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Contents

Experimental Elastomer Analysis . . . 1

Table of Contents. . . 3

CHAPTER 1 Introduction . . . 7

Course Objective: FEA & Laboratory . . . 8

Course Schedule . . . 11

About MSC.Marc Products. . . 13

About Axel Products, Inc.. . . 14

Data Measurement and Analysis. . . 15

Typical Properties of Rubber Materials . . . 17

Important Application Areas. . . 19

CHAPTER 2 The Macroscopic Behavior of Elastomers . . . 21

Microscopic Structure. . . 22

Temperature Effects, Tg . . . 23

Time Effects, Viscoelasticity. . . 24

Curing Effects (Vulcanization) . . . 26

Damage, Early Time. . . 27

Damage, Fatigue. . . 28

Damage, Chemical Causes . . . 28

Deformation States . . . 29

CHAPTER 3 Material Models, Historical Perspective . . . 31

Engineering Materials and Analysis. . . 32

Neo-Hookean Material Model . . . 33

Neo-Hookean Material Extension Deformation. . . 35

Neo-Hookean Material Shear Deformation . . . 36

Neo-Hookean Material Summary. . . 38

A Word About Simple Shear. . . 40

2-Constant Mooney Extensional Deformation . . . 41

Other Mooney-Rivlin Models. . . 43

Ogden Models. . . 45

Contents CHAPTER 4 Laboratory . . . 53 Lab Orientation. . . 54 Basic Instrumentation . . . 55 Measuring . . . 57 Measurements . . . 58

