Mar 103

Experimental Elastomer Analysis

MSC.Software Corporation

MA*2003*Z*Z*Z*SM-MAR103-NT1 1

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MA*2003*Z*Z*Z*SM-MAR103-NT1 Experimental Elastomer Analysis

ContentsTable of Contents

Experimental Elastomer Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

CHAPTER 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Course 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Viscoelastic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Model Limitations and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Determining Model Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

CHAPTER 4 Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Lab Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Experimental Elastomer Analysis 3

Contents

Basic Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Measuring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56What about Shore Hardness? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Testing the Correct Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Tensile Testing in the Lab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Compression Testing in the Lab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Equal Biaxial Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Compression and Equal Biaxial Strain States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Volumetric Compression Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Planar Tension Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Viscoelastic Stress Relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Dynamic Behavior – Testing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Friction Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Data Reduction in the Lab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Model Verification Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Testing at Non-ambient Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Loading/Unloading Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Test Specimen Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Fatigue Crack Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Experimental and Analysis Road Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

CHAPTER 5 Material Test Data Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . 81Major Modes of Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Confined Compression Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Hydrostatic Compression Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86General Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Mooney, Ogden Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Visual Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Material Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Future Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Adjusting Raw Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Consider All Modes of Deformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95The Three Basic Strain States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Curve Fitting with Mentat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

CHAPTER 6 Workshop Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Some Mentat Hints and Shortcuts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Model 1: Uniaxial Stress Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Model 1: Uniaxial Curve Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4 Experimental Elastomer Analysis

Contents

Model 1C: Tensile Specimen with Continuous Damage . . . . . . . . . . . . . . . . . . 123Model 1: Realistic Uniaxial Stress Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Model 2: Equi-Biaxial Stress Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139Model 2: Equi-Biaxial Curve Fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143Model 2: Realistic Equal-Biaxial Stress Specimen. . . . . . . . . . . . . . . . . . . . . . . 155Model 3: Simple Compression, Button Comp. . . . . . . . . . . . . . . . . . . . . . . . . . . 158Model 4: Planar Shear Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166Model 4: Planar Shear Curve Fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170Model 4: Realistic Planar Shear Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185Model 5: Viscoelastic Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188Model 5: Viscoelastic Curve Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

CHAPTER 7 Contact Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203Definition of Contact Bodies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204Control of Rigid Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206Contact Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207Bias Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208Deformable-to-Deformable Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209Potential Errors due to Piecewise Linear Description: . . . . . . . . . . . . . . . . . . . . 210Analytical Deformable Contact Bodies: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210Contact Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211Symmetry Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212Rigid with Heat Transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213Contact Table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215Contact Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217Exclude Segments During Contact Detection. . . . . . . . . . . . . . . . . . . . . . . . . . . 218Effect Of Exclude Option:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219Contacting Nodes and Contacted Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220Shell Thickness is taken into account according to: . . . . . . . . . . . . . . . . . . . . . . 220Friction Model Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221Stick-Slip Friction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222Coulomb (Sliding) Friction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223Shear (Sliding) Friction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224Coulomb Sliding Friction Model use Stresses or Forces . . . . . . . . . . . . . . . . . . 225Glued Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226Release Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228Interference Check / Interference Closure Amount . . . . . . . . . . . . . . . . . . . . . . 228Forces on Rigid Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229


Contents

APPENDIX A The Mechanics of Elastomers. . . . . . . . . . . . . . . . . . . . . . 231Deformation States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232General Formulation of Elastomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239Large Strain Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240Large Strain Viscoelasticity based on Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . 240Illustration of Large Strain Viscoelastic Behavior . . . . . . . . . . . . . . . . . . . . . . . 245

APPENDIX B Elastomeric Damage Models . . . . . . . . . . . . . . . . . . . . . . 247Discontinuous Damage Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248Continuous Damage Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

APPENDIX C Aspects of Rubber Foam Models. . . . . . . . . . . . . . . . . . . . 257Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258Measuring Material Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

APPENDIX D Biaxial & Compression Testing . . . . . . . . . . . . . . . . . . . . 263Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265Overall Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267The Experimental Apparatus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268Analytical Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277Attachment A: Compression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

APPENDIX E Xmgr – a 2D Plotting Tool. . . . . . . . . . . . . . . . . . . . . . . . . 281Features of ACE/gr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282Using ACE/gr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283ACE/gr Miscellaneous Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

APPENDIX F Notes and Course Critique . . . . . . . . . . . . . . . . . . . . . . . . 289Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292Course Critique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293


Ex

CHAPTER 1 Introduction

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 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.

Analysis 7

Chapter 1: Introduction Course Objective: FEA & Laboratory

Course Objective: FEA & Laboratory

Left Brain

W C1 I1 3–( ) C2 I2 3–( )+=

Wµn

αn------ λ1

αn λ2αn λ3

αn+ +( ) 3–[ ]n 1=

N

∑=

ComputerAnalyticalObjectiveLogical

W NkT I1 3–( ) 2⁄=

W G12--- I1 3–( ) 1

20N---------- I1

23

2–( ) …+ +=

Right BrainLaboratory

ExperimentalSubjectiveIntuitive


Course Objective: FEA & Laboratory Chapter 1: Introduction

Course Objective

Discuss the TESTCURVE FIT

ANALYSIS

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 Specimen


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?


Course Schedule Chapter 1: Introduction

Course Schedule

DAY 1

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

Macroscopic Behavior of Elastomers1, 2, 3

10:30 12:00 Laboratory Orientation 412:00 1:00 Lunch

1:00 3:00 Tensile Testing

3:15 5:00 Tensile Test Data Fitting 5FEA of Tensile Test Specimen 6

5:00 Adjourn

DAY 2 - Chapter 6 + Lab

Begin End Topic9:00 10:30 Equal Biaxial Testing, Compression, Volumetric

Equi-Biaxial Test Data Fitting, Comp., Volumetric

10:45 12:00 FEA of Biaxial Specimen, Comp., Volumetric

12:00 1:00 Lunch

1:00 3:00 Planar Shear Testing

3:15 5:00 Planar Shear Test Data Fitting

Data Fitting with All Test Modes

FEA of Planar Test Specimen

5:00 Adjourn


Chapter 1: Introduction Course Schedule

Course Schedule (cont.)

Keep Involved:

Tell Me and I’ll ForgetShow Me and I’ll Remember

Involve Me and I’ll Understand

DAY 3

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

Viscoelastic Data Fitting 610:45 12:00 FEA of Viscoelastic Test Specimen

12:00 1:00 Lunch

1:00 3:00 Contact and Case StudiesSpecimen Test, FEA, Part Test Correlation

7

3:15 5:00 Concluding Remarks 5:00 Adjourn


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 www.marc.com.


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.


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.


Chapter 1: Introduction Data Measurement and Analysis

Data Measurement and Analysis (cont.)

Linear Material, How is Young’s modulus, E, measured?

❑ Tension/Compression

❑ Torsion

❑ Bending

❑ Wave Speed

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

EP A⁄

∆L( ) L⁄-------------------

=

E 2 1 υ+( ) Tc J⁄φ

------------- =

E PL3

3δI---------=

E c2ρ=

T φ,

P δ,

P ∆L,


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.


Chapter 1: Introduction Typical Properties of Rubber Materials

Typical Extension of Rubber Materials (cont.)

0 200 400 600Extension (per cent)

0

1

2

3

4

Ten

sile

str

ess

(N/m

m2 )


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


Chapter 1: Introduction Important Application Areas


Ex

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

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) 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.


Temperature Effects, Tg Chapter 2: The Macroscopic Behavior of Elastomers

Temperature Effects, Tg

❑ All polymers have a spectrum of mechanical behavior, frombrittle, 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.


Chapter 2: The Macroscopic Behavior of Elastomers Time Effects, Viscoelasticity

Time Effects, Viscoelasticity

❑ Temp. & Time effects derive from long molecules slidingalong and around each other during deformation.

❑ A plot of shear modulus vs. test time:

❑ Material behavior related to molecule sliding (friction):

strain-rate effectscreep, stress-relaxationhysteresisdamping


Time Effects, Viscoelasticity Chapter 2: The Macroscopic Behavior of Elastomers

Time Effects, Viscoelasticity (cont.)

