msc marc training mar103

Experimental Elastomer Analysis

MSC.Software CorporationMA*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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ContentsExperimental Elastomer Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Table of Contents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3CHAPTER 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

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

CHAPTER 2 The Macroscopic Behavior of Elastomers . . . . . . . . . . . . . 21Microscopic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Temperature Effects, Tg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Time Effects, Viscoelasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Curing Effects (Vulcanization) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Damage, Early Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Damage, Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Damage, Chemical Causes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Deformation States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

CHAPTER 3 Material Models, Historical Perspective . . . . . . . . . . . . . . . 31Engineering Materials and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Neo-Hookean Material Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Neo-Hookean Material Extension Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . 35Neo-Hookean Material Shear Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Neo-Hookean Material Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38A Word About Simple Shear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402-Constant Mooney Extensional Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Other Mooney-Rivlin Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Ogden Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Foam Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Model Limitations and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Viscoelastic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Determining Model Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Table of ContentsExperimental Elastomer Analysis 3

ContentsCHAPTER 4 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Lab Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Basic Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Measuring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58What about Shore Hardness? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Testing the Correct Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Tensile Testing in the Lab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Compression Testing in the Lab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Equal Biaxial Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Compression and Equal Biaxial Strain States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Volumetric Compression Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Planar Tension Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Viscoelastic Stress Relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Dynamic Behavior Testing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Friction Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Data Reduction in the Lab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Model Verification Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Testing at Non-ambient Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Loading/Unloading Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Test Specimen Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Fatigue Crack Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Experimental and Analysis Road Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

CHAPTER 5 Material Test Data Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . 83Major Modes of Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Confined Compression Test (UniVolumetric) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Hydrostatic Compression Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Summary of All Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89General Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Mooney, Ogden Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Visual Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Material Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Future Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Adjusting Raw Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Consider All Modes of Deformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98The Three Basic Strain States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Curve Fitting with MSC.Marc Mentat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004 Experimental Elastomer Analysis

ContentsCHAPTER 6 Workshop Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Some MSC.Marc Mentat Hints and Shortcuts . . . . . . . . . . . . . . . . . . . . . . . . . . 108Model 1: Uniaxial Stress Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Model 1: Uniaxial Curve Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Model 1C: Tensile Specimen with Continuous Damage . . . . . . . . . . . . . . . . . . 133Model 1: Realistic Uniaxial Stress Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . 145Model 2: Equi-Biaxial Stress Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Model 2: Equi-Biaxial Curve Fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153Model 2: Realistic Equal-Biaxial Stress Specimen. . . . . . . . . . . . . . . . . . . . . . . 165Model 3: Simple Compression, Button Comp. . . . . . . . . . . . . . . . . . . . . . . . . . . 168Model 4: Planar Shear Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176Model 4: Planar Shear Curve Fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180Model 4: Realistic Planar Shear Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195Model 5: Viscoelastic Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198Model 5: Viscoelastic Curve Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200Model 6: Volumetric Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

CHAPTER 7 Contact Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217Definition of Contact Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218Control of Rigid Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220Contact Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221Bias Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222Deformable-to-Deformable Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223Potential Errors due to Piecewise Linear Description: . . . . . . . . . . . . . . . . . . . . 224Analytical Deformable Contact Bodies: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224Contact Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225Symmetry Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226Rigid with Heat Transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227Contact Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229Contact Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231Exclude Segments During Contact Detection. . . . . . . . . . . . . . . . . . . . . . . . . . . 232Effect Of Exclude Option:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233Contacting Nodes and Contacted Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234Friction Model Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235Coulomb ArcTangent Friction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236Coulomb Bilinear Friction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237Stick-Slip Friction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238Glued Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239Release Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241Experimental Elastomer Analysis 5

Interference Check / Interference Closure Amount . . . . . . . . . . . . . . . . . . . . . . 241

ContentsForces on Rigid Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

APPENDIX A The Mechanics of Elastomers. . . . . . . . . . . . . . . . . . . . . . 245Deformation States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246General Formulation of Elastomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253Large Strain Viscoelasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254Large Strain Viscoelasticity based on Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 254Illustration of Large Strain Viscoelastic Behavior . . . . . . . . . . . . . . . . . . . . . . . 259

APPENDIX B Elastomeric Damage Models . . . . . . . . . . . . . . . . . . . . . . 261Discontinuous Damage Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262Continuous Damage Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

APPENDIX C Aspects of Rubber Foam Models . . . . . . . . . . . . . . . . . . . 271Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272Measuring Material Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

APPENDIX D Biaxial & Compression Testing . . . . . . . . . . . . . . . . . . . . 277Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279Overall Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281The Experimental Apparatus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282Analytical Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291Attachment A: Compression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

APPENDIX E Xmgr a 2D Plotting Tool. . . . . . . . . . . . . . . . . . . . . . . . . 295Features of ACE/gr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296Using ACE/gr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297ACE/gr Miscellaneous Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

APPENDIX F Notes and Course Critique . . . . . . . . . . . . . . . . . . . . . . . . 303Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306Course Critique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3076 Experimental Elastomer Analysis

Ex

CHAPTER 1 Introduction

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.perimental Elastomer 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 Analysis 7

Chapter 1: Introduction Course Objective: FEA & LaboratoryCourse Objective: FEA & LaboratoryLeft Brain

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n 2n 3

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N

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W G 12--- I1 3( )

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Right BrainLaboratory

ExperimentalSubjectiveIntuitive8 Experimental Elastomer Analysis

Course Objective: FEA & Laboratory Chapter 1: IntroductionCourse ObjectiveDiscuss the TEST

CURVE FITANALYSIS

cycle specific to rubber and elastomers.

Limit scope to material models such as Mooney-Rivlin and Ogden form strain energy models.

Test Material Specimen

Material Model (curve fit)

Test Part

?Correlation

? Analyze Part

Analyze SpecimenExperimental Elastomer Analysis 9

Chapter 1: Introduction Course Objective: FEA & LaboratoryCourse Objective (cont.)Some important topics covered are:

What tests are preferred and why?

