Mechanical-Nonlin 13.0 App7A Hyper

8/12/2019 Mechanical-Nonlin 13.0 App7A Hyper

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Customer Training Material

Appendix 7A

Hyperelasticity

Structural Nonlinearities

A7A-1ANSYS, Inc. Proprietary

2010 ANSYS, Inc. All rights reserved.Release 13.0

December 2010


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ANSYS Mechanical Appendix to Hyperelasticity

Customer Training MaterialHyperelasticity Chapter Overview

This appendix is an optional supplement

to Chapter 7, offering a more rigorous

explanation of each particular form ofthe strain energy density function (W)

to aid the user in the selection the best

strain energy density function .

Prerequisite is Chapter 7



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Customer Training Material... Particular Forms of W

The strain energy potential (W), introduced in Chapter 7, will require

certain types of parameters input as material constants.

,

energy function Wchosen.

The choice of Wwill depend on the type of elastomer analyzed, the

loading conditions, and the amount of data available.

Some very general guidelines will be presented to aid the user in the

-

factors, no guidelines can cover 100% of situations.

From the selection of W and material constants which are input, stress

an s ra n e av or are ca cu a e y e so ver.

The next slides discuss the different forms of strain energy potential W

available in ANSYS with some comments on the selection and of their



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use.


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Customer Training Material

H

... Polynomial Form

Thepolynomial form is based on the first and second strain

invariants. It is a phenomenological model of the form

( ) ( ) ( ) ==+ +=N

k

k

k

j

N

ji

iij J

dIIcW

1

22

1

1 1133

where the initial bulk modulus and initial shear modulus are

( )01102 cco +=

1

2

do =



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Customer Training Material... Polynomial Form Guidelines

Comments on the General Polynomial Form (PF):

As noted in the figure below, more terms will be required to capture any

.

ensure that enough data is supplied with inclusion of higher-order terms.Polynomial form with N=2 or N=3 may be used up to 100-300% strains

. , .

PF is a very general form, so it can produce very good curve fits. As with

all models, data of expected modes of deformation is required when

curve-fitting. If limited (e.g., uniaxial) test data exists, consider use of

Yeoh model (see Yeoh section later).



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PF with N=1 PF with N=2 PF with N=3


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Customer Training Material... Mooney-Rivlin Model

The five-term Mooney-Rivlin model is equivalent to the polynomial

form when N=2:

( ) ( ) ( )22

2

120201110

11

333

333

+++++=

cIc

IcIcIcW

The nine-term Moone -Rivlin model can also be thou ht of as the

d

polynomial form when N=3:

2111

2

120201110 33333 +++= IIcIcIcIcW

( ) ( ) ( ) ( )

232

2

2

121

3

130

2

202

1

3333 +++ IIcIcIc



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2032112 +++

dcc


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Customer Training Material... Mooney-Rivlin Model

For all of the preceding Mooney-Rivlin forms, the initial shear and

initial bulk moduli are defined as:

cc

o

o

2

0110

=

+=



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Customer Training Material... Mooney-Rivlin Guidelines

Comments on the Mooney-Rivlin (M-R) model:

Because of its equivalence to polynomial forms (N=1, 2, 3), as discussed

, .

The 2-term Mooney-Rivlin model is most commonly used. Some very

broad rules-of-thumb are presented below.

The 2-term M-R may be valid up to 90-100% tensile strains, although it will not

account for stiffening effects of the material, usually present at larger strains.

ure s ear e av or may e c aracter ze up to - . s s ecause t e -

term M-R model exhibits a constant shear modulus.

Although moderate compression behavior can be characterized well (up to

30%), significant compression response may not be captured with only 2-term

MR.



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Customer Training Material... Yeoh Model

The Yeoh model(a.k.a. reduced polynomial form) is similar to the

polynomial form but is based on first strain invariant only.

( ) ( ) == +=N

i

i

i

N

i

i

i JdIcW 1

2

1

10 11

3

The Yeoh model is commonly considered with N=3 (a.k.a. cubic

form), although solver allows for any value of N.102co =

similar to other invariant-based models:

1

2

do =



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Customer Training Material... Yeoh Model Guidelines

One may note from the previous slides that the Yeoh model is

dependent on the first invariant I1 only.*

Yeoh ro osed omittin the second invariant term. The ustification of

this comes from the observation that changes in the strain energypotential is less sensitive to changes in the second invariant than the first

i.e. W/I >> W/I . This is es eciall true for lar er strains. Also, if only limited test data is available (e.g., uniaxial test), it has been

shown that ignoring the second invariant leads to better prediction of.

As strain increases, the shear modulus (slope) decreases slightly then

increases. To reflect this, use a cubic form (N=3):

c10 is positive, equal to half of initial shear modulus value

c20 is negative (softening at small strains), ~ 0.1 to 0.01 * c10

c30 is positive (stiffening at larger strains), ~ 1e-2 to 1e-4 * c10



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* For a detailed discussion on this topic, please refer to the papers by O.H. Yeoh, Characterization of Elastic Properties of

Carbon-Black-Filled Rubber Vulcanizates, Rubber Chem. Tech. 63, 1990 and Some Forms of the Strain Energy Function

for Rubber, Rubber Chem. Tech. 66, 1993


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Customer Training Material... Neo-Hookean Form

The neo-Hookean form can be thought of as a subset of the

polynomial form for N=1, c01=0, and c10=/2:

( ) ( )21 1132

+= Jd

IW

where the initial bulk modulus is defined as

do =



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Customer Training Material... Neo-Hookean Guidelines

The neo-Hookean model is the simplest hyperelastic model, and it

may be a good way to start.

