Modeling+Rubber+and+Viscoelasticity+with+Abaqus-

Modeling Rubber and Viscoelasticity with Abaqus

Motivation • Rubber materials are found in many components. • Some of these are illustrated on the following slide. • Rubber applications include tires, gaskets, and bushings, among others. • The vast number of applications that use rubber materials necessitates a good understanding of the modeling techniques used to analyze rubber components.


1

Motivation

Tire Gasket

Deck lid

Mount Bushing Boot


Day 1 • Lecture 1

Rubber Physics

• Lecture 2

Rubber Elasticity Models

• Lecture 3

Physical Testing

• Lecture 4

Curve Fitting

• Lecture 5

Abaqus Usage

• Workshop 1


2

Day 2 • Lecture 6

Modeling Considerations and Usage Tips in Abaqus

• Workshop 2 • Lecture 7

Viscoelastic Material Behavior

• Lecture 8

Time Domain Viscoelasticity

• Lecture 9

Frequency Domain Viscoelasticity

• Lecture 10

Time-Temperature Correspondence

• Workshop 3 • Lecture 11

Modeling Advanced Behaviors


Additional Material • Appendix 1

Finite Deformations

• Appendix 2

Rubber Elasticity Models: Mathematical Forms

• Appendix 3

Linear Viscoelasticity Theory

• Appendix 4

Harmonic Viscoelasticity Theory

• Appendix 5

Suggested Reading


3

Legal Notices The Abaqus Software described in this documentation is available only under license from Dassault Systèmes and its subsidiary and may be used or reproduced only in accordance with the terms of such license. This documentation and the software described in this documentation are subject to change without prior notice. Dassault Systèmes and its subsidiaries shall not be responsible for the consequences of any errors or omissions that may appear in this documentation. No part of this documentation may be reproduced or distributed in any form without prior written permission of Dassault Systèmes or its subsidiary. © Dassault Systèmes, 2009. Printed in the United States of America Abaqus, the 3DS logo, SIMULIA and CATIA are trademarks or registered trademarks of Dassault Systèmes or its subsidiaries in the US and/or other countries. Other company, product, and service names may be trademarks or service marks of their respective owners. For additional information concerning trademarks, copyrights, and licenses, see the Legal Notices in the Abaqus 6.9 Release Notes and the notices at: http://www.simulia.com/products/products_legal.html.


Revision Status Lecture 1

3/09

Updated for 6.9

Workshop 1

3/09

Updated for 6.9

Lecture 2

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Updated for 6.9

Workshop 2

3/09

Updated for 6.9

Lecture 3

3/09

Updated for 6.9

Workshop 3

3/09

Updated for 6.9

Lecture 4

3/09

Updated for 6.9

Workshop Answers 2

3/09

Updated for 6.9

Lecture 5

3/09

Updated for 6.9

Workshop Answers 3

3/09

Updated for 6.9

Lecture 6

3/09

Updated for 6.9

Lecture 7

3/09

Updated for 6.9

Lecture 8

3/09

Updated for 6.9

Lecture 9

3/09

Updated for 6.9

Lecture 10

3/09

Updated for 6.9

Lecture 11

3/09

Updated for 6.9

Appendix 1

3/09

Updated for 6.9

Appendix 2

3/09

Updated for 6.9

Appendix 3

3/09

Updated for 6.9

Appendix 4

3/09

Updated for 6.9

Appendix 5

3/09

Updated for 6.9


4

Notes

5

Notes

6

Rubber Physics Lecture 1

L1.2

Overview • Solid Rubber • Network Structure • Vulcanization • Temperature and Time Dependence • Damage • Real Stress - Strain Response

• Anisotropy • Rubber Foams • Cellular Structure • Compressive vs. Tensile Behavior


7

Solid Rubber

L1.4

Solid Rubber: Network Structure • Network of many entangled polymer chains • “E” in the picture represents points of entanglement

• Long chains slide across each other; the network acts as a viscous fluid. • Examples: natural rubber as extracted, latex • Network has randomness of orientation.

• Behavior is isotropic • Vulcanization process (sulphur and heat) creates chemical bonds between chains at points of entanglement – we call these bonds cross-links. Changes behavior to that of viscous solid. Cross-link density changes the modulus. • Fillers such as carbon black create additional bonds and modify the mechanical behavior.

from Engineering Materials 2, by Ashby & Jones

• Fillers may also introduce microstructural changes that lead anisotropic response. Modeling Rubber and Viscoelasticity with Abaqus

8

L1.5

Solid Rubber: Vulcanization • Curing or vulcanization produces additional chemical bonds called cross-links; these bonds stiffen the elastomer network. • Cross-link density is function of degree of cure, related to curing process, cure temperature, and curing duration. • Be very careful that your real component and test specimens share the same cure history, thus the same stiffness.

• Best solution: cut test specimens from actual parts.

from Engineering Materials 2, by Ashby & Jones


L1.6

Solid Rubber: Temperature Dependence • Temperature dependence • The mobility of these long-chain molecules is strongly temperature dependent. • At extremely low temperatures (relative to the glass transition temperature) the chains are very immobile and the material behaves as a brittle or glassy solid – very stiff. • At higher temperatures the long-chain molecules are more mobile and the material exhibits what we call a “rubbery” behavior. • Even in the rubbery regime, the long-chain mobility is still quite temperature dependent and the force-displacement behavior, stressstrain behavior, or modulus becomes softer as the temperature rises.


9

L1.7

Solid Rubber: Time Dependence • Viscoelastic behavior • The sliding of entangled long-chain molecules gives rise to rubber’s time-dependent or viscoelastic behavior


L1.8

Solid Rubber: Time Dependence • Hysteresis behavior • Long-molecules “rub” against each other (dissipate energy). In a load/unload cycle this appears as hysteresis. • Dissipated energy appears as heat.


10

L1.9

Solid Rubber: Damage • Cross-link damage may result from straining. • The Mullins effect specifically references loss of stiffness damage.

Idealized behavior


L1.10

Solid Rubber: Real Stress-Strain Response • Typical uniaxial tension response • Load, unload cycles show damage, hysteresis, and permanent set • Progressive loads show progressive damage


11

L1.11

Solid Rubber: Anisotropy • Anisotropy • Certain elastomers such as fiber-reinforced or particle-filled rubbers and soft biological tissues exhibit anisotropic behavior.

axial

circumferential Iliac adventitial strips cut along the axial, circumferential, and 15º directions of an artery


Rubber Foams

12

L1.13

Rubber Foams • Porous rubbers, or elastomeric foams (which we will regularly refer to as foams), have the following properties: • Elastomeric foams are made of rubber materials that can deform elastically to very large strains (500 or more in tension, or 90 or more in compression). • Elastomeric foams are distinct from crushable foams that undergo nonrecoverable (inelastic) deformation. • The porosity of foam permits very large volumetric deformations, as opposed to solid rubbers, which are almost incompressible. • Poisson’s ratio of solid rubber → 0.5 • Poisson’s ratio of (highly voided) foam rubber → 0.0


L1.14

Rubber Foams: Structure • Foams are made up of polyhedral cells that pack in three dimensions. • The foam cells can either be open (e.g., sponge) or closed (e.g., flotation foam). • Common examples of elastomeric foam materials are cellular polymers such as cushions, padding, and packaging materials that utilize the excellent energy absorption properties of foams.


13

L1.15

Rubber Foams: Compressive Behavior • Foams are commonly loaded in compression. • A typical compressive stress-strain curve is shown at right. • Three stages can be observed: • At small strains (< 5 ) the foam deforms in a linear elastic manner as a result of cell wall bending. • This is followed by a plateau of deformation at almost constant stress. • Caused by the elastic buckling of the columns or plates that make up the cell edges or walls.

• Finally, a region of densification develops. • The cell walls crush together, resulting in a rapid increase of compressive stress. Modeling Rubber and Viscoelasticity with Abaqus

L1.16

Rubber Foams: Tensile Behavior • The tensile deformation mechanisms are similar to the compression mechanisms for small strains but differ for large strains. • A typical tensile stress-strain curve is shown at right.

• Two stages can be observed: • At small strains the foam deforms in a linear, elastic manner. • This is due to cell wall bending (similar to that in compression). • The cell walls rotate and align, resulting in rising stiffness.

• The walls are substantially aligned at a tensile strain of about 0.33. Further stretching results in increased axial strains in the walls.


14

L1.17

Rubber Foams • At small strains for both compression and tension, the average experimentally observed Poisson’s ratio of foams is about 0.33. • At larger strains it is commonly observed that Poisson’s ratio is effectively zero during compression. • The buckling of the cell walls does not result in any significant lateral deformation. • However, during tension is nonzero, which is a result of the alignment and stretching of the cell walls. • The manufacture of foams often results in cells with different principal dimensions. • This shape anisotropy results in different loading responses in different directions. • However, the hyperfoam material model in Abaqus does not take this kind of initial anisotropy into account.


15

16

Notes

17

Notes

18

Rubber Elasticity Models Lecture 2

L2.2

Overview • Introduction • Solid Rubber Models • Automatic Material Evaluation • Choosing a Strain Energy Function • Mullins Effect • Foam Rubber Model


19

Introduction

L2.4

Introduction • As discussed in the previous lecture, the behavior of rubber is characterized by many complex physical phenomena. • However, in this lecture (and throughout the bulk of this course), the focus is on the most commonly modeled rubber material behavior: elastic and isotropic. • The following basic assumptions are made: • The material behavior is elastic (permanent set is discussed in Lecture 11).

• The material is initially isotropic (anisotropy is discussed in Lecture 11). • For solid rubber, the material is approximately incompressible. • This is true only if the material has room to shear. • For foam rubber, the material is highly compressible.

• All deformation occurs instantaneously. • Viscous effects are modeled by including a separate viscoelastic or hysteresis model. Modeling Rubber and Viscoelasticity with Abaqus

20

L2.5

Introduction • The mechanical behavior of rubber (hyperelastic or hyperfoam) materials is expressed in terms of a strain energy potential

U

U ( F ), such that S

U (F ) , F

where S is a stress measure and F is a measure of deformation. • Why use an energy potential?—It guarantees reversibility (elasticity). • Assuming the material is initially isotropic, we write the strain energy potential in terms of the strain invariants I1, I 2 , and J el :

U

U ( I1 , I 2 , J el ).

I1 and I 2 are measures of deviatoric strain J el is the volume ratio, a measure of volumetric strain. • A detailed discussion of finite deformation theory and the mathematical forms of the different rubber models available in Abaqus is presented in Appendices 1 and 2. Modeling Rubber and Viscoelasticity with Abaqus

Solid Rubber Models

21

L2.7

Solid Rubber Models • Abaqus includes many different models for solid rubber. Each model defines the strain energy function in a different way. • Physically-motivated models: • Physically-motivated models consider the material response from the viewpoint of the microstructure. • The rubber is idealized as long chains of cross-linked polymeric molecules.

• Models based on phenomenological theory: • Phenomenological theory treats the problem from the viewpoint of continuum mechanics. • A mathematical framework is constructed to characterize the observed stress-strain behavior without reference to the microscopic structure.


L2.8

Solid Rubber Models • Physically-motivated models • Arruda-Boyce • Van der Waals

Material parameters (deviatoric behavior) 2 4

• Phenomenological models • Polynomial (order N)

• Mooney-Rivlin

(1st

order)

• Reduced polynomial (independent of I 2 ) • Neo-Hookean (1st order) • Yeoh (3rd order)

• Ogden (order N) • Marlow (independent of I 2 )


22

2N 2

N 1 3

2N N/A

L2.9

Solid Rubber Models • Why so many models? • Historical • Oldest models (polynomial and Ogden) based on continuum mechanics theory. • Difficult to fit data. • Physically motivated models more recent. • Easier to calibrate with limited test data. • Literature • Can find data in the literature for these models (i.e., have already been calibrated).


L2.10

Solid Rubber Models • Comparison of the solid rubber models • Gum stock uniaxial data (Gerke): • Crude data but captures essential characteristics.


23

L2.11

Solid Rubber Models • Unit-element uniaxial tension tests are performed with Abaqus. • All material parameters are evaluated automatically by Abaqus.

• Although we are considering only a specific test case, some fairly general conclusions may still be drawn.


L2.12

Solid Rubber Models • Neo-Hookean Model • Earliest rubber material model from the 1930s. • Cannot capture the ―upturn‖ in the stress-strain curve. • Good approximation at small strains. • Simple to use. • Single material shear parameter:

U

C10 ( I1 3)

1 ( J el 1)2 . D1

• Positive C10 guarantees stability but produces curves of fixed shape; D1 controls compressibility.

Note that in this figure, a better fit would result if the last two data points were omitted.


24

L2.13

Solid Rubber Models • Mooney-Rivlin Model (two term model) • Rubber material model from the 1940s. • Two-parameter shear model: U

C10 ( I1 3) C01 ( I 2 3)

1 el (J 1) 2 . D1

• Allows shape change.

• Cannot capture the ―upturn‖ in the stress-strain curve. • Reasonable fits to moderate strains. • Positive C10 and C01 guarantee stability. • Rule of thumb: C01

Note that in this figure, a better fit would result if the last two data points were omitted.

1 1 C10 to C10 10 4


L2.14

Solid Rubber Models • Full Polynomial Models • Generalized form of the Mooney-Rivlin model N

N i

U

Cij ( I1 3) ( I 2 3) i j 1

j i 1

1 ( J el 1) 2i . Di

• The Cij control the shear behavior • The Di control the bulk (hydrostatic) compressibility • Abaqus allows up to order N = 6 in the above function • Order N = 1 gives the classic two-term Mooney-Rivlin model • Order N = 2 gives 5 terms, with coefficients C10 C01 C20 C11 C02

0

2(C10 C01 )

K0

2 D1


25

L2.15

Solid Rubber Models • Reduced Polynomial Models • The reduced polynomial form does not include any dependence on I2 • There are several rationales for eliminating I2 from the strain energy function (see Yeoh, 1993): - The sensitivity of the strain energy functions to variations in I2 is generally much smaller than the sensitivity to variations in I1 . - It is difficult to measure the influence of I2 on the strain energy function, so it might be better to avoid introducing coefficients calibrated from potentially inaccurate data into the function. - It appears that eliminating the terms containing I2 from the strain energy potential improves the ability of the models to predict behavior for complex deformation states when test data are available for only a single deformation state. - Recent research supports these rationale.


L2.16

Solid Rubber Models • Reduced Polynomial Models (cont'd) • The Neo-Hookean model is a firstorder reduced polynomial model. • The Yeoh model is a third-order reduced polynomial model. • Yeoh Model • Good fit over a large strain range. • Will capture ―upturn‖ • Can be used with limited data. • Will represent other modes well. 3

3

Ci 0 ( I1 3)i

U i 1

i 1

1 ( J el 1)2i . Di


26

L2.17

Solid Rubber Models • Ogden Model • The Ogden model is also a phenomenological model. • Ogden (1972) proposed using the principal stretches instead of invariants. • This model also uses real powers (rather than integer powers); this allows a great deal of model accuracy. N

U i 1

2

N i ( 1 2 i

i

2

i

3

i

3) i 1

1 ( J el 1)2i , Di

• Abaqus allows up to N = 6 terms in the above form; up to N = 3 is common

• Do not use this model with limited test data (e.g. just uniaxial tension)


L2.18

Solid Rubber Models • Ogden Model (cont'd) • The model often models rubber accurately for large ranges of deformation. • The model is able to capture the stiffening (i.e., upturn) behavior at large strains.


27

L2.19

Solid Rubber Models • Arruda-Boyce Model • This model is also called the Arruda-Boyce 8-chain model because it was developed based on a representative volume (hexahedron) element where 8 chains emanate from the center to the corners of the volume. • This is a two-parameter shear model, based only on I1: 5

U i 1

• Positive

and

m

Ci

(I i 2i 2 1 m

i

3)

1 J el2 1 ( ln( J el )). D 2

guarantee material stability.

• With only two coefficients, there is only limited ability to change shape. • Works well with limited test data.


L2.20

Solid Rubber Models • Arruda-Boyce Model (cont'd) • Using material parameters, can scale curve along stress and strain axes. • Typical results shown below: either under-predict initial slope (left) or under-predict the ―upturn‖ slope (right).


28

L2.21

Solid Rubber Models • Van der Waals Model • In contrast to the Arruda-Boyce model, the Van der Waals model allows you to control the shape of the curve as well as scale it. The Van der Waals model is a four-parameter model: U

(

2 m

3) ln(1

)

2 I 3 a 3 2

3 2

1 J el2 1 ln( J el ) , D 2

I

(1

) I1

I 2 and

I 3 . 2 m 3

- Changing the initial shear modulus scales the curve in the vertical (stress) direction. - Changing the locking stretch m scales the curve in the horizontal (strain) direction. - Changing the interaction parameter a changes the shape of the curve. - The linear mixture parameter controls the relative shape changes of the different deformation modes. forces an I1 model


L2.22

Solid Rubber Models • Van der Waals Model (cont'd) • The Van der Waals model is able to produce a good curve fit for a wider range of rubber materials than the Arruda-Boyce model because the Van der Waals models allows more flexibility in the shape of the curve.


29

L2.23

Solid Rubber Models • Marlow Model • The Marlow model is a general first invariant model that can exactly reproduce the test data from one of the standard modes of loading (uniaxial, biaxial, or planar)

Marlow model response

• No curve fit required. • The responses for the other modes are also reasonably good.

Gum stock data

• This model should be used when limited test data are available. • The model works best when detailed data for one kind of test are available. Modeling Rubber and Viscoelasticity with Abaqus

L2.24

Solid Rubber Models • Marlow Model (cont'd) • The model is based on an additive split of the total strain energy density into deviatoric and volumetric parts:

U

U dev ( I1 ) U vol ( J el )

• The deviatoric part depends only on the first strain invariant • This is a common assumption when only limited test data are available

• Udev is determined from test data (uniaxial, biaxial, or planar) • The model allows temperature- and field-variable dependent test data input. • Note this is not the case for the other models (can specify temperature-dependent coefficients, however). Modeling Rubber and Viscoelasticity with Abaqus

30

Automatic Material Evaluation

L2.26

Automatic Material Evaluation • The previous figures underscore the importance of verifying the correlation between the predicted behavior and experimental data. • Use Abaqus/CAE to perform standard unit-element tests. • Supply experimental test data. • Specify material models and deformation modes. • X–Y plots appear for each test. • Predicted nominal stress-strain curves plotted against experimental test data.


31

L2.27


• The hyperelastic material curve fitting capability allows you to compare different hyperelastic models with the test data.

The curve fitting capability will be discussed in detail in Lecture 4. Modeling Rubber and Viscoelasticity with Abaqus

L2.28

Automatic Material Evaluation • Evaluation procedure • Unit cube datacheck

Create basic .inp file

Run .inp file through preprocessing only

Gather coefficients from .dat file

Compute response curves and plot in Abaqus/CAE

• Key features • Only batch preprocessing required • Evaluation is robust • No licensing issues

• Works with either analysis product; uses no solver tokens


32

Choosing a Strain Energy Function

L2.30

Choosing a Strain Energy Function • This depends on the availability of sufficient and ―accurate‖ experimental data: • Use data from experiments involving simple deformations: • Uniaxial tension and compression • Biaxial tension and compression • Planar tension and compression • If compressibility is important, volumetric test data must also be used. • E.g., highly confined materials (such as an O-ring). • Guidelines on selecting a strain energy function will be provided in Lecture 5.


33

Mullins Effect

L2.32

Mullins Effect • Mullins effect in elastomers: damage due to straining • This model provides an extension to the hyperelastic models described earlier to simulate the effects of damage due to straining on the structural response: • Without damage (elastic), loading and unloading occur along the same path • With damage, softened response during unloading • The Mullins effect model is discussed further in Lecture 5. • The theory is discussed in Appendix 2. Modeling Rubber and Viscoelasticity with Abaqus

34

Foam Rubber Model

L2.34

Foam Rubber Model • The foam rubber model uses an energy function similar to that used in the Ogden model, but it is designed for highly compressible elastic foams.

• The implementation in Abaqus follows the same procedure as the implementation of the Ogden material model. • The model should be calibrated using test data corresponding to the dominant deformation mode (tension or compression).


35

L2.35

Foam Rubber Model • Hyperfoam energy potential • The energy potential for the hyperfoam material model implemented in Abaqus is given by N

2

U i 1

where

i 2 i

ˆ

1

i

ˆ

2

i

ˆ

3

1

3

i

( J el

i i

1) ,

i

and

control the deviatoric behavior and

and

control the volumetric behavior.

• The user inputs

rather than

:

i i

1 2

. i


36

Notes

37

Notes

38

Physical Testing Lecture 3

L3.2

Overview • Modes of Deformation • Uniaxial Tension • Planar Tension • Uniaxial Compression • Equibiaxial Tension • Volumetric Compression

• Loading History • Test the Right Material • Summary


39

Modes of Deformation

L3.4

Modes of Deformation • What do we mean by modes of deformation? • And why will we talk so much about them? • Initially (1930s – 1950s) all focus was on uniaxial tests and fitting coefficients to uniaxial data. • Researchers observed that these uniaxial fits (for phenomenological models) did not correlate with data taken from other types of tests. • Need to perform other tests and use this data for fitting too.

• These other tests are from different strain states than uniaxial – thus the phrase ―modes of deformation.‖


40

L3.5

Modes of Deformation • Strain states, testing and curve fitting • Needs: • Simple test to perform, simple specimen to prepare. • Single state of strain/stress in the specimen, homogeneous, no gradient of strain/stress (away from grips). This is related to both analytical solution and measurement issues. ―Pure‖ state of strain. • Simple deformation mode—need analytical solution for curve fitting. • In general, one wants to perform several types of tests (modes of deformations) and curve fit a material model using multiple test data sets.


L3.6

Modes of Deformation • The common tests for rubber • Uniaxial tension (simple tension) • Uniaxial Compression • Planar Tension • Equibiaxial Tension • Volumetric Compression


41

L3.7

Modes of Deformation: Uniaxial Tension, Simple Tension • What is simple tension? •

Uniaxial loading

•

Free of lateral constraint


L3.8

Modes of Deformation: Uniaxial Tension, Simple Tension • Measure strain only in the region where a uniform state of strain exists • Do not use crosshead travel to measure strain!

• Use non-contact measurements: • Laser Extensometer • Video Extensometer


42

L3.9

Modes of Deformation: Planar Tension, aka Plane Strain Tension • What is planar tension? • Uniaxial loading • Perfect lateral constraint • All thinning occurs in one direction


L3.10

Modes of Deformation: Planar Tension, aka Plane Strain Tension • 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


43

L3.11

Modes of Deformation: Simple Compression • Requirements: • Uniaxial loading • No lateral constraint


L3.12

Modes of Deformation: Simple Compression • It is experimentally difficult to minimize lateral constraint due to friction at the specimen-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 an effect at very small strains • Easy to prepare specimen, but difficult to achieve friction-free test, thus difficult to achieve state of pure compression • Preferable to perform equibiaxial tension test instead!


44

L3.13

Modes of Deformation: Equibiaxial Tension • Why? • Same strain state as compression • Cannot do pure compression • Can do pure biaxial


L3.14

Modes of Deformation: Equibiaxial Tension • Analysis of the specimen justifies geometry


45

L3.15

Modes of Deformation: Equibiaxial Tension


L3.16

Modes of Deformation: Equibiaxial Tension • Alternate equibiaxial jigs • Here is another testing jig used to achieve an equibiaxial stress state.

• Balloon inflation can also be used to achieve equibiaxial tension.


46

L3.17

Modes of Deformation: Typical Test Data • Get acquainted with typical test data from 3 shear modes


L3.18

Modes of Deformation: Volumetric Compression • Direct measure of the stress required to change the volume of an elastomer

• Requires resolute displacement measurement at the fixture

•

Valid for (nearly) incompressible material only


47

L3.19

Modes of Deformation: Volumetric Compression • Confined compression • Technically, this test is called a confined compression test. We make use of the fact that for solid rubber the bulk modulus is much higher than the shear modulus. At very low stress the material shears to fill the rigid container, and the response is dominated by the bulk (volumetric) properties of the material.

• This test should not be used for voided (foam) materials.


L3.20

Modes of Deformation: Volumetric Compression • True volumetric compression, valid for foams (and solids). • True volume compression can be achieved using a fluid filled rigid pressure chamber. The specimen is sealed against fluid penetration. This test fixture is expensive and relatively rare. • Another alternative to gather volumetric information is to perform a uniaxial tension experiment and measure the lateral strains.


48

L3.21

Modes of Deformation: Volumetric Compression • Initial slope = bulk modulus • Typically, only highly constrained applications require an accurate measure of the entire pressure–volume relationship.


Loading History

49

L3.23

Loading History • Initial loading, typical of data from existing standards 1.2

Stress (MPa)

1.0

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Strain Modeling Rubber and Viscoelasticity with Abaqus

L3.24

Loading History 1.2

Stress (MPa)

1.0

0.8 Initial Loading

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

Strain Modeling Rubber and Viscoelasticity with Abaqus

50

1.0

L3.25

Loading History • Consistency • Be careful that the real component and the test specimen share the same load history and pre-conditioning. • Is the analysis for a 1st load condition (installation for instance)? • Then test the virgin material. • Is the analysis for a repeated use condition? • Then test the specimen after pre-conditioning. • Be careful to test at consistent strain rates for each deformation mode. • Test at strain rates consistent with the real component use situation.


L3.26

Loading History • 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


51

Test the Right Material

L3.28

Test the Right Material • Verify that the tested material is the same as the part • Processing • Color

All are same compound!

• Cure • History …


52

L3.29

Test the Right Material • Consistent within the experimental data set • Cut all specimens from the same slab Biaxial

Tensile

Volumetric

Planar tension


Summary

53

L3.31

Summary • Test in multiple pure states of strain • Understand the loading conditions • Be consistent • Test the right material


54

Notes

55

Notes

56

Curve Fitting Lecture 4

L4.2

Overview • It’s Just Curve Fitting! • Material Stability

• Curve Fitting Demonstration in Abaqus/CAE • Deviatoric (shear) curve fitting • Volumetric Curve Fitting


57

It’s Just Curve Fitting!

L4.4

It’s Just Curve Fitting! • Curve fitting the tension test data • We use linear or nonlinear least squares curve fit procedure. • Minimize relative error norm n

(1 Tith Titest )2

E i 1

• Linear least squares for polynomial and reduced polynomial forms. • Nonlinear curve fit for all others (Ogden, Arruda-Boyce, Van der Waals). • Nonlinear curve fit uses a Levenberg-Marquardt algorithm similar to that used in Twizell and Ogden (1986). • Exception: the Marlow model requires no curve fitting!


58

L4.5

It’s Just Curve Fitting! • Remember all the things learned about polynomial least squares curve fitting. • Material models, especially the phenomenological ones, have no basis in rubber physics, not material “law.” • Data points are nothing but weight points in the procedure. • To enhance fit in certain strain range, increase the number of data points in that range (or reduce the number of data points elsewhere).

• Abaqus does the curve fitting as part of the datacheck phase of a job. • The material evaluation capability of Abaqus/CAE automates this. • To manually perform the curve fits and generate stress-strain data for plotting, see Abaqus Benchmark Problem 3.1.4, Fitting of rubber test data.


L4.6

It’s Just Curve Fitting! • Visually check the curve fit response. Abaqus/CAE automates this process by automatically generating plots of force-displacement (Engineering Stress vs. Engineering Strain). If you don’t have Abaqus/CAE, you still need to do this! • You should check tensile and compressive response. • Check Stability. The curve fitting process does not guarantee a stable material model. It is up to you to check. Besides a visual check, you can check the stability limits in the Material Parameters and Stability Limit Information dialog box of Abaqus/CAE or in the data (.dat) file. • We have made some comments about model coefficients as an indicator of material stability; in general, a set of all positive coefficients guarantees stability. However, some negative coefficients do not necessarily mean the model is unstable (it just means it might be unstable). Typically lower-order models have fewer stability problems.


59

Material Stability

L4.8

Material Stability • Stability • What do we mean by material stability? • Drucker postulate:

∙

0.

• Graphically, in a stress vs. strain plot: Instability Point Stress

Strain


60

L4.9

Material Stability • Stability Checks in Abaqus • Abaqus checks the stability in the stretch range 0.1 10.0 (nominal strain range of 0.9 9) for the following deformation modes: • Uniaxial tension and compression • Equibiaxial tension and compression • Planar tension and compression


L4.10

Material Stability • Stability Checks in Abaqus • In addition, Abaqus checks the following deformation modes for foams: • Volumetric tension and compression • Simple Shear • If an instability is found, Abaqus issues a warning message in the .dat file and prints the smallest nominal strains at which the instability occurs in a particular deformation mode.

• This same information is printed in the Material Parameters and Stability Limit Information dialog box in Abaqus/CAE when the automatic material evaluation capability is used. • If no instability is found, no messages will be printed.

• If an instability is likely to occur at the strain levels expected in the analysis, the material model should be revised.


61

L4.11

Material Stability • Stability warnings in Abaqus/CAE


L4.12

Material Stability • Guaranteeing stability for various material models • The Neo-Hookean material model is always stable if C10 is positive. • In general, the full or reduced polynomial material models will be materially stable if all the Cij are positive. • However, some negative coefficients do not necessarily mean the model is unstable (it just means it might be unstable). • Typically lower-order models have fewer stability problems. • For the Yeoh model the C20 term is typically negative to help capture the S-shape feature of the stress-strain curve. • If there are stability problems, reducing the absolute value of C20 or magnifying the value of C10 will help make the Yeoh model more stable. • If all of the i terms are positive the Ogden material model is guaranteed to be stable. Modeling Rubber and Viscoelasticity with Abaqus

62

L4.13

Material Stability • Guaranteeing stability for various material models • For positive values of the initial shear modulus, , and the locking stretch, m, the Arruda-Boyce model is always materially stable. • For positive values of the initial shear modulus, , and the locking stretch, m, the stability of the Van der Waals model depends on the global interaction parameter, a. • All of the above comments are about material stability. • Realize that geometric instabilities such as buckling, collapse, or snapthrough may still occur in your analyses.


Curve Fitting Demonstration in Abaqus/CAE

63

L4.15

Curve Fitting Demonstration in Abaqus/CAE • To define a hyperelastic material: • In the Model Tree double-click the Materials container.


L4.16

Curve Fitting Demonstration in Abaqus/CAE • The Material Editor • In the Edit Material dialog box a default material name Material-1 appears. • Replace this default name by typing the name rubber.

• The material editor appears with a blank options list and option definition area. • From the menu bar in the upper portion of the editor window, select Mechanical → Elasticity → Hyperelastic.


64

L4.17

Curve Fitting Demonstration in Abaqus/CAE • The parameters and data corresponding to hyperelasticity appear in the option definition area below the option menus, and the word Hyperelastic appears in the Material Options list at the top of the dialog box.

