there is very little difference between the two fits, with and without the volumetric data. Collecting the Ogden coefficients for the two fits we have: K μ1, α1

Without Volume Data 2999.9 -0.3819 -0.4784

With Volume Data 2927.0 -0.2004 -0.7464

μ2, α2

0.6830

0.7043

1.4894

1.4862

Remember that without the volume data, the bulk modulus, K, is 2

estimated as

K = 2500

∑

μnαn

, which is very close to the measured

n=1

value using the volume data. Let’s see how the other modes are affected. SCALE AXES DATA FIT PLOT COPY TO GENERALIZED XY PLOTTER

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215

Chapter 6: Workshop Problems

Model 6: Volumetric Fit

Model 6: Volumetric Fit Removing all the experimental, simple shear, and volumetric curves from the XY plotter we have: 1.097

0 9.1

0 X (x.1) uniaxial/ogden planar_shear/ogden biaxial/ogden

biaxial/ogden uniaxial/ogden planar_shear/ogden

that shows extremely little difference in the three basic strain states.

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Experimental Elastomer Analysis

CHAPTER 7

Contact Analysis

This features allows for the automated solution of problems where contact occurs between deformable and rigid bodies. It does not require special elements to be placed at the points of contact. This contact algorithm automatically detects nodes entering contact and generates the appropriate constraints to insure no penetration occurs and maintains compatibility of displacements across touching surfaces.

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217

Chapter 7: Contact Analysis

Definition of Contact Bodies

Definition of Contact Bodies Contact Body - Any group of elements or geometric entities that may contact themselves or others. Types of Contact Bodies: Deformable – Collection of elements. Rigid – Collection of geometric entities or heat transfer elements Add elements to contact body, here 90 elements are added to contact body, cbody1. Analytic contact may be used to smooth facets of element edges or faces. By default Rigid bodies are controlled with displacement, unless specified here. Geometric curves/surfaces have to be properly oriented.

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Experimental Elastomer Analysis

Definition of Contact Bodies

Chapter 7: Contact Analysis

Definition of Contact Bodies (cont) Contact Body – Any group of elements or geometric entities that may contact themselves or others. Types of Contact Bodies: Deformable – Collection of elements. Rigid – Collection of geometric entities or heat transfer elements Add elements to contact body, here 1 surface is added to body, cbody2. This bodies position is controlled by a table.

Experimental Elastomer Analysis

219

Chapter 7: Contact Analysis

Control of Rigid Bodies

Control of Rigid Bodies Rigid bodies can be controlled by their velocity, position, or load.

Prescribe translational and/or rotational velocity as a function of time using a time table. Prescribe position/rotation as a function of time. Prescribe force on rigid body as a function of time: define force on additional node connect node to rigid contact body motion of rigid contact body is in direction of applied force; motion in perpendicular direction is constrained

220

Experimental Elastomer Analysis

Contact Procedure

Chapter 7: Contact Analysis

Contact Procedure Deformable to Rigid Body Contact Case 1: Contact not detected when Δu A • n < D – d

Rigid Body (set of curves or surfaces)

Cases 2, 3: Contact detected when Δu A • n – d ≤ D

n

Case 4: Penetration detected when Δu A • n > D + d

Δu A

A Deformable Body (set of elements)

D D

Case 1 2

3

4

d with: Δu A :incremental displacement vector of node A n : unit normal vector with proper orientation D :contact distance (Default = h/20 or t/4) F s :separation force (Default = Maximum Residual)

Case 1:Node A does not touch, no constraint applied. Case 2:Node A is near rigid body within tolerance, contact constraint pulls node to contact surface if F < F s . Case 3:Node A penetrates within tolerance, contact constrain pushes node to contact surface. Case 4:Node A penetrates out of tolerance and increment gets split (loads reduced) until no penetration. Experimental Elastomer Analysis

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Chapter 7: Contact Analysis

Bias Factor

Bias Factor By default, the contact tolerance is equally applied to both sides of a segment; this can be changed by introducing a bias factor B ( 0 ≤ B ≤ 1 ): D contact D contact

default

D contact ( 1 – B ) D contact ( 1 + B )

with bias factor

Choosing B > 0 may be useful to • reduce increment splitting, since the distance to cause penetration is increased • improve accuracy, since the distance below which a node comes into contact is reduced:

default

222

with bias factor

Experimental Elastomer Analysis

Deformable-to-Deformable Contact

Chapter 7: Contact Analysis

Deformable-to-Deformable Contact Discrete deformable contact (default) is based on piecewise linear geometry description of either 2-node edges in 2 dimensions or 4-node faces in 3 dimensions on the outer surface of all contacting meshes. actual geometry finite element approximation contacting body contact tolerance y

A x

contacted body Then the contact constraint: [ defines tying relation for displacement component of contacting node in local y -direction [ applies correction on position in local y -direction

Experimental Elastomer Analysis

223

Chapter 7: Contact Analysis

Potential Errors due to Piecewise Linear Description:

Potential Errors due to Piecewise Linear Description: Tying relation may be not completely correct due to the assumption that the normal direction is constant for a complete segment. If contacting node slides from one segment to another, a discontinuity in the normal direction may occur. The correction on the position of the contacting node may be not completely correct.

Analytical Deformable Contact Bodies: Replace 2-node linear edges by cubic splines (2D) or 4-node bi-linear patches by bi-cubic Coons surfaces (3D). You must take care of nodes (2D) and edges (3D) where the outer normal vector is discontinuous. You may wish to use extended precision. 1

Advantages are smoother contact where in 2D, C -continuity is obtained, 1 and in 3D, at least pointwise C -continuity is obtained. Analytical deformable contact must be turned on, whereas, rigid bodies default to analytic contact where the curves or surfaces are represented as NURBS during contact.

224

Experimental Elastomer Analysis

Contact Flowchart

Chapter 7: Contact Analysis

Contact Flowchart Input Initial set up of contact bodies Incremental data input Check on contact

begin iteration

begin increment

Set up of contact constraints Apply distributed loads Assemble stiffness matrix; include friction Apply contact constraints Solve set of equations

Change contact constraints

Recover stresses Update contact constraints

Split increment

No Yes Yes No

Experimental Elastomer Analysis

“Converged” solution? Yes Separation? No Penetration? No Last increment? Yes Stop

225

Chapter 7: Contact Analysis

Symmetry Body

Symmetry Body Symmetry bodies often provide an easy way to impose symmetry conditions; they may be used instead of the TRANSFORMATION and SERVO LINK options. A symmetry plane is characterized by a very high separation force, so that only a movement tangential to the contact segment is possible The symmetry plane option can only be invoked for rigid surfaces

deformable_body

symmetry_plane_1

symmetry_plane_2

none

Y

Z

226

Experimental Elastomer Analysis

Rigid with Heat Transfer

Chapter 7: Contact Analysis

Rigid with Heat Transfer 50 20o R=6 billet

20

4.75

4 35

25 channel

Model 1: Deformable-rigid (stress or coupled analysis) billet

channel

none

MARC element 10

Experimental Elastomer Analysis

geometrical entities (straight lines and a circular arc)

deformable-rigid (stress or coupled analysis)

227

Chapter 7: Contact Analysis

Rigid with Heat Transfer

Model 2: Deformable-rigid (coupled analysis)

billet

channel

none

Rigid w Heat Transfer MARC element 40

MARC element 10

deformable-rigid (coupled a

Model 3: Deformable-deformable (stress or coupled analysis) billet

channel

MARC element 10 none

MARC element 10

228

deformable-deformable (stress or coupled analysis)

Experimental Elastomer Analysis

Contact Table

Chapter 7: Contact Analysis

Contact Table

3

2

1 4

Contact Table Properties:

Single-sided Contact:

Only body 2 may contact itself Experimental Elastomer Analysis

229

Chapter 7: Contact Analysis

Contact Table

Contact Table (cont) Very useful for specifying parameters between contacting bodies.