What about Shore Hardness? . . . 59

Testing the Correct Material . . . 60

Tensile Testing in the Lab. . . 61

Compression Testing in the Lab . . . 63

Equal Biaxial Testing . . . 65

Compression and Equal Biaxial Strain States. . . 66

Volumetric Compression Test. . . 67

Planar Tension Test. . . 68

Viscoelastic Stress Relaxation. . . 70

Dynamic Behavior – Testing. . . 71

Friction Test. . . 73

Data Reduction in the Lab. . . 74

Model Verification Experiments. . . 76

Testing at Non-ambient Temperatures . . . 78

Loading/Unloading Comparison. . . 79

Test Specimen Requirements . . . 80

Fatigue Crack Growth. . . 81

Experimental and Analysis Road Map . . . 82

CHAPTER 5 Material Test Data Fitting . . . 83

Major Modes of Deformation. . . 84

Confined Compression Test (UniVolumetric). . . 87

Hydrostatic Compression Test . . . 88

Summary of All Modes. . . 89

General Guidelines . . . 90

Mooney, Ogden Limitations . . . 91

Visual Checks . . . 92

Material Stability. . . 93 94

Contents

CHAPTER 6 Workshop Problems . . . 107

Some MSC.Marc Mentat Hints and Shortcuts . . . 108

Model 1: Uniaxial Stress Specimen . . . 109

Model 1: Uniaxial Curve Fit. . . 113

Model 1C: Tensile Specimen with Continuous Damage . . . 133

Model 1: Realistic Uniaxial Stress Specimen. . . 145

Model 2: Equi-Biaxial Stress Specimen . . . 149

Model 2: Equi-Biaxial Curve Fit. . . 153

Model 2: Realistic Equal-Biaxial Stress Specimen. . . 165

Model 3: Simple Compression, Button Comp.. . . 168

Model 4: Planar Shear Specimen . . . 176

Model 4: Planar Shear Curve Fit. . . 180

Model 4: Realistic Planar Shear Specimen. . . 195

Model 5: Viscoelastic Specimen. . . 198

Model 5: Viscoelastic Curve Fit . . . 200

Model 6: Volumetric Fit . . . 213

CHAPTER 7 Contact Analysis . . . 217

Definition of Contact Bodies . . . 218

Control of Rigid Bodies . . . 220

Contact Procedure. . . 221

Bias Factor . . . 222

Deformable-to-Deformable Contact. . . 223

Potential Errors due to Piecewise Linear Description:. . . 224

Analytical Deformable Contact Bodies:. . . 224

Contact Flowchart. . . 225

Symmetry Body . . . 226

Rigid with Heat Transfer. . . 227

Contact Table . . . 229

Contact Areas . . . 231

Exclude Segments During Contact Detection. . . 232

Effect Of Exclude Option:. . . 233

Contents

Forces on Rigid Bodies. . . 242

APPENDIX A The Mechanics of Elastomers. . . 245

Deformation States . . . 246

General Formulation of Elastomers . . . 250

Finite Element Formulation . . . 253

Large Strain Viscoelasticity. . . 254

Large Strain Viscoelasticity based on Energy. . . 254

Illustration of Large Strain Viscoelastic Behavior . . . 259

APPENDIX B Elastomeric Damage Models . . . 261

Discontinuous Damage Model . . . 262

Continuous Damage Model. . . 267

APPENDIX C Aspects of Rubber Foam Models . . . 271

Theoretical Background . . . 272

Measuring Material Constants . . . 276

APPENDIX D Biaxial & Compression Testing . . . 277

Abstract. . . 278

Introduction. . . 279

Overall Approach . . . 281

The Experimental Apparatus. . . 282

Analytical Verification . . . 285

References. . . 291

Attachment A: Compression Analysis . . . 292

APPENDIX E Xmgr – a 2D Plotting Tool. . . 295

Features of ACE/gr . . . 296

Using ACE/gr . . . 297

ACE/gr Miscellaneous Plots. . . 300

APPENDIX F Notes and Course Critique . . . 303

CHAPTER 1

Introduction

This course is to provide a fundamental understanding of how material testing and finite element analysis are combined to improve your design of rubber and elastomeric products. Most courses in elastomeric analysis stop with finite

element modeling, and leave you searching for material data. This experimental

elastomer analysis course combines

performing the analysis and the material testing. It shows how the material testing has a critical effect upon the accuracy of the analysis.

Chapter 1: Introduction Course Objective: FEA & Laboratory

Course Objective: FEA & Laboratory

Left Brain W = C₁(I₁ – 3) C+ ₂(I₂ – 3) W μn α_n --- λ₁αn λ₂αn λ₃αn + + ( ) 3– [ ] n = 1 N

∑

Course Objective: FEA & Laboratory Chapter 1: Introduction

Course Objective

CURVE 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 Elastomers

1, 2, 3 10:30 12:00 Laboratory Orientation 4 12:00 1:00 Lunch

1:00 3:00 Tensile Testing

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

5:00 Adjourn

DAY 2 - Chapter 6 + Lab

Begin End Topic

9:00 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

Chapter 1: Introduction Course Schedule

Course Schedule (cont.)

•Keep Involved:

Tell Me and I’ll Forget Show 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 6 10:45 12:00 FEA of Viscoelastic Test Specimen

12:00 1:00 Lunch

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

7

3:15 5:00 Concluding Remarks

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:

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? E P A⁄ ΔL ( ) L⁄ ---⎝ ⎠ ⎛ ⎞ = E 2 1( +υ) Tc J⁄ φ ---⎝ ⎠ ⎛ ⎞ = E PL 3 3δI ---= E = c2ρ T, φ P, δ P, ΔL

Typical Properties of Rubber Materials Chapter 1: Introduction

Typical

P

roperties 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.

1. The stress strain function for the 1st time an elastomer is

strained is never again repeated. It is a unique event.

2. The stress strain function does stabilize after between 3 and

20 repetitions for most elastomers.

3. The stress strain function will again change significantly if

the material experiences strains greater than the previous

stabilized level. In general, the stress strain function is

sensitive to the maximum strain ever experienced.

4. The stress strain function of the material while increasing

strain is different than the stress strain function of the material

while decreasing strain.

Chapter 1: Introduction Typical Properties of Rubber Materials

Typical Loading of Rubber Materials (cont.)

0.0 2.0 4.0 6.0 2 1 0 3 4 5 6 7 Engineering Strain Experiment Theory Engineer ing Stress [MP a] 0.4 0.6 0.8 1.0 1.2 Stress [MPa]

Important Application Areas Chapter 1: Introduction

Important Application Areas

– Car industry (tires, seals, belts, hoses, etc.)