• In the option definition area accept Test data as the Input source selection. • Note that the Strain energy potential defaults to the value Unknown. Click Test Data, and select Uniaxial Test Data from the list that appears. Modeling Rubber and Viscoelasticity with Abaqus

L4.18

Curve Fitting Demonstration in Abaqus/CAE • The Test Data Editor appears • Note that data required are Engineering (Nominal) Stress in column 1 and Engineering Strain in column 2. • To display context-sensitive help for specific buttons, text fields, and other options in the Test Data Editor, you must select the option of interest and then press F1. • Click mouse button 3 in the first cell of the table, and select Read from File from the list that appears. Modeling Rubber and Viscoelasticity with Abaqus

65

L4.19

Curve Fitting Demonstration in Abaqus/CAE • In the Read Data from ASCII File dialog box you may type the name of the file or click Select to browse for the file.

• Click Select. • From the ASCII File Selection dialog box, chose the file named st_treloar_abq.txt. St

= simple tension

_treloar

= test data from Treloar

_abq

= stress strain format

• Click OK in the ASCII File Selection dialog box. • Click OK in the Read Data from ASCII File dialog box.


L4.20

Curve Fitting Demonstration in Abaqus/CAE • Finished reading in the experimental data • We are back in the Test Data Editor, but the dialog box may be rather small. • Grab an edge or corner of the box and expand it until you can see all the Uniaxial Test Data values. • There should be 14 data pairs, with maximum stress of 1.95 MPa and maximum strain value of 4.37 (437% strain).


66

L4.21

Curve Fitting Demonstration in Abaqus/CAE • Saving the test data set for use in Visualization • Although the test data have been read in, they have not been saved as X–Y data. • Click mouse button 3 in the first cell of the table, and select Create XY Data from the list that appears.


L4.22

Curve Fitting Demonstration in Abaqus/CAE • Naming the X–Y data set • Type in the name st_treloar in the Create XY Data dialog box. • We will use this named data set later in the Visualization module.

• Click OK in the Create XY Data dialog box. • Click OK in the Test Data Editor.


67

L4.23

Curve Fitting Demonstration in Abaqus/CAE • Done with Test Data input • We are now back at the Edit Material dialog box.

• If we wanted to read in more test data, we would repeat this process, selecting Biaxial Test Data, Planar Test Data, or Volumetric Test Data from the Test Data pull-down menu. • We are finished with test data input, so click OK in the Edit Material dialog box.


L4.24

Curve Fitting Demonstration in Abaqus/CAE • Back to the Model Tree • This puts us back to the Model Tree. Having read in the uniaxial data, we are ready to Evaluate the material. • The Evaluate option is useful in the following scenarios: • Comparing test data with the behavior predicted by a particular strain energy potential. • Evaluating multiple strain energy potentials. • Viewing behavior predicted by coefficients for a particular strain energy potential.


68

L4.25

Curve Fitting Demonstration in Abaqus/CAE • The default settings for Test Setup are shown below: • The upper area shows the available test data with which to work.

The lower area allows you to choose which standard stress-strain responses to generate and the strain limits in tension and compression.


L4.26

Curve Fitting Demonstration in Abaqus/CAE • The defaults for Strain Energy Potentials are shown here. • There are six major categories. • For Polynomial, Ogden, and Reduced Polynomial you must select the number of terms N in the energy function. • You may choose one or more models to evaluate. • In general selecting more than three models makes viewing the results difficult.


69

L4.27

Curve Fitting Demonstration in Abaqus/CAE • Evaluate Material Function • Rather than take the defaults, let’s choose 3 models: • Mooney-Rivlin (Polynomial Form, • Ogden Form,

N = 1).

N = 2.

• Neo-Hookean (Reduced Polynomial Form,

N = 1).

• Your screen should now look like this (you may need to expand the dialog box to see all models). With these models selected, click OK in the Evaluate Material dialog box. • This will launch the curve fitting process. Curve fit results will automatically be displayed when ready. Modeling Rubber and Viscoelasticity with Abaqus

L4.28

Curve Fitting Demonstration in Abaqus/CAE • Material stability • Material coefficients and stability information for each of the selected models is automatically displayed in the Material Parameters and Stability Limit Information dialog box.


70

L4.29

Curve Fitting Demonstration in Abaqus/CAE • Default results plotting of evaluate material • The default results plotting uses three new viewports to display X–Y plots of stress-strain for Uniaxial response, Equibiaxial response, and Planar Tension response. You may want to maximize each window in turn to better view the viewports.


L4.30

Curve Fitting Demonstration in Abaqus/CAE • Nondefault results plotting: Visualization • The plots are created in the Visualization module and all the response data calculated during the curve fit are available to us in this module. • Before creating additional plots, delete the three viewports labeled Uniaxial, Planar, and Biaxial. • Maximize the remaining (original) viewport.


71

L4.31

Curve Fitting Demonstration in Abaqus/CAE • The XY Data Manager • The XY Data Manager dialog box may be rather small. Grab an edge or corner and make it bigger. Grab the vertical bar that separates Name from Description and enlarge the Name area. Notice the default names given to all the response curves from the curve fit calculations. These default names are very long to ensure uniqueness. You may shorten them with the Rename option. • Notice that our named data set st_treloar (the test data) is also listed. All other data sets are simply stress-strain response curves to particular curve fits. Note that all of these data sets persist only in this Abaqus/CAE session.


L4.32

Curve Fitting Demonstration in Abaqus/CAE • The XY Data Manager: plotting • Click the st_treloar data set, then [Control]+Click all the Uniaxial data sets • Click Plot. This will plot the actual simple tension test data along with the curve fit response for all the material models (energy potentials) chosen. • You may need to move the XY Data Manager window out of the way to see the plot.


72

L4.33

Curve Fitting Demonstration in Abaqus/CAE • Curve options • The XY Curve Options can be used to modify the line and symbol attributes for each data set. Line color, line style, and line thickness can be modified. You can choose to show or not show the line. Several symbol types are available. You can choose to show symbols or not; and you can modify the symbol types, symbol size, symbol color, and symbol frequency. • Select the st_treloar data set, and click Show symbol. Set the symbol color to Orange, the symbol type to +, and the symbol size to Large. Suppress the visibility of the line. • Select the other three uniaxial data sets ([Shift]+Click), and do the following: • Choose a solid line style, increase the line thickness one level and toggle off Show symbol. • Set the line color of R_POLY_N1 to red; POLY_N1 to blue; and OGDEN_N2 to green. • Edit the legend text to remove rubber_1 at the end of the description. Modeling Rubber and Viscoelasticity with Abaqus

L4.34

Curve Fitting Demonstration in Abaqus/CAE • Axis options • Double-click the X-axis; change the scale to range from a min value of 0.0 to a max value of 4.5; change the axis font to 12 bold Arial and the title to Engineering Strain and the title font to 18 bold Arial. • Double-click the Y-axis; change the scale to range from a min value of 0.0 to a max value of 2.0; change the axis font to 12 bold Arial and the title to Engineering Stress (MPa) and the title font to 18 bold Arial. • Legend options • Double-click the legend. • In the Contents tabbed page of the Chart Legend Options dialog box, change the legend font size to 12.

• In the Area tabbed page of the dialog box, toggle on Inset. • Dismiss the dialog box. • Drag the legend over the chart. Modeling Rubber and Viscoelasticity with Abaqus

73

L4.35

Curve Fitting Demonstration in Abaqus/CAE • Uniaxial X–Y data plots, manipulated • Shown at right is the modified X-Y plot we have generated. All further X-Y plots of stress-strain curve fit responses will be shown in this style. • What material model gives the best fit to uniaxial data?


L4.36

Curve Fitting Demonstration in Abaqus/CAE • Equibiaxial curve fit response • We can repeat the plotting process using the XY Data Manager. If you have lost your Manager window, select Tools→XY Data →Manager from the main menu bar. Choose the st_treloar data set and all the curve fit response Biaxial data sets. • What can we conclude from this plot?


74

L4.37

Curve Fitting Demonstration in Abaqus/CAE • Planar curve fit response plots • Here we have selected the st_treloar data set and all the curve fit response Planar data sets • What can we conclude from this plot?


L4.38

Curve Fitting Demonstration in Abaqus/CAE • What if I have model coefficients? • In cases where you may have some model (Mooney, Ogden, etc.) coefficients, you can use the evaluate function to see the stressstrain response from those coefficients. First, double-click rubber in the Model Tree to open the Edit Material dialog box. Change Input Source to Coefficients, select the Polynomial strain energy function, and enter the values 0.8, 0.2, and 0.0 in the boxes for C10, C01, and D1, respectively. Click OK.


75

L4.39

Curve Fitting Demonstration in Abaqus/CAE • What if I have model coefficients? • Now Evaluate material again. Under Available Input Data, change the Source to Coefficients. • Change the minimum nominal strain to 0.0 and the maximum nominal strain to 1.0 for uniaxial and deselect Planar (Pure shear) and Biaxial. Take a look under the Strain Energy Potential tab; we will evaluate the Polynomial, N = 1 material model because it was the one chosen earlier in the material editor. • Click OK to perform the evaluation.


L4.40

Curve Fitting Demonstration in Abaqus/CAE • Default X–Y plot from coefficient fit • This is the default X–Y plot that appears. Notice that only one viewport is created because we deselected Planar and Biaxial. The material test data are included in the plot since they were still available for comparison purposes. If we want, we could delete the material test data prior to the evaluate and then the test data will not appear. • Conclusion: Mooney coefficients of 0.8 and 0.2 represent a material much stiffer in uniaxial tension than the material for which we have test data.


76

Volumetric Curve Fitting

L4.42

Volumetric Curve Fitting • A few notes • Recall, if you input no D values, the material is incompressible (bulk modulus is infinite). • This is okay for the vast majority of analyses where there is plenty of room for the material to shear. • Compressibility information is needed when the part is highly confined; seals, especially O-rings, can be highly confined. • Data for volumetric is in the form of Pressure, Volume Ratio • Volume ratio is J = 1 2 3. • Cannot perform volumetric curve fit alone. This is because you are always curve fitting an energy potential – you always need to calculate the deviatoric part. Must include simple tension, equibiaxial, or planar data. • Typical K = 2000 MPa

= 290,000 psi

• Typical D1 = 0.001 1/MPa

= 7E 6

K

1/psi

2 and D1 D1

2 K


77

L4.43

Volumetric Curve Fitting • A few more notes • Remember that for each energy potential the parameter N controls the number of D terms. • For Neo-Hookean (N = 1) there is by definition only a single compressibility term, D1. • For Mooney-Rivlin (N = 1) there is by definition only a single compressibility term, D1.

• For the Arruda-Boyce model there is only D1. • For the Van der Waals model there is only D1. • The term D1 should be positive. • For all other models with N = 2 or greater the curve fit process will automatically calculate 2 or more Di terms. Some of these terms may be negative, leading to unstable compressibility. Look in the data file for the curve fit Di values. You may want to revert to coefficients and enter zero values in place of the calculated negative values. Modeling Rubber and Viscoelasticity with Abaqus

L4.44

Volumetric Curve Fitting • What will we do different? • By now we assume you know the mechanics of the curve fit process: reading in the test data, saving the data for use in the Visualization module, etc. We will read in the st_treloar_abq.txt simple tension data for the deviatoric part. For demonstration of the volumetric portion, we will read in data from a file called VC_linear.txt. These are not real test data; they are constructed using K = 2000 MPa (D1 = 0.001). The volume ratio ranges in value from 1.0 (unstrained) to a minimum of 0.9 in increments of 0.01.

• Read in the Uniaxial data. • Don’t save these data. • Read in the Volumetric data set. • Save the Volumetric data. Modeling Rubber and Viscoelasticity with Abaqus

78

L4.45

Volumetric Curve Fitting • Volumetric response • The single element model created for plotting the curve fit response is a unit cube undergoing volumetric deformation. • Data is input as Pressure and Volume Strain. The default X–Y plot is generated using Pressure and Volume Strain. • For the Test Setup the input data source defaults to Test data and both Uniaxial and Volumetric will be chosen (highlighted) by default. Under the Standard Tests area select only Volumetric; the Volume Ratio range is chosen by default. • For the Strain Energy Potential de-select the defaults and select only the NeoHookean model for curve fitting. Our concern will be the calculation of the value of D1. Modeling Rubber and Viscoelasticity with Abaqus

L4.46

Volumetric Curve Fitting • The default X–Y plot generated by the volumetric curve fitting process is shown here.

• The curve fit value for the D1 coefficient is calculated to be exactly 0.001. • Here is the result from the stability information dialog box:


79

L4.47

Volumetric Curve Fitting • Using real volumetric test data • Using the Model Tree, create a new Hyperelastic material. Using the Test Data Editor, read in the volumetric test data from the file VC_S6.txt. Save these data using the Create XY Data option, and give it the name vc_test. You must also read in the simple tension data ST_S58.txt. You are now ready to perform a variety of curve fits using this volumetric data. • Try curve fitting several of the N = 1 models. We certainly expect that the D1 coefficient calculated will be the same regardless of the N = 1 deviatoric model chosen. • For this Volumetric data set, the D1 value (for all the N = 1 material models) should be 7.574E 4, or K = 2640 MPa.


L4.48

Volumetric Curve Fitting • Yeoh model volumetric response • Then try curve fitting a model such as the Yeoh model (N = 3). Set the minimum volume ratio to 0.8. Look at the coefficients. Are they all positive? • The Yeoh model allows for three D values, associated with higher powers of the volume ratio. This is important only if you really need to capture a nonlinearity in the volumetric behavior. Here we see the volumetric response and note the instability after about 0.86 Volume Ratio. • You would probably want to use only the first D value or maybe just the positive terms. To do so, you must enter as coefficients (not as test data). Modeling Rubber and Viscoelasticity with Abaqus

80

Notes

81

Notes

82

Abaqus Usage Lecture 5

L5.2

Lecture Overview • Introduction • Test Data Guidelines • Abaqus Test Data Usage • Choosing a Strain Energy Function • Defining a UHYPER user subroutine • Mullins Effect

• Hyperfoam Material Model


83

Introduction

L5.4

Introduction • Defining rubber elasticity in Abaqus/CAE: hyperelasticity

Material description


84

L5.5

Introduction • Entering test data Temperature and field variable dependence of test data available for the Marlow model

Click mouse button 3

Nominal stress and strain


L5.6

Introduction • Entering coefficients

Temperature-dependent coefficients


85

L5.7

Introduction • Abaqus keyword syntax • The rubber elasticity models are invoked in Abaqus with the *HYPERELASTIC and *HYPERFOAM keyword options. • The *HYPERELASTIC and *HYPERFOAM options must be used in conjunction with the *MATERIAL option. • With the *HYPERELASTIC option, enter the parameter for the desired type of energy function: NEO HOOKE

MOONEY-RIVLIN

POLYNOMIAL (default)

REDUCED POLYNOMIAL

YEOH

OGDEN

ARRUDA-BOYCE

VAN DER WAALS

MARLOW

• With both polynomial models and Ogden model enter the order, N, of the series expansion.


L5.8

Introduction • Abaqus keyword syntax (cont'd) • For either hyperelastic or hyperfoam models, you may input the material coefficients directly on the data lines. See the Abaqus Keywords Reference Manual for the data line syntax for each of the various material models. Material coefficients can be given as function of temperature. For instance: *MATERIAL, NAME=rubber *HYPERELASTIC, N=3, OGDEN 1.061,0.428,5.782E-2,5.712,1.591E-2,-4.597,7.25e-04,0.0 0.0, 23.0

• The data line is

m1, a1, m2, a2, m3, a3, D1, D2, D3, temperature


86

L5.9

Introduction • Abaqus keyword syntax (cont'd) • For either hyperelastic or hyperfoam models, you may use the TEST DATA INPUT parameter to indicate that Abaqus should calculate the coefficients from test data. With the exception of the Marlow model, test data cannot be given as a function of temperature. • For example: *MATERIAL, NAME=POLY_N1 *HYPERELASTIC, POLYNOMIAL, N=1, TEST DATA INPUT *UNIAXIAL TEST DATA 0.0,0.0 0.03,0.02 0.15,0.1 data lines here are 0.23,0.2 nominal stress, nominal strain 0.33,0.34 0.41,0.57 0.51,0.85 ...


L5.10

Introduction • Abaqus keyword syntax (cont'd) • The hyperelastic and hyperfoam models are intended to be used in finite-strain applications. • Set NLGEOM=YES on the *STEP option. • Alternative energy functions (and their derivative with respect to strain invariants) can be defined with the user subroutine UHYPER.


87

L5.11

Introduction • Combining with other material models • The hyperelastic and hyperfoam material models can be used alone or can be combined in the same material definition with the following material models: - Thermal expansion properties to introduce isotropic thermal volume changes. - Viscoelastic material properties to define time- or frequencydependent hyperelastic behavior. - Hysteretic material model to define nonlinear rate-dependent inelastic behavior (for cyclic loading). - Mullins effect model to account for damage due to straining (hyperelastic only)

- The hyperelastic material model can also combined with the metal plasticity material model to capture finite elastic and plastic strains.


L5.12

Introduction • Suitable elements • The hyperelastic and hyperfoam material models can be used with solid, truss, beam, rebar, and finite-strain shell elements. - They cannot be used with S4 shell elements. - They have not been implemented for use with small-strain shell elements (S4R5, S8R, S8R5, S9R5, etc.) - The hyperelastic material model, when incompressible, must be used with hybrid elements. - The hyperfoam material model is quite compressible and therefore should not be used with hybrid elements.


88

Test Data Guidelines

L5.14

Test Data Guidelines • Test availability • The availability of sufficient and accurate test data is the most significant factor in choosing a rubber material model. • Use data from as many modes of deformation as possible • Uniaxial tension and compression • Planar tension and compression • Equibiaxial tension and compression • If compressibility is important, volumetric compressibility (D’s) must also be specified. • For example, highly confined applications such as O-rings under significant compression.


89

L5.15

Test Data Guidelines • Limited Test Availability • In some cases you may have only limited test data, for instance, only uniaxial tension test data. • In this case be sure to use a material model that involves only I1 - Marlow model - Neo-Hookean, Yeoh, or other reduced polynomial models - Arruda-Boyce - Van der Waals with  set to zero - While Ogden and full polynomial models might fit the limited test data nicely, the representation of the other modes of deformation may be very poor—stiffness errors may be several orders of magnitude.


L5.16

Test Data Guidelines • Damage • It is not uncommon for elastomers to exhibit elasticity damage and hysteresis during the cyclic loading. As shown in the figure, a few cycles of loading result in a decrease in stiffness—this is termed Mullins effect. • Abaqus provides a material model to capture Mullins effect. • To calibrate the model, supply data from several loading cycles for analyses of components that are in repeated use conditions. • If you do not wish to model Mullins effect, pre-condition the test specimen prior to testing.

• Test virgin specimens for a 1st-use component analysis (installation).


90

L5.17

Test Data Guidelines • Test Data Variations • The properties of elastomers are known to change from batch to batch. • All tests done to characterize a given material should be performed on the same batch. • Taking all the test specimens from the same physical slab of material is highly recommended. • It may be necessary to validate the test specimen slab against the real component to assure they have similar cure history. • Cutting small uniaxial specimens from real components can be used for this validation. • Test at the operating temperatures expected in the application.

• It is best to obtain data from more that one type of test. • Experience shows that data from more than one mode of deformation (strain state) should be used to achieve the most accurate material model. Modeling Rubber and Viscoelasticity with Abaqus

L5.18

Test Data Guidelines • Test redundancies: solid rubber • For fully incompressible materials the superposition of a hydrostatic pressure does not alter the deformation mode. • As a result, some apparently different types of tests are equivalent and provide redundant information, such as: • Uniaxial tension  Equibiaxial compression • Uniaxial compression  Equibiaxial tension

• Planar Tension  Planar Compression • Uniaxial tension and uniaxial compression provide independent data.


91

L5.19

Test Data Guidelines • Test redundancies: solid rubber


L5.20

Test Data Guidelines • Other Test Data Guidelines • Nominal (engineering) stress and strain data are required for the deviatoric (shear) test data input. • Both tension and compression data are allowed. • Compressive stresses and strains are given as negative values. • Always use more experimental data points than unknown coefficients • Volumetric curve fit requires pressure, volume ratio test data. • Always compare your material model stress-strain response to the simple modes of deformation! • Use Abaqus/CAE or simple unit-cube analyses to compare the model response to the original test data.

• Skipping this step results in garbage in, garbage out analyses.


92

Abaqus Test Data Usage

L5.22

Abaqus Test Data Usage • Typical Usage *MATERIAL, NAME=VITON *HYPERELASTIC, POLYNOMIAL, N=1, TEST DATA INPUT *UNIAXIAL TEST DATA 0.00,0.00 0.03,0.02 data lines are 0.15,0.10 nominal stress, nominal strain 0.23,0.20 ... Suboptions of *PLANAR TEST DATA *HYPERELASTIC ... *BIAXIAL TEST DATA ... *VOLUMETRIC TEST DATA (to define optional compressibility) ... *EXPANSION (to define optional CTE) ... *VISCOELASTICITY (to define optional timedependency) ...


93

L5.23

Abaqus Test Data Usage • Volumetric *VOLUMETRIC TEST DATA (to define optional compressibility) pressure_1, volume_ratio_1 pressure_2, volume_ratio_2

• Volumetric information should be specified for cases where the material does not have room to shear – that is, cases where the material is highly confined. • For many elastomers a bulk modulus of approximately 2000 MPa is reasonable. • For highly confined applications, it is better to input a D1 of 0.001 than to leave it unspecified (and therefore incompressible)


L5.24

Abaqus Test Data Usage • Thermal Expansion *EXPANSION **(to define optional CTE) Alpha1, temp1 Alpha2, temp2 ...

• Defines the volumetric CTE (coefficient of Thermal Expansion) for the material. • Abaqus uses a total, or secant, measure from a reference temperature.


94

L5.25

Abaqus Test Data Usage • Test data smoothing • The test data input option provides a data-smoothing capability that is recommended • Useful in situations where the test data do not vary smoothly • Avoids potential convergence problems during the analysis • User can control the smoothing process

• This capability is particularly useful with the Marlow model when the data is scattered.


L5.26

Abaqus Test Data Usage • Test data usage with the Marlow model • For uniaxial, biaxial, and planar modes, either tension or compression data can be specified. • Tension data determines the strain energy potential, which in turn determines the compression behavior, and vice versa. • When used with 1-D elements (beams, rebars, and trusses), data from both tension and compression tests can be specified together.


95

L5.27

Abaqus Test Data Usage • The volumetric behavior for the Marlow model can be defined in one of the following ways: • Volumetric test data • Lateral test data in the uniaxial, biaxial, or planar mode

• These data options allow users to specify the lateral behavior along with the primary behavior. Lateral strains define the volumetric response • Effective Poisson’s ratio • Incompressibility is assumed if none of the above specified.


L5.28

Abaqus Test Data Usage • In addition, for the volumetric mode, both hydrostatic tension and hydrostatic compression data can be specified. • More commonly, only hydrostatic compression data are available. Abaqus assumes that the hydrostatic pressure is an antisymmetric function of the nominal volumetric strain, evol, about evol = 0.


96

Choosing a Strain Energy Function

L5.30

Choosing a Strain Energy Function • The importance of using multiple types of test data when calibrating the models is discussed next. • The recommended selection procedure is then summarized. • In each case, the models are listed in order of preference • The suggested approach considers physically motivated models first. • Tips:

• Use simple models first. • Keep the order, N, as low as possible to describe the data.


97

L5.31

Choosing a Strain Energy Function • Importance of Multiple Types of Tests • Generally, when data from multiple experimental tests are available, the Van der Waals and Ogden strain energy functions are more accurate in fitting the stress-strain curves. • When limited amounts of test data exist for calibration, for instance, just uniaxial test data, the use of the Van der Waals, Ogden, full polynomial models can be quite dangerous. • When using limited test data stay with the I1 only models – Marlow, Arruda-Boyce, Van der Waals with  = 0 , reduced polynomial (NeoHookean, Yeoh).


L5.32

Choosing a Strain Energy Function • Using only Uniaxial Test data • The following group of slides show a comparison of the various strain energy functions when calibrated with only uniaxial test data. The other modes of test data (planar, equibiaxial) will be shown for reference. • The test data were taken from Treloar (―Stress-strain data for vulcanized rubber under various types of deformations,‖ Trans. Faraday Society, 40, 1944) for uniaxial tension, biaxial tension, and planar tension. • For each slide we show the fit to only uniaxial tension data. By doing so we can show that I1 based models in general do ok, while I1 and I2 models can give very poor results when fit to only uniaxial tension test data. Abbreviations:

ST = Simple Tension

PT = Planar Tension EB = Equibiaxial Tension


98

L5.33

Choosing a Strain Energy Function • Treloar test data • Stress in MPa • Focus on the relationship between 3 modes. • This is a common semiquantitative relationship • PT slightly higher than ST

• EB 50% to 100% higher than ST


L5.34

Choosing a Strain Energy Function • Marlow Model (I1 based model) • Uniaxial data represented exactly; other modes are represented reasonably well. Curve Fits to only Uniaxial Test Data


99

L5.35

Choosing a Strain Energy Function • Neo-Hookean and Yeoh Models (I1 based models) • Other modes are represented reasonably well. Curve Fits to only Uniaxial Test Data


L5.36

Choosing a Strain Energy Function • Arruda-Boyce Model (I1 based model) • Prediction similar to Neo-Hookean model Curve Fits to only Uniaxial Test Data

Curve Fit in Abaqus

A-B coefficients from paper


100

L5.37

Choosing a Strain Energy Function • Full Polynomial and Ogden models (I1 and I2 based models) • Other modes are extremely overly stiff (very poor with limited test data). Curve Fits to only Uniaxial Test Data


L5.38

Choosing a Strain Energy Function • Van der Waals models ( controls the I1 and I2 bases) • For limited test data the Van der Waals model changes dramatically. Curve Fits to only Uniaxial Test Data

 set by curve fitting

 set to 0.0 (edit input file)


101

L5.39

Choosing a Strain Energy Function • Using Full Datasets • Typically the simplest models (I1 based) will not improve very much as additional test data is used in the curve fitting process. • The fewer the model parameters, the less likely the additional modes of deformation test data will improve the fit. • The higher order (N) I1 and I2 based models will improve dramatically as additional test data is used for curve fitting.

• In the following slides we will repeat the fits shown earlier, but this time all 3 sets of Treloar data will be used as a basis for the curve fits.


L5.40

Choosing a Strain Energy Function • Neo-Hookean and Yeoh Models (I1 based models) • Not much change over earlier limited data fit Curve Fits using all Data Sets


102

L5.41

Choosing a Strain Energy Function • Arruda-Boyce Model (I1 based model) Curve Fits using all Data Sets

Curve Fit in Abaqus

A-B coefficients from paper


L5.42

Choosing a Strain Energy Function • Full Polynomial and Ogden models (I1 and I2 based models) • These fits improve dramatically over limited data case Curve Fits using all Data Sets


103

L5.43

Choosing a Strain Energy Function • Van der Waals models ( controls I1 and I2 bases) • Using all data, Van der Waals gives good fit. No need to set  = 0. Curve Fits using all Data Sets

 set by curve fitting

 set to 0.0 (edit input file)


L5.44

Choosing a Strain Energy Function • Summary: Selection procedure for strain energy functions • Limited test data: small strain data • Neo-Hookean model • Limited test data: good detailed data for one kind of test (e.g., good uniaxial data) • Marlow model • Limited test data: initial modulus and stretch limit (and possibly a few extra data points) • Arruda-Boyce • Van der Waals with  = 0 • Reduced polynomial (e.g., Yeoh) model

• Predicted behavior in other modes of straining will be plausible, but not necessarily accurate. • Avoid using the Ogden and Full Polynomial models with limited test data. Modeling Rubber and Viscoelasticity with Abaqus

104

L5.45

Choosing a Strain Energy Function • Summary (cont'd) • Full test suite of data (i.e., multi-axial data) • Van der Waals model (  0) • Ogden model • The Full Polynomial model may be OK, but generally it doesn’t fit data as well as the Ogden model. • It is better to use this model with data that have already been calibrated.


UHYPER

105

L5.47

UHYPER • UHYPER syntax • You may define your own elastomer behavior through the use of a user subroutine called UHYPER. You provide Fortran code to define the energy function, U, and first and second derivatives of U with respect to Ī1 and Ī2. • To invoke its use, the Abaqus input file looks like this: *MATERIAL, NAME=... *HYPERELASTIC, USER, TYPE=..., PROPERTIES=... *EXPANSION

(to define optional CTE)

...

*VISCOELASTICITY

(to define optional time-dependency)

...


L5.48

UHYPER • Defining UHYPER in Abaqus/CAE:


106

Mullins Effect

L5.50

Mullins Effect • Stress softening in certain filled rubbers occurs due to damage associated with straining

• The results depicted in the figure show evidence of progressive damage (with cycles), with the response stabilizing after a few cycles

Dashed line is the primary curve (given by hyperelastic material model)

Progressive damage indicated by reduced stress with fixed strain loading cycles

Damage: Unloading and further reloading follows different path characterized by stress softening

• The results also show evidence of permanent set and viscoelasticity

Courtesy: Axel Physical Testing Services

e  0  permanent set

Hysteresis: loading and unloading for a given cycle follow different paths— energy is dissipated with each cycle


107

L5.51

Mullins Effect • Idealized response—Abaqus model • Does not model progressive damage during the first few cycles • Does not take into account permanent set and viscoelasticity

Energy dissipated once (damage); no subsequent hysteresis or progressive damage

No permanent set or viscoelasticity


L5.52

Mullins Effect • The material definition consists of two parts: • Define the primary behavior using a hyperelastic material model. • Test data, strain energy density function coefficients, or user subroutine UHYPER can be used to define the primary behavior. • Define the damage behavior using the *MULLINS EFFECT option.


108

L5.53

Mullins Effect • The material parameters related to damage can be specified directly • Alternatively, these parameters can be determined by Abaqus based on calibration of unloading-reloading test data • Test data from one or more of the primary modes of deformation (uniaxial, biaxial, and planar) can be specified • For a specific deformation mode, unloading-reloading test data from multiple maximum strain levels can be specified by repeated use of the appropriate test data option • User subroutine UMULLINS is available in Abaqus/Standard • This allows you to define the damage variable directly

• The Mullins effect model cannot be used with viscoelasticity or hysteresis.