Contact tables must be turned on initially in contact control, or during any loadcase to become active. With no contact tables active, all bodies can come into contact including self contact.

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Experimental Elastomer Analysis

Contact Areas

Chapter 7: Contact Analysis

Contact Areas Very useful for defining certain nodes of a body that may enter contact.

Like contact tables, contact areas must be turned on initially in contact control, or during any loadcase to become active. With no contact areas active, all nodes of all bodies can come into contact. Both contact table and contact areas can reduce the amount of node to segment checking and can save compute time.

Experimental Elastomer Analysis

231

Chapter 7: Contact Analysis

Exclude Segments During Contact Detection

Exclude Segments During Contact Detection Exclude segment will influence the searching done for nodes detected in the contact zone during self contact. Contact table, contact node and exclude affect the initial search for contact; once a node is in contact, this is not undone by these options.

Options to influence search for contact include: Contact table: define which bodies can potentially come into contact (defined per loadcase) Contact node: define which nodes of a body can potentially come into contact (defined per loadcase) Single-sided contact: searching for contact is not done with respect to bodies with a lower body number (defined for the whole analysis) Exclude: define which segments of a body can never be contacted (defined per loadcase)

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Experimental Elastomer Analysis

Effect Of Exclude Option:

Chapter 7: Contact Analysis

Effect Of Exclude Option:

Standard contact

excluded segments

With exclude option

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233

Chapter 7: Contact Analysis

Contacting Nodes and Contacted Segments

Contacting Nodes and Contacted Segments For 3D continua, an automatic check on the direction of the normal vectors is included:

Contacting body nodes

Contacted body patches

Contact will not be accepted if n node ⋅ n patch > 0.05

Shell Thickness is taken into account according to: 2D: one fourth of the shell thickness only if the body is contacted. 3D: one fourth of the shell thickness for both the contacting and the contacted body.

234

Experimental Elastomer Analysis

Friction Model Types

Chapter 7: Contact Analysis

Friction Model Types Friction coefficient is specified in contact body or contact table. Although, the coefficient is entered a specific friction model type must be selected for friction to be active. .

Experimental Elastomer Analysis

235

Chapter 7: Contact Analysis

Coulomb ArcTangent Friction Model

Coulomb ArcTangent Friction Model Implementation of this friction model has been done using nonlinear dashpots whose stiffness depend on the relative sliding velocity as: slip

Ft

stick C

vr

MARC approximation slip MSC.Marc approximation: vr 2 F t ≤ μF n --- atan ⎛ ----⎞ ⎝ C⎠ π

with: C :“relative sliding velocity below which sticking is simulated”

(Default = 1.0! is rarely correct)

236

Experimental Elastomer Analysis

Coulomb Bilinear Friction Model

Chapter 7: Contact Analysis

Coulomb Bilinear Friction Model Implementation of this friction model has been done using nonlinear dashpots whose stiffness depend on the relative sliding velocity as: slip

Ft

stick δ

ur

MARC approximation slip MSC.Marc approximation:

with: δ : slip threshold automatically set.

Friction force tolerance has a default value of 0.05.

Experimental Elastomer Analysis

237

Chapter 7: Contact Analysis

Stick-Slip Friction Model

Stick-Slip Friction Model Discovered by Leonardo da Vinci in the 15th century and verified by experiments by Charles A. Coulomb in the 18th century, this stick-slip friction model uses a penalty method to describe the step function of Columb’s Law.

μF n

αμF n

Ft

2β 2εβ Δu t

with: F t ≤ μF n static ,

F t ≤ αμF n kinetic

Δu t :incremental tangential displacement β : slip to stick transition region (default 1 ×10

–6

α : coefficient multiplier (default 1.05) e : friction force tolerance (default 0.05) –6

ε : small constant, so that εβ ≈ 0 (fixed at 1 ×10 ) 238

Experimental Elastomer Analysis

Glued Contact

Chapter 7: Contact Analysis

Glued Contact Sometimes a complex body can be split up into parts which can be meshed relatively easy: * define each part as a contact body * invoke the glue option (CONTACT TABLE) to obtain tying equations not only normal but also tangential to contact segments * enter a large separation force

cbody 1 cbody 2 none

Z

X

Y 4

Experimental Elastomer Analysis

239

Chapter 7: Contact Analysis

Glued Contact

Glued Contact (cont) Gluing rigid to deformable bodies can help simulate testing because testing of materials generally involves measuring the force and displacement of the rigid grips. Here is an example of a planar tension

(pure shear) rubber specimen being pulled by two grips. The grip force versus displacement curve is directly available on the post file and can be compared directly to the force and displacement measured.

240

Experimental Elastomer Analysis

Release Option

Chapter 7: Contact Analysis

Release Option The release option provides the possibility to deactivate a contact body: upon entering a body to be released, all nodes being in contact with this body will be released. Using the release option e.g., a springback effect can be simulated. Releasing nodes occurs at the beginning of an increment. Make sure that the released body moves away to avoid recontacting.

Interference Check / Interference Closure Amount By means of the interference check, an initial overlap will be removed at the beginning of increment 1. By means of an interference closure amount, an overlap or a gap between contacting bodies can be defined per increment: * positive value: overlap * negative value: gap

Experimental Elastomer Analysis

241

Chapter 7: Contact Analysis

Forces on Rigid Bodies

Forces on Rigid Bodies During the analysis rigid bodies have all forces and moments resolved to a single point which is the centroid shown below.

This makes rigid bodies useful to monitor the force versus displacement behavior as shown at the right.

Body 3 Force Y

242

Experimental Elastomer Analysis

Forces on Rigid Bodies

Chapter 7: Contact Analysis

Forces on Rigid Bodies (cont) Vector plotting External Force will show the forces at each node resulting from the contact constraints.