❑ Different types of tests can be used to evaluate theshort-time and long-time stress-strain behavior.

❑ Our current favorite, the Stress-relaxation test:

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


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.


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:

❑ 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 58

(The loading curve and unloading curve are not coincident).


Chapter 2: The Macroscopic Behavior of Elastomers Damage, Fatigue

Damage, Fatigue

❑ Very early stages of understanding, see Gent’s Engineeringwith 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 elastomersand 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.


Deformation States Chapter 2: The Macroscopic Behavior of Elastomers

Deformation 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 p∆V V0⁄-----------------=

KG---- 10

3∼ ∞→

υ 12---→

K G

K G

103

G


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 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 are definedfor characterization, or process control purposes. ManyASTM specs are NOT suitable for material modelcalibration.


Ex

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.

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 Elastomer Analysis

-4000 Copper, Gold

-3500 Bronze Casting

-1400 Iron Age

-1 Damascus Steel

1660 Hookean Materials

1800 Titanium 3D Elasticity

1840 Aluminum Vulcanization

1850 Parkesine

1879 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 FEA Software

1970 Foams

1975 Treloar

1980 > 200 Polymer compounds

1990 Recycle


Neo-Hookean Material Model Chapter 3: Material Models, Historical Perspective

Neo-Hookean Material Model

❑ Definitions, Stretch ratios, Engineering Strain:

❑ Incompressibility:

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

λ i

Li ∆Li+

Li-------------------- 1 εi+= = eng. strain, εi ∆Li Li⁄( )=

t1 t1

t2

t2

t3

t3

λ1L1

λ2L2λ3L3

L1

L2

L3

λ1λ2λ3 1=

W12---G λ1

2 λ22 λ3

23–+ +( )=


Chapter 3: Material Models, Historical Perspective Neo-Hookean Material Model

Neo-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

λ1 λ= λ2 λ3 1 λ⁄= =

W12---G λ2 2

λ--- 3–+

=

σ dW dλ⁄ G λ 1

λ2-----–

= = =

G 1 ε 1

1 ε+( )2-------------------–+

=

t σ1 λ⁄---------- λσ G λ2 1

λ---–

= = =


Neo-Hookean Material Extension Deformation Chapter 3: Material Models, Historical Perspective

Neo-Hookean Material Extension Deformation

Theory versus experiments:

1 2 3 4 5 6 7 8Extension ratio

0.0

2.0

4.0

6.0

En

gin

eeri

ng

str

ess

(N/m

m2 )

theory

experiment

Extension ratio

En

gin

eeri

ng

str

ess

(N/m

m2 )


Chapter 3: Material Models, Historical Perspective Neo-Hookean Material Shear Deformation

Neo-Hookean Material Shear Deformation

❑ Experimental Verification using Simple Shear:

If , then and

Equivalent shear strain :

Strain energy function:

Shear stress depends linearly on shear strain

X

Y

γatan

λ1 λ= λ21λ---= λ3 1=

γ

γ λ 1λ---–=

W12---G λ2 1

λ2----- 2–+

12---Gγ2

= =

τ γ

τ Wdγd

-------- Gγ= =


Neo-Hookean Material Shear Deformation Chapter 3: Material Models, Historical Perspective

Neo-Hookean Material Shear Deformation (cont.)

Theory versus experiments:

0 1 2 3 4 5Shear strain

0.0

0.4

0.8

1.2

1.6

Sh

ear

stre

ss (

N/m

m2 )

theory

experiment

Sh

ear

stre

ss (

N/m

m2 )

Shear strain


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

Neo-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

λ1 λ2 λ3

λ λ λ 2–

λ 1 λ 1–

λ λ 1 2⁄– λ 1 2⁄–

1 γ2

2----- γ 1 γ2

4-----++ + 1 γ2

2----- γ 1 γ2

4-----+–+ 1

W12---G λ1

2 λ22 λ3

23–+ +( )=

σλ∂

∂W σ ε( )= =

τγ∂

∂W Gγ= =


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

Neo-Hookean Material Summary (cont.)

TABLE 3. Hookean versus Neo Hookean Values of

ModeHookean

=

Hookean as Neo Hookean

=

Biaxial

Planar Shear

Uniaxial

σ G⁄

σ G⁄ υ 0→ σ G⁄

2 1 ν–( )1 2υ–( )

--------------------ε 2ε 2 1 ε 1 ε+( ) 5––+{ }

2 1 ν– ν2–( )

1 2υ–( )-------------------------------ε 2ε 1 ε 1 ε+( ) 3–

–+{ }

2 1 υ+( )ε 2ε 1 ε 1 ε+( ) 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

Eng

inee

ring

Str

ess/

She

ar M

odul

us


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 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.

λ2

11 γ2

2----- γ 1 γ2

4-----++ +=

λ22

1 γ2

2----- γ 1 γ2

4-----+–+=

λ23

1=( )

W12---G λ1

2 λ22 λ3

23–+ +( ) 1

2---Gγ2

= =

τγ∂

∂W Gγ= =

γatan

τ


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:

❑ Simple shear:

Hence

or

W C1 λ12 λ2

2 λ32

3–+ +( ) C21

λ12

----- 1

λ22

-----+ 1

λ32

----- 3–+

+=

W C1 C2+( ) λ12 1

λ12

----- 2–+

C1 C2+( )γ2= =

τ dW dγ⁄ 2 C1 C2+( )γ= =

G 2 C1 C2+( )=

σ 2 λ 1

λ2-----–

C1C2

λ------+

= σ2 λ 1 λ2⁄–( )------------------------------ C1

C2

λ------+=


Chapter 3: Material Models, Historical Perspective 2-Constant Mooney Extensional Deformation

2-Constant Mooney Extensional Deformation (cont.)

Theory versus experiments

A

B

C

DE

F

G

0.5 0.6 0.7 0.8 0.9 1.01/λ

0.1

0.2

0.3

0.4

σ/2(

λ−1/

λ2 ) (N

/mm

2 )σ/

2(λ–

1/λ2 )

(N

/mm

2 )

1/λ


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

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

❑ Incompressibility implies that , so that:

❑ Mooney material in terms of invariants:

(Mooney’s original notation)

(Mooney-Rivlin notation)

λ i

I1 λ12 λ2

2 λ32

+ +=

I2 λ12λ2

2 λ22λ3

2 λ32λ1

2+ +=

I3 λ12λ2

2λ32

=

I3 1=

W W I1 I2,( )=

W C1 I1 3–( ) C2 I2 3–( )+=

W C10 I1 3–( ) C01 I2 3–( )+=


Chapter 3: Material Models, Historical Perspective Other Mooney-Rivlin Models

Other 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–( )2C30 I1 3–( )3

+ +=

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

C20 I1 3–( )2C+ 30 I1 3–( )3


Ogden Models Chapter 3: Material Models, Historical Perspective

Ogden Models

❑ Slightly 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 .

Wµn

αn------ J

αn–3

---------

λ1αn λ2

αn λ3αn+ +( ) 3– 4.5K J

13---

1–

2

+

n 1=

N

∑=

µn αn

K

J

J λ1λ2λ3=

N 2= N 3=


Chapter 3: Material Models, Historical Perspective Foam Models

Foam Models

❑ Elastomer foams:

, and are material constants

Wµn

αn------ λ1

αn λ2αn λ3

αn+ + 3–[ ]µn

βn----- 1 J

βn–( )n 1=

N

∑+

n 1=

N

∑=

µn αn βn


Viscoelastic Models Chapter 3: Material Models, Historical Perspective

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

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 δn1 t λn⁄–( )exp–( )

n 1=

N

∑–=

δn λn


Chapter 3: Material Models, Historical Perspective Model Limitations and Assumptions

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:

When used with an Ogden model, the material may be made slightly compressible.

K 10000 C10 C01+( )=


Determining Model Coefficients Chapter 3: Material Models, Historical Perspective

Determining Model Coefficients

❑ This 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.

❑ 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.

W Eij( )

R t( )


Chapter 3: Material Models, Historical Perspective Determining Model Coefficients


Ex

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.