L5.54

Mullins Effect • Output variables: • DMENER: Damage dissipation density at an integration point • ELDMD: Damage dissipation in an element • EDMDDEN: Damage dissipation per unit volume in an element • ALLDMD: Total damage dissipation in the whole model (or over a userspecified element set)


109

L5.55

Mullins Effect • Example: Calibration of test data • Uniaxial test data to define the primary behavior • Uniaxial unloading-reloading data from three different strain levels (stabilized cycles) • The Abaqus model replaces stabilized cycle at each strain level with a single curve that represents both loading and unloading


L5.56

Mullins Effect • Example: Load-deflection of a stationary solid rubber disk • Rigid surface displaced up against fixed disk • Unloaded • Reloaded to deformation levels that are higher than the first loading • Above deformation pattern constitutes two loading cycles


110

L5.57

Mullins Effect

Unload at constant damage

Dissipate more energy

Unload/reload at constant damage


Hyperfoam Material Model

111

L5.59

Hyperfoam Material Model • Defining rubber elasticity in Abaqus/CAE: hyperfoam


L5.60

Hyperfoam Material Model • Entering test data in Abaqus/CAE: hyperfoam

Nominal stress and strain


112

L5.61

Hyperfoam Material Model • Input Syntax for Hyperfoam • Specifying the model with model parameters (not test data): *MATERIAL, NAME=my_foam *HYPERFOAM, N=2

m1 , a1 , 1 , m2 , a2 , 2 m1 , a1 , 1 , m2 , a2 , 2

, temperature1 , temperature2

*EXPANSION

(to define optional CTE)

... *VISCOELASTICITY dependency)

(to define optional time-

...

• Abaqus allows up to N = 6 terms in the above form


L5.62

Hyperfoam Material Model • Input Syntax for Hyperfoam – using test data

Suboptions of *HYPERFOAM

*MATERIAL, NAME=my_foam *HYPERFOAM, N=1, TEST DATA INPUT, [POISSON=...] *UNIAXIAL TEST DATA nom stress1, nom strain1, [nom lateral strain1] ... *SIMPLE SHEAR TEST DATA nom shear stress1, nom shear strain1, [nom transverse stress1] ... *PLANAR TEST DATA nom stress1, nom strain1, [nom transverse strain1] ... *BIAXIAL TEST DATA nom stress1, nom strain1, [nom transverse strain1] ... *VOLUMETRIC TEST DATA (to define optional compressibility) ... *EXPANSION (to define optional CTE) ... *VISCOELASTICITY (to define optional timedependency) Modeling Rubber and Viscoelasticity with Abaqus

113

L5.63

Hyperfoam Material Model • Volumetric Information • There are several ways to indicate the volumetric behavior of the foam material. • Using model parameters: • Give the model parameters i  i = • Using test data:

i

1 - 2 i

, i =

i

1 + 2 i

.

- Use the POISSON parameter to define a single Poisson’s ratio; this is commonly used to set Poisson’s ratio to zero. - Give lateral strain/stress information for one or more of the shearing mode test data. - Give the pressure, volume ratio data for the *VOLUMETRIC TEST DATA input


L5.64

Hyperfoam Material Model • Curve fitting the hyperfoam material model • For the hyperfoam model the high compressibility ( → 0) approximately reduces the different deformation modes into a ―superposition‖ of several uniaxial states at different orientations. • This is largely true for compressive states where the buckling of cell walls in one direction is quite independent from that in perpendicular directions. • Thus, it is not uncommon that a single uniaxial test (with an assumption of Poisson's ratio=0) may be sufficient to characterize the material, particularly if compression dominates. • In the following example the results of curve fitting using only uniaxial data and using uniaxial plus simple shear data are compared (solid line: test data, dashed line: Abaqus result). • We see that the shear behavior predicted by using only uniaxial data to determine the parameters for the material model is not grossly inaccurate. Modeling Rubber and Viscoelasticity with Abaqus

114

L5.65

Hyperfoam Material Model • Curve fit based on compressive data and simple shear data

N=2

N=2

N=3

N=3

• Compressive response is accurate • Simple shear response is accurate


L5.66

Hyperfoam Material Model • Curve fit based on only compressive data

N=2

N=2

N=3

N=3

• Compressive response is accurate • Simple shear response is inaccurate, but not grossly so


115

L5.67

Hyperfoam Material Model • Difference in tension and compression • For small strains (< 5%) foams behave similarly (cell wall bending) for both compression and tension • However, at large strains the deformation mechanisms differ for compression (buckling and crushing) and tension (alignment and stretching). • The experimental data used to calibrate the model should correspond to the dominant deformation of the actual problem being analyzed. • Double-check the material model response to other modes.


L5.68

Hyperfoam Material Model • Difference in tension and compression (cont'd) • Model based on compression data • Check both tension and compression response • Here the tension response is qualitatively okay (no real data available for comparison).


116

Notes

117

Notes

118

Modeling Considerations and Usage Tips in Abaqus Lecture 6

L6.2

Overview • Modeling Issues • Contact • Element Selection • Meshing Considerations • Constraints and Reinforcements • Stability • Special Features • Example: Automotive Glass Run Channel Weatherseal • Using Abaqus/Explicit for Rubber Analysis • Example: Automotive Oil Pan Seal Compression


119

Modeling Issues

L6.4

Modeling Issues • Contact • Contact occurs routinely in elastomer analyses. • It is imperative to understand contact master-slave relationships. • E.g., only slave nodes are checked for contact in a pure-master slave formulation (default formulation)!

Incorrect Master surface placed on fine mesh Gross penetration into slave surface

Correct Master surface placed on coarse mesh Minimal penetration into slave surface


120

L6.5

Modeling Issues • Contact is complex and heuristic. • Two-dimensional contact very robust. • Three-dimensional contact has many more opportunities to go astray. • Lots of contact ―rules,‖ these will help you build robust models. Consult: • Abaqus Analysis User's Manual

• Contact with Abaqus/Standard lecture notes • Abaqus/Explicit: Advanced Topics lecture notes • Obtaining a Converged Solution with Abaqus lecture notes • Complex contact is handled more readily with explicit dynamics. • No convergence issues because iteration not required.


L6.6

Modeling Issues • Other helpful contact pair hints (for Abaqus/Standard): • Rigid body motion must be restrained in a static analysis. • Master surface smoothing has large effect on convergence. • Use *CONTACT PAIR, ADJUST for initial overclosures. • For automatic shrink fit capability use: *CONTACT INTERFERENCE, SHRINK • Pay attention to the status (.sta) file

and the severe discontinuity iterations (SDIs). • Try *CONTACT CONTROLS, AUTOMATIC TOLERANCES to reduce SDIs. • Use tie constraints (*TIE) for debugging contact problems. Modeling Rubber and Viscoelasticity with Abaqus

121

L6.7

Modeling Issues • Element Selection • Abaqus offers a variety of elements for use with the hyperelastic and hyperfoam material models. For a given state of stress (plane stress, plane strain, generalized plane strain, axisymmetric, or fully threedimensional) the user faces a number of choices: • First- or second-order elements

• Full or reduced integration • Incompatible mode elements • In addition, for solid rubbers we sometimes need to use hybrid elements.

• The following remarks serve as guidelines for the element type selection.


L6.8

Modeling Issues • First- or Second-Order Elements • Abaqus offers first-order elements with linear displacement interpolation and second-order elements with quadratic displacement interpolation. • The second-order elements give better results if the elements have a regular shape; the first-order elements work better if the elements have irregular, distorted shapes. • If strain gradients and element distortions remain small, secondorder elements are preferred. If strain gradients are large and element distortions become severe, first-order elements are recommended.

• For analyses involving variable contact and/or friction, first-order elements are recommended.


122

L6.9

Modeling Issues • Full- or Reduced-Integration Elements • Abaqus offers full or reduced integration for first- and second-order elements. • Reduced-integration elements use less computer time and yield more accurate stresses than full-integration elements. • This is particularly advantageous for problems with small strain gradients that use second-order elements.

• Reduced-integration elements, in particular first-order elements, can exhibit spurious deformation modes. • These spurious modes often cause instability problems if elements become distorted. • This is likely to occur in large-strain rubber analyses; hence, Abaqus automatically invokes enhanced hourglass control when first-order, reduced-integration elements use finite-strain elasticity (hyperelasticity or hyperfoam).


L6.10

Modeling Issues • Engine mount example

rubber

steel

Nonconvergence at 27% of load Outer rim moves up under load control


123

L6.11

Modeling Issues Severe hourglassing occurs with stiffnessbased hourglass control

No hourglassing with enhanced hourglass control.


L6.12

Modeling Issues • Regular or Hybrid Elements • Hybrid elements, where the pressure stress or volume change is interpolated separately and an extended variational principle is used, are needed for incompressible and almost incompressible behavior in plane strain, axisymmetric, and three-dimensional cases. • A constant pressure is used with the first-order elements. The pressure varies linearly with the second-order elements. • Hybrid elements must be used for incompressible hyperelasticity. • For compressible hyperelasticity hybrid elements are recommended. They are strongly recommended if second-order elements are used.

• Regular elements are used for plane stress and hyperfoam analysis.


124

L6.13

Modeling Issues • Incompatible Mode Elements • Incompatible modes enhance the bending response of fully integrated first-order elements. • Incompatible mode elements work well with hyperelastic materials up to moderate strains. • They should not be used in this case for large strains (> 100%), especially if the material is loaded in compression. • Erroneous stresses may sometimes appear in incompatible mode elements with hyperelastic material models that are unloaded after having been subjected to a complex deformation history.


L6.14

Modeling Issues • Complex geometry • In general, quad and hex elements are preferred. • These elements perform well, both for stress and contact. • Their CPU performance is also good. • However, complicated three-dimensional geometry necessitates the use of automatic mesh generation algorithms. • The resulting mesh is composed of tetrahedral elements. • In two-dimensions, automatic quad mesh algorithms are generally available (e.g., Abaqus/CAE offers such a capability) • Abaqus offers CPE6(H)M, CAX6(H)M, and C3D10(H)M just for this case.

• Use CPE3(H), CAX3(H), C3D4(H), C3D6(H) only for fill-in. • Otherwise, the model will be overly stiff.


125

L6.15

Modeling Issues • Meshing Considerations • The usual meshing considerations for linear analysis apply to the analysis of elastomers as well. • Often, elastomeric components are subjected to large strains and strain gradients. • High strain gradients lead to distorted elements, particularly in incompressible materials.

• Compared to similar problems using hyperelastic materials, there will be less distortion in elements using the hyperfoam model because of the large compressibility of foams, as shown in the following example. • The hyperfoam model experiences less lateral deformation.


L6.16

Modeling Issues • Example: Hyperelastic vs. Hyperfoam element distortions • The difference in bulk compressibility will cause significantly different element distortions between a solid elastomer (hyperelastic) and an elastic foam material (hyperfoam).

Solid Elastomer


126

Elastic Foam

L6.17

Modeling Issues • Element Distortions • In mesh generation try to anticipate the distortion in the layout of the mesh. • Manual mesh rezoning is available for hyperelasticity problems in Abaqus/Standard; experience suggest only limited benefits. • Abaqus/Explicit automatically invokes distortion control for solid elements modeled with hyperelastic or hyperfoam materials. • Element distortion control prevents excessive distortion from occurring under high compressive loads.


L6.18

Modeling Issues • Constraints and Reinforcements • If an incompressible rubber component is fully constrained, the hydrostatic pressure becomes undetermined. Hence, some part of the surface almost always remains unconstrained. • The difficulty of rubber analysis is often related to the amount of surface constraint: in highly confined components the rubber has very little freedom to move, which makes it more difficult for the analysis to converge (small changes in displacement create very large changes in forces). Thus, structures such as balloons are easy to analyze, whereas reinforced elastomeric bearings are not. • Rebars, or elements such as trusses or membranes, can be used to model reinforced rubber components. The combination of incompressible hyperelasticity with inextensible reinforcement can easily lead to ―locking‖ of the finite element mesh.


127

L6.19

Modeling Issues • Rebars

• The element on the left will shear without any stress in the reinforcement. • Shearing the element on the right with slightly skewed reinforcement will result in high stresses and much too stiff element behavior.


L6.20

Modeling Issues • Stability • Elastomeric components can exhibit structural instability similar to elastic structures. • Solid rubbers can exhibit surface instabilities when high compressive stresses develop tangential to a free surface. • These instabilities cause surface wrinkles and can be very detrimental to the convergence of iterations.

• Stability problems arise more often than you might expect! • Four classes of instability: • Material • Dynamic

• Global (geometric) • Local (material, geometric)


128

L6.21

Modeling Issues • Material stability • Check coefficients, check the .dat file, visual check in Abaqus/CAE • Check stability in all modes of deformations • Check stability outside range of data • High energy release rate (dynamic instability) • As response turns dynamic, a quasi-static analysis will encounter convergence trouble. • Global (geometric) instability • Snap-through of elastomeric components can be analyzed successfully in a static analysis with the Riks procedure.


L6.22

Modeling Issues • Local instabilities • These can be caused by local buckling, wrinkling, folds, etc. • Such local instabilities can be controlled by the automated stabilization algorithm (*STATIC, STABILIZE). • With this algorithm Abaqus/Standard chooses the damping coefficients so that energy dissipated by viscous damping is a small fraction of the strain energy in the model.


129

L6.23

Modeling Issues • Local instabilities (cont'd) • Snap-through of elastomeric components involving sudden loss of contact cannot be analyzed statically and require static stabilization or a dynamic analysis: *DYNAMIC *DYNAMIC, EXPLICIT

3-D arch


Special Features

130

L6.25

Special Features: Gasket Elements • Gasket elements • Engine sealing, gasket elements • Four cylinder engine assembly (block/gasket/head)


L6.26

Special Features: Gasket Elements • Gasket elements allow you to solve problems routinely that used to be beyond the reach of finite element analysis.


131

L6.27

Special Features: Gasket Elements • Engine sealing, gasket elements, pressure closure specification


L6.28

Special Features: CAXA elements • Modeling 3-D as 2-D—CAXA, CGAX elements


132

L6.29

Special Features: CAXA elements • Steel/rubber multi-layered spring


L6.30

Special Features: CAXA elements plunger

The analysis time for the CAXA model is 32 times faster than that for the complete 3D model. rubber seal

Asymmetric motion

Installed geometry

leaking

extremely large strain


133

L6.31

Special Features: Tire Modeling • Tire modeling: complex models, experienced users

Tire footprint and steady-state rolling (Abaqus/Standard)

Tire hitting curb (Abaqus/Explicit)

*IMPORT


L6.32

Special Features: Tire Modeling • Axisymmetric to 3-D transfer capability

sidewall bead

tread

Axisymmetric model carcass Model generation and results transfer

Model generation and results transfer

Half 3-D model Modeling Rubber and Viscoelasticity with Abaqus

134

Full 3-D model

belts

L6.33

Special Features: Tire Modeling • Coupled structural-acoustics • Used to analyze the acoustic signature of a tire design.

Coupled model

Inner air

tire


L6.34

Special Features: Miscellaneous • Abaqus offers many other special features to make your work easier • Automated pressure penetration (Abaqus Example Problems Manual) • Ex 1.1.16, Pressure penetration analysis of an air duct kiss seal • Hydrostatic fluid elements (Abaqus Example Problems Manual) • Ex 1.1.9, Hydrostatic fluid elements: modeling an airspring • Recent applications of Abaqus to the analysis of automotive rubber components, by Dr. Ken Morman • See also http://www.simulia.com/events/search-ucp.html to search proceedings online


135

Example: Automotive Glass Run Channel Weatherseal

L6.36

Automotive Glass Run Channel Weatherseal • The GRC Weatherseal model • Plane strain, half-symmetric model: glass surface

• CPE4RH elements • Rubber material

rubber

• Frictional contact

weatherseal

• Two-part analysis:

1. Seal to surrounding sheet metal assembly vehicle

2. Window closing effort—glass insertion

window frame

• We will focus on the first part only • Exhibits (strong) energy release during assembly.

Model courtesy of Advanced Elastomer Systems and Manta Corporation; example courtesy of SIMULIA Great Lakes Region


136

L6.37

Automotive Glass Run Channel Weatherseal • Material model • The Polynomial, N = 2 model parameters came from the customer. We can use Abaqus/CAE evaluate feature to show us the material response for these coefficients. What can we tell about stability just from looking at the coefficients? • The material definition is: *MATERIAL,NAME=S121-67 *HYPERELASTIC,N=2 1.325, -0.1895, 2.527e-4, -1.416e-3, 5.178e-4

• This is C10, C01, C20, C11, C02 .

• D’s are not defined.


L6.38

Automotive Glass Run Channel Weatherseal This is the standard Abaqus/CAE response plot after using the material evaluation function. Hard to compare scales.

We notice unstable BIAXIAL response of the material model. Look at the data file for more material stability checks. Will use this material model for now. Suspect problem difficulties are geometric in nature.


137

L6.39

Automotive Glass Run Channel Weatherseal This is the same response data plotted in the Visualization module using the XY Data Manager. Easier to compare scales. Best way to compare relationship of the responses to different modes of deformation. Lots of control over plot style, color, scales, text, etc.


L6.40

Automotive Glass Run Channel Weatherseal • Static analysis dies at this point. Why? • Note all force-deflection response plots show RF2 vs. U2 for the window frame rigid surface.


138

L6.41

Automotive Glass Run Channel Weatherseal • Will a Riks analysis work for this case? Single Step analysis with Riks fails! Window frame rigid surface moves down. ** Single Step w/ Riks *STEP,INC=100,NLGEOM *STATIC, RIKS 0.02,1.0,,,1.0 *BOUNDARY 9999,2,2,12.5 *END STEP

Split Step into two Steps. This runs to completion. *STEP,INC=100,NLGEOM *STATIC 0.02,1.0 *BOUNDARY 9999,2,2,10.5 *END STEP ** Continue with Riks *STEP, INC=200, NLGEOM *STATIC, RIKS .020, 1.0,,,1.0 *BOUNDARY, OP=MOD 9999,2,2,12.5 *END STEP


L6.42

Automotive Glass Run Channel Weatherseal • Successful Riks analysis • The Riks analysis reverses the window frame motion at the critical energy release point and seal remains in contact with the frame. • Riks would generally not work in cases where contact separation occurs.


139

L6.43

Automotive Glass Run Channel Weatherseal • This analysis can also be done successfully using automatic stabilization (with one small issue) Split Step into two Steps.

Single Step analysis fails!

This runs to completion.

** Single Step w/ stabilization

*STEP,INC=100,NLGEOM

*STEP,INC=100,NLGEOM

*STATIC

*STATIC, STABILIZE

Stable response in the first step

0.02,1.0,,,1.0

0.02,1.0

*BOUNDARY

*BOUNDARY

9999,2,2,12.5

9999,2,2,10.5

*END STEP

*END STEP ** Continue with auto stabilization *STEP, INC=400, NLGEOM

The two step approach allows Abaqus to compute the damping factor based on a stable response.

*STATIC, STABILIZE

.020, 1.0 *BOUNDARY, OP=MOD 9999,2,2,12.5 *END STEP


L6.44

Automotive Glass Run Channel Weatherseal • Force-deflection responses from the Riks and Stabilize analyses • Agreement is excellent. • The curves lie on top of one another until the instability (or energy release) point. • Remember that the Riks gives a ―pseudo-equilibrium‖ response.

• The stabilized result is closer to the actual physical response.


140

L6.45

Automotive Glass Run Channel Weatherseal • Dynamic analysis • Motivation: The physical event becomes dynamic. • This can be more troublesome than you might think… Define two steps (more efficient): 1st step is static. 2nd step is dynamic.

*STEP,INC=100,NLGEOM *STATIC 0.02,1.0 *BOUNDARY 9999,2,2,10.5

First phase is truly static. Trying to run as a single DYNAMIC step would run much longer. Need to answer the following: How do I set HAFTOL? What about the total time period? Is some damping needed?

*END STEP ** Dynamic Portion *STEP, INC=400, NLGEOM

*DYNAMIC,HAFTOL=10,INITIAL=NO .00100, 0.20 *BOUNDARY, OP=MOD 9999,2,2,12.5 *END STEP


L6.46

Automotive Glass Run Channel Weatherseal • This dynamic analysis gets past the energy release event but has difficulties running to completion. • Dynamic step starts with the window frame at U2=10.5 (upper image at right). • 1st movie shows that energy release happens in frames 76–78. Notice all the dynamic motions in frames 100–240 and the impact with the centerline in frame 240 (best viewed by double-clicking on movie header and controlling playback manually with the slider bar). • Analysis stops at U2=11.03 (wanted 12.5), due to difficulties with impact events at the centerline. • The 2nd movie shows a close-up of the energy release area. The mass of the seal helps regulate the motion (rate of motion).


141

L6.47

Automotive Glass Run Channel Weatherseal • Compare the force-deflection curves from the Riks, stabilize, and dynamic analyses • Zoom in to look at the characteristic behavior. Notice the ―hash‖ in the later stage of the dynamic response—this is due to impact events. We may be able to reduce or eliminate this by adding damping, but this takes considerably more effort than simply obtaining the stabilized solution.


L6.48

Automotive Glass Run Channel Weatherseal • Can we achieve a dynamic solution? Yes, but with more effort. • One might try adding numerical damping through the DYNAMIC, ALPHA parameter, but that did not help this analysis. Next, one might try DAMPING with mass and/or stiffness proportional damping. Finding an appropriate value for the DAMPING parameters can be a trial and error process. • Final results are shown here for a successful run (to completion), using 0.1.


142

L6.49

Automotive Glass Run Channel Weatherseal • Can we get a solution with Abaqus/Explicit? • Just for completeness we might ask ourselves if we can get an answer to this problem using Abaqus/Explicit. • Explicit dynamics can be used to get answers to problems such as these, but … it is best used for very large problems, or

with very large contact patches.


L6.50

Automotive Glass Run Channel Weatherseal • Using explicit dynamics will pose all the additional problems discussed earlier for implicit dynamics (Abaqus/Standard), plus some new ones. • The response will be inherently ―noisy,‖ especially if impact occurs (must filter the results). • Requires learning about time- and mass-scaling techniques to use Abaqus/Explicit effectively. • This is what happens if run too fast. • Performing a static analysis with Abaqus/Standard, then importing to Abaqus/Explicit, is time consuming. • Could be used for very large problems.


143

L6.51

Automotive Glass Run Channel Weatherseal • Comparison of solution approaches: • Abaqus/Standard: STATIC has problems because analysis becomes inherently dynamic as energy release occurs; Newton-Raphson cannot converge in light of these very large configurational changes. • Abaqus/Standard: STATIC, STABILIZE is generally the best solution for local instability, local energy release problems. Easy to use, robust. Use in step close to event. • Abaqus/Standard: STATIC, RIKS can help in many global buckling and collapse problems. It may help in energy release situations, but cannot track the problem if the body separates from the driving rigid surface. Not useful for local instability. • Abaqus/Standard: DYNAMIC is appealing since the physical event truly becomes dynamic, but the analysis poses additional choices/challenges. Expensive. • Abaqus/Explicit: DYNAMIC, EXPLICIT is useful for very large meshes with lots of contact. Fine meshes will increase solution time considerably. Postprocessing results that inherently include wave propagation can be challenging. May need to filter results.


Using Abaqus/Explicit for Rubber Analysis

144

L6.53

Using Abaqus/Explicit for Rubber Analysis • What is Abaqus/Explicit? • Abaqus/Explicit solves a dynamics problem resolving wave propagation. • It is a separate piece of software, not part of Abaqus/Standard. • Originally used just for highly dynamic events—explosions, crash. • Used extensively in sheet forming to solve quasi-static problems. • Also used to solve quasi-static rubber problems.

• Input syntax similar to Abaqus/Standard, but not exactly the same. • Learn about time scaling and mass scaling techniques. • For tough three-dimensional elastomer and contact problems, try Abaqus/Explicit.


L6.54

Using Abaqus/Explicit for Rubber Analysis • What advantages does Abaqus/Explicit have? • CPU cost lower for large models • No convergence issues • Handles large contact conditions more easily


145

L6.55

Using Abaqus/Explicit for Rubber Analysis • What disadvantages does Abaqus/Explicit have? • No hybrid elements; rubber must be hyperelastic, but compressible. • High bulk modulus decreases stable time increment and increases CPU cost. • Abaqus/Explicit solves the dynamic equilibrium equation; thus, always getting some wave propagation. • Can be difficult to postprocess for quasi-static analyses.


L6.56

Using Abaqus/Explicit for Rubber Analysis • Abaqus/Explicit usage tips for quasi-static rubber problems • Use AMPLITUDE, DEFINITION=SMOOTH STEP to smooth the loading. • Use as long a time frame as you can afford (CPU), typically still a fraction of a second, 10 or 20 milliseconds is common. • Must provide material density (solving dynamic equilibrium equations). • Always check the kinetic energy (should be small compared to internal).

• Make bulk modulus, K, only 10 times the shear modulus, G, when the part has room to shear. You can set K higher, but it drives up the CPU cost.


146

L6.57

Using Abaqus/Explicit for Rubber Analysis • Example: simple compression • We compare the force-deflection response of ABABQUS/Standard to ABABQUS/Explicit in a quasi-static 50% compression analysis.

Undeformed mesh

50% compression deformation


L6.58

Using Abaqus/Explicit for Rubber Analysis • Units are N,mm. • Puck dimension is standard ASTM: .5 inch high, 1 inch2 area (radius = 0.5642 in = 14.3 mm). • Material is Ogden • Treloar data. • Fit by Ogden (see his paper with Twizell). • Bulk modulus is 2000 MPa (from literature). • Testing of vulcanized natural rubber. • Thus D1 = 2/K = 0.001. • Density is that of rubber = 1.0E 9 N·s2/mm4.


147

L6.59

Using Abaqus/Explicit for Rubber Analysis • Compression is 50%, compressed with rigid surface platens. • Friction coefficient to platens is set to 1.0. • Element type is cax4h in Abaqus/Standard and cax4r in Abaqus/Explicit. • Total time in Abaqus/Explicit is 10 milliseconds and ran in approximately 133,000 increments. • The smooth step feature is used in Abaqus/Explicit. • Kinetic energy in Abaqus/Explicit is several orders of magnitude less than strain energy.


L6.60

Using Abaqus/Explicit for Rubber Analysis • 50% simple compression Abaqus/Standard vs. Abaqus/Explicit • Force vs. deflection is compared in the figure—very good match. Peak force to compress is 2.6% lower in Abaqus/Explicit than in Abaqus/Standard. • Plots of Mises stress, S22, and hydrostatic pressure all look similar, peak values are about 11–15% lower in Abaqus/Explicit than in Abaqus/Standard.


148

L6.61

Using Abaqus/Explicit for Rubber Analysis • Another Abaqus/Standard vs. Abaqus/Explicit comparison • Several more seal analyses that show good agreement between Abaqus/Standard and Abaqus/Explicit results are shown in the paper by DeHerrera and Heim. • M. A. DeHerrera, D. R. Heim, ―Using Abaqus/Explicit to Model Behavior of Elastomeric Sealing Components,‖ 2000 ABAQUS Users’ Conference.


Example: Automotive Oil Pan Seal Compression

149

L6.63

Example: Automotive Oil Pan Seal Compression • Oil pan seal insertion model

oil pan surface rubber seal

• Full three-dimensional; originally set up in Abaqus/Standard • Groove is aluminum, C3D8 elements • Seal modeled using C3D8RH elements

• Uses the Ogden, N = 3 model for the rubber constitutive behavior • Oil pan surface is modeled as rigid, analytical. groove (aluminum) Model courtesy of GM Powertrain; example courtesy of Abaqus Great Lakes


L6.64

Example: Automotive Oil Pan Seal Compression

Locations of weak springs used to control rigid body motion.


150

L6.65

Example: Automotive Oil Pan Seal Compression • Material model—input data • The Ogden, N = 3 model parameters came from the customer. We can use Abaqus/CAE evaluate feature to show us the material response for these coefficients. What can we tell about stability just from looking at the coefficients? • The material definition is: *MATERIAL,NAME=RUB *HYPERELASTIC, N=3,OGDEN 3.753,-0.3327,0.003215,10.09,-2.615,-0.7817,8.762e-3,0.0017525 0.0,23.0

• This is

1,

1,

2,

2,

3

,

3

, D1, D2

D3, Temperature • G0 is 1.141; K0 is 2/D1 = 228 (units are MPa, mm).


L6.66

Example: Automotive Oil Pan Seal Compression • Material model— stress-strain behavior • This is the standard Abaqus/CAE response plot after using the ―evaluate‖ function. • Hard to compare scales.

• We notice unstable BIAXIAL response of the material model.

• Look at the data (.dat) file for more material stability checks. • Will use this material model for now.


151

L6.67

Example: Automotive Oil Pan Seal Compression This is the same response data plotted in the Visualization module using the XY Data Manager. Easier to compare scales. Best way to compare relationship of the responses to different modes of deformation. Lots of control over plot style, color, scales, text. etc.


L6.68

Example: Automotive Oil Pan Seal Compression • Abaqus/Explicit vs. Abaqus/Standard • Notice mesh directly under flat rigid surface

Abaqus/Explicit

Abaqus/Standard


152

L6.69

Example: Automotive Oil Pan Seal Compression • Force vs displacement • Why so different?

Abaqus/Explicit

Abaqus/Standard


L6.70

Example: Automotive Oil Pan Seal Compression • Differences between Abaqus/Explicit and Abaqus/Standard: • Usually use hybrid elements in Abaqus/Standard for elastomers; no hybrid elements available in Abaqus/Explicit • Abaqus/ Standard typically incompressible in bulk behavior; Abaqus/Explicit must be compressible (somewhat) • Can use full integration or reduced elements in Abaqus/Standard; only reduced integration available in Abaqus/Explicit

• Don’t use springs in Abaqus/Explicit (no rigid body modes) • Setting appropriate timescale very important in Abaqus/Explicit • Smoothing the load application very important in Abaqus/Explicit


153

L6.71

Example: Automotive Oil Pan Seal Compression • Make the Abaqus/Explicit and Abaqus/Standard analyses more similar: • Use reduced-integration nonhybrid elements in both. • Use compressible material model in both. • Adjust (lengthen) the timescale in Abaqus/Explicit.

• Smooth load with AMPLITUDE, DEFINITION=SMOOTH STEP. • Set up new models in two dimensions. • Use of defaults traced error to userdefined hourglass control!


L6.72

Example: Automotive Oil Pan Seal Compression • The source of the problem in the three-dimensional Abaqus/Standard model is the nondefault hourglass control: *SOLID SECTION, ELSET=SEAL, MATERIAL=RUB *HOURGLASS STIFFNESS 13872.0,

This hourglass stiffness was overconstraining the model in Abaqus/Standard. The Abaqus/Standard result was wrong!


154

L6.73

Example: Automotive Oil Pan Seal Compression • Switch to either default hourglass control or full-integration elements in Abaqus/Standard; now results match very well.

Abaqus/Explicit

Abaqus/Standard

Mesh distortion due to presence of springs (not as weak as intended).


L6.74

Example: Automotive Oil Pan Seal Compression • Very good match of vertical strain component

Abaqus/Explicit

Abaqus/Standard


155

L6.75

Example: Automotive Oil Pan Seal Compression • Force-displacement now results match

User-defined hourglass control

Default hourglass control


L6.76

Example: Automotive Oil Pan Seal Compression • What about checking ALLAE vs ALLIE to catch the problem?