Why arent 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: IntroductionCourse Schedule

DAY 1Begin End Topic Chap.9:00 10:15 Introduction,

Macroscopic Behavior of Elastomers1, 2, 3

10:30 12:00 Laboratory Orientation 412:00 1:00 Lunch1:00 3:00 Tensile Testing3:15 5:00 Tensile Test Data Fitting 5

FEA of Tensile Test Specimen 65:00 Adjourn

DAY 2 - Chapter 6 + LabBegin End Topic9:00 10:30 Equal Biaxial Testing, Compression, Volumetric

Equi-Biaxial Test Data Fitting, Comp., Volumetric10:45 12:00 FEA of Biaxial Specimen, Comp., Volumetric12:00 1:00 Lunch1:00 3:00 Planar Shear Testing3:15 5:00 Planar Shear Test Data Fitting

Data Fitting with All Test ModesFEA of Planar Test Specimen

5:00 AdjournExperimental Elastomer Analysis 11

Chapter 1: Introduction Course ScheduleCourse Schedule (cont.)

Keep Involved:

Tell Me and Ill ForgetShow Me and Ill Remember

Involve Me and Ill Understand

DAY 3Begin End Topic Chap.9:00 10:30 Viscoelastic Testing

Viscoelastic Data Fitting 610:45 12:00 FEA of Viscoelastic Test Specimen12:00 1:00 Lunch1:00 3:00 Contact and Case Studies

Specimen Test, FEA, Part Test Correlation

7

3:15 5:00 Concluding Remarks 5:00 Adjourn12 Experimental Elastomer Analysis

About MSC.Marc Products Chapter 1: IntroductionAbout MSC.Marc ProductsMSC.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=1Experimental 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: IntroductionData Measurement and AnalysisExperiment

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 AnalysisData Measurement and Analysis (cont.)Linear Material, How is Youngs modulus, E, measured?

Tension/Compression

Torsion

Bending

Wave Speed

Do you expect all of these Es to be the same for the same material?

E P AL( ) L-------------------

=

E 2 1 +( ) Tc J-------------

=

E PL3

3I---------=

E c2=

T ,

P ,

P L,16 Experimental Elastomer Analysis

Typical Properties of Rubber Materials Chapter 1: IntroductionTypical Properties of Rubber MaterialsProperties:

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 MaterialsTypical Loading of Rubber Materials (cont.)

0.0

2.0

4.0

6.0

210 3 4 5 6 7Engineering Strain

Experiment

Theory

Engine

eringSt

ress

[MPa

]

0.0 0.1 0.2 0.3 0.4 0.50.0

0.2

0.4

0.6

0.8

1.0

1.2

Engine

eringSt

ress

[MPa

]

Engineering Strain18 Experimental Elastomer Analysis

Important Application Areas Chapter 1: IntroductionImportant 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 Important Application Areas20 Experimental Elastomer Analysis

Ex

CHAPTER 2 The Macroscopic Behavior of

Time (strain-rate)

Temperature Cure History (cross-link density) Load History (damage & fatigue) Deformation Stateperimental Elastomer 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:Analysis 21

Chapter 2: The Macroscopic Behavior of Elastomers Microscopic StructureMicroscopic Structure

Long coiled molecules, with points of entanglement.Behaves like a viscous fluid.

Vulcanization creates chemical bonds (cross-links) atthese 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 ElastomersTemperature Effects, Tg All polymers have a spectrum of mechanical behavior, from

brittle, or glassy, at low temperatures, to rubbery athigh 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, -55for EPDM, and -130 for silicone rubber. Experimental Elastomer Analysis 23

Chapter 2: The Macroscopic Behavior of Elastomers Time Effects, ViscoelasticityTime 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): strain-rate effects

creep, stress-relaxationhysteresisdamping24 Experimental Elastomer Analysis

Time Effects, Viscoelasticity Chapter 2: The Macroscopic Behavior of ElastomersTime 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 samecuring history, thus same stiffness.26 Experimental Elastomer Analysis

Damage, Early Time Chapter 2: The Macroscopic Behavior of ElastomersDamage, 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.

Mullins Effect in carbon black filled NR:

Be careful that real parts and test specimens share the sameload history, Preconditioning.

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).Experimental Elastomer Analysis 27

Chapter 2: The Macroscopic Behavior of Elastomers Damage, FatigueDamage, Fatigue Very early stages of understanding, see Gents 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 Brake FluidOxidation Hydraulic FluidUltraviolet RadiationOil, Gasoline

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 ElastomersDeformation States Shearing vs. Bulk Compression

Shearing Modulus, , typical ~ 1 - 10 MPa

Bulk Modulus, , typical ~ 2 GPa

hence

and

Ordinary solid (e.g., steel): and are the same order ofmagnitude. Whereas, in rubber the ratio of to is of the order ; hence the response to a stress is effectivelydetermined solely by the shear modulus when the material can shear.

We say rubber is (nearly) incompressible in those caseswhen it is not highly confined.

G

K pV V0-----------------=

KG---- 103

12---

K GK G

103

GExperimental Elastomer Analysis 29

Chapter 2: The Macroscopic Behavior of Elastomers Deformation StatesDeformation States (cont.) FEA Material Model calibration requires certain

types of tests.

They require states of pure stress and strain, that isthat 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 specimenitself to prove homogeneity.

The button compression test is notoriously bad fromthis perspective.

Keep in mind that many ASTM test standards aredefined for characterization, or process control purposes.Many ASTM specs are NOT suitable for material modelcalibration.30 Experimental Elastomer Analysis

Ex

CHAPTER 3 Material Models, Historical Perspectiveperimental Elastomer 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.Analysis 31

Chapter 3: Material Models, Historical Perspective Engineering Materials and AnalysisEngineering Materials and AnalysisClearly 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 Elastomer Analysis-4000 Copper, Gold-3500 Bronze Casting-1400 Iron Age-1 Damascus Steel1660 Hookean Materials1800 Titanium 3D Elasticity1840 Aluminum Vulcanization1850 Parkesine1879 Rare earth metals Colloids1929 Aminoplastics1933 Polyethylene1933 PMMA1939 Nylon1940 Neo-Hookean1940 PVC1941 Polyurethanes1943 PTFE1949 Mooney-Rivlin1950 Hills Plasticity1955 Polyester1965 FEA Software1970 Foams1975 Treloar1980 > 200 Polymer compounds1990 Recycle32 Experimental Elastomer Analysis

Neo-Hookean Material Model Chapter 3: Material Models, Historical PerspectiveNeo-Hookean Material ModelDefinitions, Stretch ratios, Engineering Strain:

Incompressibility:

From Thermodynamics and statistical mechanics,First order approximation (neo-Hookean):

iLi Li+

Li-------------------- 1 i+= = eng. strain, i Li Li( )=

t1 t1

t2

t2

t3

t3

1L12L2

3L3

L1

L2

L3

123 1=

W 12---G 1

2 22 3

2 3+ +( )=Experimental Elastomer Analysis 33

Chapter 3: Material Models, Historical Perspective Neo-Hookean Material ModelNeo-Hookean Material Model (cont.)Experimental Verification using Simple Extension

Hence:

Engineering Stress:

True Stress:

Simple, one parameter material model

Positive G guarantees material model stability

1 = 2 3 1 = =

0.8 0.4 0.0 0.4 0.8Engineering Strain

25.0

15.0

5.0

5.0

Engine

eringSt

ress

/(She

arMod

ulus

)

NeoHookean BehaviorTension and Compression very Different

Hookean (nu=.45)NeoHookean

W 12---G 2 2--- 3+

=

dW d G 12-----

= = =

G 1 11 +( )2

-------------------+ =

t

1 ---------- G 2 1

---

= = =34 Experimental Elastomer Analysis

Neo-Hookean Material Extension Deformation Chapter 3: Material Models, Historical PerspectiveNeo-Hookean Material Extension DeformationTheory versus experiments:

0.0

2.0

4.0

6.0

210 3 4 5 6 7Engineering Strain

Experiment

Theory

Engine

eringSt

ress

[MPa

]Experimental Elastomer Analysis 35

Chapter 3: Material Models, Historical Perspective Neo-Hookean Material Shear DeformationNeo-Hookean Material Shear DeformationExperimental Verification using Simple Shear

:

If , then and

Equivalent shear strain :

Strain energy function:

Shear stress depends linearly on shear strain

X

Y

atan

1 = 21---= 3 1=

1---=

W 12---G 2 1

2----- 2+

12---G2= =

Wdd-------- G= =36 Experimental Elastomer Analysis

Neo-Hookean Material Shear Deformation Chapter 3: Material Models, Historical PerspectiveNeo-Hookean Material Shear Deformation (cont.)Theory versus experiments:

Shear Strain

Experiment

Theory

Shea

rStre

ss[N

/mm

]2

0.0

0.4

0.8

1.2

1.6

0 1 2 3 4 6Experimental Elastomer Analysis 37

Chapter 3: Material Models, Historical Perspective Neo-Hookean Material SummaryNeo-Hookean Material Summary

Neo Hookean

direct stresses

shear stress

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

TABLE 2. Basic Deformation Modes

Mode

Biaxial

Planar Shear

Uniaxial

Simple Shear

1 2 3

2

1 1

1 2 1 2

1 2

2----- 1

2

4-----++ + 1

2

2----- 1

2

4-----++ 1

W 12---G 1

2 22 3

2 3+ +( )=

W

( )= =

W G= =38 Experimental Elastomer Analysis

Neo-Hookean Material Summary Chapter 3: Material Models, Historical PerspectiveNeo-Hookean Material Summary (cont.)TABLE 3. Hookean versus Neo Hookean Values of

ModeHookean

=

Hookean as Neo Hookean

=

Biaxial

Planar Shear

Uniaxial

G

G 0 G

2 1 ( )1 2( )-------------------- 2 2 1 1 +( )

5+{ }

2 1 2( )1 2( )------------------------------- 2 1 1 +( )

3+{ }

2 1 +( ) 2 1 1 +( ) 2+{ }

-1.0 -0.5 0.0 0.5 1.0-10.0

-5.0

0.0

5.0

10.0

Hookean and Neo Hookean Material ModelsPoisson Ratio = 0.45

Hookean BiaxialHookean Planar ShearHookean UniaxialNew Hookean BiaxialNeo Hookean Planar ShearNeo Hookean Uniaxial

Engineering Strain

Engi

ne

erin

g St

ress

/She

ar M

odulu

sExperimental Elastomer Analysis 39

Chapter 3: Material Models, Historical Perspective A Word About Simple ShearA Word About Simple ShearThe simple shear mode of deformation is called simple shear because of two reasons: first it renders the stress strain relation linear for a Neo-Hookean material; secondly it is simple to draw.

Linear Stress Strain Relation comes from substituting the simple shear deformations modes of:

into

and then

Secondly the mode is simple to draw.

21

1 2

2----- 1

2

4-----++ +=

22 1

2

2----- 1

2

4-----++=

23 1=( )

W 12---G 1

2 22 3

2 3+ +( ) 12---G2

= =

W G= =

atan

40 Experimental Elastomer Analysis

2-Constant Mooney Extensional Deformation Chapter 3: Material Models, Historical Perspective2-Constant Mooney Extensional DeformationBasic assumptions:

(1) The rubber is incompressible and isotropic(2) Hookes law is obeyed in simple shear

Strain energy function with two constants:

Simple shear:

Hence

or

W C1 12 2

2 32 3+ +( ) C2

11

2-----

12

2-----+

13

2----- 3+

+=

W C1 C2+( ) 12 1

12

----- 2+

C1 C2+( )2

= =

dW d 2 C1 C2+( )= =

G 2 C1 C2+( )=

2 12-----

C1C2------+

=

2 1 2( )------------------------------ C1

C2------+=Experimental Elastomer Analysis 41

Chapter 3: Material Models, Historical Perspective 2-Constant Mooney Extensional Deformation2-Constant Mooney Extensional Deformation (cont)Theory versus experiments

AB

CDE

F

G

0.5 0.6 0.7 0.8 0.9 1.01/

0.1

0.2

0.3

0.4

/2

(1/

2) (

N/mm

2 )

/2(

1/2

) (N

/mm

2 )

1/42 Experimental Elastomer Analysis

Other Mooney-Rivlin Models Chapter 3: Material Models, Historical PerspectiveOther Mooney-Rivlin ModelsBasic assumptions:

(1) The rubber is incompressible and isotropic in the unstrained state

(2) The strain energy function must depend on even powers of

The three simplest possible even-powered functions (invariants):

Incompressibility implies that , so that:

Mooney material in terms of invariants:

(Mooneys original notation)

(Mooney-Rivlin notation)

i

I1 12 2

2 32

+ +=

I2 122

2 223

2 321

2+ +=

I3 122

232

=

I3 1=

W W I1 I2,( )=

W C1 I1 3( ) C2 I2 3( )+=

W C10 I1 3( ) C01 I2 3( )+=Experimental Elastomer Analysis 43

Chapter 3: Material Models, Historical Perspective Other Mooney-Rivlin ModelsOther Mooney-Rivlin Models (cont)Some other proposed energy functions:

The Signiorini form:

The Yeoh form:

Third order Deformation Form(James, Green, and Simpson):

W C10 I1 3( ) C01 I2 3( ) C20 I1 3( )2

+ +=

W C10 I1 3( ) C20 I1 3( )2 C30 I1 3( )

3+ +=

W C10 I1 3( ) C01 I2 3( ) C11 I1 3( ) I2 3( )+ + +=C20 I1 3( )

2 C+ 30 I1 3( )344 Experimental Elastomer Analysis

Ogden Models Chapter 3: Material Models, Historical PerspectiveOgden ModelsSlightly compressible rubber:

and are material constants,

is the initial bulk modulus, and

is the volumetric ratio, defined by

The order of magnitude of the volumetric changes per unitvolume should be 0.01

Usually, the number of terms taken into account inthe Ogden models is or .

The initial bulk modulus is usually estimated instead of beingmeasured in a volumetric test.

Wnn------ J

n

3---------

1n 2

n 3n+ +( ) 3 4.5K J

13---

1 2

+

n 1=

N

=

n n

K

J

J 123=

N 2= N 3=Experimental Elastomer Analysis 45

Chapter 3: Material Models, Historical Perspective Ogden ModelsOgden ModelsLets suppose we want to fit a 1-term Ogden for tension.

1.) Assume incompressible (J=1) then

2.) Strain mode is tension, thus and

3.) Compute engineering stress, ,

or

4.) Fit data, say to st_18.data that has 60 stress-strain points. Find such that , has the best fit.

5.) Panic is nonlinear. Ok, use program and

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

W --- 1

2 3

+ +( ) 3[ ]=

1 = 2 3 1 = =

W --- 2

2---

3+

=

dW d 1 2--- 1+

= =

dW d 1 +( ) 1 1 +( )2--- 1+

, ,( )= = =

and , i i, ,( )= i 1 60,=

i i, ,( )= 25.78=

0.05298=46 Experimental Elastomer Analysis

Ogden Models Chapter 3: Material Models, Historical PerspectiveOgden Models

6.) Plot .

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

Mode

Biaxial

Planar Shear

25.78 1 +( )0.05298 1 1 +( )0.05298

2------------------- 1+

=

1 2 3

2

1 1

uniaxial/ogdenuniaxial/experiment

1.357

08.8940

(x.1)biaxial/ogden planar_shear/ogden

1Experimental Elastomer Analysis 47

8. Estimate K = 2500(25.78)0.05298 = 3414.

Chapter 3: Material Models, Historical Perspective Foam ModelsFoam ModelsElastomer foams:

, and are material constants

Wnn------ 1

n 2n 3

n+ + 3[ ] nn----- 1 J

n( )

n 1=

N

+n 1=

N

=

n n n48 Experimental Elastomer Analysis

Model Limitations and Assumptions Chapter 3: Material Models, Historical PerspectiveModel Limitations and AssumptionsThis 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:

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

K 10000 C10 C01+( )=

K 2500 nnn 1=

N

=Experimental Elastomer Analysis 49

Chapter 3: Material Models, Historical Perspective Viscoelastic ModelsViscoelastic ModelsMSC.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.Marcs large strain viscoelastic material model is based on a multiplicative decomposition of the strain energy function

where is a standard Mooney-Rivlin or Ogden form strain energy function for the instantaneous deformation.

And is a relaxation function in Prony series form:

where is a nondimensional multiplier and is the associated time constant.

W Eij t,( ) W Eij( ) R t( )=

W Eij( )

R t( )

R t( ) 1 n 1 t n( )exp( )n 1=

N

=

n n50 Experimental Elastomer Analysis

Determining Model Coefficients Chapter 3: Material Models, Historical PerspectiveDetermining Model CoefficientsThis material model requires two different types of tests beconducted and two separate curve fits be performed.

The time-independent function, , is determined fromstandard 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, , 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,

is determined in Model 5: Viscoelastic Curve Fit on page 200.

W Eij( )

R t( )

R t( )Experimental Elastomer Analysis 51

Chapter 3: Material Models, Historical Perspective Determining Model Coefficients52 Experimental Elastomer Analysis

Ex

CHAPTER 4 Laboratory

How to specify a laboratory experiment as required by your product requirements.Lets understand the specimen testing better to achieve better correlation and confidence in our component analysis.perimental Elastomer Need to know:

What are the actual tests used to measure elastomeric properties.

The limitations of common laboratory tests.Analysis 53

Chapter 4: Laboratory Lab OrientationLab Orientation

SafetyTour of Lab

Laboratory Dangers

High Pressure Hydraulics

Class II Lasers

Instrument Crushing

Wear Safety Glasses

Dont Look Into Lasers

Dont Touch Specimens or Fixtures When Testing54 Experimental Elastomer Analysis

Basic Instrumentation Chapter 4: LaboratoryBasic InstrumentationElectromechanical Tensile Testers

Servo-hydraulic TestersExperimental Elastomer Analysis 55

Chapter 4: Laboratory Basic InstrumentationBasic Instrumentation (cont.)