Analysis 51

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


Basic Instrumentation Chapter 4: Laboratory

Basic Instrumentation

Electromechanical Tensile Testers

Servo-hydraulic Testers


Chapter 4: Laboratory Basic Instrumentation

Basic Instrumentation (cont.)

Wave Propagation Instrument

Automated Crack Growth System

Aging Instrumentation


Measuring Chapter 4: Laboratory

Measuring

Force

Strain Gage Load Cells

Position

Encoders and LVDT’s

Strain

Clip-on Strain GagesVideo ExtensometersLaser Extensometers

Temperature

Thermocouples


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


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!


Chapter 4: Laboratory Testing the Correct Material

Testing the Correct Material

Consistent within The Experimental Data Set

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…

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

All Are Same Compound


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

1

2

3

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


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


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 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

3


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


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

1

2

3


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–=

λ3 Λ=

λ1 Λ 1 2⁄–=

Λ λ 2–=

λ2 Λ 1 2⁄–=

λ3 λ 2–=

λ1 λ=

λ2 λ=


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 Pressure-Volume relationship.

1

2

3

Bulk Modulus = 2.1 GPa

300

250

200

150

100

50

0

Pre

ssur

e (M

Pa)

Volumetric Strain

0.02 0.04 0.06 0.08 0.100.00

VOLCOMP_B

Base Data Set


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

Similar Stress-strain Shape to Simple Tension and Biaxial Extension

Match Loadings between Strain States 1

2

3

Base Data Set

Eng

inee

ring

Str

ess

(MP

a)

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 1.00.0


Planar Tension Test Chapter 4: Laboratory

Planar Tension Test (cont.)

A Small but Significant amount of Material will Flow From the Planar Tension Grips.


Chapter 4: Laboratory Viscoelastic Stress Relaxation

Viscoelastic Stress Relaxation

Viscoelastic BehaviorCan 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!

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)

Str

ess

(MP

a)

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.00 2000 4000 6000 8000

Time (Seconds)

Str

ess

(MP

a)

Str

ess

Strain

7

6

5

4

3

2

1

0

0 500 1000 1500 2000


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


Chapter 4: Laboratory Dynamic Behavior – Testing

Dynamic Behavior – Testing (cont.)

Mean Strain and Amplitude Effects are Significant


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:

(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

Forc

e

Position

Incr

easi

ng N

orm

al F

orce

µ µ


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.

Fit for Arruda-Boyce

Adjusted Data

Raw Data


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.


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


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.


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.


Loading/Unloading Comparison Chapter 4: Laboratory

Loading/Unloading Comparison


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.


Fatigue Crack Growth Chapter 4: Laboratory

Fatigue Crack Growth

Provides Great Potential.

Not well understood.


Chapter 4: Laboratory Experimental and Analysis Road Map

Experimental and Analysis Road Map

TABLE 4. 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 Foam Planned

9 Tube Crush Planned

TABLE 5. Analysis Workshop Models

Model Description Notes

1 Uniaxial

2 Biaxial

3 Planar Shear

4 Compression Button mu=0.4

5 Viscoelastic

6 Volumetric Compression Planned

7 Friction Sled Planned

8 Foam Planned

9 Tube Crush Planned

10 Damage Planned


Ex

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.

Analysis 81

Chapter 5: Material Test Data Fitting Major Modes of Deformation

Major Modes of Deformation

Uniaxial Tension

Biaxial Tension (equivalent strain as uniaxial compression)

1

2

3

λ1 λ2 λ= = λ2 λ3 1 λ2⁄= =

1

3

2

λ1 λ2 λ= = λ3 1 λ2⁄=


Major Modes of Deformation Chapter 5: Material Test Data Fitting

Major Modes of Deformation (cont.)

Planar Tension, Planar Shear, Pure Shear

Simple Shear

λ1 λ= λ2 1= λ3 1 λ⁄=

1

2

3


Chapter 5: Material Test Data Fitting Major Modes of Deformation

Major Modes of Deformation (cont.)

Volumetric (aka Hydrostatic, Bulk Compression)

FF

Confined Hydrostatic CompressionCompression


Confined Compression Test Chapter 5: Material Test Data Fitting

Confined Compression Test

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,

λ1 1= λ2 1= λ3 L L0⁄=

σ1 σ2 σ3 F Ao⁄– p= = = =

λ1λ2λ3 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

Pre

ssur

e [M

pa]

Volumetric DataFor Mentat Curve Fitting

13---

∆V V0⁄

p

∆L L0⁄ ∆V V0⁄=

K p∆V V0⁄------------------= p

∆L L0⁄-----------------=

p

13---

∆V V0⁄

∆V V0⁄ ∆L L0⁄=


Chapter 5: Material Test Data Fitting Hydrostatic Compression Test

Hydrostatic Compression Test

Strain 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,

λ1 λ2 λ3 λ V V0⁄( )1 3⁄= = = =

σ1 σ2 σ3 F Ao⁄– p= = = =

λ 1 ∆V+ V0⁄( )1 3⁄1 1

3---

∆V V0⁄+≅=

λ 1 ∆L L0⁄+=

∆L L0⁄ 13---

∆V V0⁄=

K p∆V V0⁄------------------= p

3 ∆L L0⁄( )--------------------------=

p13---

∆V V0⁄


General Guidelines Chapter 5: Material Test Data Fitting

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


Chapter 5: Material Test Data Fitting Mooney, Ogden Limitations

Mooney, Ogden Limitations

❑ Phenomenological models – not material “law”

These 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 compressibility


Visual Checks Chapter 5: Material Test Data Fitting

Visual Checks

❑ Extrapolations can be dangerous

❑ Always visually check your model’s predictedresponse

Check it outside of the data’s range (see below)

Check it outside the test’s scope

PredictedResponse

DATA

Real Material

PredictedResponse

Real Materialσ

dσ dε 0>•

dσ dε 0<•

ε


Chapter 5: Material Test Data Fitting Material Stability

Material Stability

❑ Unstable material model -> numerical difficultiesin FEA

❑ Druckers stability postulate,

❑ Graphically:

❑ Remember effects of Newton-Raphson andstrain range

dσ dε• 0>

σ

ε

dσ11 dε11 0>• dσ11 dε11 0<•


Future Trends Chapter 5: Material Test Data Fitting

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 frommultiple modes


Chapter 5: Material Test Data Fitting Adjusting Raw Data

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

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 1.0Engineering Strain [1]

0.0

0.5

1.0

1.5

2.0

Eng

inee

ring

Str

ess

[Mpa

]

The Raw Data4 Data Points/Second

Equal BiaxialPure ShearTension


Adjusting Raw Data Chapter 5: Material Test Data Fitting

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

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

0.5

1.0

1.5

2.0

Eng

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ring

Str

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[Mpa

]

Adjusting The Raw DataShift to the Origin

Equal Biaxial ShiftedEqual BiaxialPure Shear ShiftedPure ShearTension ShiftedTension

ε ε’ εp–( ) 1 εp+( )⁄=

σ σ’ 1 εp+( )=

εp


Chapter 5: Material Test Data Fitting Adjusting Raw Data

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

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_hookean

1

1

0

0

Engineering Strain [1]

Eng

inee

ring

Str

ess

[Mpa

]

Fit to upload & download

Fit to upload


Consider All Modes of Deformation Chapter 5: Material Test Data Fitting

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.