156

L6.77

Example: Automotive Oil Pan Seal Compression • Cam cover seal compression—summary • Comparison of Abaqus/Explicit and Abaqus/Standard solutions: • Originally thought that Abaqus/Standard was correct and initial Abaqus/Explicit analysis did not match it. • Kept changing Abaqus/Explicit analysis to mirror Abaqus/Standard analysis. • Modified Abaqus/Standard analysis at times too.

• Moved to new models in two dimensions based on seal crosssection. • Use of default settings in two dimensions led to discovery of offending overconstraint with user-defined hourglass control values.

• Reran Abaqus/Standard three-dimensional models using default hourglass control and using full-integration elements. • Now Abaqus/Explicit and Abaqus/Standard results agree well.


157

158

Notes

159

Notes

160

Viscoelastic Material Behavior Lecture 7

L7.2

Overview • Time Domain Response • Linear Viscoelasticity • Temperature Dependence • Frequency Domain Response • Hysteresis and Damping


161

Time Domain Response

L7.4

Time Domain Response • Definition • Certain materials are rate-dependent and behave elastically. • When unloaded, they return to their undeformed state. • These materials are called viscoelastic. • Examples • Polymers such as plastics

• Glass • Rubber • Foams • Solid rocket propellants


162

L7.5

Time Domain Response • For prescribed stress (force), these materials creep • Creep test measures strain (displacement) response as function of time while stress (force) is held constant on the specimen. prescribed stress


L7.6

Time Domain Response • Creep • Also occurs in metals, • Typically not recoverable (inelastic) • Creep material model is viscoplastic, not viscoelastic • Significant at high temperature (with respect to the melting point) • Creep of polymers is significant starting at low temperatures (

200 oC)

• For viscoelastic materials full elastic recovery occurs upon unloading


163

L7.7

Time Domain Response • For prescribed strains these materials exhibit stress relaxation • Stress relaxation test measures the stress (force) response as function of time while strain (displacement) is held constant on the specimen prescribed strain


L7.8

Time Domain Response • Stress Relaxation and Recovery • Viscous fluids, such as glass, polymers at high temperature and unvulcanized elastomers will relax to zero stress and will not recover when the applied strain is released. • Viscoelastic solids, such as polymers at lower temperatures, and vulcanized elastomers will relax asymptotically to a nonzero stress level. Upon release of the applied strain, they will partially recover elastically (immediately) and fully recover viscously over time.


164

Linear Viscoelasticity

L7.10

Linear Viscoelasticity • One-Dimensional Idealization • Linear and finite-strain viscoelasticity are idealized as series pairs of springs and dashpots in parallel with a spring Generalized Maxwell Model

• The number of dashpots is equal to the number of terms in the Prony series representing the stress response (the number of terms needed to fit the test data for the time domain of interest). • Every “network” (spring-dashpot pair) experiences the same total strain.


165

L7.11

Linear Viscoelasticity • Linear Viscoelasticity in Abaqus • The dashpot’s strain rate is proportional to stress

cr

A , where A

1 Viscosity

• The spring response may be linear or nonlinear:

• For “classical” linear viscoelasticity the springs are linear. • This implies a linear elastic material model in Abaqus • For finite-strain viscoelasticity the springs are nonlinear. • This implies a hyperelastic or hyperfoam material model in Abaqus


L7.12

Linear Viscoelasticity • How do I know if my material exhibits “linear” viscoelasticity? • From a practical perspective, one tests the validity of “linear” viscoelasticity by testing at multiple load levels and comparing (overlaying) the normalized response plots. Data for a silicone rubber:


166

L7.13

Linear Viscoelasticity • Response of not-as-linear viscoelastic elastomer • The material shown below is tested over a bit larger range of strain and the viscoelastic response of the material is less linear as indicated by the variations in the normalized stress relaxation curves. One must make a judgment call as to which relaxation curve to use. Stress Relaxation

Stress Relaxation 0.7 0.6

Stress (MPa)

0.5 0.4 0.3 0.2

1

0.9

Stress Normalized

20% Strain 40% Strain 60% Strain 80% Strain 100% Strain

0.8

0.7

20% Strain 40% Strain 60% Strain 80% Strain 100% Strain

0.6

0.1 0

0.5

0

500

1000

1500

2000

0

500

Time (secs)

1000

1500

2000

Time (secs)


L7.14

Linear Viscoelasticity • Creep response for "linear" viscoelasticity • Here is the creep response for a perfectly linear viscoelastic material loaded to 1, 2, and 4 MPa. • If these curves were normalized by the instantaneous strain they would perfectly overlay one another.


167

L7.15

Linear Viscoelasticity • Creep response for nonlinear viscoelasticity • The dashed lines depict the creep response for a material that does not obey “linear” viscoelasticity. • This kind of general nonlinear viscoelastic cannot be modeled in Abaqus with the *VISCOELASTIC material option. • Your material may behave nearly linear over a more narrow range of loading.


L7.16

Linear Viscoelasticity • “Classical” linear viscoelasticity: • Small-strain theory with linear elastic response. • Implies use of a linear elastic material model in Abaqus. • Experiments demonstrate that this model is accurate for many materials at small strains (say 0.05). • Finite-strain linear viscoelasticity: • Finite-strain theory with nonlinear elastic response. • Implies use of a hyperelastic or hyperfoam material to model the elasticity in Abaqus. • Simplest model for viscoelasticity at large strain. • Test assumption of linearity using tests for at least two load magnitudes.


168

Temperature Dependence

L7.18

Temperature Dependence • Elastomer and polymer material properties are strongly temperature dependent • Two types of effects: • Instantaneous response can be temperature dependent, that is temperature dependence of elastic moduli, or temperature dependence of hyperelastic model coefficients. • Time dependent behavior (rate of relaxation) can be temperature dependent. • This is modeled with the concept of a time-temperature shift function and reduced time. • Materials that can be described this way are called thermorheologically simple, or TRS (details in Lecture 10).


169

Frequency Domain Response

L7.20

Frequency Domain Response • Harmonic, or sinusoidal, excitation • Consider the application of a sinusoidal strain excitation,

(t )

where

is the frequency in radians/second.

• The material responds with a stress, where

(t )

i( t 0e

is called the loss angle.


170

i t 0e

)

:

L7.21

Frequency Domain Response • Loss Angle, • The strain lags behind the stress by an angle

• Purely elastic response (no damping)  • Polymers and Elastomers 

0.

0.


L7.22

Frequency Domain Response • Complex Modulus • It is convenient to separate the viscoelastic response into “in-phase” and “out-of-phase” components.

strain

0

sin t

stress

0

sin( t

0

(sin t cos

cos t sin

0

(sin t cos

sin ( t 90º) sin

in-phase stress

)

out-of-phase stress


171

L7.23

Frequency Domain Response • Complex Modulus (cont'd) • The complex shear modulus is denoted G* or G *( ).

shear stress shear strain

G* ( ) Complex Shear Modulus 0e

G* ( )

i( t

0e 0

cos

0

Gs

i

)

i t

0

sin

0

i Gl


L7.24

Frequency Domain Response • Complex Modulus (cont'd) • Storage Modulus, Gs :

Gs

0

cos

0

• Characterizes the in-phase shear modulus • Loss Modulus, Gl :

Gl

0

sin

0

• Characterizes the out-of-phase shear modulus


172

L7.25

Frequency Domain Response • Complex Modulus (cont'd) • For a harmonic loading of elastomers the storage and loss moduli typically look something like this:


L7.26

Frequency Domain Response • Complex Modulus (cont'd) • For unfilled rubbers the storage and loss moduli are dependent on frequency only. • The ratio:

Gl Gs

tan

is commonly referred to as “tan delta”

• For unfilled rubbers this ratio is often nearly a constant (over some frequency range of interest). • Typical value for natural rubber is 0.2.


173

L7.27

Frequency Domain Response • Complex Modulus (cont'd) • For filled rubbers the storage and loss moduli are usually dependent on the strain amplitude

Storage modulus

• The X-axis in these figures is the shear amplitude.

Loss modulus


Hysteresis and Damping

174

L7.29

Hysteresis and Damping • Viscoelastic materials dissipate energy. In the case of cyclic loadings, this is termed hysteresis; it arises from the frictional sliding of the long molecules across one another. • In other cases we refer to the energy dissipation characteristic as damping.

• Energy lost due to viscoelastic behavior is output in Abaqus using: • CENER: dissipation energy; element integration point variable • ELCD: dissipation energy; whole element variable • ECDDEN: dissipation energy per unit volume; whole element variable • ALLCD: dissipation energy; whole model variable Modeling Rubber and Viscoelasticity with Abaqus

L7.30

Hysteresis and Damping • Energy dissipation through hysteresis is represented by the area between the loading and unloading curves in a load-deformation cycle, and occurs with all rubbers.

• The complementary property is resilience, which is a measure of the energy returned. Fillers in the rubber will increase hysteresis. At high elongations, the hysteresis is much greater; this is associated with crystallization. • Rapidly repeated cyclic loading ---------------------------> heat • heat -----------------------------------> rise in temperature • rise in temperature -------------------------------> fatigue failure

• Thus, natural rubber, with its low hysteresis, is the preferred material for vibration applications.


175

L7.31

Hysteresis and Damping • Damping • In many practical applications the damping characteristics of rubber are important, and are often the reason that an elastomer is chosen for the application. • Any vibration isolation application, mounts, gaskets, etc., depends upon the damping characteristic of the elastomer. • Damping characteristics can be strongly dependent upon the chemical composition. • Damping characteristics can be strongly influenced by fillers in the rubber: • Carbon black • Silica


176

Notes

177

Notes

178

Time Domain Viscoelasticity Lecture 8

L8.2

Overview • Classical Linear Viscoelasticity • Prony Series Representation • Finite-Strain Viscoelasticity • Relaxation and Creep Test Data • Prony Series Data • Automatic Material Evaluation

• Usage Hints


179

Classical Linear Viscoelasticity

L8.4

Classical Linear Viscoelasticity • Recall that classical means small-strain theory. • In Abaqus this means: • relaxation/creep behavior defined by *VISCOELASTIC • elasticity is defined by *ELASTIC • Isotropic linear viscoelasticity is implemented in Abaqus


180

L8.5

Classical Linear Viscoelasticity • To generalize the viscoelastic equations to multiaxial stress states it is best to work with shear (deviatoric) and volumetric (dilatational) behavior:

S (t )  S0 (t ) 

1 G0

p(t )  p0 (t ) 

1 K0





0



dG ( ) S0 (t   ) d d

 dK ( )

0

d

p0 (t   ) d

where S is the deviatoric stress tensor and p is the hydrostatic pressure.


L8.6

Classical Linear Viscoelasticity • Definitions • We decompose the total stress into shear and volumetric parts by:

  S  pI

where

1 1 p   tr ( )   (11   22   33 ). 3 3

• We decompose the total strain into shear and volumetric parts by:

1 3

  e   vol I

where

 vol  tr ( )  (11   22   33 ).

• The elastic stress-strain relations decompose into:

S0 (t )  G0 e(t ) and

p0 (t )  K0  vol ,

where S0 (t ) and p0 (t ) are the deviatoric and pressure stress states that would exist for the current strain state if the material were behaving purely elastically.


181

Prony Series Representation

L8.8

Prony Series Representation • In Abaqus the time-dependent behavior G( ) and K( ) are represented in terms of a Prony series:

 G ( )  G0 1     K ( )  K 0 1   

N





G gip (1  e / i )  

material coefficients are

up to N pairs of gip and  iG

 material coefficients are   / iK p ki (1  e )  up to N pairs of ki p and  iK i 1 

i 1 N



• G0 and K0 are determined from the elasticity definition. • These are simply a sum of a series of exponential decays. • For many materials, including solid elastomers, the relaxation behavior is dominated by shear relaxation. In these cases it is not necessary to specify K( ). • An exception is void filled elastomers (elastic foams) in which there is generally significant volumetric relaxation. Modeling Rubber and Viscoelasticity with Abaqus

182

L8.9

Prony Series Representation • In Abaqus G( ) and K( ) are specified in one of four different ways: • Prony series curve fit from experimental stress relaxation test data: *VISCOELASTIC, TIME=RELAXATION TEST DATA

• Prony series curve fit from experimental creep test data: *VISCOELASTIC, TIME=CREEP TEST DATA

• Prony series coefficients specified directly by the user: *VISCOELASTIC, TIME=PRONY

• Defined from frequency-dependent cyclic test data:

Frequency-dependent input discussed in Lecture 9.

*VISCOELASTIC, TIME=FREQUENCY DATA


L8.10

Prony Series Representation • The rate-independent elastic moduli are specified using *ELASTIC *ELASTIC, MODULI=INSTANTANEOUS

E0, 0 or *ELASTIC, MODULI=LONG TERM

E1, 1

• Abaqus will determine the appropriate G and K values from the user specified E and  values.


183

L8.11

Prony Series Representation • If the user specifies long-term elastic moduli, Abaqus will compute the instantaneous elastic moduli using the relaxation information

N   G  G0 1  g kp     k 1 



 K   K 0 1   

 kkp   k 1  N



where the g kp and kkp are the Prony series coefficients. • The choice of defining the elasticity in terms of instantaneous or longterm is a matter of convenience only. • However, the MODULI parameter defaults to LONG TERM.

• Thus, if you enter instantaneous data but omit the INSTANTANEOUS parameter, it will adversely affect your solution.


Finite-Strain Viscoelasticity

184

L8.13

Finite-Strain Viscoelasticity • Polynomial Strain Energy Function • Energy function has the form: N

U

 C (I ij

1

N

 3) ( I 2  3)  i

j

i  j 1

 D (J 1

i 1

el

 1) 2i .

i

• The Prony series acts as a non-dimensionalized multiplier:

Cij ( )



N



 Cij0 1   

i 1

G  gip (1  e / i )   

N K  1 1   0 1  ki p (1  e / i )   Di ( ) Di  i 1  1 where Cij0 and 0 define the instantaneous shear and volume Di response.




L8.14

Finite-Strain Viscoelasticity • Ogden’s Strain Energy Function • Energy Function has the form: N

U

2i

 i 1

2 i

i

i

i

(1  2  3  3) 

N

 D (J 1

i 1

el

 1)2i ,

i

• The Prony series acts as a non-dimensionalized multiplier:

i ( ) 



i0 1   

N

 i 1

G  gip (1  e / i )   

N K  1 1   0 1  ki p (1  e / i )   Di ( ) Di  i 1  1 0 where i and 0 define the instantaneous shear and volume Di response.




185

L8.15

Finite-Strain Viscoelasticity • Marlow Strain Energy Function • The Prony series acts as a non-dimensionalized multiplier: R U dev ( )

R U vol ( )



N

0  U dev 1 



0  U vol

 1  

 i 1

G  gip (1  e / i )   

K  ki p (1  e / i )   i 1 

N



0 0 where U dev and U vol define the instantaneous shear and volume strain energy functions.


L8.16

Finite-Strain Viscoelasticity • Arruda-Boyce and Van der Waals Strain Energy Function • The strain energy functions are shown in Appendix 2. • The Prony series acts as a non-dimensionalized multiplier:

  ( )   1    0

1 1   0 1  D( ) D 

N

 i 1

G  gip (1  e / i )   

N

k

i

i 1

p

K  (1  e / i )   

1

0 where  and 0 define the instantaneous shear and volume D response.


186

L8.17

Finite-Strain Viscoelasticity • Hyperfoam Energy Function • The energy function is: N

U

 i 1

 2i  ˆi ˆi ˆi 1 i i       3  ( J  1)  , 1 2 3 el i i2  

• Note the deviatoric and volumetric behavior is fully coupled. • *VISCOELASTIC should obey

g kp  kkp ; that is, the shear and volume

relaxation rates should be equal. • The relaxation behavior is governed by Prony series:

 i ( )   i 1    0

N

 i 1

 gip (1  e / i )   


L8.18

Finite-Strain Viscoelasticity • Hyperfoam Energy Function (cont'd) • When using viscoelasticity in conjunction with the hyperfoam material model:

• You may use *SHEAR TEST DATA to specify the viscoelastic behavior. • Abaqus will set the volume behavior equal to the shear. • You may use *VOLUMETRIC TEST DATA to specify the viscoelastic behavior. • Abaqus will set the shear behavior equal to the volumetric. • You may use *COMBINED TEST DATA to specify the viscoelastic behavior. • You should make the shear and volume data the same. • You may use TIME=PRONY and specify the Prony coefficients directly. p p • If you give just the g k terms, then the kk terms will be set to the same value (and vice-versa). Modeling Rubber and Viscoelasticity with Abaqus

187

Relaxation and Creep Test Data

L8.20

Relaxation and Creep Test Data • Relaxation Test Data • Consider a simple shear relaxation test

• Here g0 is the instantaneous (short-time) applied shear strain, and

 (t) is the measured shear stress response. • Note: Be careful that the short-time duration of the prescribed strain is consistent with the time scale of your linear elastic or hyperelastic material definition.


188

L8.21

Relaxation and Creep Test Data • Relaxation Test Data (cont'd) • The measured shear stress response is data pairs of (shear stress, time)

0 , time0 1 , time1 2 , time2 3 , time3 …

• Because this is linear viscoelasticity only one curve may be used. If your material is not exactly linear in its viscoelastic response, then test at an applied strain that is close to your component analysis strain level of interest.


L8.22

Relaxation and Creep Test Data • Relaxation Test Data (cont'd) • Test data processing for use in Abaqus: • Simply normalize all the measured stress values by 0: 0 / 0 , time0 1 / 0 , time1 2 / 0 , time2 3 / 0 , time3 : : and input this data using the *SHEAR TEST DATA suboption of *VISCOELASTIC, TIME=RELAXATION TEST DATA.


189

L8.23

Relaxation and Creep Test Data • Example 1: Relaxation Test Data Usage *MATERIAL,NAME= *ELASTIC, MODULI= or *HYPERLEASTIC, MODULI= ... (data lines) *VISCOELASTIC, TIME=RELAXATION TEST DATA, {ERRTOL = 0.01, NMAX = 13} *SHEAR TEST DATA, SHRINF = 0.5 1.0000, 0.0001 0.9695, 0.001 0.9417, 0.002 ... *VOLUMETRIC TEST DATA, VOLINF = 0.5 1.0000, 0.0001 0.9695, 0.001 0.9417, 0.002 ... nonlinear least squares fits

pairs of gi and  i p

G

pairs of ki and  i p

K


L8.24

Relaxation and Creep Test Data • Example 1: Relaxation Test Data Usage (cont'd) • Volumetric relaxation test data is optional; many solid materials exhibit insignificant volumetric relaxation behavior. • Separate fits are performed on the shear and volumetric parts and combined into one set of Prony series parameters. • SHRINF is the normalized shear stress (modulus) as time → 1; if a value for this parameter is specified it will act as a further constraint to enforce N

1

g

p i

 g R ().

i 1

• VOLINF is the normalized pressure (modulus) as time → 1; if a value for this parameter is specified it will act as a further constraint to enforce N

1

k

i

p

 k R ().

i 1


190

L8.25

Relaxation and Creep Test Data • Example 2: Relaxation Test Data Usage (combined test data) • If both the shear and volumetric relaxation tests are performed and the same time intervals are used in each, then the normalized experimental data can be specified using a single keyword: *COMBINED TEST DATA


L8.26

Relaxation and Creep Test Data • Example 2: Relaxation Test Data Usage (combined test data, cont'd) *MATERIAL,NAME= *ELASTIC, MODULI= or *HYPERLEASTIC, MODULI= or *HYPERFOAM ... (data lines) *VISCOELASTIC, TIME=RELAXATION TEST DATA, {ERRTOL = 0.01, NMAX = 13} *COMBINED TEST DATA, SHRINF = 0.5, VOLINF = 0.5 0.99256, 0.99256, 0.1 0.98525, 0.98525, 0.2 0.97805, 0.97805, 0.3 ...

single curve fit

groups of gi , ki , and  i p

p


191

L8.27

Relaxation and Creep Test Data • Creep Test Data • Consider a simple shear creep test

• Here 0 is the instantaneous (short-time) applied shear stress, and

g (t) is the measured shear strain response. • Note: Be careful that the short-time duration of the prescribed stress is consistent with the time scale of your linear elastic or hyperelastic material definition.


L8.28

Relaxation and Creep Test Data • Creep Test Data (cont'd) • The measured shear strain response is data pairs of (shear strain, time)

g0 , time0 g1 , time1 g2 , time2 g3 , time3 …

• Because this is linear viscoelasticity only one curve may be used. If your material is not exactly linear in its viscoelastic response then test at an applied stress that is close to your component analysis stress level of interest.


192

L8.29

Relaxation and Creep Test Data • Creep Test Data (cont'd) • Test data processing for use in Abaqus: • Simply normalize all the measured strain values by g0 :

g0 / g0 , time0 g1 / g0 , time1 g2 / g0 , time2 g3 / g0 , time3 … and input this data using the *SHEAR TEST DATA suboption of *VISCOELASTIC, TIME=CREEP TEST DATA.


L8.30

Relaxation and Creep Test Data • Example 3: Creep Test Data Usage *MATERIAL, NAME= *ELASTIC, MODULI= OR *HYPERLEASTIC, MODULI= ... (data lines) *VISCOELASTIC, TIME=CREEP TEST DATA, {ERRTOL=0.01, NMAX=13} *SHEAR TEST DATA, SHRINF=2 1.00747, 0.1 1.01487, 0.2 ... 1.99619, 100.0 *VOLUMETRIC TEST DATA, VOLINF=2 1.00747, 0.1 1.01487, 0.2 ... 1.99619, 100.0

nonlinear least squares fits

pairs of gi and  i p

G

pairs of ki and  i p

K


193

L8.31

Relaxation and Creep Test Data • Example 3: Creep Test Data Usage (cont'd) • Volumetric creep test data is optional; many solid materials exhibit insignificant volumetric creep/relaxation behavior. • Separate fits are performed on the shear and volumetric parts and combined into one set of Prony series parameters. • SHRINF is the normalized shear strain (compliance) as time → 1; if a value for this parameter is specified it will act as a further constraint to enforce N

1

g

p i

 g R ().

i 1

• VOLINF is the normalized volume strain (compliance) as time → 1; if a value for this parameter is specified it will act as a further constraint to enforce N

1

k

i

p

 k R ().

i 1


L8.32

Relaxation and Creep Test Data • Example 4: Creep Test Data Usage (combined test data) • If both the shear and volumetric creep tests are performed and the same time intervals are used in each, then the normalized experimental data can be specified using a single keyword: *COMBINED TEST DATA


194

L8.33

Relaxation and Creep Test Data • Example 4: Creep Test Data Usage (combined test data, cont'd) *MATERIAL, NAME= *ELASTIC,MODULI= or *HYPERLEASTIC,MODULI= or *HYPERFOAM ... (data lines) *VISCOELASTIC, TIME = CREEP TEST DATA, {ERRTOL = 0.01, NMAX = 13} *COMBINED TEST DATA, SHRINF = 2, VOLINF = 2 1.00747, 1.00747, 0.1 1.01487, 1.01487, 0.2 ... 1.99619, 1.99619, 100.0

single curve fit

groups of gi , ki , and  i p

p


L8.34

Relaxation and Creep Test Data • Prony Series Curve Fit Notes • The proper number of terms in the Prony series should be used • Too few terms will yield a poor fit. • Too many terms may cause ill-conditioning • Typically you need about the same number of Prony terms as you have decades of time data. • The ERRTOL parameter on the *VISCOELASTIC option controls the accuracy of the fit, and thus the number of terms generated. • It is the allowable average RMS error in the least squares fit. • The default value is 0.01.


195

L8.35

Relaxation and Creep Test Data • The NMAX parameter on the *VISCOELASTIC option controls specifies the maximum number of terms in the Prony series. • Fit is performed from N1 to NNMAX until convergence is achieved for the lowest N with respect to ERRTOL . • Visually check your fit using unit-cube type analyses.


Prony Series Data

196

L8.37

Prony Series Data • Prony Series Data • An alternative to specifying test data is to enter the Prony coefficients directly. *MATERIAL, NAME=... *ELASTIC,MODULI=... Or *HYPERLEASTIC,MODULI=... Or *HYPERFOAM ... (data lines) *VISCOELASTIC, TIME=PRONY,

g1p , k1p , 1

1   2   3

g 2p , k2p ,  2 g3p , k3p ,  3


L8.38

Prony Series Data • Prony Series Data • Rule of thumb is one-two logarithmic decades of time per i . • For example, suppose we want to model the relaxation over the time span from 0.1 seconds to 7200 seconds. 0.1 – 1.0 seconds is one decade of time 1 – 10 is 2 decades 10 – 100 is 3 decades 100 – 1000 is 4 decades 1000 – 10,000 is 5 decades • We can estimate that it will take 3-5 Prony series terms for a good fit to 5 decades of relaxation.


197

L8.39

Prony Series Data • Changing number of Prony Terms: N= 2, N= 3, N= 5

N=3 Better

N=2 Very Poor

N=5 Best



198

L8.41

Automatic Material Evaluation • Abaqus offers a material evaluation capability for viscoelastic material models. • Similar to the capability for hyperelastic materials. • Use Abaqus/CAE to perform standard tests. • Supply experimental test data. • Specify relaxation or creep response (or both). • X–Y plots appear for each test. • Predicted normalized moduli curves plotted against experimental test data.


L8.42


• The material curve fitting capability allows you to view the behavior predicted by a viscoelastic material and compare it with the test data.


199

Usage Hints

L8.44

Usage Hints • Interpolation / Extrapolation • Experimental data must cover the time domain of interest in the analysis.

• Prony series can only represent behavior over the fitted time domain. • Extrapolation does not work.


200

L8.45

Usage Hints • Viscoelastic assumption for solid rubber • For many solid rubber (incompressible or nearly incompressible) materials it is reasonable to assume that the viscoelastic behavior is entirely a shearing action; that is, there is no appreciable viscoelastic action in the volumetric deformation. • With the above assumption one can perform UNIAXIAL creep/relaxation tests and after normalizing provide this data as *SHEAR TEST DATA. *VOLUMETRIC TEST DATA is not given in this case.

• To test this assumption one might perform a volumetric relaxation test, or perform and compare a simple shear vs. uniaxial relaxation test.


L8.46

Usage Hints • Procedures • Time-domain viscoelasticity can be used with: *STATIC (viscoelastic behavior ignored) *VISCO *DYNAMIC *COUPLED TEMPERATURE-DISPLACEMENT *STEADY STATE TRANSPORT

• In the *VISCO procedure the CETOL parameter controls the automatic time incrementation. • It limits Dt such that Dcreep  CETOL


201

L8.47

Usage Hints • Procedures (cont'd) • Instantaneous elastic response • The *STATIC procedure can be used to apply loads instantaneously: *STATIC • The instantaneous modulus characterizes the response *STATIC, LONG TERM • The long term (long-time) modulus characterizes the response

• In the *COUPLED TEMPERATURE-DISPLACEMENT procedure the CREEP=NONE parameter setting can be used to force Abaqus to use only the instantaneous modulus.


L8.48

Usage Hints • Procedures (cont'd) • In the *COUPLED TEMPERATURE-DISPLACEMENT procedure the CETOL and/or DELTMX parameters control the automatic time incrementation. • In a *DYNAMIC procedure the HAFTOL parameter controls the automatic time incrementation and CETOL cannot be used. • In general, the *STEADY STATE TRANSPORT procedure allows for timedependent viscoelastic behavior. • Use *STEADY STATE TRANSPORT, LONG TERM to indicate that there is no viscoelastic material response during this step and that the solution must be based on the long-term elastic moduli.


202

Notes

203

Notes

204

Frequency Domain Viscoelasticity Lecture 9

L9.2

Overview • Classical Isotropic Linear Viscoelasticity • Tabular Data • Formula Data • Isotropic Finite-Strain Viscoelasticity • Procedures


205

Classical Isotropic Linear Viscoelasticity

L9.4

Classical Isotropic Linear Viscoelasticity • Isotropic linear viscoelasticity is implemented in Abaqus. • Independent storage and loss moduli for deviatoric and volumetric behavior. • The user must supply G , K , G* ( ), and K * ( ).

G  Gs  iGl *

K  K s  iK l *

G0

K0 K

G




206



L9.5

Classical Isotropic Linear Viscoelasticity • The complex shear and bulk moduli are defined as

G* ( )  Gs ( )  iGl ( ) and

K * ( )  K s ( )  iKl ( ), respectively. • The storage moduli Gs and Ks and the loss moduli Gl and Kl are measured as a function of frequency f = /2p.


L9.6

Classical Isotropic Linear Viscoelasticity • The following dimensionless moduli can be obtained by using the long term moduli G and K:

m1 ( f )  Gl G m2 ( f )  1  Gs G m3 ( f )  K l K  m4 ( f )  1  K s K  • The advantage of dimensionless viscoelastic moduli is that they can be defined independently of the elastic material data. • If data are unavailable for a given material, one can try using dimensionless data from similar materials.


207

L9.7

Classical Isotropic Linear Viscoelasticity • Long-term moduli G and K are determined from E and . • Defined with the *ELASTIC option. • Storage and loss moduli can be supplied (indirectly) in three different ways: • Tabular input of nondimensional moduli: *VISCOELASTIC, FREQUENCY=TABULAR

• Formula parameters for nondimensional moduli: *VISCOELASTIC, FREQUENCY=FORMULA

• Prony series expression for the relaxation moduli (discussed in Lecture 8) *VISCOELASTIC, FREQUENCY=CREEP TEST DATA *VISCOELASTIC, FREQUENCY=RELAXATION TEST DATA *VISCOELASTIC, FREQUENCY=PRONY


L9.8

Classical Isotropic Linear Viscoelasticity • Tabular data • Abaqus usage: *MATERIAL, NAME=... *ELASTIC

E,  *VISCOELASTIC, FREQUENCY=TABULAR

m1( f1 ), m2( f1 ), m3( f1 ), m4( f1 ), f1 m1( f2 ), m2( f2 ), m3( f2 ), m4( f2 ), f2 :

m1( fn ), m2 ( fn ), m3 ( fn ), m4 ( fn ), fn where fi = i/2p … frequency in cycles per time.