– Biomechanics (tubes, pumps, valves, implants, etc.) – Packaging industry

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

Chapter 2: The Macroscopic Behavior of Elastomers Microscopic Structure

Microscopic Structure

• Long coiled molecules, with points of entanglement.

Behaves like a viscous fluid.

• Vulcanization creates chemical bonds (cross-links) at

these entanglement points.

Now behavior is that of a rubbery viscous solid.

• Initial orientation of molecules is random.

Behavior is initially isotropic.

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

Temperature Effects, T

• All polymers have a spectrum of mechanical behavior, from

brittle, or glassy, at low temperatures, to rubbery at high temperatures.

• The properties change abruptly in the glass transition region.

• The center of this region is known as the T_g, the

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

Time Effects, Viscoelasticity

• Temp. & Time effects derive from long molecules sliding

along and around each other during deformation.

• A plot of shear modulus vs. test time:

• Material behavior related to molecule sliding (friction): • strain-rate effects

creep, stress-relaxation hysteresis

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

Time Effects, Viscoelasticity (cont.)

• Different types of tests can be used to evaluate the

short-time and long-time stress-strain behavior.

Chapter 2: The Macroscopic Behavior of Elastomers Curing Effects (Vulcanization)

Curing Effects (Vulcanization)

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

• Be careful that real parts and test specimens share the same

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 same

load history, Preconditioning.

This is a textbook

idealization. Real material behavior looks like:

“Progressively Increasing Load History…” on

page 60

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

Chapter 2: The Macroscopic Behavior of Elastomers Damage, Fatigue

Damage, Fatigue

• Very early stages of understanding, see Gent’s Engineering

with Rubber, Chapter 6, Mechanical Fatigue.

http://www.amazon.com/exec/obidos/ASIN/1569902992/ref%3Ded%5Foe%5Fh/002-1221807-2520837

• Beyond scope of this course.

Damage, Chemical Causes

• Many other chemicals are known to damage elastomers

and degrade the mechanical behavior:

Ozone Brake Fluid

Oxidation Hydraulic Fluid Ultraviolet Radiation

Oil, Gasoline

• Sometimes preconditioning of test specimens can be

helpful in gauging these 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 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 when the material can shear.

• We say rubber is (nearly) incompressible in those cases

when it is not highly confined. G K p ΔV V⁄ ₀ ---= K G ---- ∼ 103 →∞ υ 1 2 ---→ 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 is

that the stress/strain state be homogeneous.

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

• Or at least homogeneous throughout a large area/volume

of the test specimen (minimize end effects).

• It is good practice to model and analyze the test specimen

itself to prove homogeneity.

• The “button compression” test is notoriously bad from

this perspective.

• Keep in mind that many ASTM test standards are

defined for characterization, or process control purposes. Many ASTM specs are NOT suitable for material model calibration.

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

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

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 + ΔLi L_i --- 1 +ε_i = = eng. strain, ε_i = (ΔL_i ⁄ L_i) t₁ t₁ t₂ t₂ t₃ t₃ λ₁L₁ λ₂L₂ λ3L3 L₁ L₂ L₃ λ₁λ₂λ₃ = 1 W 1 2 ---G(λ₁2 + λ₂2+ λ₃2 – 3) =

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 0.8 0.4 0.0 0.4 0.8 Engineering Strain 25.0 15.0 5.0 5.0 Engineering Stress/(Shear Modulus) NeoHookean Behavior

Tension and Compression very Different

Hookean (nu=.45) NeoHookean W 1 2 ---G λ2 2 λ --- – 3 + ⎝ ⎠ ⎛ ⎞ = σ dW d⁄ λ G λ 1 λ2 ---– ⎝ ⎠ ⎛ ⎞ = = = G 1 ε 1 1 +ε ( )2 ---– + ⎝ ⎠ ⎛ ⎞ = t σ 1 ⁄λ --- λσ G λ2 1 λ ---– ⎝ ⎠ ⎛ ⎞ = = =