Chapter 5: Material Test Data Fitting The Three Basic Strain States

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

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 be come 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

0.5

1.0

1.5

2.0

Eng

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ring

Str

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[Mpa

]

The Three Basic Strain StatesGeneral Elastomer Trends

Equal BiaxialPure ShearTension


Curve Fitting with MSC.Marc Mentat Chapter 5: Material Test Data Fitting

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 PROPERTIESTABLESREAD

RAW(name of file)TABLE TYPEexperimental_dataOKRETURN

EXPERIMENTAL DATA FITTINGUNIAXIAL(pick table1)OK

ELASTOMERSNEO-HOOKEAN

UNIAXIALCOMPUTEAPPLYOKSCALE AXES


Chapter 5: Material Test Data Fitting Curve Fitting with MSC.Marc Mentat

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 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 AXES




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 AXES




❑ 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 default

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

errorR

1σcalc

i

σimeasured

------------------------– 2

i

Ndata

∑= or

errorA σi

measured σcalci

–( )2

i

Ndata

∑=

errorR

errorA

Ndata

σcalci

σimeasured




❑ Ogden fitting is nonlinear, uses downhill-simplexmethod

❑ Downhill-simplex method is an iterative method

Uses 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+( )---------------------------------------------------------------- tol

2-------<

tol

errormin




❑ Viscoelastic fitting is linear, uses least squares fitting

❑ A Prony series (exponential decay) is fit to thetest data

❑ The least squares error is given by:

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

errorR

1σcalc

i

σimeasured

------------------------– 2

i

Ndata

∑=


Ex

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.

Please note the directory hierarchy of:

eea/wkshops_A/ or eea/wkshops_B/

uniaxial

biaxial

planarcomp

visco

test_data (raw data)

Analysis 103

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.<cr> 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 currentactivity and prompts for input. Check this regionfrequently 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 modelRight – zooms in/outMiddle – rotates in 3D

Use RESET VIEW and FILL to return to original view.Be sure to turn off DYNAMIC VIEW before pickingin the graphics area.

7. CTRL P/N recall Previous/Next commands entered.


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:

UTILSPROCEDURES

EXECUTEpick the file named uni_neo05.procOKOK

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.


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 onx=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.


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.

MAINRESULTS

OPEN DEFAULTDEF & ORIGSKIP TO INC10 <cr>PLOT

SURFACES WIREFRAMEREGENRETURN

CONTOUR BANDSCALAR

Displacement Z, OKSCALAR PLOT SETTINGS

#LEVELS5 <cr>, RETURN

SKIP TO INC30 <cr>FILLREWINDMONITOR


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 stress-strain curve should match the curve fit response exactly.

HISTORY PLOT

COLLECT GLOBAL DATANODE/VARIABLES

ADD GLOBAL CRV.Pos Z cbody3Force Z cbody2FIT, 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 MODELOK, MAIN


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 PROPERTIESEXPERIMENTAL DATA FITTING

TABLESREAD

RAWFILTER: type *.data pick file st_18.data, OK


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 TYPEexperimental_data, OK, RETURN

UNIAXIALtable2



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.


UNIAXIALCOMPUTE, OK

SCALE AXESPLOT OPTIONS

SIMPLE SHEAR, RETURN (this turns off simple shear)



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 neo-Hookean 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 stress-strain 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.



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-HOOKEANEXTRAPOLATION

EXTRAPOLATELEFT BOUND, enter -0.5, <cr>RIGHT BOUND, enter 2.0, <cr>, OK

COMPUTE, OKSCALE AXES



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 (we want to turn it off)COMPUTE, OK

SCALE AXES



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 3-term). Which one is correct?

MOONEY(3)COMPUTE, OK

SCALE AXES

MOONEY(3), EXTRAPOLATIONEXTRAPOLATE, OK

COMPUTE, OKSCALE AXES (after viewing this turn extrapolate back off)



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)



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 and the exponents are . If each and have the same sign then stability is guaranteed. If a is positive and its corresponding 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.

OGDENCOMPUTE, OK

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

µi

α i µi α i

µi α i



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 post-process and show the analysis calculated stress-strain curve.

OGDEN# OF TERMS = 3, OKCOMPUTE, APPLY, OKPLOT OPTIONS (turn off all – leave uniaxial only)> XY (sends to generalized xy plotter)

RETURN (thrice)MECH. MATERIALS TYPE, MOREOGDEN (look at the material properties)

OKFILES

SAVE AS ogden3, OKMAIN

JOBSRUN

SUBMIT1MONITOR



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.

MAINRESULTS

OPEN DEFAULTHISTORY PLOT

COLLECT DATA14 30 1 <cr> (this collects just the tensile part)NODE/VARIABLES

ADD GLOBAL VAR.Pos Z cbody 3Force Z cbody 2FIT, RETURN> XY (send results to generalized 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).



To compare the two stress-strain curves, we will use MSC.Marc Mentat’s generalized plotter feature.

UTILSGENERALIZED XY PLOTFITSHOW IDS = 0



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

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

Data

Ogden

(3) f

it

Respon

se

Eng

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Str

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Engineering Strain



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.


Model 1C: Tensile Specimen with Continuous Damage Chapter 6: Workshop Problems

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.

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


0.0

0.2

0.4

0.6

0.8

1.0

Eng

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[Mpa

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Tensile DataContinuous Damage


Chapter 6: Workshop Problems 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.

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.

0.0 2.0 4.0 6.0 8.0 10.0Cycle Number

0.90

0.95

1.00

1.05

1.10

Eng

inee

ring

Str

ess

[Mpa

]

Tensile DataContinuous Damage for Engineering Strain = 1.00



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:

UTILSPROCEDURES

EXECUTEpick the file named uni_neo05.procOKMAIN

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.


TABLESREAD

NORMALFILTER: type st* pick file st_1st.tab, OK (different data from st_18.data)



Your screen should look similar to the one below

While we are here let’s read some more tables.

READNORMALFILTER: type st*pick file st_cont.tabpick file st_load.tabRETURN

Now we are ready to start curve fitting the data.

UNIAXIALtable2

CONSTANTpick st_const table



ELASTOMERSMORECONTINUOUS DAMAGE

CONSTANTNUMBER OF TERMS = 2FREE ENERGY = 1.07 (this is just the 1st peak stress)

COMPUTEAPPLY, OK, RETURN

OGDENUNIAXIALNUMBER OF TERMS = 1COMPUTE, APPLY, OK


SIMPLE SHEAR (this turns off simple shear)



RETURN (twice)

Let’s review the material properties to check that the curve fit has been properly applied to the selected material.

MAINMATERIAL PROPERTIES

MOREOGDEN, DAMAGE EFFECTS - RUBBER, OKOK



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.

MAINCONTACT

CONTACT BODIESEDIT (pick cbody3)

RIGIDPOSITION(Z) TABLE (pick table st_load)OK (twice)

MAINLOADCASE

MECHANICAL



STATICTOTAL LOADCASE TIME = 940# STEPS = 20OK

MAINFILES

SAVE AS ogden_damage OKMAIN

JOBSRUN, SUBMIT1, MONITOR, OK

MAINRESULTS


COLLECT DATA1 19 2NODE/VARIABLES

ADD GLOBAL VAR.TimeForce Z cbody 2FIT, RETURN> XY (send results to generalized xy plotter)

SAVE type ogden_damage.tab



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.

MAINRESULTS

CLOSEMAIN

JOBSRUN

ADVANCED JOB SUBMISSIONEDIT INPUT

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



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 10

time step

4.700000000000000+1

motion change

2

2 0

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

3 -1

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

continue

auto load

1 0 10

time step

4.700000000000000+1

motion change

2

2 0

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

3 -1

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

continue

end sequence

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



Save the input file and run the job by selecting the execute button, namely:

OKRUN, EXECUTE1, MONITOR, OK

MAINRESULTS


COLLECT DATA1 1999 2NODE/VARIABLES

ADD GLOBAL VAR.TimeForce Z cbody 2FIT, RETURN



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.

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.


0.0

0.2

0.4

0.6

0.8

1.0

Eng

inee

ring

Str

ess

[Mpa

]

Tensile Simulation - Continuous Damage1-Term Ogden and Original Data

1-Term OgdenOriginal Data


Model 1: Realistic Uniaxial Stress Specimen Chapter 6: Workshop Problems

Model 1: Realistic Uniaxial Stress Specimen

Extra Credit Problem Statement:

This problem is in the subdirectory named ./uniaxial/big.

Geometry is 55 L x 4 H x 2 W (mm) between grips, 10 mm length under grip.

Elements are 1 mm x 1mm in XY, 0.5 mm in Z.

Read model from file uni_f4.mud or uni_f6.mud.