208

L9.9

Classical Isotropic Linear Viscoelasticity G  Gs  iGl *

• Abaqus Usage Examples *MATERIAL, NAME= ELASTMAT

storage

G0

*ELASTIC, ... G

:

loss

: *VISCOELASTIC, FREQUENCY=TABULAR

Gl /G1, 1 – Gs /G1, Kl /K , 1 –Ks /K, freq1

Gl /G1, 1 – Gs /G1, Kl /K , 1 –Ks /K, freq2

K  K s  iK l *

K0

: :

VOLUMETRIC

K

SHEAR


L9.10

Classical Isotropic Linear Viscoelasticity *material, name=rubber

Test data

*elastic :

Long term modulus G1= 486.6 psi from (long-term) elastic constants

:

*viscoelastic, frequency=tabular 0.0020552, 0.0184966, 0.0369931, 0.0822069, 0.1233104, 0.1644139,

0.0011858, 0.0340686, 0.0751721, 0.1368273, 0.0957238, 0.0237927,

0, 0, 0, 0, 0, 0,

0, 0, 0, 0, 0, 0,

1.0 10.0 100.0 200.0 400.0 500.0

frequency Gl /G1

1 – Gs /G1 Volumetric

Long term modulus

storage modulus loss modulus

frequency (Hz) 1 10 100 200 400 500

Gs (psi) Gl (psi) 486 1.0 470 9.0 450 18.0 420 40.0 440 60.0 475 80.0


209

L9.11

Classical Isotropic Linear Viscoelasticity • Defining tabular frequency data in Abaqus/CAE


L9.12

Classical Isotropic Linear Viscoelasticity • Formula data • Dimensionless relaxation moduli are approximated by a power law formula:

m1 ( f )  2p f  s1  f  a m2 ( f )  2p f  s2  f  a m3 ( f )  2p f  s3  f b m4 ( f )  2p f  s4  f b where a and b are real constants. • Abaqus usage: *MATERIAL, NAME=... *ELASTIC

E, 

*VISCOELASTIC, FREQUENCY=FORMULA

s1

s1, s2, a, s3, s4, b Modeling Rubber and Viscoelasticity with Abaqus

210

s2

a

L9.13

Classical Isotropic Linear Viscoelasticity • Special case: structural damping • The case with s2 = 0 and a = 1 and s4 = 0 and b = 1 is attractive since the damping becomes frequency independent (GS = G and KS = K):

GL G  2p  s1   s K L K   2p  s3   b .


L9.14

Classical Isotropic Linear Viscoelasticity • Prony series data • The relaxation moduli can be defined in terms of Prony series data. • The input is identical to that for time domain response. • See Lecture 8 for usage details.


211

Isotropic Finite-Strain Viscoelasticity

L9.16

Isotropic Finite-Strain Viscoelasticity • As with isotropic linear viscoelasticity, independent moduli for deviatoric and volumetric behavior are required. • Moduli are effective tangent moduli: Force-displacement curve

Storage modulus with preload

P

Ds D

D

D

D

without preloads





G1 should be determined from and self-consistent with the hyperelastic constants.


212

L9.17

Isotropic Finite-Strain Viscoelasticity • Two methods for specifying data

with preload

• Long-term elastic moduli

Ds

• Data are based on measurements at a single state. For example: • The undeformed state, the prestrain level about which the response is required, etc.

D

D

without preloads



• If use data at prestrain levels other than the one at which they were measured, are assuming properties are independent of prestrain level. • Direct specification of storage and loss moduli as functions of frequency and prestrain.

• More general; allows for data at multiple prestrain levels; does not assume independence of data and prestrain level.


L9.18

Isotropic Finite-Strain Viscoelasticity • Specifying long-term elastic properties • Appropriate for cases where properties are independent of prestrain or data are measured at desired prestrain level. • Frequency-dependent behavior is specified in the same way as described earlier; long-term moduli specification depends on material type: • Hyperelasticity:   • Polynomial energy functions  Cij , Di .







• Ogden energy function  i ,  i , Di .   • Marlow energy function  U dev , U vol .





• Arruda-Boyce and Van der Waals energy function   , D .

• Hyperfoam: 





• Hyperfoam energy function  i ,  i ,  i .


213

L9.19

Isotropic Finite-Strain Viscoelasticity • Direct specification of storage and loss moduli • Direct specification from uniaxial and volumetric tests: *VISCOELASTIC, PRELOAD=UNIAXIAL *VISCOELASTIC, PRELOAD=VOLUMETRIC

• Abaqus converts the data to ratios of shear (or bulk) storage and loss moduli to long-term elastic moduli. • E.g., for uniaxial data: • Enter Elnom and Esnom • Abaqus converts to Gl and Gs

and then computes Gl /G1and Gs/G1

Elnom

Esnom

Properties are a function of frequency and prestrain


L9.20

Isotropic Finite-Strain Viscoelasticity • How are the data measured? • Consider uniaxial text

l0 Nominal area A0 Force

DF

Fp

time

1.

Preload to Fp

2.

Cycle load relative to Fp

3.

Measure Du (relative to preloaded u) and lag d

4.

Compute moduli according to

DF E

nom


Du

A0 l0

nom

 E nom cos(d )

nom

 E nom sin(d )

Es El

214



Procedures

L9.22

Procedures • Frequency-domain viscoelasticity can be used only with *STEADY STATE DYNAMICS, DIRECT *STEADY STATE DYNAMICS, SUBSPACE PROJECTION *FREQUENCY

*COMPLEX FREQUENCY

• For steady state dynamic procedures: • Loads and nonzero boundary conditions specified prior to the procedure are kept constant. • Loads and nonzero boundary conditions specified within the procedure are applied as harmonic loads.


215

L9.23

Procedures • For frequency extraction procedures: • Applied loads are ignored but the load stiffness determined at the end of the previous general analysis step is included. • The PROPERTY EVALUATION parameter is required when using frequency-domain viscoelasticity with these procedures.


216

Notes

217

Notes

218

Time-Temperature Correspondence Lecture 10

L10.2

Overview • Reduced Time • Measuring Temperature Dependence • Input Data for Temperature Effects • WLF Examples


219

Reduced Time

L10.4

Reduced Time • At right is the relaxation modulus, E(t), of a rubber specimen subjected to a relaxation test at room temperature (20 C). • Let this temperature be the reference temperature 0.

E t E0

t1

t2


220

t3

t

L10.5

Reduced Time • At right is the relaxation modulus, E70ºC(t), of the same rubber specimen subjected to a relaxation test at an elevated temperature of 70ºC.

E t E0

.03 t1

.03 t 2

.03 t3

t


L10.6

Reduced Time • Observe that these curves are the same except for the time scales: • Modulus at 10 minutes at 20ºC = modulus at 10 0.03 minutes at 70ºC.

E t E0 20ºC

t1

t2

t3

t

E0 70ºC

.03 t1 .03 t 2 .03 t3

t


221

L10.7

Reduced Time • In general for this material

E 70 C (t )

E 20 C (t 0.03).

• This is called the reduced-time concept:

ET0 (t A( )).

ET (t )

• A( ) is the time reduction factor at temperature reference temperature

relative to the

0.

• A( ) decreases with increasing . • A( ) is material dependent.


L10.8

Reduced Time • A material obeying the above law is called thermo-rheologically simple. • Viscous flow mechanisms are sped up by a constant factor for a given temperature rise. • The general response of a thermo-rheologically simple material at temperature is t

(t , )

E

0

t d ( ) d . A( ) d


222

L10.9

Reduced Time • Shift function • This time-temperature correspondence is often represented by the logarithmic time shift:

h( )

E (t ) 0 0

h( 0

)

log( A( )) extrapolated

0

measured

extrapolated log t

• As shown in this figure, when E is plotted for various temperatures against the log of t, then h( is the horizontal shift of the curve at temperature from the curve at the reference temperature 0.


L10.10

Reduced Time • Note that: • When

>

0

A( ) < 1 and h( ) > 0 (viscous flow speed up).

• When

=

0

A( ) = 1 and h( ) = 0.

• When

<

0

A( ) >1 and h( ) < 0 (viscous flow slow down).

• For thermo-rheologically simple materials, the viscoelastic properties at 0 and logarithmic time shift h( ) characterize viscoelastic properties for all temperatures .


223

Measuring Temperature Dependence

L10.12

Measuring Temperature Dependence • The time shift property can be used to extrapolate the relaxation data to very long or very short times: • Instead of testing for very long times, test at high temperature. • Instead of testing for very short times, test at low temperature. • The extrapolation procedure works as follows: 1. Relaxation tests are carried out for a given time range and for a given temperature range.

• For example, t between 1 sec and 1000 sec. • For example, between 0 and 0 + (where 0 is the reference temperature).


224

L10.13

Measuring Temperature Dependence 2.

The results are plotted on a logarithmic scale: E (t ) 0 h( 0

0

)

0 extrapolated

measured

extrapolated log t

3.

With h( 0) = 0, the measured curves make it possible to determine the shifts h( 0 ) and h( 0 + ).

4.

Then, the logarithmic time shift, h( ), can be calibrated and used to extrapolate the relaxation curves well beyond the measured domain. Modeling Rubber and Viscoelasticity with Abaqus

L10.14

Measuring Temperature Dependence 5. By carrying out relaxation experiments over a wide enough range of temperatures, a complete relaxation curve spanning many decades in time can be obtained.

• See Mercier et al. (listed in Appendix 5) for a detailed example of determining the shift function.


225

Input Data for Temperature Effects

L10.16

Input Data for Temperature Effects • Abaqus supports the following forms of the shift function h( • Williams-Landell-Ferry (WLF) • Arrhenius form • User-defined forms


226

L10.17

Input Data for Temperature Effects • Williams-Landell-Ferry (WLF) • The shift functions for materials about a reference temperature commonly fit the Williams-Landell-Ferry (WLF) form

h( )

log10 ( A)

C1 ( C2 (

0) 0)

.

• C1 and C2 are material constants at the reference temperature • Any convenient

0

0.

can be chosen.

• Curve-fit of measured values of h( ) is used to calibrate C1 and C2. • Since relative temperatures are used, any appropriate temperature scale can be chosen. • The WLF function is the default shift function in Abaqus


L10.18

Input Data for Temperature Effects • If the reference temperature is the material’s glassy transition temperature, g, then C1 and C2 are close to the “universal” values that are available for many materials:

WLF parameters (after Ferry, 1980) Polymer

C1g

C2g (K)

g (K)

Polyisobutylene

16.6

104

202

Natural rubber (Hevea)

16.7

53.6

200

Polystyrene

14.5

50.4

373

Polyethyl mathacrylate

17.6

65.5

335

“Universal" constants

17.4

51.6


227

L10.19

Input Data for Temperature Effects g

g

• The “universal” constants C1 and C2 are related to C1 and C2 (for some convenient reference temperature 0) as follows:

C1

g C1

g 1 ( 0 g ) C2 g C2 C2 0 g


L10.20

Input Data for Temperature Effects • Abaqus usage for WLF shift function: *MATERIAL, NAME=... *ELASTIC, MODULI=...

E, *VISCOELASTIC,... ... ...(Data at temperature ... *TRS, DEFINITION=WLF 0,

0)

C1, C2


228

L10.21

Input Data for Temperature Effects • Arrhenius shift function • The Arrhenius appropximation is commonly used for semi-crystalline polymers.

h( )

ln( A)

E0 R

1

1 Z

0

Z

• E0 is the activation energy, R is the universal gas constant,

is the absolute zero in the temperature scale being used, and 0 is the reference temperature at which the relaxation data are given. Z

• Usage: *PHYSICAL CONSTANTS, ABSOLUTE ZERO=..., UNIVERSAL GAS CONSTANT=... : *TRS, DEFINITION=ARRHENIUS 0,

E0 Modeling Rubber and Viscoelasticity with Abaqus

L10.22

Input Data for Temperature Effects • User-defined shift function • User subroutine (V)UTRS can be used to define other forms of the shift function. • Usage: *TRS, DEFINITION=USER


229

WLF Examples

L10.24

WLF Examples • Example 1: • Calculate the time reduction factor of rubber at 293 K (20 C) relative to its glassy transition temperature of 200 K. Use the data in the table. • Resolution: • From the table: C1g

16.7 and C2g

53.6 K.

• So, h(293 K) 16.7(293 200) (53.6 293 200) 10.594. • Thus, A(293 K) 10

10.594

2.546 10

11

.


230

L10.25

WLF Examples • Example 2: • Calculate the time reduction factor of rubber at 343 K (70 C) relative to room temperature (293 K, 20 C). • Resolution A: • Recall that h( ) = log (A( )). • Then, relative time shift from rubber at 293 K to rubber at 343 K is given by

h

h(343 K) h(293 K) log( A(343 K)) log( A(293 K)) log

A(343 K) A(293 K)


L10.26

WLF Examples • Therefore, using the procedure outlined in Example 1, the relative time reduction factor is 293 K

A

(343 K)

A(343 K) A(293 K)

7.128 10 2.546 10

13 11

0.028 0.03.

• Resolution B (using conversion formulae on page L10.18):

C1293 K

16.7 1 (293 200) 53.6

C2293 K

53.6 293 200 146.6 K

6.106

h293 K (343 K) 6.106(343 293) (146.6 343 293) 1.553. A293 K (343 K) 10

1.553

0.028.


231

232

Notes

233

Notes

234

Modeling Advanced Behaviors Lecture 11

L11.2

Overview • Hysteresis in Elastomers • Modeling Permanent Set in Elastomers • Anisotropic Hyperelasticity


235

Hysteresis in Elastomers

L11.4

Hysteresis in Elastomers • “Classical” linear viscoelasticity: small-strain theory in which the instantaneous stress is proportional to the strain. • Experiments demonstrate that this model is accurate for many materials at small strains ( 0.01). • “Finite-strain” linear viscoelasticity: hyperelastic or hyperfoam theory in which the relaxation rate is proportional to the stress. • Simplest model for viscoelasticity at large strains.

• Small amount of experimental data required to calibrate model. • For many materials the relaxation rate is proportional to the stress and the viscoelastic models are appropriate; however, there are many materials that do not exhibit this proportional behavior.


236

L11.5

Hysteresis in Elastomers • In filled and some unfilled rubbers the creep or relaxation rate is not proportional to the stress. • Typically, creep and stress relaxation are more pronounced at higher stress levels. • In addition, at higher stress levels creep and stress relaxation occur faster initially and reach a plateau more slowly than with viscoelasticity. • This leads to hysteresis-type behavior in cyclic loading, where the amount of hysteresis increases with loading amplitude but is relatively independent of the cycling frequency. • This kind of general nonlinear, finite-strain, time-dependent behavior is what the hysteresis model attempts to capture.


L11.6

Hysteresis in Elastomers • The figure at right shows a typical hysteresis response (uniaxial compression at constant strain rate) for a filled rubber subjected to different final strains (from Bergstrom and Boyce1).

• Response is rate-dependent and exhibits hysteresis upon cyclic loading.

1. Bergstrom, J.S., and M.C. Boyce, “Constitutive Modeling of the Large Strain TimeDependent Behavior of Elastomers,” Journal of the Mechanics and Physics of Solids, vol. 46, pp. 931-954, 1998.


237

L11.7

Hysteresis in Elastomers • The data show:

• No permanent set after one completed load cycle

• The figure at right shows a typical strain rate dependence during uniaxial compression to a fixed strain level.

2.00

True stress ( MPa)

• Repeatability of the results

strain rates: 0.001/s, 0.01/s, 0.05/s, 0.2/s

1.60

increasing strain rate 1.20

0.80

Chloroprene rubber (15 pph)

0.40

0.0 0.0

0.20

0.40

0.60

0.80

True strain (compressive)


L11.8

Hysteresis in Elastomers • Bergstrom and Boyce developed a large strain (400 is not uncommon), time-dependent constitutive model for elastomeric materials. • They observed the following in experiments with carbon-black-filled Chloroprene rubber subjected to different time-dependent strain histories: • Both filled and unfilled elastomers show significant amounts of hysteresis during cyclic loading.

• The amount of carbon black particles does not strongly influence the normalized amount of hysteresis. • Both filled and unfilled elastomers are strain-rate dependent, and the rate dependence is higher during loading than unloading. • At fixed strain the stress approaches the same equilibrium level with relaxation time whether loading or unloading. • They then derived a phenomenological constitutive model, which is implemented in Abaqus ( HYSTERESIS).


238

L11.9

Hysteresis in Elastomers • Components in the model

network B

• Elastic and creep strains are large and of comparable magnitude. • Creep response only for shear distortional behavior; the volumetric response is purely elastic.

network A

• Nonlinear dependence on strain rate. • The hysteresis model decomposes the mechanical behavior into two parts: an equilibrium or purely elastic response (network A) and a time-dependent deviation from equilibrium (network B). • The figure shows a one-dimensional idealization. Modeling Rubber and Viscoelasticity with Abaqus

L11.10

Hysteresis in Elastomers

• The strain for each case is normalized with respect to the instantaneous strain. • The material reaches a strain plateau much more slowly than with viscoelasticity.

2.00 nominal stress - 4 nominal stress - 3 nominal stress - 2 nominal stress - 1

1.50

Normalized strain

• Creep test: The following plot shows the normalized strain versus time for four different stresses using the hysteresis model.

1.00

0.50

0.0 0.0

0.50

1.00

1.50

2.00

2.50

3.00

Time


239

L11.11

Hysteresis in Elastomers • Stress relaxation test: The following plot shows the normalized stress versus time for four different strains using the hysteresis model.

1.00 nominal strain - 4 nominal strain - 3 nominal strain - 2 nominal strain - 1

Normalized stress

0.80

• The stress for each case is normalized with respect to the instantaneous stress.

0.60

0.40

0.20

• The stress reaches a plateau much more slowly than with viscoelasticity.

0.0 0.0

0.50

1.00

1.50

2.00

2.50

3.00

Time


L11.12

Hysteresis in Elastomers • The HYPERELASTIC option defines the response of network A; the spring response is nonlinear. • The HYSTERESIS option defines the response of network B; the effective creep strain rate in network B is given by the expression

 cr

A(

cr

1)C

m

.

• The positive exponent m, generally greater than 1, characterizes the (scalar) effective stress dependence of the effective creep strain rate. • The exponent C, restricted to the interval [ 1, 0], characterizes the creep strain dependence (through the creep stretch cr) on the creep strain rate. • The nonnegative constant A maintains dimensional consistency in the equation.


240

L11.13

Hysteresis in Elastomers • In addition to these material constants the hysteresis model is characterized by a stress scaling factor, S, that defines the ratio of the stress carried by network B to the stress carried by network A under instantaneous loading; i.e., identical elastic stretching in both networks. • Typical values of the constants above (Bergstrom and Boyce, 1998):

S 1.6, A

5 (sec) 1 (MPa) m ( 3)

m

, m 4, C

1.0.

• Usage: the above four values in the given order are entered on the data line for the HYSTERESIS option.


L11.14

Hysteresis in Elastomers • Restrictions • Hysteresis is active in the following procedures only: STATIC VISCO DYNAMIC • The model requires the HYPERELASTIC option to define the elastic behavior. • Hysteresis can be used only with elements that permit hyperelastic materials; thus, is can be used only in large-strain problems. • Hybrid elements can be used only when the accompanying hyperelasticity definition is incompressible.


241

L11.15

Hysteresis in Elastomers • Restrictions (cont'd) • The hysteresis material properties cannot be temperature dependent; however, the elastic material properties can be temperature dependent. • The model does not model “Mullin’s effect” or the softening of an elastomer when it is first subjected to loading. • Before material properties are measured, the rubber should be stretched repeatedly to operating strain levels.


L11.16

Hysteresis in Elastomers • Abaqus usage • The elasticity of the model is defined by using the HYPERELASTIC option. • The MODULI parameter may be set to either LONG TERM (to define the long-term behavior of the material; default setting) or INSTANTANEOUS (to define the instantaneous behavior). • The stress scaling factor and the creep parameters for network B are input directly on the data line of the HYSTERESIS option. • Both the HYPERELASTIC option and HYSTERESIS option must be used together in the material definition. • The hysteresis material model creates unsymmetric stiffness matrices, so Abaqus/Standard uses unsymmetric matrix storage and solution by default. • Typical values of the material parameters are given in the Abaqus Analysis User’s Manual.


242

L11.17

Hysteresis in Elastomers • Defining hysteresis in Abaqus/CAE


L11.18

Hysteresis in Elastomers • Example • This example is taken from the Abaqus Verification Manual. • The material being modeled is Chloroprene rubber (15 pph carbon black filler). • Material model • The rubber is modeled with the Arruda-Boyce hyperelasticity model with the following values for the model’s parameters:

0.6 MPa,

m

8, D 0.01

• The hysteresis behavior is modeled with the following values for the parameters:

S 1.6, A 0.5556(MPa) 4 s 1, m 4.0, C

1.0


243

L11.19

Hysteresis in Elastomers • Loading • The test specimen is subjected to this compressive loading history. • The constant strain rate loading is interrupted by relaxation segments during the loading and unloading phases of the test. 0.0

Applied strain

0.20

A

0.40

B 0.60

0.0

50.00

100.00

Time Modeling Rubber and Viscoelasticity with Abaqus

L11.20

Hysteresis in Elastomers • Results

End 0.0

unloading

Start

0.40

Stress

A 0.80

At strain level A, the stress decreases (becomes less compressive) during "loading" relaxation segments.

B loading

1.20

At strain level B, the stress increases (becomes more 1.60 compressive) during "unloading" relaxation segments.

0.60

0.40

0.20

0.0

Strain


244

Modeling Permanent Set in Elastomers

L11.22

Modeling Permanent Set in Elastomers • Motivation: Test data Loading

Unloading / reloading (Mullins' effect)

Permanent set


245

L11.23

Modeling Permanent Set in Elastomers • Approach

plastic part of the deformation gradient

Fe F p

F

Lee (1969)

elastic part of the deformation gradient

• Multiplicative split of the deformation gradient (motivated by crystal plasticity) • Plasticity is modeled with isotropic hardening Mises plasticity • Hyperelasticity can be modeled with any isotropic hyperelastic models available in Abaqus • Can be combined with Mullins’ effect to capture damaged response during unloading after initial loading


L11.24

Modeling Permanent Set in Elastomers • There two applications of this capability: • The material (usually a rubber compound) clearly exhibits permanent set. • All strains are not recovered after the load has been removed even after sufficient time lapses. • The component being modeled is in service under cyclic loading and the material exhibits viscoelastic behavior.

• Given sufficient time after removal of load, one recovers almost all the strains. • In this case the user may want to model viscoelastic strains (when the component is in service under cyclic loading) using permanent set.


246

L11.25

Modeling Permanent Set in Elastomers • Defining permanent set • The primary hyperelastic behavior can be defined by using any of the hyperelastic material models. • Permanent set can be defined through an isotropic hardening function in terms of the yield stress and the equivalent plastic strain. • You can specify permanent set and the Mullins effect using the hyperelastic and Mullins effect coefficients and the hardening data.

• However, if you have uniaxial and biaxial test data, you can include these data in a material model by using the FeFp Data Processor plug-in for Abaqus/CAE to calibrate hyperelastic, plastic, and Mullins effect data (SIMULIA Answer 3522).


L11.26

Modeling Permanent Set in Elastomers • Abaqus/CAE plug-in to calibrate test data (SIMULIA Answer 3522) • The plug-in automatically extracts loading, unloading and permanent set data from uniaxial and biaxial test data • Edit the data (remove any kinks, sudden jumps) Experimental data

Abaqus/CAE plug-in

Loading, unloading and permanent set

Calibration script

Material (*Hyperelastic, *Plastic, *Mullins) Modeling Rubber and Viscoelasticity with Abaqus

247

L11.27

Modeling Permanent Set in Elastomers • Abaqus/CAE plug-in (cont’d)


L11.28

Modeling Permanent Set in Elastomers • How the plug-in works • Processes two modes of test data, namely, uniaxial and biaxial. • For each mode, the GUI will help users identify loading, permanent set and optionally, unloading / reloading data in their test data. • Creates data for the following keyword options: *Hyperelastic, Test Data *Plastic *Mullins Test Data

• For detailed instructions on using the plug-in, consult SIMULIA Answer 3522 in the SIMULIA Online Support System (SOSS)


248

L11.29

Modeling Permanent Set in Elastomers • Validation


L11.30

Modeling Permanent Set in Elastomers • Example: Axial / Torsion loading of a specimen

• All parameters based on pure axial and torsional response of the specimen • Reduced polynomial strain energy function used for hyperelasticity • Linear hardening function used for plasticity

y

y o

H

p

• All units are MPa except r and , which are dimensionless

c10

146.74, c20

6.5252, c30

r

3, m 56.282,

0.1

y 0

29.6679, H

0, c 40

0.028648

8168.04


249

L11.31

Modeling Permanent Set in Elastomers


L11.32

Modeling Permanent Set in Elastomers • Loading paths


250

L11.33

Modeling Permanent Set in Elastomers • Results – Path H


L11.34

Modeling Permanent Set in Elastomers • Summary • Multiplicative split of deformation gradient • Following keywords can be combined • *HYPERELASTIC • *MULLINS EFFECT (optional) • *PLASTIC, TYPE=ISOTROPIC • Can be used to model filled elastomers and thermoplastics that show

• Permanent strain upon removal of load and/or • Damaged unloading behavior • Process test data through an Abaqus/CAE plug-in • Current limitations

• Cannot include rate effects such as hysteresis or viscoelasticity • Available only for rate-independent isotropic hardening plasticity


251

Anisotropic Hyperelasticity

L11.36

Anisotropic Hyperelasticity • Overview • Provides a capability for modeling materials that exhibit highly anisotropic and nonlinear elastic behavior, such as biomedical soft tissues and fiber-reinforced elastomers • Two forms of strain energy potentials are available: • Generalized Fung form • Holzapfel-Gasser-Ogden form

• User-defined forms of the strain energy potential supported via two sets of user subroutines: • (V)UANISOHYPER_STRAIN for strain-based formulations • (V)UANISOHYPER_INV for invariant-based formulations

• These models can be combined with • Mullins effect to include stress softening (damage) behavior • Viscoelasticity to include rate effects (Abaqus/Explicit only) Modeling Rubber and Viscoelasticity with Abaqus

252

L11.37

Anisotropic Hyperelasticity • Applications • Biomedical • E.g., modeling arterial walls in simulations of balloon angioplasty and implantation of Nitinol stents

Schematic of a healthy elastic artery

Typical uniaxial stress-strain curves for circumferential arterial strips in passive condition


L11.38

Anisotropic Hyperelasticity • Applications (cont’d) • Consumer products • Fiber reinforced molded plastics • Fibrous polymers, paper, cloth, etc. • Others

• Reinforced rubber and polymers, composites, etc. • General capability to model fiber-induced anisotropy


253

L11.39

Anisotropic Hyperelasticity • Two formulations are commonly used for anisotropic hyperelasticity • Strain-based formulation (e.g., Generalized Fung) • Strain energy given as an anisotropic function of the Green strain:

U

G

U(

, J)

• Invariant-based (fiber-based) formulation (e.g., Holzapfel-GasserOgden) • Strain energy given as a function of preferred material directions:

U

1,..., N

U (C , A )

• Invariant representation:

U

U ( I1 , I 2 , J , I 4

, I5

;

)

,

1,..., N


L11.40

Anisotropic Hyperelasticity • Some details regarding the invariant-based models • Invariant representation:

U

U ( I1 , I 2 , J , I 4

, I5

;

)

,

( I1

tr ( C 2 ));

1,..., N

Invariants:

I1

tr ( C);

I2

1 2

J

Pseudo-invariants:

I 4(

)

A

C A

I 5(

)

A

C2 A

Geometrical constants (independent of deformation):

A

A


254

det F

L11.41

Anisotropic Hyperelasticity • Generalized Fung form • Phenomenological model for modeling soft biological tissue expressed in terms of the Green strain:

c 1 (exp(Q) 1) ( J 2 1 2ln J ) 2 2D G G G G :b: ij bijkl kl

U Q

• User interface for orthotropic and anisotropic cases *ANISOTROPIC HYPERELASTIC, FUNG-ORTHOTROPIC, DEPENDENCIES = b1111 , b1122 , b2222 , b1133 , b2233 , b3333 , b1212 , b1313

b2323 , c , D ,Temp, FVs *ANISOTROPIC HYPERELASTIC, FUNG-ANISOTROPIC, DEPENDENCIES = b1111 , b1122 , b2222 , b1133 , b2233 , b3333 , b1112 , b2212

b3312 , b1212 , b1113 , b2213 , b3313 , b1213 , b1313 , b1123 b2223 , b3323 , b1223 , b1323 , b2323 , c , D ,Temp FVs Modeling Rubber and Viscoelasticity with Abaqus

L11.42

Anisotropic Hyperelasticity • Holzapfel-Gasser-Ogden form • Constitutive model for arterial walls • Includes the effects of dispersion in the fiber directions

U E

C10 ( I1

k1 1 3) ( J 2 1 2ln J ) 2D 2k 2

( I1 3) (1 3 )( I 4(

)

N

exp k2 E

2

1

1

1)

=0

→ perfectly aligned fibers

= 1/3

→ randomly distributed fibers

• User interface: *ANISOTROPIC HYPERELASTIC, HOLZAPFEL, LOCAL DIRECTIONS=N

C10 , D , k1 , k 2 , , Temp, FVs Modeling Rubber and Viscoelasticity with Abaqus

255

L11.43

Anisotropic Hyperelasticity • Definition of local directions: *ORIENTATION, NAME=Ori_name, LOCAL DIRECTIONS=N

x2 A12 A1

Usual data for *ORIENTATION A11 , A12 , A13 … AN1 , AN2 , AN3

Beginning on third data line, define local material directions with respect to the orthonormal system at the material point

A11 x1

A13

x3

*SOLID SECTION,MATERIAL=Mat_name, ORIENTATION=Ori_name *MATERIAL, NAME=Mat_name *ANISOTROPIC HYPERELASTIC, HOLZAPFEL, LOCAL DIRECTIONS=N


L11.44

Anisotropic Hyperelasticity • Local directions are written to the output database (.odb) file and can be visualized in Abaqus/Viewer using symbols plots

Local directions before deformation

Local directions after deformation Modeling Rubber and Viscoelasticity with Abaqus

256

L11.45

Anisotropic Hyperelasticity • User-defined models formulated in terms of Green strain *ANISOTROPIC HYPERELASTIC, USER, FORMULATION=GREEN STRAIN, PROPERTIES=Num_props User defined properties

• The components of the Green strain are referred to the material basis in the reference configuration (specified with *ORIENTATION). • Inside (V)UANISOHYPER_STRAIN, user defines

U

U(

G

, J)

• See section 1.2.9 of the Abaqus User Subroutines Reference Manual for an example


L11.46

Anisotropic Hyperelasticity • User-defined models formulated in terms of pseudo-invariants *ANISOTROPIC HYPERELASTIC, USER, FORMULATION=INVARIANT, PROPERTIES=Num_props, LOCAL DIRECTIONS=N User defined properties

• The fiber directions are defined by the local directions specified with the orientation definition for the section • Inside (V)UANISOHYPER_INV, user defines

U

U ( I1 , I 2 , J , I 4

, I5

;

)

,

1,..., N

• See section 1.2.8 of the Abaqus User Subroutines Reference Manual for an example


257

L11.47

Anisotropic Hyperelasticity • Example: Anisotropic hyperelastic modeling of arterial layers • Simulation of the mechanical response of the adventitial layer of human iliac arteries • Numerical analysis of simple tension tests of iliac adventitial strips • Based on a paper by Gasser, Ogden and Holzapfel (2006)

= 49.98º

Iliac adventitial strips cut along the axial, circumferential, and 15º directions of the artery Modeling Rubber and Viscoelasticity with Abaqus

L11.48

Anisotropic Hyperelasticity • Results for specimen with dispersed fibers • Results correspond to an applied load of 2.0 N and the dispersion of collagen fibers is included ( )

Model uses C3D8H elements

Strip cut in axial direction

Stress in the direction of applied load.