Wave Propagation Instrument

Automated Crack Growth System

Aging Instrumentation56 Experimental Elastomer Analysis

Measuring Chapter 4: LaboratoryMeasuring

ForceStrain Gage Load Cells

PositionEncoders and LVDTs

StrainClip-on Strain GagesVideo ExtensometersLaser Extensometers

TemperatureThermocouplesExperimental Elastomer Analysis 57

Chapter 4: Laboratory MeasurementsMeasurements

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)Thermocouple58 Experimental Elastomer Analysis

What about Shore Hardness? Chapter 4: LaboratoryWhat 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.comExperimental Elastomer Analysis 59

Chapter 4: Laboratory Testing the Correct MaterialTesting the Correct MaterialConsistent within The Experimental Data Set

Cut All Specimens from the Same Slab

Verify that The Tested Material is the Same as the Part

ProcessingColor Cure Progressively Increasing Load History

Cut Specimens from Same Material150mm x 150mm x 2mm Sheet

All Are Same Compound60 Experimental Elastomer Analysis

Tensile Testing in the Lab Chapter 4: LaboratoryTensile Testing in the Lab

What is Simple Tension?Uniaxial LoadingFree of Lateral ConstraintGage Section: Length: Width >10:1Measure Strain only in the Region where a Uniform State of Strain ExistsNo Contact

1

2

3

Cut Specimens from Same Material150mm x 150mm x 2mm SheetExperimental Elastomer Analysis 61

Chapter 4: Laboratory Tensile Testing in the LabTensile 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 Data62 Experimental Elastomer Analysis

Compression Testing in the Lab Chapter 4: LaboratoryCompression Testing in the LabIt is Experimentally Difficult to Minimize Lateral Constraint due to Friction at the Specimen Loading Platen Interface

Friction Effects Alter the Stress-strain Curves

The Friction is Not Known and Cannot be Accurately Corrected

Even Very Small Friction Levels have a Significant Effect at Very Small Strains

1

2


Chapter 4: Laboratory Compression Testing in the LabCompression 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: LaboratoryEqual Biaxial Testing

Why?Same Strain State as CompressionCannot Do Pure CompressionCan Do Pure Biaxial

Analysis of the Specimen justifies Geometry

1

2


Chapter 4: Laboratory Compression and Equal Biaxial Strain StatesCompression and Equal Biaxial Strain StatesThere 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 buttons strain state, simply substitute . 2=

3 =

1 1 2

=

2=

2 1 2

=

3 2

=

1 =

2 =66 Experimental Elastomer Analysis

Volumetric Compression Test Chapter 4: LaboratoryVolumetric 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 ModulusTypically, only highly constrained applications require an accurate measure of the entire Pressure-Volume relationship.

1

2

3

Bulk Modulus = 2.1 GPa

300

250

200

150

100

50

0

Pres

sure

(MPa

)

Volumetric Strain0.02 0.04 0.06 0.08 0.100.00

VOLCOMP_B

Base Data SetExperimental Elastomer Analysis 67

Chapter 4: Laboratory Planar Tension TestPlanar Tension TestUniaxial LoadingPerfect Lateral ConstraintAll Thinning Occurs in One DirectionStrain Measurement is Particularly CriticalSome Material Flows from the GripsThe Effective Height is Smaller than Starting Height so >10:1 Width:Height is NeededSimilar Stress-strain Shape to Simple Tension and Biaxial ExtensionMatch Loadings between Strain States 1

2

3

Base Data Set

Engi

ne

erin

g St

ress

(M

Pa)

Planar Tension

Engineering Strain

0.6

0.5

0.4

0.3

0.2

0.1

0.0

PT23C_B

0.2 0.4 0.6 0.8 1.00.068 Experimental Elastomer Analysis

Planar Tension Test Chapter 4: LaboratoryPlanar 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 RelaxationViscoelastic Stress RelaxationViscoelastic Behavior

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

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

Strain = 30 %

Strain = 50 %

Time (s)

Stre

ss (M

Pa)

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.00 2000 4000 6000 8000

Time (Seconds)

Stre

ss (M

Pa)

Stre

ss

Strain

7

6

5

4

3

2

1

0

0 500 1000 1500 200070 Experimental Elastomer Analysis

Dynamic Behavior Testing Chapter 4: LaboratoryDynamic Behavior TestingTypes of Dynamic Behavior

Large strains at high velocity

Small sinusoidal strains superimposed on large mean strainsExperimental Elastomer Analysis 71

Chapter 4: Laboratory Dynamic Behavior TestingDynamic Behavior Testing (cont.)Mean Strain and Amplitude Effects are Significant72 Experimental Elastomer Analysis

Friction Test Chapter 4: LaboratoryFriction TestFriction is the force that resists the sliding of two materials relative to each other. The friction force is:

(1) approximately independent of the area of contact over a wide limit and

(2) is proportional to the normal force between the two materials.

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.

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.

In order to model friction in finite element analysis, one needs to measure the aforementioned proportionally factor or coefficient of friction, . The measurement of 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.

Friction Test

Fric

tion

Fo

rce

Position

Incr

ea

sin

g No

rma

l Fo

rce

Experimental Elastomer Analysis 73

Chapter 4: Laboratory Data Reduction in the LabData Reduction in the LabThe 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.

Fit for Arruda-Boyce

Adjusted Data

Raw Data74 Experimental Elastomer Analysis

Data Reduction in the Lab Chapter 4: LaboratoryData 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 ExperimentsModel Verification Experiments

Attributes of a Good Model Verification ExperimentThe geometry is realistic.All Relevant Constraints are Measurable.The Analytical Model is Well Understood76 Experimental Elastomer Analysis

Model Verification Experiments Chapter 4: LaboratoryModel 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 TemperaturesTesting 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: LaboratoryLoading/Unloading ComparisonExperimental Elastomer Analysis 79

Chapter 4: Laboratory Test Specimen RequirementsTest 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: LaboratoryFatigue Crack Growth

Provides Great Potential.

Not well understood.Experimental Elastomer Analysis 81

Chapter 4: Laboratory Experimental and Analysis Road MapExperimental and Analysis Road MapTABLE 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 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 Planned82 Experimental Elastomer Analysis

Ex

CHAPTER 5 Material Test Data Fittingperimental Elastomer The experimental determination of elastomeric material constants depends greatly on the deformation state, specimen geometry, and what is measured.Analysis 83

Chapter 5: Material Test Data Fitting Major Modes of DeformationMajor Modes of DeformationUniaxial Tension

Biaxial Tension (equivalent strain as uniaxial compression)

1

2

3

1 2 = = 2 3 1 2= =

1

3

2

1 2 = = 3 1 2=84 Experimental Elastomer Analysis

Major Modes of Deformation Chapter 5: Material Test Data FittingMajor Modes of Deformation (cont.)Planar Tension, Planar Shear, Pure Shear

Simple Shear

1 = 2 1= 3 1 =

1

2


Chapter 5: Material Test Data Fitting Major Modes of DeformationMajor Modes of Deformation (cont.)Volumetric (aka Hydrostatic, Bulk Compression)

FF

Confined Hydrostatic CompressionCompression86 Experimental Elastomer Analysis

Confined Compression Test (UniVolumetric) Chapter 5: Material Test Data FittingConfined Compression Test (UniVolumetric)Strain State:

Stress State:

For this deformation state we have

,

and the uniaxial strain is equal to the volumetric strain or

.