Grips are modeled as discrete rigid surfaces, with a friction coefficient of 0.4 (uni_f4) and 0.6 (uni_f6).

Run analysis with Mooney 1-term model and plot engineering stress-strain, compare with original test data. Use princa.f usersub if possible.


Chapter 6: Workshop Problems Model 1: Realistic Uniaxial Stress Specimen

Results: uni_f4.mud

Mooney 1 Term C = 0.265

1 Element

600 Element

Engineering Strain

Eng

inee

ring

Str

ess

L

A=4mm^2

d

L=27.5mm


Model 1: Realistic Uniaxial Stress Specimen Chapter 6: Workshop Problems

Results: uni_f4.mud

0.0 0.5 1.0 1.5 2.0Engineering Strain

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Eng

inee

ring

Str

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[Mpa

]

Model 1 Stress Strain DiagramSpecimen, Data, Test Element Mooney C1=0.265

Specimen (Grip Force)/4 versus (Grip Disp)/27.5Uniaxial DataTest Element


Chapter 6: Workshop Problems Model 1: Realistic Uniaxial Stress Specimen

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?

________________________________________________________________________________________________________________________________________________________________________________________________________________


Model 2: Equi-Biaxial Stress Specimen Chapter 6: Workshop Problems

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.


UTILSPROCEDURES

EXECUTEpick the file named eb_neo05.procOKOK

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.


Chapter 6: Workshop Problems Model 2: Equi-Biaxial Stress Specimen


Here is a brief summary of the biaxial model we have created:


• Boundary conditions on y=0 face to prevent free translation in space.



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.


Model 2: Equi-Biaxial Stress Specimen Chapter 6: Workshop Problems


MAINRESULTS

OPEN DEFAULTDEF & ORIGSKIP TO INC10 <cr>CONTOUR BANDSCALAR

Displacement Z, OK

SCALAR PLOT SETTINGS#LEVELS5 <cr>, RETURN

SKIP TO INC30 <cr>REWINDMONITOR


Chapter 6: Workshop Problems Model 2: Equi-Biaxial Stress Specimen


HISTORY PLOTSET NODES(pick node 8 shown)END LISTCOLLECT DATA0 30 1 <cr>NODE/VARIABLES

ADD VARIABLEDisplacement ZForce Z cbody 2FIT, RETURN

RETURNCLOSE, MAIN

Since the original area is one, and since the original length in the z-direction 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.

Pick


Model 2: Equi-Biaxial Curve Fit Chapter 6: Workshop Problems

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.


TABLESREAD



Chapter 6: Workshop Problems 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.

NAMEuniaxial


UNIAXIALuniaxial

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.



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.


USE ALL DATACOMPUTE, OK





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 111). 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.



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 114) 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.


SCALE AXES



To see the old (uniaxial data only) fit response plotted along with the new data, use the EVALUATE feature.

MOONEY(2)EVALUATEtype in the old coeff’s as prompted at the command lineENTER C10: 0.074 <cr>ENTER C01: 0.280 <cr>All coefficients entered. Continue? y <cr>

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).



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.


SCALE AXES



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.

OGDENCOMPUTE, OK



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) COMPUTEAPPLY, OKPLOT OPTIONS, > XY, RETURN

RETURN (twice)MECHANICAL MATERIALS TYPE, MOREMOONEY look at the material properties

OKFILES

SAVE AS moon2, OKRETURN (twice)

JOBSRUN

SUBMIT1MONITOR



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.

MAINRESULTS


SET NODES(pick node 8 shown)END LISTCOLLECT DATA14 30 1 <cr>NODE/VARIABLES

ADD VARIABLEDisplacement ZForce Z cbody2FIT, RETURN

> XYSAVE type moon2.tab

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

Pick



To compare the two stress-strain curves we will use MSC.Marc Mentat’s generalized plotter feature.

UTILSGENERALIZED XY PLOTFITSHOW IDS = 0

Eng

inee

ring

Str

ess

[Mpa

]

Engineering Strain

Biaxial Fit

Biaxial Data

Biaxial Response

Uniaxial Fit

Uniaxial Data



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.


Model 2: Realistic Equal-Biaxial Stress Specimen Chapter 6: Workshop Problems

Model 2: Realistic Equal-Biaxial Stress Specimen


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.


Chapter 6: Workshop Problems Model 2: Realistic Equal-Biaxial Stress Specimen

Results: bi_glue.mud

0.0 0.2 0.4 0.6 0.8 1.0Engineering Strain

0.0

0.5

1.0

1.5

2.0

Eng

inee

ring

Str

ess

[Mpa

]

Model 2 Equal-BiaxialSpecimen Model versus Data

Specimen DataSpecimen Model


Model 2: Realistic Equal-Biaxial Stress Specimen Chapter 6: Workshop Problems

Model 2: Realistic Equal-Biaxial Stress Specimen (cont.)


________________________________________________________________________________________________________________________________________________________________________________________________________________

How does the specimen model compare to the test data?

________________________________________________________________________________________________________________________________________________________________________________________________________________


________________________________________________________________________________________________________________________________________________________________________________________________________________


________________________________________________________________________________________________________________________________________________________________________________________________________________


Chapter 6: Workshop Problems 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, OKSAVE AS type in comp, OK

PLOTELEMENTS SOLIDREGEN, RETURN

VIEWLOAD VIEW

(select file OBL.VIEW from list), OKRETURN

MAIN


Model 3: Simple Compression, Button Comp. Chapter 6: Workshop Problems

CONTACTCONTACT BODIESDEFORMABLE, OKELEMENTS ADD(pick top elements)NAME, uniaxialNEWDEFORMABLE, OKELEMENTS ADD(pick bottom elems)NAME, buttonNEWRIGIDDISCRETE, OKSURFACES ADD(pick z=30 surface)NAME, uni_botNEWRIGIDDISCRETE, OKSURFACES ADD(pick z=43 surface)NAME, uni_topID BACKFACES

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

SAVE

z=0

z=13

z=30

z=43



CONTACT (cont’d)NEWRIGIDDISCRETE, OKSURFACES ADD(pick z=0 surface)NAME, but_botNEWRIGIDDISCRETE, OKSURFACES 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, OKRIGID

VELOCITY PARAMETERSVELOCITY Z=-6OK (twice)

(repeat the above sequence for the but_top contact surface)

SAVE, RETURN



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 TABLENEW

PROPERTIES

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

OK, MAIN



LOADCASESMECHANICAL

STATICSTEPPING PROCEDURE FIXED PARAMETNERS

# OF STEPS=12 <cr>, OK (twice), MAINJOBS

MECHANICALlcase1ANALYSIS OPTIONS

LARGE DISPLACEMENT, OKJOB RESULTS

CAUCHY STRESSTOTAL STRAIN, OKOK

INITIAL LOADSxsymysym

CONTACT CONTROLINITIAL CONTACT

CONTACT TABLEctable1OK (3 times)

JOBSSAVERUNSUBMIT1MONITOROK, MAIN



RESULTSOPEN DEFAULTDEF & ORIGSKIP TO INC12 <cr>PLOT

SURFACES WIREFRAMEREGENRETURN



POST PROCESSINGHISTORY 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)



POST PROCESSINGHISTORY PLOTCOLLECT GLOBAL DATANODES/VARIABLESADD GLOBAL CURVEPOS Z UNI_TOPFORCE Z UNI_TOPADD GLOBAL CURVEPOS Z BUT_TOPFORCE Z BUT_TOP


Chapter 6: Workshop Problems Model 4: Planar Shear Specimen

Model 4: Planar Shear Specimenalso 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 thewkshops_A/planar or the wkshops_B/planar directory.


UTILSPROCEDURES

EXECUTEpick the file named ps_neo05.procOKOK

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.


Model 4: Planar Shear Specimen Chapter 6: Workshop Problems

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:


• Boundary conditions on y=0 face to prevent free translation in space.



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.