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

Neo-Hookean Material Extension Deformation

0.0 2.0 4.0 6.0 Experiment Theory Engineer ing Stress [MP a]

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

Neo-Hookean Material Shear Deformation

:

If , then and Equivalent shear strain :

Strain energy function:

Shear stress depends linearly on shear strain X Y γ atan τ λ₁ = λ λ₂ 1 λ ---= λ₃ = 1 γ γ = λ – _λ---1 W 1 2 ---G λ2 1 λ2 --- – 2 + ⎝ ⎠ ⎛ ⎞ 1 2 ---Gγ2 = = τ γ

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

Neo-Hookean Material Shear Deformation (cont.)

Shear Strain Experiment Theory Shear Stress [N/mm ] 2 0.0 0.4 0.8 1.2 1.6 0 1 2 3 4 6

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

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 γ 2 2 --- γ 1 γ 2 4 ---+ + + 1 γ 2 2 --- γ 1 γ 2 4 ---+ – + 1 W 1 2 ---G(λ₁2 +λ₂2 + λ₃2 – 3) = σ = _∂∂W_λ = σ ε( ) τ = _∂∂W_γ = Gγ

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

Neo-Hookean Material Summary (cont.)

TABLE 3. Hookean versus Neo Hookean Values of

Mode Hookean

=

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} -5.0 0.0 5.0 10.0

Hookean and Neo Hookean Material Models

Poisson’ Ratio = 0.45 Hookean Biaxial

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

n g ine er in g S tr e ss /S hea r Mod u lu s

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 1 1 γ2 2 --- γ 1 γ 2 4 ---+ + + = ⎝ ⎠ ⎛ ⎞ _λ2 2 1 γ2 2 --- γ 1 γ 2 4 ---+ – + = ⎝ ⎠ ⎛ ⎞ _λ2 3 = 1 ( ) W 1 2 ---G(λ₁2 + λ₂2 +λ₃2 – 3) 1 2 ---Gγ2 = = τ = _∂∂W_γ = Gγ γ atan 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 + λ₂2 +λ₃2 – 3) C₂ 1 λ₁2 --- 1 λ₂2 ---+ 1 λ₃2 --- – 3 + ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ + = W (C₁ +C₂) λ₁2 1 λ₁2 --- – 2 + ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ C₁ +C₂ ( )γ2 = = τ = dW d⁄ γ = 2 C( ₁ +C₂)γ G = 2 C( ₁ +C₂) σ 2 λ 1 λ2 ---– ⎝ ⎠ ⎛ ⎞ _C 1 C₂ λ ---+ ⎝ ⎠ ⎛ ⎞ = σ 2(λ 1 λ– ⁄ 2) --- 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 D E F G 0.5 0.6 0.7 0.8 0.9 1.0 1/λ 0.1 0.2 0.3 0.4 σ /2( λ −1/ λ 2 ) (N/mm 2 ) σ /2( λ –1/ λ 2 ) ( N /m m 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) λi I₁ = λ₁2+ λ₂2 + λ₃2 I₂ = λ₁2λ₂2 + λ₂2λ2₃ + λ₃2λ₁2 I₃ = λ₁2λ₂2λ₃2 I₃ = 1 W = W I( ₁, I₂) W = C₁(I₁ – 3) C+ ₂(I₂ – 3) W = C₁₀(I₁ – 3) C+ ₀₁(I₂ – 3)

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

Other Mooney-Rivlin Models (cont)

The Signiorini form:

The Yeoh form:

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

W = C₁₀(I₁ – 3) C+ ₀₁(I₂ – 3) C+ ₂₀(I₁ – 3)2

W = C₁₀(I₁ – 3) C+ ₂₀(I₁ – 3)2+ C₃₀(I₁ – 3)3

W = C₁₀(I₁ – 3) C+ ₀₁(I₂– 3) C+ ₁₁(I₁ – 3) I( ₂ – 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 unit volume should be 0.01