Strip cut in circumferential direction


258

L11.49

Anisotropic Hyperelasticity • Results for specimen with perfectly aligned fibers • Results correspond to an applied load of 2.0 N and the collagen fibers are perfectly aligned ( )

Model uses C3D8H elements

Strip cut in axial direction

Stress in the direction of applied load.

Strip cut in circumferential direction


L11.50

Anisotropic Hyperelasticity • Load-displacement results

Load-displacement response of circumferential and axial specimens


259

L11.51

Anisotropic Hyperelasticity • Example: Stent deployment • Stent: 12432 C3D8I elements, linear elasticity • Vessel: 21120 C3D8H elements, anisotropic hyperelasticity • Rigid balloon: 1280 surface elements • Surface-to-surface contact with penalty enforcement • Two steps: Pressurize vessel then expand balloon • Keyword edits required to define anisotropic hyperelasticity; Python script required to map stent mesh • Approximately 400,000 DOF


L11.52

Anisotropic Hyperelasticity • Results


260

L11.53

Anisotropic Hyperelasticity • Limitations • Cannot model compressible material behavior with • Hybrid elements • Plane-stress elements • Initial stress conditions cannot be defined • Results can only be transferred into Abaqus/Explicit (not Abaqus/Standard) • Cannot be used with viscoelasticity in Abaqus/Standard


261

262

Notes

263

Notes

264

Finite Deformations Appendix 1

A1.2

Overview • Motions and Displacements • Extension of a Material Line Element • The Deformation Gradient Tensor • Finite Deformations and Strain Tensors • Decomposition of a Deformation • Principal Stretches and Principal Axes of Deformation

• Strain Invariants • Summary


265

Motions and Displacements

A1.4

Motions and Displacements • A body occupies the material within R0 at t = 0. • This is the reference configuration. • The configuration at time t is the current configuration.


266

A1.5

Motions and Displacements • The motion of the body takes the reference configuration R0 into the current configuration R. • An essential assumption of continuum mechanics is that the motion can be described as

x

x( X , t )

for every X in R0 for every x in R

• In above expression, X act as independent variables; this is a Lagrangian (material) description of the problem.


A1.6

Motions and Displacements • The motion can be described in terms of the displacement vector u:

x

X

u

u

x X.

or

• Lagrangian description:

u( X , t )

x( X , t ) X .


267

Extension of a Material Line Element

A1.8

Extension of a Material Line Element • A deformation is a motion in which a change of shape can occur. • For the purposes of stress analysis we need to separate that part of the motion that corresponds to a rigid-body motion from that part that involves deformation.

A and a are unit vectors.


268

A1.9

Extension of a Material Line Element • Given the motion x = x(X, t): • We are interested in determining the length and orientation of the material line element after the motion. • Straightforward analysis gives

ai

where

xi AR , XR

is the stretch ratio and FiR

xi is the deformation gradient. XR


The Deformation Gradient Tensor

269

A1.11

The Deformation Gradient Tensor • The nine quantities, gradient tensor, F :

xi , are the components of the deformation XR

FiR

xi . XR

• They describe how a particle moves in relation to neighboring particles.


A1.12

The Deformation Gradient Tensor • Our previous results for a material line element oriented in direction a in the current configuration and in direction A in the reference configuration can be summarized as follows:

a 2

A 2

1

F A

A FT F A F

1

a

a ( F 1 )T F

1

a


270

A1.13

The Deformation Gradient Tensor • Remarks: • If there is no motion, x = X, and so F = I (identity).

• F is important in the analysis of deformation, but it is not a measure of deformation only (the motion includes rotation). • We need measures that do not change when no deformation takes place; i.e., we want them to remain unchanged under rigid body motions:

x

Q X

c

QT Q c

Q QT

I

rotation

translation (does not vary with position)

• For a rigid body motion F = Q.


Finite Deformations and Strain Tensors

271

A1.15

Finite Deformations and Strain Tensors • Consider the tensor:

C

F T F.

• Recall the result from the line extension: 2

A FT F A

A C A;

stretch of material line element with direction A in reference configuration. • Knowledge of C at a point determines the local deformation in the vicinity of that point.


A1.16

Finite Deformations and Strain Tensors • Moreover, for rigid body motions F = Q, so C = QT ∙ Q = I. • C is constant throughout a rigid body motion.

• C is connected with deformation and not with rigid body motion; therefore, it is a suitable measure of deformation. • C is called the right Cauchy-Green deformation tensor. • Note that C is not a unique measure of deformation; there are many other candidates. • But C is convenient because it is easy to calculate from F.


272

A1.17

Finite Deformations and Strain Tensors • Recall the result from the line extension: 2

a F

T

F

1

a,

stretch of material line element with direction a in current configuration. • Let B = F ∙ FT ; then B

1

2

= F T ∙ F 1 , and so a B

1

a.

• B is called the left Cauchy-Green deformation tensor.


A1.18

Finite Deformations and Strain Tensors • The Lagrangian strain tensor E (Green-Lagrange) is defined by

1 (C I ). 2 • A nice feature is that E = 0 for rigid body motions. E

• C, B, and E are symmetric second-order tensors, so they have real principal values and orthogonal principal directions.


273

Decomposition of a Deformation

A1.20

Decomposition of a Deformation • The deformation gradient tensor F can be expressed as same

F RU V R describes rotation of body

right stretch tensor

left stretch tensor

• U and V are symmetric and unique for a given F. • J = det(F) is the ratio of volume in the current configuration to dV . volume in the reference configuration: J dV0 • J > 0 for physically realistic deformations.


274

A1.21

Decomposition of a Deformation • The tensors U and V are related to the deformation tensors C and B through:

C

FT F

U2

B

F FT

V2

• Therefore, U and C are equivalent measures of deformation.

• For a given F, however, calculation of U is inconvenient, whereas the computation of C is straightforward. • Similar remarks apply to V and B.


Principal Stretches and Principal Axes of Deformation

275

A1.23

Principal Stretches and Principal Axes of Deformation • Recall

2

= A ∙ C ∙ A.

• Find directions A for which

takes extreme values.

• Find the minimum and maximum of constraint A ∙ A = 1.

2

= A ∙ C ∙ A under the

• Results in eigenvalue problem:

C A*

2

A*.

• The extreme values of 2 are the eigenvalues of C and occur in the directions of the eigenvectors (A* ) of C. • Alternatively, the extreme values of C = U2).

are eigenvalues of U (recall


A1.24

Principal Stretches and Principal Axes of Deformation • Since U is symmetric and positive-definite, its principal values are real and positive: 1 2 3 

principal stretches

• Moreover, U has 3 orthogonal principal directions:

A , A2 , A3 1  principal axes of U


276

A1.25

Principal Stretches and Principal Axes of Deformation • The motion that corresponds to F = R ∙ U consists of three extensions of magnitude 1, 2, 3 along the three directions A1, A2, A3, followed by the rotation R. • A similar interpretation can be given for the motion F = V ∙ R. • It can be shown that: • The principal values of

A , A2 , A3 1  principal directions of U

1,

2,

3

are also the principal values of V.

a1 R A1 , a2 R A2 , a3 R A3    principal directions of V


A1.26

Principal Stretches and Principal Axes of Deformation • Since C = U 2 and E = ½ C

I , the principal directions of C and E

coincide with those of U. • The principal values of C are • The principal values of E are

2 2 2 1 , 2, 3.

1 ( 2

2 i

1) i 1, 2, 3.

• Likewise, the principal directions of B and V coincide. • The principal values of B are

2 2 2 1 , 2, 3.


277

Strain Invariants

A1.28

Strain Invariants • The strain invariants are defined by

I1 I2 J

tr( B )

tr( F F T ),

1 2 ( I1 tr( B B )), 2 det( F ).

• In terms of the principal stretches these invariants are

I2

2 2 1 2 2 2 2 1 2 2

J

1 2 3.

I1

2 3, 2 3

2 2 3 1,

• Without deformation B = I, so I1 = I2 = 3, J=1.


278

A1.29

Strain Invariants • In Abaqus revised invariants are used to separate deviatoric and volumetric effects in solid rubbers: 13

J 1 3F ,

F, F

F

J

I1

tr( B )

tr( F F T ),

I2

1 2 ( I1 2

tr( B B )).

• In terms of principal deviatoric stretches, invariants have the form

I1

2 1

2 2

I2

2 2 1 2

J

i

13

i,

the revised

2 3, 2 2 2 3

2 2 3 1

where 1 2 3

1

1

1

2 1

2 2

2 3

,

1.


Summary

279

A1.31

Summary • F=3

3 deformation gradient tensor which contains all information

about the motion in the vicinity of a point in the material. • We take F = F(X, t), where X is the position in the reference configuration. • This is called a Lagrangian description. • We need to separate rigid body motion and deformation. This can be done as F R U or F V R,

where R is a pure rigid body motion (so R represent deformation.

1

= RT) and U and V


A1.32

Summary • We can write U in terms of its principal values, 1, 2, 3, (the “principal stretch ratios”) and the corresponding principal directions, A1, A2, A3, (which are given in the reference configuration):

U

1 A1 A1

2 A2 A2

3 A3 A3 .

• Likewise, we can write as:

V

1a1a1

aI

2 a2 a2

3a3a3 ,

R AI .

• The AI (and ai) are orthogonal unit vectors.


280

A1.33

Summary • This is really all we need to know about deformation. However, many materials such as ceramics or concrete cannot undergo large deformation ( I cannot be much different from 1.0), while others yield inelastically at small amounts of deformation (in metals, yield typically happens when I 1 0.01). • For convenience we introduce the idea of “strain” to have a measure of deformation that is 0.0 when there is no deformation (that is, when I = 1.0). • Useful strains are: • Nominal strain: • Thus, • Log strain:

N I ln I

N I

I

1.

= change in length per unit initial length.

ln( I ). 1 2 G ( I 1). • Green strain: I 2


A1.34

Summary • We easily construct three-dimensional strain tensors from the principal stretch directions:

εN

nominal strain

(

1

1) A1 A1 (

ε ln

log strain ln 1 A1 A1 or, in the current configuration,

εN and so on.

(

1

1)a1a1 (

2

ln

2

1) A2 A2 ( 2 A2 A2

1)a2a2 (

3

3

1) A3 A3 ,

3 A3 A3 ,

1)a3a3 ,

• Such strains are convenient for output. Abaqus provides them for this purpose. • Rubber constitutive models in Abaqus are written directly in terms of deformation. Strain is just given for output purposes.


281

282

Notes

283

Notes

284

Rubber Elasticity Models: Mathematical Forms Appendix 2

A2.2

Overview • Energy Functions for Solid Rubbers (Isotropic) • Polynomial Model • Mooney-Rivlin Model • Reduced Polynomial Model • Neo-Hookean Model • Yeoh Model

• Ogden Model • Marlow Model • Arruda-Boyce Model • Van der Waals Model

• Foam Rubber Model • Mullins Effect


285

Energy Functions for Solid Rubbers

A2.4

Energy Functions for Solid Rubbers • General form of strain energy function (assuming isotropy):

U

U ( I1 , I 2 , I 3 ).

• In general, the response of rubber is completely different to volumetric or deviatoric deformations. • This suggests an additive split of the strain energy function. • In Abaqus we write this modified strain energy function as

U U1 ( I1 3, I 2 3) U 2 ( J el 1).


286

A2.5

Energy Functions for Solid Rubbers • There are several forms of the strain energy function for solid rubber in Abaqus. • Most forms are expressed in terms of series expansions. • For all strain energy functions expressed in terms of a series expansion, some terms are common:

• N is the order of the strain energy function. • The Di coefficients introduce compressibility into the material behavior. • When the material is incompressible, the terms with Di are ignored.


A2.6

Energy Functions for Solid Rubbers • Jel is the elastic volume ratio,

J ; J th

J el

and the thermal volume ratio, Jth, follows from the linear thermal expansion, th, with

J th

(1

th )

3

,

where th follows from the temperature and the thermal expansion coefficient. • Abaqus assumes that the thermal expansion coefficients define nominal thermal strains. Usually thermal strains are small enough that this distinction is not important. • Only isotropic thermal expansion can be used with the hyperelastic material models in Abaqus.


287

A2.7

Energy Functions for Solid Rubbers • Polynomial model • The polynomial strain energy function has the following form: N

N i

U

Cij ( I1 3) ( I 2 3)

j

i j 1

i 1

1 ( J el 1) 2i . Di

• The constants Cij and Di are calibrated from experimental test data. • Abaqus allows up to N = 6 terms in the above function.

• The initial shear modulus and bulk modulus are given by 0

2 . D1

2(C10 C01 ), K0

• If D1 is equal to zero, Abaqus requires that all Di must be zero.


A2.8

Energy Functions for Solid Rubbers • Mooney-Rivlin model • This form is obtained when N = 1 in the full polynomial form:

U

C10 ( I1 3) C01 ( I 2 3)

1 el (J 1) 2 . D1

• If D1 is equal to zero, the material is fully incompressible. • The initial shear and bulk moduli are given by 0

2(C10 C01 ), K0

2 . D1


288

A2.9

Energy Functions for Solid Rubbers • Reduced polynomial model • The reduced polynomial strain energy function has the following form: N

U

N

Ci 0 ( I1 3)

i

i 1

i 1

1 el (J 1)2i . Di

• Curve fitting with experimental test data for polynomial models with this parameter can be performed up to N = 6.


A2.10

Energy Functions for Solid Rubbers • Neo-Hookean model • The simplest form of the strain energy function, U, proposed by Treloar in 1943, is

U

1 el (J 1)2 , D1

C10 ( I1 3)

where C10 is a calibration constant.

C10

1 2

0,

where

0

is the initial shear modulus.


289

A2.11

Energy Functions for Solid Rubbers • Yeoh model • The Yeoh strain energy function is a special case of the general reduced polynomial model with N = 3: 3

U

3

Ci 0 ( I1 3)

i

i 1

i 1

1 el (J 1)2i . Di

• The initial shear and bulk moduli are given by 0

2C10 , K0

2 . D1

• The following relationships are usually seen between the Ci0:

C20 is negative and 1–2 orders of magnitude smaller than C10.

C30 is positive and 3–4 orders of magnitude smaller than C10.


A2.12

Energy Functions for Solid Rubbers • Ogden model • The Ogden strain energy function is based on the principal stretch ratios, I: N

U i 1

2

N i ( 1 2 i

i

2

i

3

i 1

where i

and

i,

i,

3)

i

J

1 3

1 el (J 1)2i , Di

i

and Di are to be determined from experimental test data.

• Abaqus allows up to N = 6 terms in the above form. • Up to N = 3 is common.


290

A2.13

Energy Functions for Solid Rubbers • Only if i = 2 or i = 2 can the first part of the strain energy function be expressed explicitly in terms of I1 and I 2 . • The Mooney-Rivlin form is also a special case of the Ogden form, for which N = 2, 1 = 2C10, 2 = 2C01, 1 = 2 , and 2 = 2.


A2.14

Energy Functions for Solid Rubbers • Marlow Model • The Marlow strain energy function has the following form

U U dev ( I1 ) Uvol ( J el ) • The deviatoric part of the potential is defined by providing either • uniaxial,

• equibiaxial, or • planar test data. • The volumetric part is defined by providing • volumetric test data,

• defining the Poisson's ratio, or • specifying the lateral strains together with the uniaxial, equibiaxial, or planar test data.


291

A2.15

Energy Functions for Solid Rubbers • Arruda-Boyce model • The Arruda-Boyce strain energy function has the following form: 5

Ci

U i 1

2i m

( I i 3i ) 2 1

1 J el2 1 ln( J el ) . D 2

• The first term in the above equation is a function of I1 only.

• Three material parameters: , which is the initial shear modulus. m,

which is the locking stretch at which the model’s stress-strain curve will rise (stiffen) significantly.

D, which is related to the initial bulk modulus through K 0

2 . D


A2.16

Energy Functions for Solid Rubbers • If experimental test data are specified, and m are calculated by Abaqus by using a nonlinear least-squares-fit procedure based on all the test data provided.

• These values will be output to the data (.dat) file. • The strain energy function can be viewed as a polynomial of order 5, with the constants Ci arising out of the statistical treatment of the material.

C1

1 , C2 2

1 , C3 20

11 , C4 1050

19 , C5 7000

519 . 673750

• Calculation of D requires volumetric test data.

• If D is zero, the material is fully incompressible.


292

A2.17

Energy Functions for Solid Rubbers • Van der Waals model • The Van der Waals strain energy function has the following form:

U

(

2 m

3) ln(1

2 I 3 a 3 2

)

3 2

1 J el2 1 ln( J el ) , D 2

where

I

(1

) I1

I 3 . 2 3 m

I 2 and


A2.18

Energy Functions for Solid Rubbers • Four material parameters: ,

m,

a, and .

is the initial shear modulus at low strains. m

is the locking stretch. • The Van der Waals strain energy function limits the deformation of the material to stretches less than m.

a is the interaction parameter, which accounts for the interaction between chains in the representative volume. • It is difficult to estimate the value of a; a reasonable approximation is

a

2 m 3 m

• Typical values are a

1

.

0.1 to 0.3.


293

A2.19

Energy Functions for Solid Rubbers represents the linear mixture parameter used to combine the two strain invariants into Ĩ. When using test data to calibrate the Van der Waals model, it may not be possible to calculate a value of that is admissible (0 1.0). In these situations Abaqus will attempt the fitting procedure again with = 0.

= 0 is the recommended value when only one type of test data is available.


Foam Rubber Model

294

A2.21

Foam Rubber Model • The energy function for the foam rubber model has the following form: N

U i 1

2

i 2 i

ˆ

1

ˆ

i

2

ˆ

i

3

1

3

i

( J el

i i

1) ,

i

where we have defined

ˆ

J th1 3

i

i

with the thermal volume change,

J th

(1

th )

3

,

and the elastic volume change,

J J th

J el

ˆ ˆ ˆ. 1 2 3


A2.22

Foam Rubber Model • The coefficients are related to the initial shear modulus N i,

0 i 1

while the initial bulk modulus follows from N

K0

2 i 1

i

1 3

i

.


295

A2.23

Foam Rubber Model • For each term in the energy function the coefficient i determines the degree of compressibility. i is related to the Poisson’s ratio, i, by the expressions i i

• If

i

1 2

i

,

i

1 2

i

. i

is the same for all terms, we have a single effective Poisson’s ratio,

. • This Poisson’s ratio is valid for finite values of the logarithmic principal strains e1, e2, e3 ; in uniaxial tension e2 = e3 = e1.


A2.24

Foam Rubber Model • If we choose

i

= 0 (i.e.,

i

= 0), there is no Poisson’s effect.

• With the Taylor series expansion,

ax

( x ln a)2 2!

1 x ln a

( x ln a)3 3!

,

we obtain for this case the energy function, N

U i 1

2

i 2 i

ˆ

1

i

ˆ

2

i

ˆ

3

i

3

i

ln J el .

• The implementation in Abaqus follows the same procedure as the implementation of the Ogden hyperelastic model.


296

Mullins Effect

A2.26

Mullins Effect • The Abaqus model is based on the model developed by Ogden and Roxburgh (1999). • The model is an extension of the classical theory of isotropic incompressible elasticity, modified by the addition of a damage variable

U dev

U dev ( i , ).

• Equilibrium provides an additional equation for evolution of damage variable

U dev

0.


297

A2.27

Mullins Effect • The damage variable may be either active or inactive or may switch from active to inactive; it always varies continuously. • When it is inactive, it is set to the constant value of 1. • In this case the energy density reduces to the “primary strain energy density function” given by

U dev ( i ,1) U dev ( i ). • The primary strain energy density function defines the response of the material under monotonic straining (usual hyperelastic potential)


A2.28

Mullins Effect • Ogden and Roxburgh use the following modified energy function:

U dev ( i , ) • The function

U dev ( i )

( )

( ) is called the damage function ( (1) 0).

• The above modified energy function leads to the following expression for the deviatoric stress tensor:

S

S


298

A2.29

Mullins Effect • Modified deviatoric stresses can be obtained by simply scaling the “primary” deviatoric stresses with the damage variable. • For stress softening • Require

1.

0 so that stresses remain nonzero until zero deformation.


A2.30

Mullins Effect • Abaqus uses the following form of the damage variable (which satisfies all the required properties): max ever U dev U 1 1 erf max ever r m U dev r, m, and are material parameters

max ever

• U dev is the maximum strain energy density experienced on the primary curve during the loading history


299

A2.31

Mullins Effect • Error function:

erf(x)

2

x

exp( w2 ) dw 0


A2.32

Mullins Effect • The damage variable, , varies monotonically from a maximum of 1 to a minimum of m. is 1 on the “primary curve”; plane.

=

m

at the origin of the stress-strain


300

A2.33

Mullins Effect • The Ogden-Roxburgh model is modified to account for compressibility:

U dev ( i )

U( i, )

( ) U vol ( J el )

• The stresses are now given by

S ( i )

( i, )

p ( J el ) I .

• Only the deviatoric part of the deformation is associated with damage.

• A purely volumetric deformation will not exhibit the Mullins effect.


A2.34

Mullins Effect • At the zero-deformation state the energy density has the residual value of ( m ). This quantity represents the energy dissipated due to damage. • The recoverable part of the energy is given by

U re ( i , ) U ( i , )

(

m

)


301

302

Notes

303

Notes

304

Linear Viscoelasticity Theory Appendix 3

A3.2

Overview • Classical Linear Viscoelasticity


305


A3.4

Classical Linear Viscoelasticity • The stress at time t is characterized by:

 (t ) =



t

-

E (t - )

d e ( )

where e (- ) = 0.

d

d

• E ( t ) is the relaxation modulus.


306

A3.5

Classical Linear Viscoelasticity • The relaxation modulus can be obtained from a standard stress relaxation test. • The specimen has prescribed constant displacement (strain). • The measured response is the force (stress) over time.


A3.6

Classical Linear Viscoelasticity • In this case e ( t ) = e0 H ( t ) and

or

t

 (t ) =



E (t ) =

 (t ) e0

-

E ( t -  ) e 0 d ( ) d = E (t ) e 0

where H is the Heaviside (step) function and d is the Dirac delta function.


307

A3.7

Classical Linear Viscoelasticity • The stress-strain relation can be inverted and strain at time t is characterized by

e (t ) =



t

-

J (t - )

d ( ) d d

where  (- ) = 0.

• J ( t ) is the creep function, or creep compliance.


A3.8

Classical Linear Viscoelasticity • The creep compliance can be obtained form a standard creep test. • The specimen is loaded with a prescribed constant force (stress) and the measured response is the changing displacement (strain) over time.

• In this case  ( t ) = 0 H ( t ) and

e (t ) =



t

-

J ( t -  )  0 d ( ) d = J (t )  0

or

J (t ) =

e (t ) . 0


308

A3.9

Classical Linear Viscoelasticity • E ( t ) and J ( t ) are related through t

 J (t -  ) E ( ) d = t 0

• Abaqus uses this relation to convert user-supplied creep test data into relaxation data. • This is valid only for linear viscoelasticity.


A3.10

Classical Linear Viscoelasticity • For a finite-strain viscoelasticity formulation it is important that the stress relaxation equation be written purely in terms of stress.


309

A3.11

Classical Linear Viscoelasticity • Use integration by parts to obtain



 ( t ) = E0 e (t ) 

 dE ( )

d

0

e ( t -  ) d

or

 ( t ) =  0 (t ) 

1 E0



 dE ( )

0

d

 0 ( t -  ) d

where 0 ( t )  E0 e ( t ) is the stress that would exist at the current

strain state if the specimen were purely elastic.


A3.12

Classical Linear Viscoelasticity • A similar expression holds for the shear stress S in terms of the shear strain g ( t ) for a specimen in a time history of pure shear:

S ( t ) = S0 (t ) 

1 G0



 dG ( )

0

d

S0 ( t -  ) d .

• Here S0 ( t )  G0 g ( t ) is the shear stress that would exist at the current shear strain state if the specimen were purely elastic. • The function G ( t ) is the shear relaxation modulus, and its physical meaning is analogous to that of E ( t ).


310

Notes

311

Notes

312

Harmonic Viscoelasticity Theory Appendix 4

A4.2

Overview • Classical Linear Viscoelasticity • Harmonic Excitation


313


A4.4

Classical Linear Viscoelasticity • Recall the stress relaxation equation: t

(t )

E (t

)

d ( ) d . d

• This equation is inadequate for Fourier transform methods since (for solids) E(t) 0 as t . • Introduce e(t )

E (t ) 1 E

(dimensionless relaxation function).

• Here E is the long-term modulus.

• We see that e(t)

0 as t

.


314

A4.5

Classical Linear Viscoelasticity • Substituting for e(t) in the stress relaxation equation yields t

(t )

E

(t ) E

e(t

t

(t ) • Letting

t

e(t

)

) d d

d ( ) d d ( ) d .

′ yields

(t )

(t )

e( ) 0

d

(t d

)

d .


Harmonic Excitation

315

A4.7

Harmonic Excitation • Consider the application of a sinusoidal strain,

(t ) where

exp(i t ).

is the angular frequency.

• Substituting (t) in the long-term stress relaxation equation yields

(t )

E

(t )

e( ) 0

E

1 i

d (E d

exp(i (t

e( )exp( i

)d

))) d

(t ).

0


A4.8

Harmonic Excitation • In other words, where E* and e*

t

E*

t ,

E∞ 1 i e*

is the complex modulus

is the Fourier transform of e t .

• Since e*

Re e*

i Im e* :

E * ( ) E (1 Im(e* ))  Es ( ) storage modulus

i

E Re(e* )  El ( ) loss modulus


316

A4.9

Harmonic Excitation • Previous formulas relating storage and loss moduli to the Fourier transform of the dimensionless relaxation function define frequency domain viscoelasticity data:

Im( e* ) 1 *

Re( e )

Es ( ) E El ( ) E

data required by Abaqus


A4.10

Harmonic Excitation • The storage and loss moduli can be measured with excitation tests in the frequency domain. • The data produced can be used not only in frequency domain analyses but also to derive short-term time domain data. • If the time periods of interest are of the same order as the response time of the equipment used to measure relaxation or creep data, this may be the only way to obtain such data. • Consider the complex modulus E* intermediate function:

eˆ( )

known (measured). Calculate an

E* ( ) 1. E


317

A4.11

Harmonic Excitation • The Fourier transform of the relaxation function is then

e* ( )

eˆ( ) , i

from which we can do an inverse Fourier transform to obtain e(t):

e(t )

F

1

eˆ( ) i

1 2

eˆ( ) i t e d . i

• Then

E (t )

E (1 e(t )).


318

Notes

319

Notes

320

Suggested Reading Appendix 5

A5.2

Suggested Reading • Introductory • Aklonis, J. J., et al., Introduction to Polymer Viscoelasticity, 2nd ed., Wiley, New York, 1982. • Mathematical • Flugge, W., Viscoelasticity, 2nd ed., Springer-Verlag, New York, 1980. • Pipkin, A. C., Lectures on Viscoelasticity Theory, 2nd ed., SpringerVerlag, New York, 1986. • Standard reference • Ferry, J. D., Viscoelastic Properties of Polymers, 3rd ed., Wiley, New York, 1980.


321

A5.3

Suggested Reading • Permanent set in elastomers • Lee, E.H. 1969. Elastic-plastic deformation at finite strain. Journal of Applied Mechanics 36:1-6. • Weber, G. & Anand, L. 1990. Finite deformation constitutive equations and a time integration procedure for isotropic hyperelastic-viscoplastic solids. Computer Methods in Applied Mechanics and Engineering 79: 173-202. • Simo, J.C. 1992. Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Computer Methods in Applied Mechanics and Engineering 99: 61-112. • Govindarajan, S.M. Hurtado, J.A. & Mars, W.V. 2007. Simulation of Mullins effect and permanent set in filled elastomers using multiplicative decomposition. Proceedings of the 5th European Conference of Constitutive Models of Rubber, Paris, France 5:249254. Modeling Rubber and Viscoelasticity with Abaqus

A5.4

Suggested Reading • Detailed example on time-temperature correspondence • Mercier, J. P., et al., “Viscoelastic Behavior of the Polycarbonate of Bisphenol A,” Journal of Applied Polymer Science, vol. 9, pp. 447–459, 1965.


322

Notes

323

Notes

324

Workshop Preliminaries Setting up the workshop directories and files If you are taking a public seminar, the steps in the following section have already been done for you: skip to Basic Operating System Commands, (p. WP.2). If everyone in your group is familiar with the operating system, skip directly to the workshops. The workshop files are included on the Abaqus release CD. If you have problems finding the files or setting up the directories, ask your systems manager for help. Note for systems managers: If you are setting up these directories and files for someone else, please make sure that there are appropriate privileges on the directories and files so that the user can write to the files and create new files in the directories. Workshop file setup (Note: UNIX is case-sensitive. Therefore, lowercase and uppercase letters must be typed as they are shown or listed.) 1. Find out where the Abaqus release is installed by typing UNIX and Windows NT: abqxxx whereami where abqxxx is the name of the Abaqus execution procedure on your system. It can be defined to have a different name. For example, the command for the 6.9–1 release might be aliased to abq691. This command will give the full path to the directory where Abaqus is installed, referred to here as abaqus_dir. 2. Extract all the workshop files from the course tar file by typing UNIX: abqxxx perl abaqus_dir/samples/course_setup.pl Windows NT: abqxxx perl abaqus_dir\samples\course_setup.pl Note that if you have Perl and the compilers already installed on your machine, you may simply type: UNIX:

abaqus_dir/samples/course_setup.pl

Windows NT: abaqus_dir\samples\course_setup.pl 3. The script will install the files into the current working directory. You will be asked to verify this and to choose which files you wish to install. Choose “y” for the appropriate lecture series when prompted. Once you have selected the lecture series, type “q” to skip the remaining lectures and to proceed with the installation of the chosen workshops.