Chapter 6: Workshop Problems Model 4: Planar Shear Specimen


MAINRESULTS

OPEN DEFAULTDEF & ORIGSKIP TO INC10 <cr>CONTOUR BANDSCALAR

Displacement Z, OK

SETTINGS#LEVELS5 <cr>, RETURN

SKIP TO INC30 <cr>REWINDMONITOR


Model 4: Planar Shear Specimen Chapter 6: Workshop Problems


HISTORY PLOTSET NODES(pick node 8 shown)END LISTCOLLECT DATA0 30 1 <cr>NODE/VARIABLES

ADD VARIABLEDisplacement ZForce Z cbody2FIT, RETURN

RETURN

Since the original area is one, and since the original length in the z-direction 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

Pick


Chapter 6: Workshop Problems 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.


TABLESREAD



Model 4: Planar Shear Curve Fit Chapter 6: Workshop Problems

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.

NAMEuniaxial


UNIAXIALuniaxial

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.



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.


USE ALL DATACOMPUTE, OK





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 111), and the uniaxial+biaxial fit on (page 145). 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.



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 147. There is very little difference in the fit and the coefficients have changed only slightly.


SCALE AXES



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 Data

Uniaxial + Biaxial Data

Uniaxial+Biaxial+Planar Shear Data

C10 0.074 0.247 0.244

C01 0.280 0.027 0.027



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.


SCALE AXES



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.



Uniaxial Data



C10 -0.735 0.246 0.239

C01 1.21 0.029 0.035

C11 0.194 -0.0004 -0.0015

Uniaxial Data



C10 0.074 0.247 0.244

C01 0.280 0.027 0.027



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.

OGDENCOMPUTE, OK



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.



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

Uniaxial Data



-3.01 -2.55E-6 -0.353

0.733 -10.5 -.582

-0.861 1.00 0.592

-4.91 1.18 1.60

µ1

α1

µ2

α2



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) COMPUTEAPPLY, OK

RETURN (twice)MECHANICAL MATERIALS TYPE, MOREMOONEY (look at the material properties)

OKFILES

SAVE AS moon2, OKRETURN (twice)

JOBSRUN

SUBMIT1MONITOR, OK



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.

MAINRESULTS


SET NODES(pick node 8 shown)END LISTCOLLECT DATA14 30 1 <cr>NODE/VARIABLES

ADD VARIABLEDisplacement ZForce Z cbody 2 FIT, RETURN> XY

SAVE type moon2.tab

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

Pick



For the moment, we shall use the generalized xy plotter to compare the response of the model to the curve fit.

MAINRESULTSCLOSE, RETURN

UTILSGENERALIZED XY PLOT

DATA FITFIT, FILL

Planar Shear Response

Planar Shear Curve Fit

Planar Shear Data



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


Model 4: Realistic Planar Shear Specimen Chapter 6: Workshop Problems

Model 4: Realistic Planar Shear Specimen


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 stress-strain, compare with original test data. Use princa.f usersub if possible.


Chapter 6: Workshop Problems Model 4: Realistic Planar Shear Specimen

Results: pt_45.mud

0.0 0.2 0.4 0.6 0.8 1.0Engineering Strain

0.0

0.5

1.0

1.5

Eng

inee

ring

Str

ess

[Mpa

]

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

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


Model 4: Realistic Planar Shear Specimen Chapter 6: Workshop Problems

Model 4: Realistic Planar Stress Specimen (cont.)


________________________________________________________________________________________________________________________________________________________________________________________________________________

How does the specimen model compare to the test data?

________________________________________________________________________________________________________________________________________________________________________________________________________________


________________________________________________________________________________________________________________________________________________________________________________________________________________


________________________________________________________________________________________________________________________________________________________________________________________________________________


Chapter 6: Workshop Problems 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.


UTILSPROCEDURES

EXECUTEpick the file named visco.procOKOK

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.


Model 5: Viscoelastic Specimen Chapter 6: Workshop Problems


Here is a brief summary of the uniaxial model we have created:


• 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.


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).


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.

MAINMATERIAL PROPERTIES

EXPERIMENTAL DATA FITTINGTABLES

NEWADD POINT0, 0 <cr>0.30, 0.7524 <cr>0.50, 1.1695 <cr>SAVE


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.


UNIAXIALtable2



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.


UNIAXIALCOMPUTE, APPLY, OK

SCALE AXES



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.



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.



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?



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).

RETURNTABLES

READRAWFILTER: type *.datapick file 50percent.data, OK


ENERGY RELAX.table3

ELASTOMERSENERGY RELAXATION

RELAXATIONCOMPUTE, 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.



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



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.

SAVEMAINJOBS

RUNSUBMIT1MONITOROK (when finished)



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.

MAINRESULTS


COLLECT DATA1 60 1 <cr>NODE/VARIABLES

ADD GLOBAL CRVTimeForce Z cbody 2FIT, RETURN> XY

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).



We can now use the GENERALIZED XY PLOTTER to compare the response with the data.

CLOSEUTILSGENERALIZED XY PLOT

DATA FIT (this get the data fit curves)



To compare the two stress-strain curves we will use 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?



Q: Why is there a difference between the two lines?

A: Recall from (page 197) 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 neo-Hookean model. Remember (page 192) 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.


Ex

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.

Analysis 203

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.


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 curve is added to body, cbody2.

This bodies position is controlled by a table.


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


Contact Procedure Chapter 7: Contact Analysis

Contact Procedure

Deformable to Rigid Body Contact

with:

:incremental displacement vector of node

: unit normal vector with proper orientation:contact distance (Default = h/20 or t/4):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 .

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.

D DA

n∆uA

Rigid Body

Deformable Body 32Case 1 4(set of elements)

(set of curvesor surfaces)

∆uA n d–• D≤

d

Cases 2, 3: Contact detected when

Case 1: Contact not detected when∆uA n• D d–<

Case 4: Penetration detected when∆uA n• D d+>

∆uA A

nDFs

F Fs<


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 ( ):

Choosing 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:

B 0 B 1≤ ≤

Dcontact

DcontactDcontact 1 B+( )

Dcontact 1 B–( )

default with bias factor

B 0>

default with bias factor


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.

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

y


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

C1


Contact Flowchart Chapter 7: Contact Analysis

Contact Flowchart

Input

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 ite

ratio

n

Splitincrement

Changecontact

constraints


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 that WILL CAUSE PROBLEMS in contact as these nodes come into contact. 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

none


Rigid with Heat Transfer Chapter 7: Contact Analysis

Rigid 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 10


Chapter 7: Contact Analysis Rigid with Heat Transfer

Model 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 10


Contact Table Chapter 7: Contact Analysis

Contact Table

Contact Table Properties:

Single-sided Contact:

Only body 2 may contact itself

1

23

4


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.


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.


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.

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.


Effect Of Exclude Option: Chapter 7: Contact Analysis

Effect Of Exclude Option:

Standard contact

excluded segments

With exclude option


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:

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>


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.

.


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.

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

2β

2εβ

µFn

αµF

n

Ft µFn≤ static Ft αµFn≤, kinetic

∆ut

β 16–×10

α

e

ε εβ 0≈ 16–×10


Coulomb (Sliding) Friction Model Chapter 7: Contact Analysis

Coulomb (Sliding) Friction Model

Implementation 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!)

slip

slip

MARC approximation

Ft

vr

stick

C

Ft µFn2π---

vr

C----

atan≤

C


Chapter 7: Contact Analysis Shear (Sliding) Friction Model

Shear (Sliding) Friction Model

This friction model has been done using nonlinear dashpots whose stiffness depend on the relative sliding velocity as:

However, the friction force depends upon a fraction of the shear strength of the material, not the normal force:

with:

: “relative sliding velocity below which sticking is simulated”(Default = 1.0!)

slip

slip

MARC approximation

σt

vr

stick

C

σt µ Y

3-------

2π---

vr

C----

atan≤

C


Coulomb Sliding Friction Model use Stresses or Forces Chapter 7: Contact Analysis

Coulomb Sliding Friction Model use Stresses or Forces

Nodal stresses: (Applies to continuum elements)

define distributed load in opposite direction of relative sliding velocity: ,where, , friction coefficient and, , normal stress in contact point

Nodal forces: (Always used for shell elements)

define nodal load in opposite direction of relative sliding velocity, , where, , friction coefficient and, , normal force in

contact point

pt

pt µσn= µ σn

vr

pt

Ft

Ft µFn= µ Fn

Ft

vr

Ft


Chapter 7: Contact Analysis Glued Contact

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

Z

body 1

body 2


Glued Contact Chapter 7: Contact Analysis

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.