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

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

W μn α_n --- J αn – 3 ---λ₁αn λ₂αn λ₃αn + + ( ) 3– 4.5K J 1 3 ---1 – ⎝ ⎠ ⎜ ⎟ ⎛ ⎞2 + n = 1 N

∑

Chapter 3: Material Models, Historical Perspective Ogden Models

Ogden Models

Let’s suppose we want to fit a 1-term Ogden for tension. 1.) Assume incompressible (J=1) then

2.) Strain mode is tension, thus and

3.) Compute engineering stress, ,

or

4.) Fit data, say to st_18.data that has 60 stress-strain points. Find such that , has the “best fit.” 5.) Panic is nonlinear. Ok, use program and

W μ α ---[(λ₁α + λ₂α + λ₃α) 3– ] = λ₁ = λ λ₂ = λ₃ = 1 ⁄ λ W μ α --- λα 2λ α 2 ---– 3 – + ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ = σ dW d⁄ λ μ λα 1– λ α 2 ---+1 ⎝ ⎠ ⎛ ⎞ – – ⎝ ⎠ ⎜ ⎟ ⎜ ⎟ ⎛ ⎞ = = σ dW d⁄ λ μ 1 ε( + )α 1– (1 +ε) α 2 ---+1 ⎝ ⎠ ⎛ ⎞ – – ⎝ ⎠ ⎜ ⎟ ⎜ ⎟ ⎛ ⎞ σ μ α ε( , , ) = = = μ and α, σ_i = σ μ α ε( , , _i) i = 1 60, σ_i = σ μ α ε( , , _i) μ 25.78=

Ogden Models Chapter 3: Material Models, Historical Perspective

Ogden Models

6.) Plot .

7.) Repeat plot of engineering stress versus engineering strain for biaxial and planar shear where:

TABLE 4. Basic Deformation Modes

Mode Biaxial Planar Shear σ 25.78 (1 + ε)0.05298–1 (1 + ε) 0.05298 2 --- +1 ⎝ ⎠ ⎛ ⎞ – – ⎝ ⎠ ⎜ ⎟ ⎜ ⎟ ⎛ ⎞ = λ₁ λ₂ λ₃ λ λ _λ–2 λ 1 λ–1 1.357

Chapter 3: Material Models, Historical Perspective Foam Models

Foam Models

, and are material constants

W μn αn --- λ₁αn λ₂αn λ₃αn + + – 3 [ ] μn βn --- 1 Jβn – ( ) n = 1 N

∑

∑

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

Model Limitations and Assumptions

This material model assumes that the rate of relaxation is independent of the load magnitude. For instance, for relaxation tests at 20%, 50%, and 100% strain, the percent reduction in stress at any time point should be the same.

The relaxation is purely deviatoric, there is no relaxation associated with the dilatational (bulk) behavior.

When used with a Mooney-Rivlin form model, the material is assumed to be incompressible. In MSC.Marc some small

compressibility is introduced for better numerical behavior, namely if no bulk modulus is specified, then MSC.Marc computes the

following for the bulk modulus:

When used with an Ogden model, the material may be slightly compressible, and if a bulk modulus is not supplied, it is estimated as: K = 10000 C( ₁₀ + C₀₁) K 2500 μ_nα_n n = 1 N

∑

Chapter 3: Material Models, Historical Perspective Viscoelastic Models

Viscoelastic Models

MSC.Marc has the capability to perform both small strain and large strain viscoelastic analysis. The focus here will be on the use of the large strain viscoelastic material model.

MSC.Marc’s large strain viscoelastic material model is based on a multiplicative decomposition of the strain energy function

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 E( _ij, t) = W E( ) R t_ij × ( ) W E( )_ij R t( ) R t( ) 1 δn(1–exp(–t ⁄ λn)) n = 1 N

∑

Determining Model Coefficients Chapter 3: Material Models, Historical Perspective

Determining Model Coefficients

This material model requires two different types of tests be conducted and two separate curve fits be performed.

The time-independent function, , is determined from standard uniaxial, biaxial, etc., stress-strain tests. These tests are covered in more detail in Chapter 5 and demonstrated in

Chapter 6.