© Dassault Systèmes, 2009

Preliminaries for Abaqus Workshops

325

WP.2

Basic operating system commands (You can skip this section and go directly to the workshops if everyone in your group is familiar with the operating system.) Note: The following commands are limited to those necessary for doing the workshop exercises. Working with directories 1. Start in the current working directory. List the directory contents by typing UNIX: ls Windows NT:

dir

Both subdirectories and files will be listed. On some systems the file type (directory, executable, etc.) will be indicated by a symbol. 2. Change directories to a workshop subdirectory by typing Both UNIX and Windows NT: cd dir_name 3. To list with a long format showing sizes, dates, and file, type UNIX: ls -l Windows NT:

dir

4. Return to your home directory: UNIX:

cd

Windows NT: cd home-dir List the directory contents to verify that you are back in your home directory. 5. Change to the workshop subdirectory again. 6. The * is a wildcard character and can be used to do a partial listing. For example, list only Abaqus input files by typing UNIX: ls *.inp Windows NT: dir *.inp Working with files Use one of these files, filename.inp, to perform the following tasks: 1. Copy filename.inp to a file with the name newcopy.inp by typing UNIX:

cp filename.inp newcopy.inp

Windows NT: copy filename.inp newcopy.inp 2. Rename (or move) this new file to newname.inp by typing UNIX: mv newcopy.inp newname.inp Windows NT: rename newcopy.inp newname.inp (Be careful when using cp and mv since UNIX will overwrite existing files without warning.)


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3. Delete this file by typing UNIX:

rm newname.inp

Windows NT: erase newname.inp 4. View the contents of the files filename.inp by typing UNIX:

more filename.inp

Windows NT:

type filename.inp | more

This step will scroll through the file one page at a time. Now you are ready to start the workshops.



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Workshop 1 Curve Fitting Hyperelastic Material Models from Test Data Goals When you complete this workshop, you will be able to: Use experimental data to derive coefficients for hyperelastic material models within Abaqus/CAE. See how very limited test data (such as using only uniaxial data) can yield poor material model predictions. Use Abaqus/CAE to visualize the accuracy of any given material model response and compare the accuracy of various material models against each other. Create and compare hyperelastic material models. Use the correct keywords in Abaqus/Standard to define a material model for hyperelasticity.

Introduction For this workshop you will use experimental data gathered by L.R.G. Treloar based on his work with lightly vulcanized natural rubber (The Physics of Rubber Elasticity, 1949). Treloar’s data is presented in Table W1–1. Note that it is presented in the form of engineering stress–strain values (also called nominal stress–strain). This is consistent with the form required by Abaqus.



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Table W1–1 Uniaxial tension data from Treloar. Engineering Stress, MPa

Engineering Strain

0

0

0.03

0.02

0.15

0.10

0.23

0.20

0.33

0.34

0.41

0.57

0.51

0.85

0.59

1.13

0.67

1.40

0.86

1.98

1.04

2.55

1.22

3.00

1.59

3.77

1.95

4.37

Problems A text file named st_treloar_abq.txt is provided which contains the data given in the above table (note that “st” stands for Simple Tension). Problem 1: Viewing stress–strain data in the Visualization module 1. Enter the working directory for this workshop: ../rubber_visco/workshop1

2. Start an Abaqus/CAE session and switch to the Visualization module. 3. In the Results Tree, double-click XYData. 4. In the Create XY Data dialog box, choose ASCII file as the source, and click Continue. 5. In the XY Data From ASCII File dialog box, click Select to browse for the test data file. In the ASCII File Selection dialog box, select the file st_treloar_abq.txt and click OK. 6. Since you will want strain plotted along the X-axis, the X-values should be read from field 2 and the Y-values (stress) should be read from field 1. Make these changes in the XY Data From ASCII File dialog box. 7. Set the X-axis quantity type to Strain and the Y-axis quantity type to Stress.


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8. In the lower left corner of the dialog box, click Save As. Name the X-Y data ST_TRELOAR and click OK. 9. In the XY Data From ASCII File dialog box, click Plot and then click Cancel. 10. From the main menu bar, select Options→XY Options→Curve (or click in the toolbox). 11. Toggle on Show symbol, and set the symbol size to Large. Dismiss the dialog box. 12. By default, the X- and Y-axes are labeled Strain and Stress, respectively. You can specify alternate axis titles using the Axis Options (Options→XY Options→Axis or simply double-click an axis). For example, double-click the Xaxis, switch to the Title tabbed page of the Axis Options dialog box and type Engineering Strain as the title. Similarly, specify the Y-axis title Engineering Stress (MPa). Change the axis title font size for each axis to 18. 13. In the Axis Options dialog box, switch to the Axes tabbed page. Change the axis label font size for each axis to 12. 14. In the Scale tabbed page, specify an X-axis major increment size of 0.5. The resulting stress–strain curve is shown below in Figure W1–1.

Figure W1–1 Stress-strain curve. Notice that this test data from Treloar goes out to 437% strain. For the engineering design of many products (such as seals) a more typical maximum strain of interest is perhaps 4050% strain. Use engineering judgment when testing your material. If your component (product) sees maximum strains of 25%, then test out to about 40-50% strain. If your component sees a maximum strain of 50%, then test your material specimen out to about 75-100% strain.



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W1.4

In this workshop you will use only the simple tension (uniaxial tension) test data, but it is important to understand what the typical responses are for the planar tension and equibiaxial tension modes of deformation. Figure W1–2 shows the stress-strain curves for all three modes of deformation. These data are taken from Treloar’s work. However, the general trend you see here is common for a broad variety of elastomers. For instance, even in the absence of equibiaxial test data, we know that the equibiaxial stress–strain response should be about 1.5 to 2 times higher than the uniaxial response. This rule of thumb allows us to have a reasonable expectation for the approximate equibiaxial response even when the data are not available.

Figure W1–2 Treloar Test Data, Comparison of 3 tests.

Problem 2: Obtaining a Hyperelastic material model curve fit You will use Abaqus/CAE to curve fit the test data and derive coefficients for several different hyperelastic material models. The file st_treloar_abq.txt is the basis for the workshop hyperelastic material models. You can follow the directions below, and in addition you may want to refer to curve fitting demonstration presented in Lecture 4. 1. In the Model Tree, double-click the Materials container to create a new material definition. Name the material Treloar. 2. In the Edit Material dialog box, select Mechanical→Elasticity→Hyperelastic. 3. Click Test Data and select Uniaxial Test Data. The Test Data Editor appears. Click mouse button 3 in the first cell of the table and select Read from File from the list that appears.


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W1.5

4. In the Read Data from ASCII File dialog box, click Select to browse for the test data file. In the ASCII File Selection dialog box, select the file st_treloar_abq.txt and click OK. In the Read Data from ASCII File dialog box, click OK. In the Test Data Editor, click OK. 5. You should now be back in the material editor. If you wanted to read in more experimental data you would repeat this process, selecting Biaxial Test Data, Planar Test Data, or Volumetric Test Data from the Test Data pull-down menu. You have finished importing test data for this workshop; thus, click OK in the material editor. 6. In the Model Tree, click mouse button 3 on the material named Treloar; in the menu that appears, select Evaluate. The Evaluate Material dialog box has two tabbed pages: Test Setup and Strain Energy Potentials. 7. In the Test Setup tabbed page, you will accept most of the defaults. However, you will change the nominal strain values. Enter the value 0.0 in the Min Strain field and 4.0 in the Max Strain field for the Uniaxial, Biaxial, and Planar tests. This simply changes the range over which the material model response will be plotted. 8. Switch to the Strain Energy Potentials tabbed page. Notice that the default choices for the energy potentials are (full) Polynomial (N=2) and Ogden (N=3). In addition to these two material models, also select the neo-Hookean model (expand the Reduced Polynomial list and select N=1 (Neo-Hooke)). 9. When you click OK in the lower left corner of the Evaluate Material dialog box, Abaqus performs a datacheck analysis to extract the material constants; then the material response is calculated using a simple set of equations within Abaqus/CAE. Once the evaluation is complete, the coefficients (such as MooneyRivlin coefficients C10 and C01) and stability limit information are given in the Material Parameters and Stability Limit Information dialog box. For example, the Polynomial, N=2 data for this material are shown in Figure W1–3:



335

W1.6

Figure W1–3 Material Parameters and Stability Limit Information (Polynomial, N=2). All of the data in the Material Parameters and Stability Limit Information dialog box are also written to the data (.dat) file produced by the material evaluation analysis. Click Dismiss to close the dialog box. The test results from the material evaluation are automatically displayed in the Visualization module of Abaqus/CAE, as shown in Figure W1–4. Each of the deformation modes is displayed in a separate viewport (you can maximize each viewport to see the results more clearly). The first thing to notice, by looking at the uniaxial results, is that the neo-Hookean, Polynomial N=2, and Ogden N=3 models all fit the uniaxial test data reasonably well. However, by looking at the biaxial and planar results you can see that the Polynomial N=2 response in these deformation modes is off the scale compared to the responses predicted by the other two models. The Polynomial N=2 model is in fact very inaccurate in this case and should not be used. In general, full polynomial models should not be used when only limited test data are available (for instance, when only uniaxial tension data are available).


336


W1.7

Figure W1–4 Material evaluation results.

Next you will use the X-Y plotting capability of the Visualization module to look further at the Ogden N=3 and neo-Hookean material models. 10. Maximize the viewport with the uniaxial test results. 11. In the Results Tree, expand the XYData container. 12. Use [Ctrl]+Click to select the three data objects associated with the OGDEN_N3 model, the three data objects associated the R_POLY_N1 model and the original uniaxial test data (Test Data UNIAXIAL Treloar_1). Click mouse button 3, and from the menu that appears, select Plot. The resulting X-Y plot is shown in Figure W1–5.



337

W1.8

Biaxial, Ogden (N=3)

Figure W1–5 Reduced polynomial (neo-Hookean) and Ogden (N=3) results. Notice that the Ogden N=3 biaxial material model response is overly stiff. We conclude that the Ogden N=3 material model based only on uniaxial test data should not be used. Like the conclusion drawn earlier for full polynomial models, this conclusion is indeed generally true. 13. In the XYData container of the Results Tree, select just the three R_POLY_N1 (neo-Hookean) curves and the uniaxial test data. Plot these curves. The resulting plot is shown in Figure W1–6.


338


W1.9

Uniaxial neo-Hookean

Uniaxial test data

Figure W1–6 Reduced polynomial (neo-Hookean) results. While the uniaxial neo-Hookean response is not as good as we might like, the neo-Hookean material model predictions in the planar tension and equibiaxial modes are much better than the Ogden and full polynomial predictions. 14. Using a text editor, open the input (.inp) file that this curve fitting exercise produced. In this file search for the string *MATERIAL. You will see a fragment of text such as: *MATERIAL, NAME=OGDEN_N3 *HYPERELASTIC, OGDEN, N=3, TEST DATA INPUT *UNIAXIAL TEST DATA 0.0,0.0 0.03,0.02 0.15,0.1 0.23,0.2 0.33,0.34 0.41,0.57 0.51,0.85 0.59,1.13 0.67,1.4 0.86,1.98 1.04,2.55 1.22,3.0 1.59,3.77 1.95,4.37

This is the appropriate syntax for using experimental data to define the Ogden material model in an Abaqus input file. Following this text you will see the definitions for the POLYNOMIAL, N=2 and the REDUCED POLYNOMIAL, N=1



339

W1.10

(neo-Hookean) material models. In addition to the UNIAXIAL TEST DATA option, there are also options available (but not used in this input file) for entering equibiaxial and planar test data to define the hyperelastic material. In general, using all three types of test data and allowing Abaqus to simultaneously curve fit all the data creates the best material model. 15. Try curve fits using the Yeoh, Arruda-Boyce, Van der Waals, and Marlow models.


340


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341

Notes

342

Workshop 2 Inflation of a Spherical Balloon Interactive Version Note: This workshop provides instructions in terms of the Abaqus GUI interface. If you wish to use the Abaqus Keywords interface instead, please see the “Keywords” version of these instructions. Please complete either the Keywords or Interactive version of this workshop.

Goals In this workshop you will: Use experimental data from a uniaxial tension test to calibrate various hyperelastic models in Abaqus/Standard. Use the Visualization module of Abaqus/CAE to create X-Y plots.

Introduction Ogden (1972) computed the inflation pressure vs. radial displacement of a spherical balloon assuming a material stress-strain relationship based on a 3-term fit to Treloar’s rubber data (1944):

Initial radius = 10 cm Thickness = 0.4 mm

Figure W2–1 Inflation pressure vs. radial displacement of a spherical balloon.



343

W2.2

References: 1) Ogden, R.W., “Large Deformation Isotropic Elasticity: on the Correlation of Theory and Experiment for Incompressible Rubberlike Solids,” Proceedings of the Royal Society of London, Series A, Vol. 326, pp. 565-584, 1972 2) Treloar, L.R.G., “Stress-Strain Data for Vulcanized Rubber under Various Types of Deformations,” Trans. Faraday Soc., Vol. 40, pp. 59-70, 1944

Modeling the Balloon 1. Enter the working directory for this workshop: ../rubber_visco/workshop2/interactive 2. Run the script ws_rubber_balloon.py using the following command: abaqus cae startup=ws_rubber_balloon.py The above command creates an Abaqus/CAE database named balloon.cae in

the current directory. It contains a quarter-symmetry shell model suitable for a simulation along with Treloar’s test data for vulcanized natural rubber. The uniaxial, biaxial, and planar test data are all included in the hyperelastic material definition; however, the hyperelastic material definition is incomplete because the form of the strain energy potential has not been specified. The model is meshed with S4R elements. The initial quarter-symmetric mesh is shown in Figure W2–2.

Figure W2–2 Rubber balloon mesh. The purpose of this workshop is to simulate the inflation of a spherical balloon and to compare the FEA results with Ogden’s solution, which can be found in the file balloon_cur.inp.


344


W2.3

Figure W2–3 Inflation of the rubber balloon.

Fitting Case 1 First, perform the simulation with the best possible fit to the experimental data: 1. Edit the material definition so that the Ogden strain energy potential of order 3 is used. a. In the Model Tree, expand the Materials container and double-click the material named rubber. b. Select Ogden as the material strain energy potential. c. Increase the strain energy potential order to 3. d. Click OK. Question W2–1: How do you pressurize the shell model, given that as the balloon is inflated the applied pressure increases to a maximum and then further inflation is achieved with a reduced pressure? 2. In the Step module, activate the DOF monitor (Output→DOF Monitor). Monitor the radial displacement (degree of freedom 1) of the set monitor (click Points in the prompt area). 3. Create a history output request to write the radial displacement (U1) of the set monitor to the output database (.odb) file (in the Model Tree, double-click History Output Requests). 4. Apply a pressure load to the inner surface of the balloon (in the Model Tree, double-click Loads). Choose an arbitrary magnitude for the pressure load (e.g., 1.0). The magnitude is arbitrary because the problem is solved using the modified Riks method, and the analysis terminates only when the displacement in the radial direction passes a specified value: 70 cm in this case.



345

W2.4

5. Edit the job case1 so that model definition data will be printed during preprocessing (in the Model Tree, double-click the job named case1; activate the option in the General tabbed page of the job editor). With this option Abaqus will write detailed model information, including the coefficients used for the hyperelastic material model, to the data (.dat) file. 6. In the Model Tree, click mouse button 3 on the job named case1; in the menu that appears select Submit to run the analysis job. 7. While the job is running, create a data object containing the Ogden data. a. In the Results Tree, double-click XYData. b. Select ASCII file as the source and select the file balloon_cur.inp. c. Set the X-axis quantity type to Displacement. This will facilitate comparison with the curves that will be created later. d. Save the data as Ogden. 8. When the job completes, open the output database file (case1.odb) in the Visualization module and use the X-Y plotting capability to compare the simulation results with the Ogden results. A reasonable comparison can be made by plotting the pressure versus the radial displacement. The pressure at any time in the analysis is equal to the product of the load proportionality factor and the magnitude of the distributed load. Detailed instructions to create the pressure versus the radial displacement curve from the Riks analysis results are provided below. Your plot should look similar to Figure W2–4. (Note the axis labels in this figure have been customized.) a. In the Results Tree, expand the History Output container underneath the output database named case1.odb. Select the radial displacement (U1) variable for the set MONITOR. Click mouse button 3, and from the menu that appears, select Save As. Name the X-Y data OgdenN3-U. b. Click mouse button 3 on the load proportionality factor (LPF) variable; from the menu that appears, select Save As. Name the X-Y data OgdenN3-LPF. c. In the Results Tree, double-click XYData. d. Choose Operate on XY data as the source and click Continue. e. In the Operate on XY Data dialog box, select combine(X, X) from the list of operators. Select OgdenN3-U and click Add to Expression. Repeat this for OgdenN3-LPF. If necessary, edit the expression to use the load magnitude specified earlier for the model (this is not necessary if you used a load magnitude of 1.0). The final expression is: combine( "OgdenN3-U", "OgdenN3-LPF"* load_magnitude ) where load_magnitude is the magnitude of the pressure load you applied.

f. Click Save As and name the X-Y data OgdenN3-PvU. g. Plot both curves (Ogden and OgdenN3-PvU) simultaneously.


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W2.5

Figure W2–4 Inflation pressure vs. radial displacement of rubber balloon.

Fitting Case 2 Ogden’s analysis reveals that for a single-term Ogden strain energy function with 23 3, a maximum inflation pressure exists but no minimum pressure exists. For such values the pressure reaches a maximum and then decreases eventually to zero. Question W2–2:

Can you confirm this numerically?

1. Reduce the strain energy potential order of the material name rubber to 1. 2. Create a new job named case2 and rerun the analysis. Check the value of (run the job with model data printed during preprocessing and look for ALPHA_I in the .dat file). 3. Compare the results with case1 using the Visualization module.



347

W2.6

Fitting Case 3 Suppose that not all three types of test data are available and all we have is the uniaxial test data: 1. Copy the model named case1 to a new model named case3 (in the Model Tree, click mouse button 3 on the model named case1; in the menu that appears, select Copy Model). 2. In the new model, delete the biaxial and shear test data. a. In the Model Tree, expand the Materials container and double-click rubber. b. At the top of the material editor, select Biaxial Test Data and click Delete. c. Select Planar Test Data and click Delete. d. Set the strain energy potential to Ogden N = 3. e. Click OK. 3. Create a job named case3 and run the analysis. Are the results of the simulation realistic? Are the results different if we fit the hyperelastic constants to the biaxial test data only? Why? Note: With the biaxial test data only, Abaqus fails to converge on the material coefficients when Ogden N = 3; use Ogden N = 2. 4. Calibrate the following material models with just the uniaxial test data: a. Yeoh b. Reduced Polynomial, N = 2 c. Arruda-Boyce d. Van der Waals with = 0 e. Marlow Note: The Marlow model requires that the test data include a data point corresponding to zero stress at zero strain. You must add this point to the uniaxial test data to use this material model. Rerun each to simulate the balloon inflation. Question W2–3:

Question W2–4: What can we conclude from all this?

Note: A script that creates the complete model described in these instructions is available for your convenience. Run this script if you encounter difficulties following the instructions outlined here or if you wish to check your work. The script is named ws_rubber_balloon_answer.py and is available using the Abaqus fetch utility.


348


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349

Notes

350

Workshop Answers 2 Inflation of a Spherical Balloon Interactive Version Question W2–1:

How do you pressurize the shell model, given that as the balloon is inflated the applied pressure increases to a maximum and then further inflation is achieved with a reduced pressure?

Answer:

The Riks method must be used when a structure has a negative slope in the global force-deflection curve (a global instability). With the Riks method both the displacement and load applied to a structure are considered unknowns.

Question W2–2:

Can you confirm this numerically?

Answer:

Yes. When the Ogden model with N = 1 is calibrated the value of  is set to 2.16. With this value of  no minimum pressure is attained. Instead, the pressure continues to drop asymptotically to zero. This confirms Ogden’s result (at least at one value of ).



351

WA2.2

Question W2–3a: Are the results of the simulation realistic? Answer:

No. The Ogden N = 3 model calibrated with only uniaxial test data produces results that are much too stiff.


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WA2.3

Question W2–3b: Are the results different if we fit the hyperelastic constants to the

biaxial test data only? Why? Answer:

Yes. When we use just the biaxial test data to curve fit our Ogden hyperelastic material model the pressure vs. radial displacement curve looks good out to about 40 cm of radial displacement. The reason for this is coincidental―the deformation mode in spherical balloon inflation just happens to be almost exactly that of equibiaxial tension. In fact, some researchers use a disk inflation experiment to measure equibiaxial tension stress-strain relationships.



353

WA2.4

Question W2–4: Answer:

What can we conclude from all this? Simulating complex multiaxial behavior with hyperelastic models that are calibrated with only uniaxial data can be very difficult, and using the wrong energy potential can give very inaccurate results. As a general rule, when only limited test data is available, one should use a simple material model, preferably an I1-based model. All of the I1-based models tested here (Arruda-Boyce, Yeoh, reduced polynomial N = 2, Van der Waals, and Marlow) do a reasonable job in this case out to about 40 cm of radial displacement (see the figure below). The results past 40 cm of radial displacement vary widely due to extrapolation past the end of the uniaxial experimental data.


354


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355

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356

Workshop 3 Time Domain Viscoelasticity Interactive Version Note: This workshop provides instructions in terms of the Abaqus GUI interface. If you wish to use the Abaqus Keywords interface instead, please see the “Keywords” version of these instructions. Please complete either the Keywords or Interactive version of this workshop.

Goals When you complete this workshop, you will be able to: Obtain an N-term Prony series fit to given relaxation data. Examine this fit numerically and graphically for N = 1, N = 2, and N = 3. Modify a model to conduct a shear-relaxation analysis in Abaqus/Standard. View the shear relaxation modulus versus time in the Visualization module for N = 1 and N = 3. Define the temperature-dependent viscoelastic properties, and demonstrate the effect that raising the temperature has on the relaxation curve. Attempt gross time integration of viscoelastic equations by using a large viscoelastic strain error tolerance (CETOL). Simulate a simple shear creep test and study the effects of using a large viscoelastic strain error tolerance.



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W3.2

Introduction The following is a normalized shear relaxation modulus for polycarbonate of bisphenol taken approximately from Mercier (1965)1:

Shear relaxation modulus

Time (seconds)

1.0000

.01

0.8913

.1

0.6310

1.

0.1995

3.981

0.0631

12.589

1.585E-2

31.622

7.943E-3

100.

3.548E-3

398.1

1.995E-3

10000.

Table W3–1 Normalized relaxation modulus for polycarbonate of bisphenol.

Figure W3–1 Normalized relaxation modulus vs. time. 1

Mercier, J. P., Journal of Applied Polymer Science, vol. 9, pp 447-459, 1965.


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W3.3

Problems Problem 1: Viewing relaxation data 1. Enter the working directory for this workshop: ../rubber_visco/workshop3/interactive

2. The normalized shear relaxation data for polycarbonate of bisphenol is included in the file bis_cur.inp. Look at the contents of this file in a text editor. 3. Start an Abaqus/CAE session and switch to the Visualization module. 4. Create an X-Y plot of the relaxation data in file bis_cur.inp. Detailed instructions are provided below. a. In the Results Tree, double-click XYData. b. Select ASCII file as the source, and click Continue. c. Next to the File field, click Select to browse for the file bis_cur.inp. The X-values (time) should be read from field 2, and the Y-values (shear relaxation modulus) should be read from field 1. d. Set the X-axis quantity type to Time and the Y-axis quantity type to Stress. e. In the XY Data From ASCII File dialog box, click Save As. Name the X-Y data BIS, and click OK. f. In the XY Data From ASCII File dialog box, click Plot and then click Cancel. g. From the main menu bar, select Options→XY Options→Curve (or click in the toolbox). h. Toggle on Show symbol, and set the symbol size to Large. Dismiss the dialog box. Question W3–1:

What is wrong with the curve?

5. Use logarithmic scales for both of the X-Y plot axes. a. Double-click the X-axis to open the Axis Options dialog box. b. In the Scale tabbed page of the dialog box, choose the Log scale type with 8 minor ticks per decade. c. Repeat for the Y-axis (simply select the Y-axis in the Axis Options dialog box, and make the changes). d. Enter Normalized Relaxation Modulus as the Y-axis title.



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Problem 2: Obtaining a Prony series fit Next, you will evaluate the material to obtain a Prony series fit to the above data. 1. Run the script ws_visco_plate.py (File→Run Script). The script creates an Abaqus/CAE database named visco.cae in the current directory. This model is the basis for a shear relaxation simulation in Abaqus/Standard, although it is not complete. For now you will use the model to evaluate the Prony series parameters to fit the BIS curve. 2. In the Model Tree, expand the Materials container and double-click Material-1. 3. Add elastic properties to Material-1. The material has an instantaneous shear modulus of 100 and a Poisson’s ratio of 0.499. a. From the material editor's menu bar, select Mechanical→Elasticity→ Elastic. b. Select Instantaneous as the moduli time scale. c. Enter the appropriate Young's modulus and Poisson’s ratio in the data table. Note: The relationship between the shear modulus G, Young's modulus E, and Poisson’s ratio ν is G E 2 (1 v) . You can use the command line interface (CLI) of Abaqus/CAE as a simple calculator. Click the tab in the bottom left corner of the Abaqus/CAE window to activate the CLI. 4. Add viscoelastic properties to Material-1. Use the normalized shear relaxation modulus data for polycarbonate of bisphenol provided in Table 3–1 and the file bis_cur.inp. a. From the material editor's menu bar, select Mechanical→Elasticity→ Viscoelastic. b. Select Time in the Domain field and Relaxation test data in the Time field. c. Decrease the maximum number of terms in the Prony series to 3 and set the allowable average root-mean-square error to 0.05. Note: a fit with an order N that is larger than 3 will not attempted; fewer

terms may be used if the error tolerance is satisfied. d. Click Test Data and select Shear Test Data. The Test Data Editor appears. You will read in the normalized shear relaxation modulus (gR) and time data pairs from the file bis_cur.inp as follows. e. In the Test Data Editor, click mouse button 3 in the first cell of the data table and select Read from File.


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f. In the Read Data from ASCII File dialog box, select the file bis_cur.inp and click OK. g. In the Test Data Editor, click, click OK. h. In the Edit Material dialog box, click OK. 5. In the Model Tree, click mouse button 3 on the material named Material-1. In the menu that appears, select Evaluate. 6. In the Evaluate Material dialog box, select Stress Relaxation as the response mode (i.e., deselect Creep) and specify a time period of 10000. Click OK. 7. When you click OK in the lower left corner of the Evaluate Material dialog box, Abaqus performs a datacheck analysis to extract the material constants; then the material response is calculated using a simple set of equations within Abaqus/CAE. Once the evaluation is complete, the Prony series terms are given in the Material Parameters and Stability Limit Information dialog box. 8. The test results from the material evaluation are automatically displayed in the Visualization module of Abaqus/CAE. By default, linear scales are used for the plot. Customize the plot as follows: In the Results Tree, expand the XYData container. Rename the SHEARRELAXATION* curve to N1-10000. Delete the first data point of curve N1-10000 and set the X-axis type to Time and the Y-axis type to Stress (click mouse button 3 on the curve name and select Edit from the menu that appears). Plot N1-10000 together with BIS. Using the method indicated earlier, change both axis types to logarithmic. Figure W3–2 (left) shows how the one-term Prony series fits the normalized relaxation curve. The data is skewed towards times greater than 250 and shows a constant value. To get a more complete representation of the fit over the entire timescale, re-evaluate the material using time periods of 2, 75, and 1000. After each evaluation, expand the XYData container of the Results Tree, and rename the corresponding SHEARRELAXATION* curve to N1-2, N175, or N1-1000 (according to the time period). Delete the first data point of the N1-75 and N1-1000 curves. Set the X-axis type of each N1-* curve to Time and the Y-axis type of each to Stress. Figure W3–2 (right) shows the complete fit using the four evaluations.



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Test data

N=1

N=1

Test data

Figure W3–2 Comparison of Abaqus model with N=1 to experimental data.

The plot seems to indicate that the fit is good for the first four data points, but after that the fit is poor. However, the logarithmic scale for the Y-axis (which represents the stress in a relaxation test) tends to exaggerate the misfit for very small values of gR at large times and understate the misfit at large values of gR.

Question W3–2:

How much should the total stress change as a fraction of the initial stress after t=100 in a relaxation test?

9. Try making the Y-axis linearly spaced, and assess the fit again. a. Double-click the Y-axis. b. In the Scale tabbed page of the Axis Options dialog box, choose the Linear scale type. Click OK. The fit is not very good even for the second data point. 10. Modify the viscoelastic properties of Material-1 (in the Model Tree, double-click Material-1) to decrease the allowable error to 0.025. Evaluate the material using time periods of 250 and 10000. Rename each curve N2-250 and N2-10000, respectively. Delete the first data point of each N2-* curve; set the X-axis type of each to Time and the Y-axis type of each to Stress. Question W3–3: What effect does restricting the allowable error have on the

number of terms used?


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W3.7

11. Modify the viscoelastic properties of Material-1 to further decrease the allowable error to 0.01. This will force Abaqus to use three terms in the curve fit. Evaluate the material using time periods of 250 and 10000. Rename each curve N3-250 and N3-10000, respectively. Delete the first data point of each N3-* curve; set the X-axis type of each to Time and the Y-axis type of each to Stress. 12. Display all the curves in an X-Y plot, as shown in Figure W3–3.

N=1

N=2 N=3

Figure W3–3 Comparison of Abaqus models with N = 1, 2, and 3 (log scale for normalized modulus).

The long-term behavior is somewhat off for all of the Prony series fits. We can specify the long-term modulus exactly at the top of the shear test data editor, although this will not necessarily produce a better overall fit. Question W3–4: Is the curve for N = 3 really better than the curve for N = 2? Is

the fit for N = 3 good enough after t = 100? (View it with a linear Y-axis scale.) We had a crude set of data points for the original curve. The curve-fitting procedure for the Prony parameters only fits to the given data points; i.e., those marked by a square in the graph.



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W3.8

Problem 3: Simulating a relaxation test

Figure W3–4 Shear relaxation test. 1. Copy the model relax to relax1 (in the Model Tree, click mouse button 3 on the model named relax and select Copy Model in the menu that appears), and make the following modifications: a. Modify the viscoelastic properties of Material-1. Set the maximum number of terms in the Prony series to 1 and adjust the average root-meansquare error tolerance in such a way that this single Prony series parameter will be accepted (e.g., set to 0.03). b. In the Model Tree, double-click the Steps container to create a Visco step after Step-1 with a total time of 1000 and an initial time increment of 1E2. Specify a viscoelastic strain error tolerance (CETOL) of 0.05E-2 to allow automatic time incrementation. (This value was arrived at as follows: the total creep strain when the material relaxes fully will be 0.01, which is equal to the instantaneously applied shear strain. We can force at least 20 increments by taking CETOL to be 0.01/20.) c. In the Model Tree, double-click the History Output Requests container to create a history output request in Step-2. Request shear stress S12 and shear strain E12 for the set plate. d. In the Model Tree, double-click the BCs container to create a Displacement/Rotation boundary condition in Step-1 named moveTop. Apply the boundary condition to the set top and specify a U1 displacement of 0.01. This will put the model into a state of simple shear with a positive shear strain of 0.01 during the first step.


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W3.9

The shear modulus of 100 was chosen so that the shear stress in this test will coincide with the normalized shear relaxation modulus:

G (t ) We have contrived G0 gR ( t ) as .