Experimental Elastomer Analysis

243

Chapter 7: Contact Analysis

244

Forces on Rigid Bodies

Experimental Elastomer Analysis

APPENDIX A

The Mechanics of Elastomers

The macroscopic behavior of elastomers depends greatly upon the deformation states because the material is nearly incompressible.

Experimental Elastomer Analysis

245

Appendix A: The Mechanics of Elastomers

Deformation States

Deformation States t3

L1

t2

L3 L2

λ1 L1 λ3 L3

t1

λ2 L2

t1

t2 t3

Stretch ratios: L i + ΔL i λ i = -------------------- = 1 + ε Li

engineering strain = ( ΔL i ⁄ L i ) = ε

Incompressibility: λ1 λ2 λ3 = 1

First order approximation (Neo-Hookean): 1 2 2 2 W = --- G ( λ 1 + λ 2 + λ 3 – 3 ) 2

Eliminate λ 3 : ⎞ 1 ⎛ 2 1 2 W = --- G ⎜ λ 1 + λ 2 + ----------– 3 ⎟ 2 2 2 ⎝ ⎠ λ λ 1 2

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Experimental Elastomer Analysis

Deformation States

Appendix A: The Mechanics of Elastomers

Two-dimensional extension:

F1

L2

dL 2

F2

F1

dL 1

L1

F2

dW = F 1 dL 1 + F 2 dL 2 = σ 1 dλ 1 + σ 2 dλ 2 ∂W ∂W dW = --------- dλ 1 + --------- dλ 2 ∂λ 1 ∂λ 2 ⎛

1 ⎞

⎛

λ 1 λ 2⎠

⎝

1 ⎞

Hence: σ 1 = G ⎜ λ 1 – ----------⎟ , σ 2 = G ⎜ λ 2 – ----------⎟ 3 2 2 3 ⎝

λ 1 λ 2⎠

,

σ3 = 0

Engineering stresses σ i : forces per unit undeformed area True stresses t i : forces per unit deformed area

Experimental Elastomer Analysis

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Appendix A: The Mechanics of Elastomers

Deformation States

Two-dimensional extension: t1 = σ1 ⁄ ( λ2 λ3 ) = λ1 σ1

or: 2

2

2

2

t1 = G ( λ1 – λ3 )

and: t2 = G ( λ2 – λ3 ) t3 = 0

Constant volume implies that a hydrostatic pressure p cannot have an effect on the state of strain, so that the stresses are indeterminate to the extent of the hydrostatic pressure

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Experimental Elastomer Analysis

Deformation States

Appendix A: The Mechanics of Elastomers

(Nearly) incompressible material: Bulk Modulus K 2(1 + ν) ------------------------------------------ = -----------------------Shear Modulus G 3 ( 1 – 2υ )

1 K υ → --- , hence ---- → ∞ 2 G

Ordinary solid (e.g. steel): G and K are the same order of magnitude. Whereas, in rubber the ratio of G to K is of the order –4 10 ; hence the response to a stress is effectively determined solely by the shear modulus G

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Appendix A: The Mechanics of Elastomers

General Formulation of Elastomers

General Formulation of Elastomers Material points in undeformed configuration: X i ; material points in deformed configuration: x i Lagrange description: xi = xi ( Xj ) ∂x i dx i = F ij dX j with F ij = -------∂X j F ij is the deformation gradient tensor

Green-Lagrange strain tensor: 2

2

( dx ) – ( dX ) = 2E ij dX i dX j

Right Cauchy-Green strain tensor: 2

( dx ) = C ij dX i dX j

Some additional relations: C ij = δ ij + 2E ij ∂x k ∂x k C ij = -------- -------- = F ki F kj ∂X i ∂X j 1 ∂x k ∂x k 1 E ij = --- -------- -------- – δ ij = --- [ F ki F kj – δ ij ] 2 ∂X i ∂X j 2

250

Experimental Elastomer Analysis

General Formulation of Elastomers

Appendix A: The Mechanics of Elastomers

Introduce displacement vector u i : xi = Xi + ui 1 E ij = --- ( u i, j + u j, i + u k, i u k, j ) 2 C ij = ( δ ki + u k, i ) ( δ kj + u k, j )

With respect to principal directions: 2

λ1 0 0 C i'j' =

2

0 λ2 0 2

0 0 λ3

Invariants of C ij : I 1 = C ii 1 I 2 = --- ( C ii C jj – C ij C ij ) 2 I 3 = det C ij

Strain energy function: *

W = W ( I 1, I 2 ) + h ( I 3 – 1 )

Experimental Elastomer Analysis

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Appendix A: The Mechanics of Elastomers

General Formulation of Elastomers

Second-Piola Kirchhoff stresses: ∂I 3 ∂W ∂W S ij = 2 -------- δ ij + 2 -------- [ δ ij C kk – C ij ] + 2h ---------∂C ij ∂I 1 ∂I 2

True or Cauchy stresses: ρ t ij = ----- ( δ ik + u i, k )S kl ( δ jl + u j, l ) ρ0

Zero deformation: ∂W 0 S ij = ⎛ 2 -------⎝ ∂I 1

0

∂W + 4 -------∂I 2

0

+ 2h⎞ δ ij ⎠

hence: ∂W p = – 2 -------∂I 1

0

0 ∂W ------– 2h –4 ∂I 2

so that the stresses can be expressed in terms of displacements and the hydrostatic pressure

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Experimental Elastomer Analysis

Finite Element Formulation

Appendix A: The Mechanics of Elastomers

Finite Element Formulation Modified virtual work equation:

∫ Sij δEij dV – ∫ Qi δui dV – ∫- Ti δui dA + ∫ δλ ( I3 – 1 ) dV = 0

V

V

A

V

In addition to the displacements, within an element we need to interpolate the pressure: α

∑ Nα ( Xi )ui

ui ( Xi ) =

and

p ( Xi ) =

∑ hα ( Xi )p

α

α

α

The incremental stresses are related to the linear strain increment by: ΔSij =

n D ijkl ΔE kl

n –1

– Δp ( C ij )

The incremental set of equations to be solved reads: [K

(0)

+K

–[ H ]

(1)

] – [ H ] Δu

T

[0]

Δp

α α

= P–R α g

α

with: [K

(0)

] : the linear stiffness matrix

[K

(1)

] : the geometric stiffness matrix

[ H ] : the nodal pressure coupling matrix P : nodal load vector R : internal stress vector g : vector quantity representing the incompressibility constraint Experimental Elastomer Analysis

253

Appendix A: The Mechanics of Elastomers

Large Strain Viscoelasticity

Large Strain Viscoelasticity The behavior of rubber is in most cases considered to be time independent elastic. This approximation is no longer valid, if specific hysteresis effects need to be taken into account. The theory of linear visco-elasticity cannot be applied directly since there is no linear relation between the applied strain and the resulting stress. Various forms are proposed in literature to describe nonlinear visco-elasticity. In MSC.Marc, a rather simple form, based on an extension of the elastic energy function as proposed by Simo, is used. The model is based in the observation that for short time loading more energy is required then in a long term loading. Also if one loads at a high rate and keeps the deformation constant for a specific period of time, part of the elastic energy is released.