The time-dependent function, , is determined from one or more stress relaxation tests. This is a test at constant strain, where one measures the stress over a period of time. For example,

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

W E( )_ij

R t( )

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.

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

Basic Instrumentation Chapter 4: Laboratory

Basic Instrumentation

Electromechanical Tensile Testers

Chapter 4: Laboratory Basic Instrumentation

Basic Instrumentation (cont.)

Automated Crack Growth System

Measuring Chapter 4: Laboratory

Measuring

Force

Strain Gage Load Cells

Position

Encoders and LVDT’s

Strain

Clip-on Strain Gages

Video Extensometers

Laser Extensometers

Temperature

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!

“The Shore Round Style Durometer was introduced in 1944. It is a general purpose device that is considered the most widely used instrument throughout the world for the hardness testing of cellular,

Chapter 4: Laboratory Testing the Correct Material

Testing the Correct Material

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 Material 150mm x 150mm x 2mm Sheet

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 Material 150mm 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

Compression Testing in the Lab Chapter 4: Laboratory

Compression Testing in the Lab

Minimize Lateral Constraint due to Friction at the Specimen Loading Platen Interface

Friction Effects Alter the Stress-strain Curves

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

1 2

Chapter 4: Laboratory Compression Testing in the Lab

Compression Testing in the Lab (cont.)

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

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⁄ – = λ1 = 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 Pres s u re (M Pa) VOLCOMP_B

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

2

3

Base Data Set

En gi nee ring S tres s (M Pa ) Planar Tension 0.6 0.5 0.4 0.3 0.2 0.1 0.0 PT23C_B

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 Behavior

Can be Assumed to Reasonably Follow Linear Viscoelastic Behavior in Many Cases

Is not the same as aging!

Describes the short term reversible behavior of elastomers.

Tensile, shear and biax have similar viscoelastic properties!

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) S tres s (M Pa ) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 2000 4000 6000 8000 S tress (MPa) St re s s Strain 7 6 5 4 3 2 1 0

Dynamic Behavior – Testing Chapter 4: Laboratory

Dynamic Behavior – Testing

Types of Dynamic Behavior

Large strains at high velocity

Chapter 4: Laboratory Dynamic Behavior – Testing

Dynamic Behavior – Testing (cont.)

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

Friction Test Fric tio n Fo rce Position In c rea s ing Norm al F o rc e 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

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

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 5. Experimental Tests

Test Description Notes

1 Uniaxial

1a Uniaxial - Rate Effects

2 Biaxial

2a Biaxial - Temperature Effects 3 Planar Shear

4 Compression Button 5 Viscoelastic

6 Volumetric Compression 7 Friction Sled

8 Viscoelastic Damper Planned

9 Foam Planned

TABLE 6. Analysis Workshop Models

Model Description Notes

1 Uniaxial 2 Biaxial 3 Planar Shear 4 Compression Button 5 Viscoelastic 6 Volumetric Compression

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.

Chapter 5: Material Test Data Fitting Major Modes of Deformation

Major Modes of Deformation

Uniaxial Tension

Biaxial Tension

2

3 λ₁ = λ₂ = λ λ₂ = 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

Chapter 5: Material Test Data Fitting Major Modes of Deformation

Major Modes of Deformation (cont.)

Volumetric (aka Hydrostatic, Bulk Compression)

F F

Confined

Hydrostatic

Compression

Compression

Confined Compression Test (UniVolumetric) Chapter 5: Material Test Data Fitting

Confined Compression Test (UniVolumetric)

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

F L,

λ₁ = 1 λ₂ = 1 λ₃ = L L⁄ ₀

σ₁ = σ₂ = σ₃ = – F A⁄ _o = p

λ₁λ₂λ₃ = V V⁄ ₀ = L L⁄ ₀

0.000 0.010 0.020 0.030 0.040

Equivalent Uniaxial Strain [1] 0.0 100.0 200.0 300.0 400.0 P re ssur e [M pa ] Volumetric Data