0

0

G0 g

R

(t )

0

.

= 1 so that we can read the normalized shear modulus

2. In the Model Tree, double-click the Jobs container to create a job named relax1 for the model relax1. Submit the job for analysis. 3. When the job completes, create a plot of the shear stress (S12) versus the step time, using the history data reported for the second step of the relax1 analysis. a. In the Model Tree, click mouse button 3 on the job named relax1; in the menu that appears, select Results to open relax1.odb in the Visualization module. b. In the Results Tree, expand the History Output container underneath the output database named relax1.odb. c. From the list of output variables, select Stress components: S12. d. Click mouse button 3; from the menu that appears, select Save As. Name the X-Y data S1 and click OK. e. In the Results Tree, expand the XYData container, and double-click S1 to plot the curve. 4. Copy the model relax1 to relax3. 5. Modify the viscoelastic properties of Material-1. Set the maximum number of terms in the Prony series to 3 and adjust the average root-mean-square error tolerance in such a way that the Prony series will be accepted for N = 3, but not N = 1 and N = 2 (e.g., set to 0.01). 6. Create a job named relax3 for the model relax3. Submit the job for analysis. 7. Repeat the procedure given above to define curve S3 using the data in relax3.odb. 8. Plot curves N1-* and S1 together (select in the XYData container of the Results Tree, click mouse button 3, and then select Plot from the menu that appears). Question W3–5:

Are the results from the viscoelasticity analysis true to the Prony series defined for it?

9. Similarly, compare curve S3 with curves N3-*. The integration of the viscoelastic equations is very accurate.



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W3.10

Problem 4: Simulating the effects of temperature-dependent relaxation data 1. Copy the model relax3 to relax_hot. You will use relax_hot to model a relaxation test under the following conditions: the relaxation data are assumed to be applicable to 20 C, the relaxation test is carried out at 70 C, the material has the time-temperature transformation properties of the rubber material referred to in Lecture 10, and the time constants for the WLF shift function have already been worked out to refer to 20 C ( 0 = 20, C1=6.106, C2=146.6). Make the following modifications to the model: a. Modify the viscoelastic properties of Material-1 to define the required constants for the temperature-time shift. Add these constants using the viscoelastic suboption Trs (under the viscoelastic properties, click Suboptions and select Trs). b. Edit Step-2 and specify a minimum time increment of 1.E-10 to prevent premature termination of the analysis due to small time increment sizes. c. In the Model Tree, double-click the Predefined Fields container to create a temperature field in the Initial step to assign an initial temperature of 20 to the set plate. d. Edit the temperature field defined above to change the model temperature to 70 in Step-1. This can be done by selecting the appropriate cell in the Predefined Field Manager and clicking Edit. In the field editor that appears, change the temperature magnitude to 70. 2. Create a job named relax_hot for model relax_hot. Submit the job for analysis. 3. Once the job completes, save the shear stress history data as S-HOT. Plot this against the curve of your choice. Question W3–6:

What do you notice? What is the value of the time shift? Does it correspond to the calculated value in the notes?

Figure W3–5 shows the normalized relaxation moduli versus time.


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W3.11

Figure W3–5 Effect of temperature on viscoelastic response. Problem 5: Determining the effects of time integration in a relaxation analysis 1. Copy the model relax3 to relax_coarse. 2. Edit Step-2. Set the initial time increment to 100 seconds and increase the viscoelastic strain error tolerance to a value that is so large it will never restrict the time step. 3. Create a job named relax_coarse for the model relax_coarse. Submit the job for analysis. 4. Once the job completes, save the shear stress history data as S3-COR and compare it to curve N3 or S3. The viscoelastic equations were integrated very well despite the coarseness of the integration and the complexity of the relaxation response. Question W3–7:


Why did the integration procedure produce good results with such large time increments?


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W3.12

Problem 6: Determining the effects of time integration in a creep analysis 1. Copy the model relax3 to creep. 2. Modify the model so that it conducts a creep test rather than a relaxation test; that is, it subjects an element to constant shear stress of 0.01. a. Delete the boundary condition named moveTop. b. Create a concentrated force load in Step-1 named loadTop (in the Model Tree, double-click the Loads container). Apply the load to the set topCorners and specify a load magnitude of 0.005 in the CF1 field. 3. Set the Step-2 viscoelastic strain error tolerance to 0.05E-4. Use the same time limits as before. 4. Create a job named creep for the model creep and submit the job for analysis. 5. Copy the model creep to creep_coarse. Modify this model so that the Step-2 time integration is very coarse. (Use a very large viscoelastic strain error tolerance, and specify an initial time increment of 100.) 6. Create a job named creep_coarse for the model creep_coarse and submit the job for analysis. 7. View the shear strain response history data (E12) and compare the results of the two analyses. You will see small differences in the strain response at the output times because the total strain no longer varies linearly with time. Nevertheless, the creep integration is still quite good. Your results should produce curves similar to the ones shown in Figure W3–6 (with both scales linear).


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W3.13

Figure W3–6 Effect of time integration on viscoelastic response in Abaqus.

Note: A script that creates the complete model described in these instructions is available for your convenience. Run this script if you encounter difficulties following the instructions outlined here or if you wish to check your work. The script is named ws_visco_plate_answer.py and is available using the Abaqus fetch utility.



369

370

Notes

371

Notes

372

Workshop Answers 3 Time Domain Viscoelasticity Interactive Version Question W3–1: What is wrong with the curve? Answer:

It has linear scales. It should have logarithmic scales.

Question W3–2: How much should the total stress change as a fraction of the initial

stress after t=100 in a relaxation test?

Answer:

The normalized relaxation modulus is reduced to 0.0079 % at 100 seconds. Therefore, the stress in a relaxation test should also be 0.0079 % of the instantaneous value after 100 seconds.

Question W3–3: What effect does restricting the allowable error have on the number

of terms used? Answer:

More terms are required. With a 2.5% allowable error, two Prony series terms are used.

Question W3–4: Is the curve for N=3 really better than the curve for N=2? Is the fit

for N=3 good enough after t=100? (View it with a linear Y-axis scale.)

Answer:

The distinction between the N=3 curve and the N=2 curve can be misleading on the logarithmic scale. When viewed on a linear scale, the differences appear to be very minor.

Question W3–5: Are the results from the viscoelasticity analysis true to the Prony

series defined for it? Answer:

Yes.



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WA3.2

Question W3–6: What do you notice? What is the value of the time shift? Does it

correspond to the calculated value in the notes?

Answer:

The relaxation modulus at the elevated temperature is much smaller than the value at 20 C. The time shift is 0.03, which corresponds to the value calculated in the lecture on Time-Temperature correspondence. You can verify this using the Operate on XY data Visualization functionality (Results Tree: double-click XYData). Use the following expression to multiply the time data of the S3 curve by 0.03: swap(swap("S3")*0.03). The resulting curve is very similar to S-HOT, as expected.

Question W3–7: Why did the integration procedure produce good results with such

large time increments? Answer:

The viscoelastic equations are integrated exactly when the total strain varies linearly over an increment. In this case the strain is constant. Thus a large viscoelastic strain error tolerance (CETOL) will not affect the accuracy of a simple stress-relaxation problem.


374


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375

Notes

376

Workshop 2 Inflation of a Spherical Balloon Keywords Version Note: This workshop provides instructions in terms of the Abaqus Keywords interface. If you wish to use the Abaqus GUI interface instead, please see the “Interactive” version of these instructions. Please complete either the Keywords or Interactive version of this workshop.

Goals In this workshop you will: Use experimental data from a uniaxial tension test to calibrate various hyperelastic models in Abaqus/Standard. Use the Visualization module of Abaqus/CAE to create X-Y plots.

Introduction Ogden (1972) computed the inflation pressure vs. radial displacement of a spherical balloon assuming a material stress-strain relationship based on a 3-term fit to Treloar’s rubber data (1944):

Initial radius = 10 cm Thickness = 0.4 mm

Figure W2–1 Inflation pressure vs. radial displacement of a spherical balloon.



377

W2.2

References: 1) Ogden, R.W., “Large Deformation Isotropic Elasticity: on the Correlation of Theory and Experiment for Incompressible Rubberlike Solids,” Proceedings of the Royal Society of London, Series A, Vol. 326, pp. 565-584, 1972 2) Treloar, L.R.G., “Stress-Strain Data for Vulcanized Rubber under Various Types of Deformations,” Trans. Faraday Soc., Vol. 40, pp. 59-70, 1944

Modeling the Balloon Go to the rubber_elasticity/workshop2 directory. The Abaqus/Standard input file is named balloon.inp. It contains a quartersymmetry shell-element mesh suitable for a simulation along with Treloar’s test data for vulcanized natural rubber. Open this file in a text editor and scan the file. Note the shell element definitions using S4R elements and the experimental data listed under the *UNIAXIAL TEST DATA, *BIAXIAL TEST DATA, and *PLANAR TEST DATA options. This input file is incomplete (the *HYPERLEASTIC option is incomplete). The initial quartersymmetric mesh looks like this:

Figure W2–2 Rubber balloon mesh. The purpose of this workshop is to simulate the inflation of a spherical balloon and to compare the FEA results with Ogden’s solution, which can be found in the file balloon_cur.inp.


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W2.3

Figure W2–3 Inflation of the rubber balloon.

Fitting Case 1 First, perform the simulation with the best possible fit to the experimental data: 1. Add the parameters TEST DATA INPUT, OGDEN, N=3 to the *HYPERELASTIC option, so that the entire line reads: *HYPERELASTIC, TEST DATA INPUT, OGDEN, N=3 Question W2–1: How do you pressurize the shell model, given that as the

balloon is inflated the applied pressure increases to a maximum and then further inflation is achieved with a reduced pressure? 2. Define a distributed pressure load across all elements of the shell. Choose an arbitrary magnitude for this load (e.g., 1.0). The magnitude is arbitrary because we are using the modified Riks method, and the analysis terminates only when the displacement in the radial direction passes a specified value: 70 cm in this case. 3. Monitor the radial displacement of node 512 with the *MONITOR history option. 4. Run the analysis. 5. While the job is running, open Abaqus/Viewer and create a data object containing the Ogden data. a. In the Results Tree, double-click XYData. b. Select ASCII file as the source and select the file balloon_cur.inp. c. Set the X-axis quantity type to Displacement. This will facilitate comparison with the curves that will be created later. d. Save the data as Ogden.



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W2.4

6. When the job completes, open the output database file (balloon.odb) in Abaqus/Viewer and use the X-Y plotting capability to compare the simulation results with the Ogden results. A reasonable comparison can be made by plotting the pressure versus the radial displacement. The pressure at any time in the analysis is equal to the product of the load proportionality factor and the magnitude of the distributed load. Detailed instructions to create the pressure versus the radial displacement curve from the Riks analysis results are provided below. Your plot should look similar to Figure W2–4. (Note the axis labels in this figure have been customized.) a. In the Results Tree, expand the History Output container underneath the output database named balloon.odb. Select the radial displacement (U1) variable for the set N_OUT. Click mouse button 3, and from the menu that appears, select Save As. Name the X-Y data OgdenN3-U. b. Click mouse button 3 on the load proportionality factor (LPF) variable; from the menu that appears, select Save As. Name the X-Y data OgdenN3-LPF. c. In the Results Tree, double-click XYData. d. Choose Operate on XY data as the source and click Continue. e. In the Operate on XY Data dialog box, select combine(X, X) from the list of operators. Select OgdenN3-U and click Add to Expression. Repeat this for OgdenN3-LPF. If necessary, edit the expression to use the load magnitude specified earlier for the model (this is not necessary if you used a load magnitude of 1.0). The final expression is: combine( "OgdenN3-U", "OgdenN3-LPF"* load_magnitude ) where load_magnitude is the magnitude of the pressure load you applied.

f. Click Save As and name the X-Y data OgdenN3-PvU. g. Plot both curves (Ogden and OgdenN3-PvU) simultaneously.

Figure W2–4 Inflation pressure vs. radial displacement of rubber balloon.


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W2.5

Fitting Case 2 Ogden’s analysis reveals that for a single-term Ogden strain energy function with 23 3, a maximum inflation pressure exists but no minimum pressure exists. For such values the pressure reaches a maximum and then decreases eventually to zero. Question W2–2: Can you confirm this numerically?

1. Set N=1 on the *HYPERELASTIC option and check the value of “ALPHA_I” in the .dat file).

(look for

2. Rerun the analysis. 3. Compare the results using the Visualization module.

Fitting Case 3 Suppose that not all three types of test data are available and all we have is the uniaxial test data: 1. Make a copy of the input file. 2. In the new file, delete the biaxial and shear test data. 3. Fit the hyperelastic constants with OGDEN, N=3. 4. Rerun the analysis. Question W2–3: Are the results of the simulation realistic? Are the results different if we fit the hyperelastic constants to the biaxial test data only? Why? Note: With the biaxial test data only, Abaqus fails to converge on the material coefficients when Ogden N = 3; use Ogden N = 2. 5. Calibrate the following material models with just the uniaxial test data: a. Yeoh b. Reduced Polynomial, N = 2 c. Arruda-Boyce d. Van der Waals with = 0 e. Marlow Note: The Marlow model requires that the test data include a data point corresponding to zero stress at zero strain. You must add this point to the uniaxial test data to use this material model. Rerun each to simulate the balloon inflation. Question W2–4: What can we conclude from all this?



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W2.6

Note: Complete input files are available for your convenience. You may consult these files if you encounter difficulties following the instructions outlined here or if you wish to check your work. The input files are named ball_ogden1.inp ball_ogden3.inp ball_uni.inp

and are available using the Abaqus fetch utility.


382


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383

Notes

384

Workshop Answers 2 Inflation of a Spherical Balloon Keywords Version Question W2–1: How do you pressurize the shell model, given that as the balloon is

inflated the applied pressure increases to a maximum and then further inflation is achieved with a reduced pressure? Answer:

The Riks method must be used when a structure has a negative slope in the global force-deflection curve (a global instability). With the Riks method both the displacement and load applied to a structure are considered unknowns.

Question W2–2: Can you confirm this numerically? Answer:

Yes. When the Ogden model with N = 1 is calibrated the value of  is set to 2.16. With this value of  no minimum pressure is attained. Instead, the pressure continues to drop asymptotically to zero. This confirms Ogden’s result (at least at one value of ).



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WA2.2

Question W2–3a: Are the results of the simulation realistic? Answer:

No. The Ogden N = 3 model calibrated with only uniaxial test data produces results that are much too stiff.


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WA2.3

Question W2–3b: Are the results different if we fit the hyperelastic constants to the

biaxial test data only? Why? Answer:

Yes. When we use just the biaxial test data to curve fit our Ogden hyperelastic material model the pressure vs. radial displacement curve looks good out to about 40 cm of radial displacement. The reason for this is coincidental―the deformation mode in spherical balloon inflation just happens to be almost exactly that of equibiaxial tension. In fact, some researchers use a disk inflation experiment to measure equibiaxial tension stress-strain relationships.



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WA2.4

Question W2–4: What can we conclude from all this? Answer:

Simulating complex multiaxial behavior with hyperelastic models that are calibrated with only uniaxial data can be very difficult, and using the wrong energy potential can give very inaccurate results. As a general rule, when only limited test data is available, one should use a simple material model, preferably an I1-based model. All of the I1-based models tested here (Arruda-Boyce, Yeoh, reduced polynomial N = 2, Van der Waals, and Marlow) do a reasonable job in this case out to about 40 cm of radial displacement (see the figure below). The results past 40 cm of radial displacement vary widely because we are extrapolating past the end of our uniaxial experimental data.


388


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389

Notes

390

Workshop 3 Time Domain Viscoelasticity Keywords Version Note: This workshop provides instructions in terms of the Abaqus Keywords interface. If you wish to use the Abaqus GUI interface instead, please see the “Interactive” version of these instructions. Please complete either the Keywords or Interactive version of this workshop.

Goals When you complete this workshop, you will be able to: Obtain an N-term Prony series fit to given relaxation data. Examine this fit numerically and graphically for N = 1, N = 2, and N = 3. Modify an input file to conduct a shear-relaxation analysis in Abaqus/Standard. View the shear relaxation modulus versus time in Abaqus/Viewer for N = 1 and N = 3. Define the temperature-dependent viscoelastic properties, and demonstrate the effect that raising the temperature has on the relaxation curve. Attempt gross time integration of viscoelastic equations by using a large CETOL. Simulate a simple shear creep test and the effects of using a large CETOL.



391

W3.2

Introduction The following is a normalized shear relaxation modulus for polycarbonate of bisphenol taken approximately from Mercier (1965)1: Shear relaxation modulus

Time (seconds)

1.0000

.01

0.8913

.1

0.6310

1.

0.1995

3.981

0.0631

12.589

1.585E-2

31.622

7.943E-3

100.

3.548E-3

398.1

1.995E-3

10000.

Table W3–1 Normalized relaxation modulus for polycarbonate of bisphenol.

Figure W3–1 Normalized relaxation modulus vs. time.

1

Mercier, J. P., Journal of Applied Polymer Science, vol. 9, pp 447-459, 1965.


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W3.3

Problems Problem 1: Viewing relaxation data with Abaqus/Viewer Go to the viscoelasticity/workshop directory. 1. The normalized shear relaxation data for polycarbonate of bisphenol is located in the file bis_cur.inp. Look at the contents of this file. 2. Start an Abaqus/Viewer session. 3. Create an X-Y plot of the relaxation data in file bis_cur.inp. Detailed instructions are provided below. a. In the Results Tree, double-click XYData. b. Select ASCII file as the source, and click Continue. c. Next to the File field, click Select to browse for the file bis_cur.inp. The X-values (time) should be read from field 2, and the Y-values (shear relaxation modulus) should be read from field 1. d. Set the X-axis quantity type to Time and the Y-axis quantity type to Stress. e. In the XY Data From ASCII File dialog box, click Save As. Name the X-Y data BIS, and click OK. f. In the XY Data From ASCII File dialog box, click Plot and then click Cancel. g. From the main menu bar, select Options→XY Options→Curve (or click in the toolbox). a. Toggle on Show symbol, and set the symbol size to Large. Dismiss the dialog box. Question W3–1: What is wrong with the curve?

4. Use logarithmic scales for both of the X-Y plot axes. a. Double-click the X-axis to open the Axis Options dialog box. b. In the Scale tabbed page of the dialog box, choose the Log scale type with 8 minor ticks per decade. c. Repeat for the Y-axis (simply select the Y-axis in the Axis Options dialog box, and make the changes). d. Enter Normalized Relaxation Modulus as the Y-axis title.



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W3.4

Problem 2: Obtaining a Prony series fit Next, you will run datacheck analyses to obtain a Prony series fit to the above data. The file relax0.inp is the basis for a shear relaxation simulation in Abaqus/Standard, although it is not complete. For now we will run the file to find the Prony series parameters to fit the BIS curve. 1. Copy relax0.inp to the file relax_data1.inp, and modify the file to give the material an instantaneous shear modulus of 100 and a Poisson’s ratio of 0.499 (must set MODULI=INSTANTANEOUS on the *ELASTIC option). Note that there is a dummy static step in the file. Note: The relationship between the shear modulus G, Young's modulus E, and Poisson’s ratio ν is G E 2 (1 v) . 2. Add the viscoelastic properties to relax_data1.inp using the VISCOELASTIC, TIME=RELAXATION TEST DATA option and *SHEAR TEST DATA suboption. Add the shear test data according to the table listing the normalized shear relaxation modulus for polycarbonate of bisphenol. Specify NMAX=1 so that a fit with N larger than 1 is not attempted. Be careful to place the data into the correct table columns. You can copy the data from bis_cur.inp and paste it into your input file. 3. Conduct a datacheck analysis (we do not need the analysis results). Look at the file relax_data1.dat. There is an error because the Prony series was not good enough within the required tolerance. Search for “ROOT” to get to the right place in relax_data1.dat. Notice the RMS percentage error. Abaqus still lists the Prony parameters, however. You will now look at how the given one-term Prony series fits the normalized relaxation curve. 4. The file prony_prog.f (prony_prog.for on Windows systems) is a FORTRAN program that takes Prony series parameters and writes the normalized shear relaxation modulus versus time to a file. Compile this program with the Abaqus make utility, abaqus make job=prony_prog

and then execute the program: abaqus prony_prog

Now enter the Prony series parameters for the N=1 fit we obtained (these parameters will come from the data file relax_data1.dat where they are labeled “GP(TI)” and “TI”). Then, specify some values of time in the table and see how good the fit was. Observe, in particular, the fit for t=12.589. The program does nothing more than evaluate the Prony series form for the normalized shear relaxation modulus in terms of time:


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W3.5

N g R (t ) 1

i 1

gi (1 e

t i . )

5. Enter a negative value of t to continue with the program, and write out the normalized shear relaxation modulus for 30 intervals of time in the range 1E-2 seconds to 1E4 seconds. Write the data to the file n1.cur. 6. You can now compare the results of the Abaqus model with N=1 and the experimental data using Abaqus/Viewer. a. In the Results Tree, double-click XYData. b. Select ASCII file as the source, and click Continue. c. Next to the File field, click Select to browse for the file n1.cur. The Xvalues (time) should be read from field 1, and the Y-values (shear relaxation modulus) should be read from field 2. Note: The order of the data pairs in n1.cur (time then shear relaxation modulus) is the opposite of the data pair order in bis_cur.inp.

d. Set the X-axis quantity type to Time and the Y-axis quantity type to Stress. e. In the lower left corner of the dialog box, click Save As. Name the X-Y data N1, and click OK. f. In the XY Data From ASCII File dialog box, click Cancel. g. Expand the XYData container of the Results Tree, select both BIS and N1, and click mouse button 3. From the menu that appears, select Plot. The output should look like the graph below:

Figure W3–2 Comparison of Abaqus model with N=1 to experimental data. © Dassault Systèmes, 2009


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W3.6

The plot seems to indicate that the fit is good for the first four data points, but after that the fit is poor. However, the logarithmic scale for the Y-axis (which represents the stress in a relaxation test) tends to exaggerate the misfit for very small values of gR at large times and understate the misfit at large values of gR. Question W3–2: How much should the total stress change as a fraction of the

initial stress after t=100 in a relaxation test? 7. Try making the Y-axis linearly spaced, and assess the fit again. a. Double-click the Y-axis. b. In the Scale tabbed page of the Axis Options dialog box, choose the Linear scale type. Click OK. The fit is not very good even for the second data point. 8. Modify relax_data1.inp to remove the NMAX restriction (a default restriction of 13 points will be applied). Run another datacheck analysis. Question W3–3: Does this procedure find a satisfactory set of Prony

parameters? Abaqus tries N = 1 and N = 2 before going to N = 3. 9. Run the program prony_prog again, enter the data for N = 2, and write the data to a file named n2.cur. Then run prony_prog with the data for N = 3; write the data to a file named n3.cur.


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W3.7

10. Now read curves N2 and N3 into Abaqus/Viewer, and display all the curves again.

Figure W3–3 Comparison of Abaqus models with N = 1, 2, and 3 (log scale for normalized modulus). The long-term behavior is somewhat off for all of the Prony series fits. We can specify the long-term modulus exactly by using the SHRINF parameter on the SHEAR TEST DATA option, although this will not necessarily produce a better overall fit. Question W3–4: Is the curve for N = 3 really better than the curve for N = 2? Is

the fit for N = 3 good enough after t = 100? (View it with a linear Y-axis scale.) We had a crude set of data points for the original curve. The curve-fitting procedure for the Prony parameters only fits to the given data points; i.e., those marked by a circle in the graph. 11. Try to obtain a better fit by tightening the tolerance with the ERRTOL parameter (try ERRTOL = 0.001). The parameters for N = 7 to N = 13 are not listed because they contain negative moduli. Here is a graph of the various fits with a linear Y-axis scale:



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W3.8

Figure W3–4 Comparison of Abaqus models with N = 1, 2, and 3 (linear scale for normalized modulus). Problem 3: Simulating a relaxation test

1. Copy the file relax_data1.inp to relax1.inp, and modify the file so that the first step puts the element into simple shear with a positive shear strain of .01 (move only nodes 3 and 4). Now add a VISCO step with an initial time increment of 1E-2 and a total time of 1000. Use a CETOL parameter of 0.05E-2 to allow automatic time incrementation. (This value for CETOL was arrived at as follows: the total creep strain when the material relaxes fully will be .01, which is


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W3.9

equal to the instantaneously applied shear strain. We can force at least 20 increments by taking CETOL to be .01/20.) Finally, on the *VISCOELASTIC option set NMAX=1, and adjust the ERRTOL parameter in such a way that this single Prony series parameter will be accepted (e.g., set to 0.1). The shear modulus of 100 was chosen so that the shear stress in this test will coincide with the normalized shear relaxation modulus:

G (t ) We have contrived G0 gR ( t ) as .

0

0

G0 g

R

(t )

0

.

= 1 so that we can read the normalized shear modulus

2. Run relax1. 3. When the job completes, create a plot of the shear stress S12 in element 1 versus the step time, using data only from the second step. a. In the Results Tree, expand the History Output container underneath the output database named relax1.odb. b. From the list of output variables, select Stress components: S12. c. Click mouse button 3; from the menu that appears, select Save As. Name the X-Y data S1 and click OK. d. In the Results Tree, expand the XYData container, and double-click S1 to plot the curve. 4. Copy relax1.inp to relax3.inp, and remove the NMAX and ERRTOL parameters from *VISCOELASTIC. Run relax3. 5. Repeat the procedure given above to define curve S3 using the data in relax3.odb. 6. Plot curves N1 and S1 together (select both in the XYData container of the Results Tree, click mouse button 3, then select Plot from the menu that appears). Question W3–5: Are the results from the viscoelasticity analysis true to the

Prony series defined for it? 7. Similarly, compare curve S3 with curve N3. The data points for curve S3 can be viewed by using the XY Curve Options in Abaqus/Viewer. The integration of the viscoelastic equations is very accurate.



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W3.10

Problem 4: Simulating the effects of temperature-dependent relaxation data 1. Copy the file relax3.inp to relax_hot.inp. You will use this file to model a relaxation test under the following conditions: the relaxation data are assumed to be applicable to 20 C, the relaxation test is carried out at 70 C, the material has the time-temperature transformation properties of the rubber material referred to in Lecture 10, and the time constants for the WLF shift function have already been worked out to refer to 20 C ( 0 = 20, C1=6.106, C2=146.6). The input file must be modified to include the TRS option (after the SHEAR TEST DATA option/data), and the required constants must be added. In addition, assign an initial temperature of 20 C to all nodes with the INITIAL CONDITIONS, TYPE=TEMPERATURE option, and raise the temperature to 70 C in the static step. The VISCO step is as before, except that you will have to specify a minimum time increment of 1.E 10 to prevent premature termination of the analysis due to small time increment sizes. 2. Run the job, and read the shear stress into the curve S-HOT. Plot this against the curve of your choice. Question W3–6: What do you notice? What is the value of the time shift? Does

it correspond to the calculated value in the notes? Figure W3–5 shows the normalized relaxation moduli versus time.

Figure W3–5 Effect of temperature on viscoelastic response.


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W3.11

Problem 5: Determining the effects of time integration in a relaxation analysis Copy the file relax3.inp to relax_coarse.inp, and modify the new file so that the CETOL parameter is so large it will never restrict the time step. Set the initial time increment to 100 seconds, and run the analysis. Now read the shear relaxation modulus into curve S3-COR, and compare it to curve N3 or S3. The viscoelastic equations were integrated very well despite the coarseness of the integration and the complexity of the relaxation response.

Question W3–7:

Why did the integration procedure produce good results with such large time increments?

Problem 6: Determining the effects of time integration in a creep analysis 1. Copy the file relax3.inp to creep.inp, and modify this file so that it conducts a creep test rather than a relaxation test; that is, it subjects an element to constant shear stress. Subject the element to a constant shear stress of 0.01, using an appropriate CLOAD, and use CETOL=.05E-4. Use the same time limits as before. 2. Copy creep.inp to creep_coarse.inp. Modify this file so that the time integration is very coarse. (Use a very large CETOL, and specify an initial time increment of 100.) 3. View the shear strain response (E12) in Abaqus/Viewer, and compare the results of the two analyses. You will see small differences in the strain response at the output times because the total strain no longer varies linearly with time. Nevertheless, the creep integration is still quite good. Your results should produce curves similar to the ones shown in Figure W3–6 (with both scales linear).



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W3.12

Figure W3–6 Effect of time integration on viscoelastic response in Abaqus.

Note: Complete input files are available for your convenience. You may consult these files if you encounter difficulties following the instructions outlined here or if you wish to check your work. The input files are named relax_data1.inp

relax_hot.inp

relax_data3.inp

relax_coarse.inp

relax_data_err.inp

creep.inp

relax1.inp

creep_coarse.inp

relax3.inp

and are available using the Abaqus fetch utility.


402


Notes

403

Notes

404

Workshop Answers 3 Time Domain Viscoelasticity Keywords Version Question W3–1: What is wrong with the curve? Answer:

It has linear scales. It should have logarithmic scales.

Question W3–2: How much should the total stress change as a fraction of the initial

stress after t=100 in a relaxation test?

Answer:

The normalized relaxation modulus is reduced to 0.0079 % at 100 seconds. Therefore, the stress in a relaxation test should also be 0.0079 % of the instantaneous value after 100 seconds.

Question W3–3: Does this procedure find a satisfactory set of Prony parameters?

Abaqus tries N=1 and N=2 before going to N=3. Answer:

Yes, a reasonably satisfactory set of parameters is calculated.

Question W3–4: Is the curve for N=3 really better than the curve for N=2? Is the fit

for N=3 good enough after t=100? (View it with a linear Y-axis scale.)

Answer:

The distinction between the N=3 curve and the N=2 curve can be misleading on the logarithmic scale. When viewed on a linear scale, the differences appear to be very minor.

Question W3–5: Are the results from the viscoelasticity analysis true to the Prony

series defined for it? Answer:

Yes.



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WA3.2

Question W3–6: What do you notice? What is the value of the time shift? Does it

correspond to the calculated value in the notes?

Answer:

The relaxation modulus at the elevated temperature is much smaller than the value at 20 C. The time shift is 0.03, which corresponds to the value calculated in the lecture on Time-Temperature correspondence. You can verify this using the Operate on XY data Abaqus/Viewer functionality (Results Tree: double-click XYData). Use the following expression to multiply the time data of the S3 curve by 0.03: swap(swap("S3")*0.03). The resulting curve is very similar to S-HOT, as expected.

Question W3–7: Why did the integration procedure produce good results with such

large time increments? Answer:

The viscoelastic equations are integrated exactly when the total strain varies linearly over an increment. In this case the strain is constant. Thus a large CETOL will not affect the accuracy of a simple stress-relaxation problem.


406


Notes

407

Notes

408

Modeling+Rubber+and+Viscoelasticity+with+Abaqus-

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