Chapter 7: Contact Analysis Release Option

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 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 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


Forces on Rigid Bodies Chapter 7: Contact Analysis

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


Chapter 7: Contact Analysis Forces on Rigid Bodies

Forces on Rigid Bodies (cont.)

Vector plotting External Force will show the forces at each node resulting from the contact constraints.


Ex

APPENDIX A The Mechanics of Elastomers


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

Analysis 231

Appendix A: The Mechanics of Elastomers Deformation States

Deformation States

❑ Stretch ratios:

❑ Incompressibility:

❑ First order approximation (Neo-Hookean):

❑ Eliminate :

t1 t1

t2

t2

t3

t3

λ1L1

λ2L2λ3L3

L1

L2

L3

λ i

Li ∆Li+

Li-------------------- 1 ε+= = engineering strain ∆Li Li⁄( ) ε==

λ1λ2λ3 1=

W12---G λ1

2 λ22 λ3

23–+ +( )=

λ3

W12---G λ1

2 λ22 1

λ12λ2

2----------- 3–+ +

=


Deformation States Appendix A: The Mechanics of Elastomers

❑ Two-dimensional extension:

Hence: , ,

❑ Engineering stresses : forces per unit undeformed area

❑ True stresses : forces per unit deformed area

dL1L1L2

dL

2F1

F1

F2

F2

dW F1dL1 F2dL2+ σ1dλ1 σ2dλ2+= =

dW∂W∂λ1---------dλ1

∂W∂λ2---------dλ2+=

σ1 G λ11

λ13λ2

2-----------–

=

σ3 0=

σ2 G λ21

λ12λ2

3-----------–

=

σi

ti


Appendix A: The Mechanics of Elastomers Deformation States

❑ Two-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 σ1 λ2λ3( )⁄ λ1σ1= =

t1 G λ12 λ3

2–( )=

t2 G λ22 λ3

2–( )=

t3 0=

p


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 ν+( )

3 1 2υ–( )------------------------=

υ 12---→ K

G---- ∞→

G KG K

104–

G


Appendix A: The Mechanics of Elastomers General Formulation of Elastomers

General Formulation of Elastomers

❑ Material 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:

Xi

xi

xi xi Xj( )=

dxi FijdXj with Fij

xi∂Xj∂

--------= =

Fij

dx( )2dX( )2

– 2EijdXidXj=

dx( )2CijdXidXj=

Cij δij 2Eij+=

Cij

∂xk

∂Xi--------

∂xk

∂Xj-------- FkiFkj= =

Eij12---

∂xk

∂Xi--------

∂xk

∂Xj-------- δij–

12--- FkiFkj δij–[ ]= =


General Formulation of Elastomers Appendix A: The Mechanics of Elastomers

❑ Introduce 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 δki uk i,+( ) δkj uk j,+( )=

Ci’j’

λ12

0 0

0 λ22

0

0 0 λ32

=

Cij

I1 Cii=

I212--- CiiCjj CijCij–( )=

I3 det Cij=

W*

W I1 I2,( ) h I3 1–( )+=


Appendix A: The Mechanics of Elastomers General Formulation of Elastomers

❑ Second-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 2∂W∂I1--------δij 2

∂W∂I2-------- δijCkk Cij–[ ] 2h

∂I3

∂Cij----------+ +=

tijρρ0----- δik ui k,+( )Skl δjl uj l,+( )=

Sij0

2∂W∂I1-------- 4

∂W∂I2--------+

02h+

0

δij=

p 2∂W∂I1--------–

04

∂W∂I2--------–

02h–=


Finite Element Formulation Appendix A: The Mechanics of Elastomers

Finite Element Formulation

❑ Modified 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

: vector quantity representing the incompressibility constraint

Sij

V∫ δEijdV Qiδui Vd

V∫– Tiδui Ad

A-∫– δλ I3 1–( ) Vd

V∫ 0=+

ui Xi( ) Nα Xi( )uiα

α∑= p Xi( ) hα Xi( )p

α

α∑=and

∆Sij Dijkln ∆Ekl

-∆p Cij

n( )1–

–=

K0( )

K1( )

+[ ] H[ ]–

H[ ] T– 0[ ]

∆uα

∆pα

P Rα

–

gα

=

K0( )[ ]

K1( )[ ]

H[ ]

P

R

g


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 visco-elastic 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:

W Eij,Qijn( ) W

0Eij( ) Qij

nEij

n 1=

N

∑–=


Large Strain Viscoelasticity based on Energy Appendix A: The Mechanics of Elastomers

where are the components of the Green-Lagrange strain tensor,

internal variables and the elastic strain energy density for instantaneous deformation.

In MSC.Marc, it is assumed that 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:

The energy function can also be written in terms of the long term moduli

resulting in a different set of internal variables :

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

Eij Qijn

W0

W0

Sij∂W∂Eij--------- ∂W

0

∂Eij---------- Qij

n

n 1=

N

∑–= =

Tijn

W Eij Tijn,( ) W

∞Eij( ) Tij

nEij

n 1=

N

∑+=

W∞

Sij∂W

∞E( )

∂Eij-------------------- Tij

n

n 1=

N

∑+=


Appendix A: The Mechanics of Elastomers Large Strain Viscoelasticity based on Energy

Observing the similarity with the equations for small strain visco-elasticity the internal variables can be obtained from a convolution expression:

where are internal stresses following from the time dependent part of the energy functions.

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

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 term

is of similar form to the short time elastic energy and only different by

a scalar multiplier .

This equation can now be rewritten as:

Tijn

Sijn τ( ) t τ–( ) λn⁄–[ ] τ

.dexp

0

t

∫=

Sijn

Sijn ∂W

n

∂Eij----------=

W W∞

Wn

t λn⁄–( )exp

n 1=

N

∑+=

W0

Wn δW

0=

W W∞ δn

W0

t λn⁄–( )exp

n 1=

N

∑+=


Large Strain Viscoelasticity based on Energy Appendix A: The Mechanics of Elastomers

where is a scalar multiplier for the energy function based on the short term values.

The stress strain relation is now given by:

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:

δn

Sij t( ) Sij∞

t( ) Tijn

t( )n 1=

N

∑+=

Sij∞ ∂W

∞

∂Eij----------- 1 δn

n 1=

N

∑– ∂W

0

∂Eij----------= =

Tijn δn

Sij0

t( ) t τ–( ) λn⁄–[ ] τ.

dexp

0

t

∫=

∆Sij tm( ) ∆Sij∞

tm( ) ∆Sijn

tm( )n 1=

N

∑+=

∆Sij∞

tm( ) Sij∞

tm( ) Sij∞

tm( )–=

∆Sijn

tm( ) βnh( ) Sij

ntm( ) Sij

ntm h–( )–[ ] α n

h( )Sijn

tm h–( )n 1=

N

∑–=


Appendix A: The Mechanics of Elastomers Large Strain Viscoelasticity based on Energy

The functions and are a function of the time step h in the time interval :

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:

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.