For Mentat Curve Fitting

1 3 ---⎝ ⎠ ⎛ ⎞_{ΔV V} 0 ⁄ p ΔL L⁄ 0 = ΔV V⁄ 0 K p ΔV V⁄ 0 ---= p ΔL L⁄ 0 ---= p 1 ⎛ ⎞_{ΔV V⁄}

Chapter 5: Material Test Data Fitting Hydrostatic Compression Test

Hydrostatic Compression Test

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

F L, λ₁ = λ₂ = λ₃ = λ = (V V⁄ ₀)1 3⁄ σ1 = σ2 = σ3 = – F A⁄ o = p λ (1 + ΔV ⁄ V₀)1 3⁄ 1 1 3 ---⎝ ⎠ ⎛ ⎞_{ΔV V} 0 ⁄ + ≅ = λ = 1 +ΔL L⁄ ₀ ΔL L⁄ ₀ 1 3 ---⎝ ⎠ ⎛ ⎞_{ΔV V} 0 ⁄ = K p ΔV V⁄ ₀ ---= p 3(ΔL L⁄ ₀) ---= p

Summary of All Modes Chapter 5: Material Test Data Fitting

Summary of All Modes

Mode: X x₁ x₂ x₃ = F = λi, i = 1, 2, 3 b–λ2_i1 = 0 Uniaxial λX1 X₂ λ ---X₃ λ ---λ 0 0 0 1 λ --- 0 0 0 1 λ ---λ2 0 0 0 1 λ --- 0 0 0 1 λ ---λ 1/ λ 1/ λ Biaxial λX1 λX2 X₃ λ2 ---λ 0 0 0 λ 0 0 0 1 λ2 ---λ2 0 0 0 λ2 0 0 0 1 λ4 ---λ λ 1/λ2 b = F FT Planar λX1 X₂ λ ---X₃ λ 0 0 0 1 λ --- 0 0 0 1 λ2 0 0 0 1 λ2 --- 0 0 0 1 λ 1/λ 1 Simple Shear X₁+γX₂ X₂ X₃ 1 γ 0 0 1 0 0 0 1 1+γ2 γ 0 0 0 γ 1 0 1 1 γ γ γ γ γ γ 2 2 ---- 1 2 4 ----+ + + 1 2 2 ---- 1 2 4 ----+ – + 1 UniVolumetric X₁ X₂ λX3 1 0 0 0 1 0 0 0 λ 1 0 0 0 1 0 0 0 λ2 1 1 λ Maping Shape Deformation Gradient Figer Tensor Principal Stretch Ratios γ _τ Volumetric λX1 λX2 λX3 λ 0 0 0 λ 0 0 0 λ λ2 0 0 0 λ2 0 0 0 λ2 λ λ λ

Chapter 5: Material Test Data Fitting General Guidelines

General Guidelines

Its just curve fitting!

No Polymer physics as basis Don’t use too high order fit

Remember polynomial fit lessons (high school?)

Number of Data Points

Don’t use too many Regularize if needed Add/Subtract points if needed

Weighting of Points

Range and Scope of Data

Check fit outside range of data

Mooney, Ogden Limitations Chapter 5: Material Test Data Fitting

Mooney, Ogden Limitations

Phenomenological models – not material “law”

These models are mathematical forms, nothing more

Summary of phenomenological models given by

Yeoh (1995)

“Rivlin and Saunders (1951) have pointed out that the agreement between experimental tensile data and the Mooney-Rivlin

equation is somewhat fortuitous. The Mooney-Rivlin model obtained by fitting tensile data is quite inadequate in other modes of deformation, especially compression.”

Using only uniaxial tension data is dangerous!