The bulk modulus becomes

MSC.Marc Mentat uses the pressure, , versus a uniaxial equivalent of

the volumetric strain namely, , to determine the bulk

modulus as shown on the right. Take care to divide the volumetric strain by 3, because you may forget.

F L,

1 1= 2 1= 3 L L0=

1 2 3 F Ao p= = = =

123 V V0 L L0= =

0.000 0.010 0.020 0.030 0.040Equivalent Uniaxial Strain [1]

0.0

100.0

200.0

300.0

400.0

Pres

sure

[M

pa]

Volumetric DataFor Mentat Curve Fitting

13--- V V0

p

L L0 V V0=

K pV V0------------------

=p

L L0-----------------

=

p

13--- V V0

V V0 L L0=Experimental Elastomer Analysis 87

Chapter 5: Material Test Data Fitting Hydrostatic Compression TestHydrostatic Compression TestStrain State:

Stress State:

For this strain state we have

and since

the uniaxial strain becomes one third the volumetric strain or

.

The bulk modulus becomes

Again MSC.Marc Mentat uses the pressure, , versus a uniaxial equivalent of the volumetric strain namely, , to determine the bulk modulus.

F L,

1 2 3 V V0( )1 3

= = = =

1 2 3 F Ao p= = = =

1 V+ V0( )1 3 1 13---

V V0+=

1 L L0+=

L L013--- V V0=

K pV V0------------------

=p

3 L L0( )--------------------------

=

p13--- V V088 Experimental Elastomer Analysis

Summary of All Modes Chapter 5: Material Test Data FittingSummary of All ModesMode:

Xx1

x2

x3

=

F =

i , i = 1, 2, 3

b i2 1 0=

Uniaxial

X1X2

-------

X3

-------

0 0

0 1

------- 0

0 0 1

-------

2 0 0

0 1--- 0

0 0 1---

1 /1 /

Biaxial

X1X2X32------

0 00 0

0 0 12-----

2 0 00 2 0

0 0 14-----

1 2/

b = F FT

Planar

X1X2------

X3

0 0

0 1--- 0

0 0 1

2 0 0

0 12----- 0

0 0 1

1/1

Simple Shear

X1 X2+X2X3

1 00 1 00 0 1

1 2+ 0

00 1 0

1

1

2

2---- 12

4----++ +

12

2---- 12

4----++

1

UniVolumetric

X1X2

X3

1 0 00 1 00 0

1 0 0

0 1 0

0 0 2

11

Maping

Shape

DeformationGradient

FigerTensor

PrincipalStretch Ratios

Volumetric

X1X2X3

0 00 00 0

2 0 00 2 00 0 2


Chapter 5: Material Test Data Fitting General GuidelinesGeneral Guidelines

Its just curve fitting!No Polymer physics as basis

Dont use too high order fit

Remember polynomial fit lessons (high school?)

Number of Data PointsDont use too many Regularize if needed

Add/Subtract points if needed

Weighting of Points

Range and Scope of DataCheck fit outside range of data

Check fit in other modes of deformation scope90 Experimental Elastomer Analysis

Mooney, Ogden Limitations Chapter 5: Material Test Data FittingMooney, Ogden Limitations

Phenomenological models not material lawThese models are mathematical forms, nothing more

Summary of phenomenological models given byYeoh (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 compressibilityExperimental Elastomer Analysis 91

Chapter 5: Material Test Data Fitting Visual ChecksVisual Checks

Extrapolations can be dangerous

Always visually check your models predictedresponse

Check it outside of the datas range (see below)Check it outside the tests scope

PredictedResponse

DATA

Real Material

PredictedResponse

Real Material

d d 0>

d d 0

Material Stability Chapter 5: Material Test Data FittingMaterial Stability

Unstable material model -> numerical difficultiesin FEA

Druckers stability postulate,

Graphically:

Remember effects of Newton-Raphson andstrain range

d d 0>

d11 d11 0> d11 d11 0

Chapter 5: Material Test Data Fitting Future TrendsFuture Trends

Statistical Mechanics ModelsBased 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 frommultiple modes94 Experimental Elastomer Analysis

Adjusting Raw Data Chapter 5: Material Test Data FittingAdjusting Raw DataThe stress strain response of the three modes of deformation are shown below 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.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

Equal Biaxial

Engin

eering

Stre

ss[M

pa]

The Raw Data (4 points/sec)

Engineering Strain [1]

Pure Shear

TensionExperimental Elastomer Analysis 95

Chapter 5: Material Test Data Fitting Adjusting Raw DataAdjusting 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

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.

0.0 0.2 0.4 0.6 0.8 1.0Engineering Strain [1]

0.0

0.5

1.0

1.5

2.0

Engine

eringSt

ress

[Mpa

]

Adjusting The Raw DataShift to the Origin

Equal Biaxial ShiftedEqual BiaxialPure Shear ShiftedPure ShearTension ShiftedTension

' 1 p+( )= '

p

p

( ) 1 p+( )=96 Experimental Elastomer Analysis

Adjusting Raw Data Chapter 5: Material Test Data FittingAdjusting 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

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

0 1

0

1

uniaxial/experiment uniaxial/neo_hookean1

1

0

0 Engineering Strain [1]

Engi

nee

ring

Str

ess

[Mpa

]

Fit to upload & download

Fit to uploadExperimental Elastomer Analysis 97

Chapter 5: Material Test Data Fitting Consider All Modes of DeformationConsider All Modes of DeformationThe 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 FittingThe Three Basic Strain StatesAfter 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

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.