Large Strain Viscoelasticity based on Energy For an elastomeric time independent material the constitutive equation is expressed in terms of an energy function W. For a large strain viscoelastic material Simo generalized the small strain visco-elasticity material behavior to a large strain visco-elastic material using the energy function. The energy functional is now a time dependent function and is written in the following form: N n

0

W ( E ij ,Q ij ) = W ( E ij ) –

∑

n

Q ij E ij

n=1

254

Experimental Elastomer Analysis

Large Strain Viscoelasticity based on Energy

Appendix A: The Mechanics of Elastomers

where E ij are the components of the Green-Lagrange strain tensor, Q nij internal variables and W 0 the elastic strain energy density for instantaneous deformation. In MSC.Marc, it is assumed that W 0 is the energy density for instantaneous deformations is given by the third order James Green and Simpson form, or the energy function as defined by Ogden. The components of the second Piola-Kirchhoff stress then follow from: N

0

∂W ∂W S ij = --------- = ---------- – ∂E ij ∂E ij

∑ Qij n

n=1

The energy function can also be written in terms of the long term moduli resulting in a different set of internal variables T nij : N ∞

n

W ( E ij, T ij ) = W ( E ij ) +

∑

n

T ij E ij

n=1

where W ∞ is the elastic strain energy for long term deformations. Using this energy definition the stresses are obtained from: ∞

∂W ( E ) S ij = -------------------- + ∂E ij

Experimental Elastomer Analysis

N

∑ Tij n

n=1

255

Appendix A: The Mechanics of Elastomers

Large Strain Viscoelasticity based on Energy

Observing the similarity with the equations for small strain viscoelasticity the internal variables can be obtained from a convolution expression: t n

T ij =

n . n S ( τ ) exp [ – ( t – τ ) ⁄ λ ] dτ ∫ ij 0

where S nij are internal stresses following from the time dependent part of the energy functions. n S ij

n

∂W = ---------∂E ij

Let the total strain energy be expressed as a Prony series expansion: N ∞

W = W +

∑

n

n

W exp ( – t ⁄ λ )

n=1

Observing the difficulty in finding accurate expressions for the multiaxial aspect of the elastic energy in time independent rubber a further simplification is used. We assume that the energy expression for each 0 term is of similar form to the short time elastic energy W and only n 0 different by a scalar multiplier W = δW . This equation can now be rewritten as: N ∞

W = W +

∑δ

n

0

n

W exp ( – t ⁄ λ )

n=1

256

Experimental Elastomer Analysis

Large Strain Viscoelasticity based on Energy

Appendix A: The Mechanics of Elastomers

where δ n is a scalar multiplier for the energy function based on the short term values. The stress strain relation is now given by: N ∞

∑

S ij ( t ) = S ij ( t ) +

n

T ij ( t )

n=1 N ⎛ ⎞ 0 ∂W n⎟ ∂W ⎜ = ----------- = 1 – ∑ δ ---------⎜ ⎟ ∂E ij ∂E ij ⎝ n=1 ⎠ ∞

∞

S ij

t n

T ij =

n 0 . n δ S ( t ) exp [ – ( t – τ ) ⁄ λ ] dτ ij ∫ 0

Analogue to the derivation for small strain visco-elasticity, a recurrent relation can be derived expressing the stress increment as a function of the strain increment and the internal stresses at the start of the increment: N ∞

ΔS ij ( t m ) = ΔS ij ( t m ) +

∑

n

ΔS ij ( t m )

n=1 ∞

∞

∞

ΔS ij ( t m ) = S ij ( t m ) – S ij ( t m ) N n

n

n

n

ΔS ij ( t m ) = β ( h ) [ S ij ( t m ) – S ij ( t m – h ) ] –

∑α

n

n

( h )S ij ( t m – h )

n=1

Experimental Elastomer Analysis

257

Appendix A: The Mechanics of Elastomers

Large Strain Viscoelasticity based on Energy

The functions α and β are a function of the time step h in the time interval [ t m – 1, t m ] : n

α ( h ) = 1 – exp ( – h ) ⁄ λ

n

n

λ β ( h ) = α ( h ) ⋅ ----h n

n

The equations above are based on the long term moduli. Since in the MSC.Marc program always the instantaneous values of the energy function are given on the MOONEY option, the equations are reformulated in terms of the short time values of the energy function: N

ΔS ij ( t m ) = 1 –

∑

n

[ 1 – β ( h ) ]δ

n

0

0

S ij ( t m ) – S ij ( t m – h )

n=1 N

–

∑

n n

α S ij ( t m – h )

n=1 n

n

n

0

n

ΔS ij ( t m ) = β ( h )δ [ S ij ( t m ) – S ij ( t m – h ) ] n

n

– α ( h )S ij ( t m – h )

It is assumed that the visco-elastic behavior in MSC.Marc acts only on the deviatoric behavior. The incompressible behavior is taken into account using special Herrmann elements.

258

Experimental Elastomer Analysis

Illustration of Large Strain Viscoelastic Behavior

Appendix A: The Mechanics of Elastomers

Illustration of Large Strain Viscoelastic Behavior A large strain visco-elastic material is characterized by the following time dependent elastic energy function: N ∞

∑

W(t) = W +

n

n

W exp ( – t ⁄ λ )

n=1 ∞

n

where W is the energy function for very slow processes. W is an extra amount of energy necessary for time dependent processes. To each amount W n , a characteristic time is associated. n

At time zero (or for time processes: t < λ ), the elastic energy reduces to: N ∞

0

∑W

W(0) = W = W +

n

n=1

If we assume that the energy function for each time dependent part is different only by a scalar constant: n

n

W = δ W

0

the equations reduce to: N 0

∞

W = W +W

0

∑ n=1

Experimental Elastomer Analysis

δ

n

or

W

∞

N ⎛ ⎞ n 0 = ⎜1 – ∑ δ ⎟ W ⎜ ⎟ ⎝ n=1 ⎠

259

Appendix A: The Mechanics of Elastomers

Illustration of Large Strain Viscoelastic Behavior

The time dependent energy is then given by: N 0

W(t) = W – W

N

∑δ

0

n

n=1

+W

0

∑δ

n

n

exp ( – t ⁄ λ )

n=1

N 0

= W 1–

∑

n

n

δ ( 1 – exp ( – t ⁄ λ ) )

n=1

If we restrict ourselves for simplicity of the discussion to the case N = 1 we have: W

∞

= ( 1 – δ )W 0

0 n

W ( t ) = W [ 1 – δ ( 1 – exp ( – t ⁄ λ ) ) ]

260

Experimental Elastomer Analysis

APPENDIX B

Elastomeric Damage Models

Under repeated application of loads, elastomers undergo damage by mechanisms involving chain breakage, multi-chain damage, micro-void formation, and micro-structural degradation due to detachment of filler particles from the network entanglement. Two types of phenomenological models namely, discontinuous and continuous, exists to simulate the phenomenon of damage.