α βtm 1 tm,–[ ]

αnh( ) 1 h–( ) λn⁄exp–=

βnh( ) αn

h( ) λn

h-----⋅=

∆Sij tm( ) 1 1 βnh( )–[ ]δn

n 1=

N

∑– Sij0

tm( ) Sij0

tm h–( )–=

αnSij

ntm h–( )

n 1=

N

∑–

∆Sijn

tm( ) βnh( )δn

Sij0

tm( ) Sijn

tm h–( )–[ ]=

αnh( )Sij

ntm h–( )–


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:

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

At time zero (or for time processes: ), the elastic energy reduces to:

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

the equations reduce to:

or

W t( ) W∞

Wn

t λn⁄–( )exp

n 1=

N

∑+=

W∞

Wn

Wn

t λn<

W 0( ) W0

W∞

Wn

n 1=

N

∑+= =

Wn δn

W0

=

W0

W∞

W0 δn

n 1=

N

∑+= W∞

1 δn

n 1=

N

∑–

W0

=


Appendix A: The Mechanics of Elastomers Illustration of Large Strain Viscoelastic Behavior

The time dependent energy is then given by:

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

W t( ) W0

W0 δn

W0 δn

t λn⁄–( )exp

n 1=

N

∑+

n 1=

N

∑–=

W0

1 δn1 t λn⁄–( )exp–( )

n 1=

N

∑–=

W∞

1 δ–( )W0

=

W t( ) W0

1 δ 1 t λn⁄–( )exp–( )–[ ]=


Ex

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.

Analysis 247

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.

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.

0.0 0.5 1.0Time

0.0

0.2

0.4

0.6

0.8

1.0

Eng

inee

ring

Str

ain

Strain HistoryFor Discontinuous Damage


Discontinuous Damage Model Appendix B: Elastomeric Damage Models

The free energy, W, can be written as

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

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

Mathematically, the discontinuous damage model has a structure very similar to that of strain space plasticity. Hence, if a damage surface is defined as:

The loading condition for damage can be expressed in terms of the Kuhn-Tucker conditions:

The consistent tangent can be derived as:

W K α β,( )W0

=

W0

α max W0( )=

S 2K α β,( ) ∂W

0

∂C----------=

Φ W α 0≤–=

Φ 0≤ α· 0≥ α· Φ 0=

C 4 K ∂2W

0

∂C∂C---------------

∂K

∂W0

---------- ∂W

0

∂C---------- ∂W

0

∂C----------⊗+=



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 , the stress level. A loading history is thus:

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 stress-strain plot closely.

λ0

time

λ

λ0λ0

σ

12

34

5

12

3

λ


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.

σ1

σ2

σn

σ2a

σ3a

σna

1σ1

σ1----=

σ1

σ2----

σ1

σn----

w1a

w1--------- 1–

w

2aw1

---------- 1– wna

w1---------- 1–

σ1 σ1a

=

wia

12---σiaε

iaSi,≅ 1 2 3…n, ,=

wia

12---σiaε

iaSi,≅ 1 2 3…n, ,=



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

0 0.60

0.4165

Engineering Strain [1]

Eng

inee

ring

Str

ess

[Mpa

]


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:

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:


0.0

0.2

0.4

0.6

0.8

1.0

Eng

inee

ring

Str

ess

[Mpa

]

Tensile DataContinuous Damage

β ∂∂s ′-------W

0s ′( ) s′d

0

t

∫=


Appendix B: Elastomeric Damage Models Continuous Damage Model

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

You specify the phenomenological parameters , while

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.

K α β,( )

K α β,( ) d∞

dnα

αηn------–

dmβ

β

λm------–

exp

m 1=

2

∑+exp

n 1=

2

∑+=

K α β,( ) d∞

dn α δnβ+

ηn-------------------–

exp

n 1=

2

∑+=

dnα

dnβ ηn λm dm

β δn, , , , ,

d∞

time

λ1 2 3 4

λ

σ 1

2σ2

σ1 W1

1 σ1

2 σ2

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


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.

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.


0.0

0.2

0.4

0.6

0.8

1.0

Eng

inee

ring

Str

ess

[Mpa

]

Tensile Simulation - Continuous Damage1-Term Ogden and Original Data

1-Term OgdenOriginal Data


Appendix B: Elastomeric Damage Models Continuous Damage Model


Ex

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.

Analysis 257

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:

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

, and are material constants and J is defined as

.

Densification

Plateau (Elastic Buckling)

σcr

E

Cell Wall Alignment

Wµn

αn------ λ1

αn λ2αn λ3

αn+ + 3–[ ]µn

βn----- 1 J

βn–( )n 1=

N

∑+

n 1=

N

∑=

µi α i βi

J λ1λ2λ3=


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 , 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 must satisfy:

Where , and are Lame’s constants. The initial bulk modulus K and the shear modulus G can be derived from the above as:

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

βi 0=

W W λ1 λ2, λ3,( )=

W 1 1, 1,( ) 0=

λ i∂∂W

1 1, 1,( ) 0= i, 1 2 3, ,=

λ i λ j∂

2

∂∂ W

1 1, 1,( ) λ 2µδij+= i j,( ), 1 2 3, ,=

λ µ

K13--- µi α i 3βi–( )

i∑= G

12--- µiα i

i∑=

νβi–

α i 2βi–( )------------------------

i∑=


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:

where:

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:

Wµf2----- I1 3

1 2ν–ν

--------------- I3

2ν–1 2ν–---------------

1–

+– +=

µ 1 f–( )2

------------------- I2 31 2ν–

ν--------------- I3

2ν–1 2ν–---------------

1–

+–

I1 λ12 λ2

2 λ32

+ +=

I2 λ12– λ2

2– λ32–

+ +=

I3 λ12λ2

2λ32

=

λT

i1 ∆Tα+= i, 1 2 3, ,=


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:

where , is the second Piola-Kirchhoff stress tensor, is the Green-Lagrange strain tensor, is the body force per unit undeformed volume, and is the prescribed surface tractions per unit undeformed area. All elements in MSC.Marc except Herrmann elements and be used in the foam model.

Sij

V∫ δEijdV Qiδui Vd

V∫– Tiδui Ad

A-∫– 0=

Sij Eij

Qi

Ti


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, , and engineering strain with corresponding stretch, , table can then be constructed from specimen measurements. The material constants are found to satisfy the following two equations:

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

σ1

λ1

σ1

µi

λ1----- λ

α i

1Jβi–

i∑=

0 µiJλ1-----

12--- α i 1–( )

Jβi

12---–

λ1

12---

–i

∑=


Ex

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.

Analysis 263

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


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 278).

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.


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 allows the


Overall Approach Appendix D: Biaxial & Compression Testing

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.


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 264). 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


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


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:

where: Di = Diameter as measured between punched holesF = Sum of radial forcest = Original thickness

= Engineering stress

σ F ΠDit( )⁄=

σ


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.


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.

where: D = Original outside diameterF = Sum of radial forcest = Original thickness

= Engineering stress

This formula can now be used in a testing environment since all the parameters are known.

σ F ΠDt( )⁄=

σ



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.



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.



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



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.


References Appendix D: Biaxial & Compression Testing

References1. 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


Appendix D: Biaxial & Compression Testing 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.


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


Appendix D: Biaxial & Compression Testing Attachment A: Compression Analysis


Ex

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/

Analysis 281

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/auto-correlation, . . .

• Support for dynamic module loading.

• Hardcopy support for PostScript, HP-GL, FrameMaker, and InterLeaf formats.

An example of ACE/gr is shown below:


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


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:

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

Pick x column

Pick y column

Pick this to Auto Scale the plot.


Using ACE/gr Appendix E: Xmgr – a 2D Plotting Tool

Here is the resulting plot:

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.

Title Area

X-Axis Area

Y-A

xis

Are

a


Appendix E: Xmgr – a 2D Plotting Tool ACE/gr Miscellaneous Plots

ACE/gr Miscellaneous Plots

Multiple Graphs:

Menus:


ACE/gr Miscellaneous Plots Appendix E: Xmgr – a 2D Plotting Tool

Axis Summary:

Symbol Summary:


Appendix E: Xmgr – a 2D Plotting Tool ACE/gr Miscellaneous Plots

Log Plots:

Bar Charts:


Ex

APPENDIX F Notes and Course Critique


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

Analysis 289

Appendix F: Notes and Course Critique Notes

Notes


Notes Appendix F: Notes and Course Critique

Notes


Appendix F: Notes and Course Critique Notes

Notes


Course Critique Appendix F: Notes and Course Critique

Course CritiquePlease 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?


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?


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):_____________________________________


Appendix F: Notes and Course Critique Course Critique


Mar 103

Documents

technical

leonardo da

generalized

prevent free

experimental

excellent

max princ

effectively