0.0 0.2 0.4 0.6 0.8 1.0Engineering Strain [1]

0.0

0.5

1.0

1.5

2.0

Engine

eringSt

ress

[Mpa

]

The Three Basic Strain StatesGeneral Elastomer Trends

Equal BiaxialPure ShearTensionExperimental Elastomer Analysis 99

Chapter 5: Material Test Data Fitting Curve Fitting with MSC.Marc MentatCurve Fitting with MSC.Marc MentatObjective: Fit experimental data of Mooney or Ogden materials with MSC.Marc Mentat. Begin at the main menu.

MATERIAL PROPERTIESTABLESREAD

RAW(name of file)TABLE TYPEexperimental_dataOKRETURN

EXPERIMENTAL DATA FITTINGUNIAXIAL(pick table1)OK

ELASTOMERSNEO-HOOKEAN

UNIAXIALCOMPUTEAPPLYOKSCALE AXES100 Experimental Elastomer Analysis

Curve Fitting with MSC.Marc Mentat Chapter 5: Material Test Data FittingCurve 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 pop-up 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 OPTIONSSIMPLE SHEAR (this toggles it off)PLANAR SHEAR (this toggles it off)RETURN

SCALE AXESExperimental Elastomer Analysis 101

Chapter 5: Material Test Data Fitting Curve Fitting with MSC.Marc MentatCurve Fitting with MSC.Marc Mentat (cont)Objective: Fit experimental data of Viscoelastic materials with MSC.Marc Mentat. Begin at the main menu.

MATERIAL PROPERTIESTABLESREAD

RAW(name of file)TABLE TYPEexperimental_dataOKRETURN

EXPERIMENTAL DATA FITTING

ENERGY RELAX(pick table1),OK

ELASTOMERSENERGY RELAX

RELAXATION# OF TERMS 3COMPUTEAPPLY, OKSCALE AXES102 Experimental Elastomer Analysis

Curve Fitting with MSC.Marc Mentat Chapter 5: Material Test Data FittingCurve Fitting with MSC.Marc Mentat (cont)

Mooney-Rivlin fitting is linear, uses least squares fitting

The least squares error is given by either:

The and are relative or absolute respectively is the total number of data points

is the calculated stress

is the measured engineering stress

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

errorR 1

calci

imeasured

------------------------

2

i

Ndata

= or

errorA

imeasured calc

i( )2

i

Ndata

=

errorR

errorA

Ndata

calci

imeasuredExperimental Elastomer Analysis 103

Chapter 5: Material Test Data Fitting Curve Fitting with MSC.Marc MentatCurve Fitting with MSC.Marc Mentat (cont)

Ogden fitting is nonlinear, uses downhill-simplexmethod

Downhill-simplex method is an iterative methodUses a number of start points

Continues until:

is set using CONVERGENCE TOLERANCE

can be set with the ERROR LIMIT button

abs errormax errormin( )abs errormax errormin+( )----------------------------------------------------------------

tol2

-------

default with bias factor222 Experimental Elastomer Analysis

Deformable-to-Deformable Contact Chapter 7: Contact AnalysisDeformable-to-Deformable ContactDiscrete 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.

Then the contact constraint:

[ defines tying relation for displacement component of contacting node in local -direction

[ applies correction on position in local -direction

actual geometry

finite element approximation

contacted body

contacting bodycontact tolerance

yx

A

y

yExperimental 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.

Advantages are smoother contact where in 2D, -continuity is obtained, and in 3D, at least pointwise -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.

C1

C1224 Experimental Elastomer Analysis

Contact Flowchart Chapter 7: Contact AnalysisContact FlowchartInput

Initial set up of contact bodies

Incremental data input

Check on contact

Set up of contact constraints

Apply distributed loads

Assemble stiffness matrix; include friction

Apply contact constraints

Solve set of equations

Recover stresses

Converged solution?

Separation?

Penetration?

Last increment?

Stop

No

No

Yes

Yes

Yes

Update contact constraints

No

No

Yes

begi

n in

crem

ent

begi

n it

eratio

n

Splitincrement

Changecontact

constraintsExperimental Elastomer Analysis 225

Chapter 7: Contact Analysis Symmetry BodySymmetry BodySymmetry 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

Y

Z

deformable_body

symmetry_plane_1

symmetry_plane_2

none226 Experimental Elastomer Analysis

Rigid with Heat Transfer Chapter 7: Contact AnalysisRigid with Heat Transfer

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

billet

channel 35

20o

44.75

50

R = 6

25

20

billet

channel

none

deformable-rigid (stress or coupled analysis)

geometrical entities(straight lines and acircular arc)

MARC element 10Experimental Elastomer Analysis 227

Chapter 7: Contact Analysis Rigid with Heat TransferModel 2: Deformable-rigid (coupled analysis)

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

billet

channel

none

deformable-rigid (coupled a

MARC element 40

MARC element 10

Rigid w Heat Transfer

billet

channel

none

deformable-deformable (stress or coupled analysis)MARC element 10

MARC element 10228 Experimental Elastomer Analysis

Contact Table Chapter 7: Contact AnalysisContact Table

Contact Table Properties:

Single-sided Contact:

Only body 2 may contact itself

1

23


Chapter 7: Contact Analysis Contact TableContact 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.230 Experimental Elastomer Analysis

Contact Areas Chapter 7: Contact AnalysisContact AreasVery 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 DetectionExclude Segments During Contact DetectionExclude segment will influence the searching done for nodes detected in the contact zone during self contact.

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)

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.232 Experimental Elastomer Analysis

Effect Of Exclude Option: Chapter 7: Contact AnalysisEffect Of Exclude Option:

Standard contact

excluded segments

With exclude optionExperimental Elastomer Analysis 233

Chapter 7: Contact Analysis Contacting Nodes and Contacted SegmentsContacting Nodes and Contacted SegmentsFor 3D continua, an automatic check on the direction of the normal vectors is included:

Contact will not be accepted if

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.