Experimental Elastomer Analysis

261

Appendix B: Elastomeric Damage Models

Discontinuous Damage Model

Discontinuous Damage Model Discontinuous damage denotes the phenomenon where progressively increasing strain levels, the material regains its original stiffness (as in a single pull) until subsequent reloading as shown in the stress-strain plot below.

Strain History For Discontinuous Damage 1.0

Engineering Strain

0.8

0.6

0.4

0.2

0.0 0.0

0.5 Time

1.0

The higher the maximum attained strain, the larger is the loss of stiffness upon reloading. Hence, there is a progressive stiffness loss with increasing maximum strain amplitude. Also, most of the stiffness loss takes place in the few earliest cycles provided the maximum strain level is not increased. This phenomenon is found in both filled as well as natural rubber although the higher levels of carbon black particles increase the hysteresis and the loss of stiffness.

262

Experimental Elastomer Analysis

Discontinuous Damage Model

Appendix B: Elastomeric Damage Models

The free energy, W, can be written as W = K ( α, β )W

0

where W 0 is the nominal (undamaged) strain energy function, and 0

α = max ( W )

determines the evolution of the discontinuous damage. The reduced form of Clausius-Duhem dissipation inequality yields the stress as: 0

∂W S = 2K ( α ,β ) ---------∂C

Mathematically, the discontinuous damage model has a structure very similar to that of strain space plasticity. Hence, if a damage surface is defined as: Φ = W–α≤0

The loading condition for damage can be expressed in terms of the KuhnTucker conditions: Φ≤0

· α≥0

· αΦ = 0

The consistent tangent can be derived as: 2

0

0

0

∂K ∂W ∂W ∂ W C = 4 K --------------- + ----------0 ---------- ⊗ ---------∂C∂C ∂W ∂C ∂C

Experimental Elastomer Analysis

263

Appendix B: Elastomeric Damage Models

Discontinuous Damage Model

The parameters required for the damage model can be obtained using the experimental data fitting option MSC.Marc Mentat. To calibrate the Kachanov factor for the discontinuous damage mode, one measures at a stretch amplitude λ 0 , the stress level. A loading history is thus: λ

5

σ 3

4

λ0 1

2

3

2 1

time

λ0

λ

The model is hyperelastic and assumes that unloading from say state 2 to the undeformed state, and subsequent reloading, occur along the same path. Viscoelastic effects tend to cause the reloading path to reside above the unloading path. Secondary damage effects tend to cause the reloading path to reside below the unloading path. We will now examine the stressstrain plot closely.

264

Experimental Elastomer Analysis

Discontinuous Damage Model

Appendix B: Elastomeric Damage Models

A procedure to get the discontinuous damage increasing strain table is shown below. The bottom curve is used to compute the damage parameters in MSC.Marc Mentat using a Prony series. σ σ

σ

na

3a

2a

σ1 σ2 σ

1

σ

-1 = --σ

σ1 = σ

1

w σ

n

2

w

---σ

1a

1 --- σ ε ,S i = ≅ 2 ia ia ia

1, 2, 3…n

≅ 1--2- σ ia ε ia ,S i =

1, 2, 3…n

ia

1

σ

n ---σ 1

⎛ w1 a ⎞ ⎜ --------- – 1⎟ ⎝ w1 ⎠

⎛ w 2a ⎞ ⎜ ---------- – 1⎟ ⎝ w2 ⎠

Experimental Elastomer Analysis

⎛ wn a ⎞ ⎜ ---------- – 1⎟ ⎝ wn ⎠

265

Appendix B: Elastomeric Damage Models

Discontinuous Damage Model

The results from the analysis show how the damage model works below.

Engineering Stress [Mpa]

0.4165

0 0

266

Engineering Strain [1]

0.6

Experimental Elastomer Analysis

Continuous Damage Model

Appendix B: Elastomeric Damage Models

Continuous Damage Model The continuous damage model can simulate the damage accumulation for strain cycles for which the values of effective energy is below the maximum attained value of the past history as shown below: Tensile Data Continuous Damage

Engineering Stress [Mpa]

1.0

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4 0.6 0.8 Engineering Strain [1]

1.0

This model can be used to simulate fatigue behavior. More realistic modeling of fatigue would require a departure from the phenomenological approach to damage. The evolution of continuous damage parameter is governed by the arc length of the effective strain energy as: t

β =

∫

∂- 0 -----W ( s′ ) ds′ ∂s′

0

Experimental Elastomer Analysis

267

Appendix B: Elastomeric Damage Models

Continuous Damage Model

Hence, β accumulates continuously within the deformation process. The Kachanov factor K ( α, β ) is implemented in MSC.Marc through both an additive as well as a multiplicative decomposition of these two effects as: 2 ∞

K ( α, β ) = d +

α

∑ dn n=1 2

∞

K ( α, β ) = d +

∑ n=1

α exp ⎛ – ------⎞ + ⎝ η n⎠

2

∑ m=1

β β d m exp ⎛ – ------⎞ ⎝ λ m⎠

α + δ n β⎞ ⎛ d n exp – ------------------⎝ ηn ⎠

You specify the phenomenological parameters d αn , dβn , η n, λ m, d βm, δn , while ∞

d is enforced to be such that at zero damage, K assumes a value of 1.

To calibrate the Kachanov factor for the continuous damage mode, one applies the following loading history to get the input file shown. λ 1

2

3

σ σ1

4

σ2

1

W1 2

time

For the MSC.Marc Mentat implementation, the user needs to know the value of the Free Energy Function at point 1, W1.

268

1 σ1

λ

2 σ2

Experimental Elastomer Analysis

Continuous Damage Model

Appendix B: Elastomeric Damage Models

Below is a sample of the continuous damage simulation using a 1-term Ogden model superimposed onto the original data. Tensile Simulation - Continuous Damage 1-Term Ogden and Original Data

Engineering Stress [Mpa]

1.0

1-Term Ogden Original Data

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4 0.6 Engineering Strain [1]

0.8

1.0

The above damage model is available for deviatoric behavior and is flagged by means of the OGDEN and DAMAGE model definition options. If, in addition, viscoelastic behavior is desired, the VISCELOGDEN option can be included. Finally, a user subroutine UELDAM can be used to define damage functions different from the above.

Experimental Elastomer Analysis

269

Appendix B: Elastomeric Damage Models

270

Continuous Damage Model

Experimental Elastomer Analysis

APPENDIX C

Aspects of Rubber Foam Models

Elastomeric foams (e.g., rubber foam) are widely used in industry. They exhibit linear elasticity at low stress followed by a long collapse plateau, truncated by a regime of densification in which the stress rises steeply. Furthermore, when loading is compressive, the plateau is associated with the collapse of the cells by elastic buckling. Unlike conventional rubber, foam can undertake large amounts of volumetric compression.