Mooney model in MSC.Marc allows no

compressibility

Chapter 5: Material Test Data Fitting Visual Checks

Visual Checks

Extrapolations can be dangerous

Always visually check your model’s predicted

response

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

Predicted Response DATA Real Material Predicted Response Real Material σ dσ dε 0• > dσ dε 0• < ε

Material Stability Chapter 5: Material Test Data Fitting

Material Stability

Unstable material model -> numerical difficulties

in FEA

Druckers stability postulate,

Graphically:

Remember effects of Newton-Raphson and

strain range

dσ dε• > 0

σ

ε

Chapter 5: Material Test Data Fitting Future Trends

Future Trends

Statistical Mechanics Models

Based on single-chain polymer chain physics

Build up to network level using non-gaussian statistics

8 Chain model by Arruda-Boyce (1993)

2 parameter model, can be expressed in terms of I₁

Paper: “A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials”, J. Mech. Phys. Solids, V41 N2, pp 389-412.

Also similar is the Gent model (1996)

Paper: “A new Constitutive Relation for Rubber”, Rubber Chem. and Technology, v. 69, pp 59-61.

Claim: alleviates need to gather test data from

multiple modes

Adjusting Raw Data Chapter 5: Material Test Data Fitting

Adjusting Raw Data

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

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

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 Equal Biaxial Engineering Str ess [Mpa]

The Raw Data (4 points/sec)

Engineering Strain [1] Pure Shear

Chapter 5: Material Test Data Fitting Adjusting Raw Data

Adjusting Raw Data (cont.)

The raw data is adjusted as shown below by taking the 18th upload cycle. In doing this Mullins effect is ignored. This 18th upload cycle

then needs to be shifted such that the curve passes through the origin. Remember hyperelastic models must be elastic and have their stress

0.0 0.2 0.4 0.6 0.8 1.0 Engineering Strain [1] 0.0 0.5 1.0 1.5 2.0 Engineering Stress [Mpa]

Adjusting The Raw Data

Shift to the Origin

Equal Biaxial Shifted Equal Biaxial

Pure Shear Shifted Pure Shear Tension Shifted Tension σ σ₌ '_{(1 εp+ )} ε ε ' ε ε p p – ( _{) 1 εp}⁄( ₊ ) =

Adjusting Raw Data Chapter 5: Material Test Data Fitting

Adjusting Raw Data (cont.)

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

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] Engi neering S tr es s [Mpa] Fit to upload & download Fit to upload

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

Consider All Modes of Deformation

The plot below illustrates the danger in curve fitting only the tensile data, namely the other modes may become too stiff. This is why MSC.Marc Mentat always draws the other modes even when no experimental data is present.

Below, a 3-term Ogden provides a great fit to the tensile data, but spoils the other modes. This can be avoided by looking for a balance between the various deformation modes.

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

The Three Basic Strain States

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

increasing stress for the same strain, respectively. Knowledge of this and the actual shape above where say at a strain of 80%, the ratio of equal biaxial to uniaxial stress is about 2 (i.e., 1.3/0.75 = 1.73) will become very

0.0 0.2 0.4 0.6 0.8 1.0 Engineering Strain [1] 0.0 0.5 1.0 1.5 2.0 Engineering Stress [Mpa]

The Three Basic Strain States

General Elastomer Trends

Equal Biaxial Pure Shear Tension

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

Curve Fitting with MSC.Marc Mentat

Objective: Fit experimental data of Mooney or Ogden materials with

MSC.Marc Mentat. Begin at the main menu.

MATERIAL PROPERTIES TABLES READ RAW (name of file) TABLE TYPE experimental_data OK RETURN

EXPERIMENTAL DATA FITTING UNIAXIAL (pick table1) OK ELASTOMERS NEO-HOOKEAN UNIAXIAL COMPUTE APPLY OK

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

Curve Fitting with MSC.Marc Mentat (cont)

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 OPTIONS

SIMPLE SHEAR (this toggles it off) PLANAR SHEAR (this toggles it off) RETURN

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

Curve Fitting with MSC.Marc Mentat (cont)

Objective: Fit experimental data of Viscoelastic materials with

MSC.Marc Mentat. Begin at the main menu.

MATERIAL PROPERTIES TABLES READ RAW (name of file) TABLE TYPE experimental_data OK RETURN EXPERIMENTAL DATA FITTING ENERGY RELAX (pick table1),OK ELASTOMERS ENERGY RELAX RELAXATION # OF TERMS 3 COMPUTE APPLY, OK