Contacting body nodes Contacted body patches

nnode npatch 0.05>234 Experimental Elastomer Analysis

Friction Model Types Chapter 7: Contact AnalysisFriction Model TypesFriction 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 ModelCoulomb ArcTangent Friction ModelImplementation of this friction model has been done using nonlinear dashpots whose stiffness depend on the relative sliding velocity as:

MSC.Marc approximation:

with:

:relative sliding velocity below which sticking is simulated(Default = 1.0! is rarely correct)

slip

slip

MARC approximation

Ft

vr

stick

C

Ft Fn2---

vr

C---- atan

C236 Experimental Elastomer Analysis

Coulomb Bilinear Friction Model Chapter 7: Contact AnalysisCoulomb Bilinear Friction ModelImplementation of this friction model has been done using nonlinear dashpots whose stiffness depend on the relative sliding velocity as:

MSC.Marc approximation:

with:

: slip threshold automatically set.

Friction force tolerance has a default value of 0.05.

slip

slip

MARC approximation

Ft

ur

stick


Chapter 7: Contact Analysis Stick-Slip Friction ModelStick-Slip Friction ModelDiscovered 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 Columbs Law.

with:

:incremental tangential displacement

: slip to stick transition region (default : coefficient multiplier (default 1.05): friction force tolerance (default 0.05)

: small constant, so that (fixed at )

ut

Ft

22

Fn

F

n

Ft Fn static Ft Fn, kinetic

ut

1 610

e

0 1 610238 Experimental Elastomer Analysis

Glued Contact Chapter 7: Contact AnalysisGlued ContactSometimes 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

cbody1

cbody2

none

Z

YX4Experimental Elastomer Analysis 239

Chapter 7: Contact Analysis Glued ContactGlued 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 AnalysisRelease OptionThe 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 spring-back 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 AmountBy 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: gapExperimental Elastomer Analysis 241

Chapter 7: Contact Analysis Forces on Rigid BodiesForces on Rigid BodiesDuring 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 Y242 Experimental Elastomer Analysis

Forces on Rigid Bodies Chapter 7: Contact AnalysisForces 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 Forces on Rigid Bodies244 Experimental Elastomer Analysis

Ex

APPENDIX A The Mechanics of Elastomersperimental Elastomer The macroscopic behavior of elastomers depends greatly upon the deformation states because the material is nearly incompressible.Analysis 245

Appendix A: The Mechanics of Elastomers Deformation StatesDeformation States

Stretch ratios:

Incompressibility:

First order approximation (Neo-Hookean):

Eliminate :

t1 t1

t2

t2

t3

t3

1L1

2L23L3

L1

L2

L3

iLi Li+

Li-------------------- 1 += = engineering strain Li Li( ) ==

123 1=

W 12---G 1

2 22 3

2 3+ +( )=

3

W 12---G 1

2 22 1

122

2----------- 3+ +

=246 Experimental Elastomer Analysis

Deformation States Appendix A: The Mechanics of ElastomersTwo-dimensional extension:

Hence: , ,

Engineering stresses : forces per unit undeformed area

True stresses : forces per unit deformed area

dL1L1L2

dL2

F1F1

F2

F2

dW F1dL1 F2dL2+ 1d1 2d2+= =

dW W1---------d1

W2---------d2+=

1 G 11

132

2-----------

=

3 0=

2 G 21

122

3-----------

=

i

tiExperimental Elastomer Analysis 247

Appendix A: The Mechanics of Elastomers Deformation StatesTwo-dimensional extension:

or:

and:

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

t1 1 23( ) 11= =

t1 G 12 3

2( )=

t2 G 22 3

2( )=

t3 0=

p248 Experimental Elastomer Analysis

Deformation States Appendix A: The Mechanics of Elastomers(Nearly) incompressible material:

, hence

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

; hence the response to a stress is effectively determined solely by the shear modulus

Bulk Modulus KShear Modulus G------------------------------------------

2 1 +( )3 1 2( )------------------------=

12---

KG----

G KG K

10 4

GExperimental Elastomer Analysis 249

Appendix A: The Mechanics of Elastomers General Formulation of ElastomersGeneral Formulation of ElastomersMaterial points in undeformed configuration: ; material points in deformed configuration:

Lagrange description:

is the deformation gradient tensor

Green-Lagrange strain tensor:

Right Cauchy-Green strain tensor:

Some additional relations:

Xixi

xi xi Xj( )=

dxi FijdXj with FijxiXj

--------= =

Fij

dx( )2 dX( )2 2EijdXidXj=

dx( )2 CijdXidXj=

Cij ij 2Eij+=

CijxkXi--------

xkXj-------- FkiFkj= =

Eij12---

xkXi--------

xkXj-------- ij

12--- FkiFkj ij[ ]= =250 Experimental Elastomer Analysis

General Formulation of Elastomers Appendix A: The Mechanics of ElastomersIntroduce displacement vector :

With respect to principal directions:

Invariants of :

Strain energy function:

ui

xi Xi ui+=

Eij12--- ui j, uj i, uk i, uk j,+ +( )=

Cij ki uk i,+( ) kj uk j,+( )=

Ci'j'

12 0 0

0 22 0

0 0 32

=

Cij

I1 Cii=

I212--- CiiCjj CijCij( )=

I3 det Cij=

W* W I1 I2,( ) h I3 1( )+=Experimental Elastomer Analysis 251

Appendix A: The Mechanics of Elastomers General Formulation of ElastomersSecond-Piola Kirchhoff stresses:

True or Cauchy stresses:

Zero deformation:

hence:

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

Sij 2WI1--------ij 2

WI2-------- ijCkk Cij[ ] 2h

I3Cij----------+ +=

tij0----- ik ui k,+( )Skl jl uj l,+( )=

Sij0 2WI1

-------- 4WI2--------+

02h+0 ij=

p 2WI1--------

0 4WI2--------

02h=252 Experimental Elastomer Analysis

Finite Element Formulation Appendix A: The Mechanics of ElastomersFinite Element FormulationModified virtual work equation:

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

The incremental stresses are related to the linear strain

increment by:

The incremental set of equations to be solved reads:

with:

: the linear stiffness matrix

: the geometric stiffness matrix

: the nodal pressure coupling matrix: nodal load vector: internal stress vector

SijV EijdV Qiui Vd

V Tiui Ad

A- I3 1( ) Vd

V 0=+

ui Xi( ) N Xi( )ui

= p Xi( ) h Xi( )p

=and

Sij Dijkln Ekl

-

p Cijn( ) 1=

K 0( ) K 1(

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