Experimental Elastomer Analysis

271

Appendix C: Aspects of Rubber Foam Models

Theoretical Background

Theoretical Background Foams and convention rubber behave differently in tension and compression, with foams have a much larger difference as shown in the figure below:

Cell Wall Alignment E

σ cr Plateau (Elastic Buckling) Densification

Elastomer foams are modeled as a compressible Ogden Model with the strain energy density of: N

W =

∑ n=1

μ α α α -----n- [ λ 1 n + λ 2 n + λ 3 n – 3 ] + αn

N

∑ n=1

μn β ----- ( 1 – J n ) βn

μ i , α i and β i are material constants and J is defined as J = λ1 λ2 λ3 .

272

Experimental Elastomer Analysis

Theoretical Background

Appendix C: Aspects of Rubber Foam Models

The last term of the strain energy equation is the volumetric change, which can be as high as 90% engineering strain for foams in compression. For β i = 0 , there are no lateral effects. For the general theory of isotropic elasticity to be consistent with the classical theory in the linear approximation, the strain-energy function W = W ( λ 1 ,λ 2 ,λ 3 ) must satisfy: W ( 1 ,1 ,1 ) = 0 ∂W ( 1 ,1 ,1 ) = 0 ∂ λi

,i = 1, 2, 3

2

∂W ( 1 ,1 ,1 ) = λ + 2μδ ij ∂ λ i ∂λ j

,( i, j ) = 1, 2, 3

Where λ , and μ are Lame’s constants. The initial bulk modulus K and the shear modulus G can be derived from the above as: 1 K = --- ∑ μ i ( α i – 3β i ) 3 i

1 G = --- ∑ μ i α i 2 i

The initial Poisson’s ratio can be derived from above as: ν =

–β i ----------------------∑ ( αi – 2βi -) i

Experimental Elastomer Analysis

273

Appendix C: Aspects of Rubber Foam Models

Theoretical Background

Blatz and Ko proposed a material model for rubber foams with the strain energy function defined as: – 2ν

--------------μf 1 – 2ν ⎛ 1 – 2ν ⎞ W = ----- I 1 – 3 + --------------- ⎜ I 3 – 1⎟ + 2 ν ⎝ ⎠ – 2ν

--------------⎛ μ(1 – f) 1 – 2ν ⎛ 1 – 2ν ⎞ ⎞ – 1⎟ ⎟ ⎜ ------------------- I 2 – 3 + --------------- ⎜ I 3 2 ν ⎝ ⎝ ⎠ ⎠

where: 2

2

2

I1 = λ1 + λ2 + λ3 –2

–2

–2

I2 = λ1 + λ2 + λ3 2 2 2

I3 = λ1 λ2 λ3

By using the two-term MSC.Marc foam model, the generalized compressible Ogden model can be reduced to the Blatz-Ko model. For temperature effects, the thermal principal stretches follow the temperature and the isotropic thermal expansion coefficient. The thermal principal stretches are defined as: λ

274

T i

= 1 + ΔTα

,i = 1, 2, 3

Experimental Elastomer Analysis

Theoretical Background

Appendix C: Aspects of Rubber Foam Models

The total Lagrange method with conventional elements is used in MSC.Marc for the foam model. The virtual work equation can be formulated as:

∫ Sij δEij dV – ∫ Qi δui dV – ∫- T i δui dA

V

V

= 0

A

where S ij , is the second Piola-Kirchhoff stress tensor, E ij is the GreenLagrange strain tensor, Q i is the body force per unit undeformed volume, and T i is the prescribed surface tractions per unit undeformed area. All elements in MSC.Marc except Herrmann elements and be used in the foam model.

Experimental Elastomer Analysis

275

Appendix C: Aspects of Rubber Foam Models

Measuring Material Constants

Measuring Material Constants Currently, only uniaxial testing is available in the experimental curve fitting option in MSC.Marc Mentat. An engineering stress, σ 1 , and engineering strain with corresponding stretch, λ 1 , table can then be constructed from specimen measurements. The material constants are found to satisfy the following two equations: σ1 =

μi ⎛ αi β ⎞ ---∑ λ1- ⎝ λ 1 – J i⎠ i

0 =

∑ μi i

J⎞ ⎛ ---⎝ λ 1⎠

1 --- ( α i – 1 ) 2

–J

⎛ – 1---⎞ ⎝ β 2⎠ i

λ1

1--2

The specimen should be measured at different load levels. This makes a table of stress, strain, and cross sectional area for these load levels.

276

Experimental Elastomer Analysis

APPENDIX D

Biaxial & Compression Testing

Equibiaxial Stretching of Elastomeric Sheets, An Analytical Verification of Experimental Technique by: Jim Day, GM Powertrain Kurt Miller, Axel Products, Inc.

Experimental Elastomer Analysis

277

Appendix D: Biaxial & Compression Testing

Abstract

Abstract Constitutive models for hyperelastic materials may require multiple complimentary strain states to get an accurate representation of the material. One of these strain states is pure compression. Uniaxial compression testing in the laboratory is inaccurate because small amounts of friction between the specimen and the loading fixture cause a mixed state of compressive, shear, and tensile strain. Since uniaxial compression can also be represented by equibiaxial tension, a test fixture was developed to obtain compressive strain by applying equibiaxial tensile loads to circular sheets while eliminating the errors due to friction. This paper outlines an equibiaxial experiment of elastomeric sheets while providing analytical verification of its accuracy.

Figure 1. Biaxial Stretching Apparatus

278

Experimental Elastomer Analysis

Introduction

Appendix D: Biaxial & Compression Testing

Introduction Constitutive models for hyperelastic materials are developed from strain energy functions and require nominal stress vs. nominal strain data to fit most models available. In general, it is desirable to represent the three major strain states which are: uniaxial tension, uniaxial compression, and pure shear. If compressibility is a concern, then bulk compressibility information is also recommended. The uniaxial tension strain state is easily obtained and the pure shear test can be performed using a planar tension test with excellent, repeatable accuracy. However, the uniaxial compression test is difficult to perform without introducing other strain states that will affect the accuracy. The main cause of the inaccuracy is the friction between the specimen and the loading platens. The friction can also vary as the compressive load (normal force) increases. To characterize the friction effect, an analysis of a standard ASTM D395, type 1 button under uniaxial compression loading was performed. A plot of compressive stress vs. compressive strain with varying coefficients of friction shows the variation caused by friction (see “Attachment A: Compression Analysis” on page 292). The analysis of the standard button indicates that for small levels of friction the deviation from the pure uniaxial compressive strain state causes significant errors. An equibiaxial testing fixture is examined to determine if a pure compressive strain could be obtained accurately because an equibiaxial tension state of strain is equivalent to an uniaxial compressive strain.

Experimental Elastomer Analysis

279

Appendix D: Biaxial & Compression Testing

Introduction

The equibiaxial straining apparatus described in this paper also has other advantages with respect to specimen availability and load control. These advantages include: 1. Achieving a strain condition equivalent to simple compression while avoiding the inherent experimental errors associated with compression. 2. Being able to perform strain and load control experiments as well as look at equilibrium behavior. 3. Testing on readily available test slabs. 4. Performing a test at the loading rates equivalent to tension and shear loading rates. Several other experimental approaches to the biaxial straining of elastomers have been developed. In general, two approaches have been used. The first involves the expansion of a thin elastomer membrane using air pressure. Strain control is difficult to obtain with this procedure making it difficult to create conditions that compliment the other strains states required to get a full set of data for fitting hyperelastic constitutive equations. The other problem is that the thickness of the sheets needs to be much thinner than the typical sheet thickness that is created. The second approach involves the gripping of a rectangular specimen around the perimeter and stretching the specimen with multiple arms or cable bearing systems. This approach has been used with great success by several investigators. Difficulties arise with the measurement of strain and the calculation of stress. The advantage of this approach is that while somewhat complex, it 280

Experimental Elastomer Analysis

Overall Approach

Appendix D: Biaxial & Compression Testing

allows the investigator to examine elastomer deformation in unequal biaxial deformation states. Since the objectives herein do not involve the need for unequal biaxial straining, the mechanical aspects of the experimental approach can be greatly simplified and the relations between forces and stresses in the specimen can be ascertained with greater certainty by restricting the apparatus to equal biaxial straining.

Overall Approach The overall approach is to strain a circular specimen radially. Constant stress and strain around the periphery of the disk will create an equibiaxial state of stress and strain in the disk independent of thickness or radial position.

Experimental Elastomer Analysis

281

Appendix D: Biaxial & Compression Testing

The Experimental Apparatus

The Experimental Apparatus Applying Radial Forces

In the apparatus, 16 small grips mechanically attach to the perimeter of an elastomer disk using spring force attachment. The grips are moved radially outward by pulling with thin flexible cables which are redirected around pulleys to a common loading plate (Figure 1 on page 278). When the loading plate is moved all of the attachment points move equally in a radial direction and a state of equal biaxial strain is developed in the center of the disk shaped specimen, Figure 2.

Figure 2. Biaxial Apparatus Schematic

282

Experimental Elastomer Analysis

The Experimental Apparatus

Appendix D: Biaxial & Compression Testing

The Specimen

The actual shape of the specimen is not a simple disk as shown in Figure 3. There are radial cuts introduced into the disc specimen so that there are no tangential forces between the grips. This is necessary because the grips are not attached to the outer edge of the specimen. They are attached to the top and bottom surfaces of the specimen which does not allow material to flow within the grip. Small holes are introduced at the ends of the radial cuts so that the specimen is less likely to tear.

Figure 3. Biaxial Test Specimen Outline

Experimental Elastomer Analysis

283

Appendix D: Biaxial & Compression Testing

The Experimental Apparatus

Strain Measurement

The relationship between grip travel and actual straining in the center area of the specimen is not known with certainty because of the unknown strain field around the grips and the compliance that may exist in the loading cables and the material flowing from the grips. To determine the strain, a laser non contacting extensometer is used to measure the strain on the surface of the specimen away from the grips. Force Measurement

The total force transmitted by the 16 grips to the common loading plate is measured using a strain gage load cell. Relating Force Measured to Stress: The nominal equibiaxial stress contained inside the specimen inner diameter (Di) is calculated as follows: σ = F ⁄ ( ΠD i t )

where: Di = Diameter as measured between punched holes F = Sum of radial forces t = Original thickness σ = Engineering stress

284

Experimental Elastomer Analysis

Analytical Verification

Appendix D: Biaxial & Compression Testing

Analytical Verification Once the closed form solution has shown that a circular disk pulled with a uniform circumferential load produces a biaxial stress and strain field we then need to verify that pulling the disk from 16 discrete grip locations is an acceptable approximation. The following analytical procedure will examine the effects of the boundary conditions imposed by the experimental approach on the ideal closed form solution. The experimental aspects of concern are: A. The specimen is not gripped continually around the circumference. B. Cuts are introduced between the grips that alter the strain field. C. The relationship between force and stress is based on the “inside” diameter indicated in Figure 3. First finite element analysis is used to verify the closed form solution on a representative specimen model. The following steps will show how the proposed specimen will be compared to the closed form solution.

Experimental Elastomer Analysis

285

Appendix D: Biaxial & Compression Testing

Analytical Verification

Closed Form Solution Comparison

The disk specimen finite element model used to verify the closed form solution is shown in Figure 4. Radial loads are applied at every node around the perimeter.

Figure 4. FEA model of uncut specimen with radial loads applied at every perimeter node.

The nominal finite element stress calculated within each element was compared to the stress calculated with the formula below and found to be equivalent. σ = F ⁄ ( ΠDt )

where: D = Original outside diameter F = Sum of radial forces t = Original thickness σ = Engineering stress This formula can now be used in a testing environment since all the parameters are known.

286

Experimental Elastomer Analysis

Analytical Verification

Appendix D: Biaxial & Compression Testing

Analysis of the Experimental Condition

The next step needs to show that using a cut specimen with 16 grips (FEA model shown in Figure 5) will accurately represent the “ideal” loading condition of the previous finite element analysis.

Figure 5. FEA model of specimen with slits and punched holes, radial loads applied at 16 grip locations.

The original outside diameter used in the above stress formula will be equal to the diameter measured at the inside edges of the punched holes at the ends of the radial slits between the grips. For the proposed configuration, this dimension is 50 mm.

Experimental Elastomer Analysis

287

Appendix D: Biaxial & Compression Testing

Analytical Verification

A deformed shape sequence of this configuration under loads is shown in Figure 6.

Figure 6. Specimen Deformed Shape

A nominal stress vs. nominal strain comparison of this configuration vs. FEA “closed form” results is shown for two hyperelastic material representations.

288

Experimental Elastomer Analysis

Analytical Verification

Appendix D: Biaxial & Compression Testing

The first (Figure 7) represents a simple 2nd order polynomial approximation and the second (Figure 8) represents an Ogden 5-term approximation. Both show excellent correlation between the proposed test configuration and the theoretical results.

Figure 7. 2nd Order Polynomial Fit

Figure 8. 5-term Ogden Fit

Experimental Elastomer Analysis

289

Appendix D: Biaxial & Compression Testing

Analytical Verification

Summary

The equibiaxial experiment as proposed in this paper does an excellent job of obtaining the pure strain state required for hyperelastic constitutive models. The error due to the boundary condition approximations are small but consistent as opposed to the uniaxial compression test where the experimental error depends on friction which is unknown and varies as a function of the test material and the normal force. The testing done in this manner can provide excellent consistent and accurate compression strain states while using standard ASTM slabs and a minor amount of specimen preparation to perform.

290

Experimental Elastomer Analysis

References

Appendix D: Biaxial & Compression Testing

References 1. Kao, B. G. and Razgunas, L.,”On the Determination of Strain Energy Functions of Rubbers”, SAE Paper 860816, (1986) 2. Treloar, L. R. G., “Stresses and Birefringence in Rubber Subjected to General Homogeneous Strain,” Proc. Phys. Soc., London, 60, 135-144 (1948) 3. Rivlin, R. S. and Saunders, D. W., “Large Elastic Deformations of Isotropic Materials, VII, Experiments on the Deformation of Rubber,” Phil. Trans. Roy. Soc., London, 243 (Pt. A), 251-288 (1951) 4. Zapas, L. J., “Viscoelastic Behaviour Under Large Deformations,” J. Res. Natl. Bureau of Standards, 70A (6), 525-532 (1966) 5. Blatz, P. J. and Ko, W. L., “Application of Finite Elastic Theory to the Deformation of Rubbery Materials,” Trans. Soc. Rheol., 6, 223-251 (1962) 6. Ko, W. L., “Application of Finite Elastic Theory to the Behavior of Rubberlike Materials.” PhD Thesis, California Ins. Tech., Pasadena, California (1963) 7. Hutchinson, W. D., Becker, G. W. and Landel, R. F., “Determination of the Strain Energy Function of Rubberlike Materials,” Space Prams Summary No. 37-31, Jet Propulsion Laboratory, Pasadena, California, IV, 34-38 (Feb. 1965) 8. Becker, G. W., “On the Phenomenological Description of the Nonlinear Deformation Behavior of Rubber-like High Poymers,” Jnl Polymer Sci., Part C (16), 2893-2903 (1967) 9. Obata, Y., Kawabata, S. and Kawai, H., “Mechanical Properties of Natural Rubber Vulcanizates in Finite Deformation,” J. Polymer Sci. (Part A-2), 8, 903-919 (1970) 10. Burr, A., Mechanical Analysis and Design, Elsevier, New York, 1981, p.315 11. Timoshinko, S.P., Goodier, J.N., Theroy of Elasticity, p 69, 3rd Ed, McGraw hill, New York, 1951 12. ABAQUS v5.8 User’s Manual Vol. 1, Section10.5.1 Experimental Elastomer Analysis

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Attachment A: Compression Analysis

Attachment A: Compression Analysis The effect of friction between the compression loading platens and the specimen under test is examined analytically. The ASTM D395, type 1 button which is used in ASTM 575 Standard Test Methods for Rubber Properties in Compression was modeled and analytically strained. The coefficient of friction was altered to see the effect of friction on the resulting stress-strain data.

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Attachment A: Compression Analysis

Appendix D: Biaxial & Compression Testing

A coefficient of friction value of zero corresponds to a perfect state of simple uniaxial compression (Figures A1 and A2). From the analysis, one can conclude even very small levels of friction significantly effect the measured stiffness and this effect is apparent at both low and high strains.

Figures A1 and A2 Friction Effects on Stress

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Attachment A: Compression Analysis

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APPENDIX E

Xmgr – a 2D Plotting Tool

ACE/gr is a 2D plotting tool for X Window System. It uses an Motif based user interface, which is the reason why it’s also known as Xmgr. For more detail see: http://plasma-gate.weizmann.ac.il/Xmgr/

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Appendix E: Xmgr – a 2D Plotting Tool

Features of ACE/gr

Features of ACE/gr • User defined scaling, tick marks, labels, symbols, line styles, colors. • Batch mode for unattended plotting. • Read and write parameters used during a session. • Regressions, splines, running averages, DFT/FFT, cross/autocorrelation, . . . • Support for dynamic module loading. • Hardcopy support for PostScript, HP-GL, FrameMaker, and InterLeaf formats. An example of ACE/gr is shown below:

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Using ACE/gr

Appendix E: Xmgr – a 2D Plotting Tool

Using ACE/gr The use of ACE/gr or xmgr will be to read in from a file existing xy data (Block Data) and overlay plots. To read in block data click on File, and select Read, then Block Data. This brings up the file browser below:

Here you can select the data you have stored from test data or MSC.Marc Mentat history plots. Let’s suppose that we have two Block Data files that look like: file1

file2

0

1

0

1.1382

1.66667

3.77778

1.66667

3.39864

3.33333

12.1111

3.33333

10.1483

5

26

5

30.3025

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Appendix E: Xmgr – a 2D Plotting Tool

Using ACE/gr

Using the file browser, select file1 and identify from which column you want x and y to come from in the menu below:

Pick x column Pick y column

Clicking Accept will bring in the first curve then autoscale by picking the icon below:

Pick this to Auto Scale the plot.

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Using ACE/gr

Appendix E: Xmgr – a 2D Plotting Tool

Here is the resulting plot:

Y-Axis Area

Title Area

X-Axis Area To place symbols on the plot, simply click on a curve and select a symbol desired. To place a Title or Axis Labels, click in the Title area or Axis area and fill in the menu.

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Appendix E: Xmgr – a 2D Plotting Tool

ACE/gr Miscellaneous Plots

ACE/gr Miscellaneous Plots Multiple Graphs:

Menus:

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ACE/gr Miscellaneous Plots

Appendix E: Xmgr – a 2D Plotting Tool

Axis Summary:

Symbol Summary:

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Appendix E: Xmgr – a 2D Plotting Tool

ACE/gr Miscellaneous Plots

Log Plots:

Bar Charts:

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APPENDIX F

Notes and Course Critique

The purpose of this appendix is to provide pages for notes and the course critique.

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Notes

Notes

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Notes

Appendix F: Notes and Course Critique

Notes

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Notes

Notes

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Course Critique

Appendix F: Notes and Course Critique

Course Critique Please use this form to provide feedback on your training program. Your comments will be reviewed, and when possible included in the remainder of your course. Lecture Materials

excellent

average

poor

_ _ _

_ _ _

_ _ _

Is the level of technical detail appropriate? Are the format and presentation correctly paced? Are the discussions clear and easy to follow? What changes do you suggest?

What additional information would you like? ____________________________________________________________________________ ____________________________________________________________________________ Workshop

excellent

average

poor

_ _ _

_ _ _

_ _ _

Are the available problems relevant? Was the technical assistance prompt and clear? Was the equipment satisfactory? What changes do you suggest?

What additional information would you like? ____________________________________________________________________________ ____________________________________________________________________________ Laboratory

excellent

average

poor

_ _ _

_ _ _

_ _ _

Are the available specimens relevant? Was the technical assistance prompt and clear? Was the equipment satisfactory? What changes do you suggest?

What additional information would you like? ____________________________________________________________________________ ____________________________________________________________________________ General How would you change the balance of time spent on theory, workshop, and laboratory

_ no change

_ more theory

_ more workshop

_ more laboratory

Your Name:______________________________________ Date: ___________________ Instructor(s):_____________________________________

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Course Critique

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