Althou h it will robabl not redict moderate/lar e strains well for small

strain applications, it may be suitable. Similar considerations apply to the neo-Hookean model as to the 2-

erm ooney- v n mo e scusse ear er :

The neo-Hookean form may be valid up to 30-40% tensile strains, and it

will not account for stiffening effects of the material, usually present at

larger strains.

Pure shear behavior may be characterized up to 70-90%. This is because

the neo-Hookean model exhibits a constant shear modulus.

Although moderate compression behavior can be characterized well (up

to 30%), significantcompression response may not be captured with neo-

Hookean.



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Customer Training Material... Arruda-Boyce Model

The initial shear modulus is In the Arruda-Boyce paper,* the rubbery modulus (shear modulus) is defined as

, , ,temperature (). In ANSYS, = nk

The limiting network stretch L is the chain stretch at which stress startsto increase without limit.

Note that as L becomes infinite, the Arruda-Boyce form becomes the Neo-Hookean form.

Also in the paper,* the equation references the locking stretch (limiting network

stretch) as N. In ANSYS, L= N. The initial bulk modulus is defined as usual by 2/d.



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* For a detailed discussion on this topic, please refer to the paper by M.C. Boyce and E.M. Arruda, A Three-Dimensional

Constitutive Model for the Large Stretch Behavior of Rubber Elastic Materials, J. Mech. Phys. Solids, Vol 41 No 2, 1993.


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Customer Training Material... Arruda-Boyce Guidelines

A few comments on the Arruda-Boyce (A-B) model:

It is apparent that the A-B model has some similarities to the Yeoh model,

although the coefficients are fixed, predefined functions of the limiting

network stretchL

.

This means that discussion of the Yeoh model and consideration of I1-

de endenc are a licable here as well.

From a physical standpoint, the use of I1 only means that the eight chains are

equally stretched under any deformation state, i.e., I1= 12+ 22+ 32 represents.

Additional usefulness of the Arruda-Boyce model stem from the fact that the

material behavior can be characterized well even with limited test data (uniaxial

es , an ewer ma er a parame ers are requ re . owever, s s a xeformulation, which may limit its applicability for any material.

Generally speaking, suited for large strain ranges.



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No stress softening but only stress stiffening with increasing strain.


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Customer Training Material... Arruda-Boyce Guidelines

More comments on the Arruda-Boyce (A-B) model:

The A-B equation used here is actually the first five terms of the strain

.

contains an inverse Langevin function. This equation needs to beconverted to a series expansion and numerically integrated to get W (i.e.,

.

Because only the first five terms of W are commonly used, this may cause the

limiting network stretch to be slightly less pronounced.

This does not invalidate the model but is simply mentioned in case one does

an academic exercise to stretch a model near the limiting stretch value. The

stress will rise dramaticall but will not ex erience an limitin stretch value.

It is important to note that, in this discussion, one is referring to the chain

stretch, which is defined as:1



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33 321chain =++=


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Customer Training Material... Gent Model

The Gent modelis a micromechanical model, similar to Arruda-

Boyce, which also utilizes the concept of limiting network stretch:

2

where the constants , Jm, and d are input. is the initial shear

+

= J

-

dJW

m

m ln

2

1ln

2

1

modulus. Jm is the limiting value of (I1-3), analogous to L for Arruda-Boyce.

In Gents paper,* the tensile modulus E is defined instead, with = E/3.

Jm is the limiting value of (I1-3) where stresses become infinitely large.

o e a as m ecomes n n e, e en mo e approac es e eo- oo eanmodel.

The initial bulk modulus is defined as usual by 2/d.



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* For a detailed discussion on the model, please refer to A. N. Gent, A New Constitutive Relation for

Rubber, Rubber Chem. Tech. 69, 1996.


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Customer Training Material... Gent Guidelines

Some comments on the Gent model:

If a series expansion of the natural logarithm is performed, the resulting

. , ,

arepredefined functions of Jm.

It is quite clear that there are many similarities between the Gent and A-B

models.*

Jm is the limiting value of (I1-3) in Gent, analogous to L being the limiting valueof chain stretch for A-B.

As stated in Gents paper, the value of Jm should be on the order of 100.

Because of the fact that Gents strain energy function is used exactly, Jm

is the limiting value of (I1-3) where stresses will increase without bounds.

Like the Yeoh and Arruda-Boyce models, the Gent model is applicable for



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.* For a detailed discussion on the comparison of the two models, see M.C. Boyce, Direct Comparison of the Gent and the

Arruda-Boyce Constitutive Models of Rubber Elasticity, Rubber Chem. Tech. 69, 1996


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Customer Training Material... Ogden Model

The Ogden form, another phenomenological model, is directly based

on the principal stretch ratios rather than the strain invariants:

( ) ( ) == +++= ii

ii i

i

JdW iii

1

2

3211

13

where the initial bulk and shear moduli are defined as

N

1

1 2

2 d

oi

o == =

The model is equivalent to the (two-term) Mooney-Rivlin form ifN=2 1=2c10 1=2 2=-2c01 2=-2



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N=1 1= 1=2


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Customer Training Material... Ogden Guidelines

Some comments on the Ogden model:

Since Ogden is based on principal stretch ratios directly, it may be more

accurate and often provides better curve fitting of data. However, it may

also be a little more computationally expensive.

Note that if limited test data exists and multiple modes of deformation are

ex ected curve-fittin uniaxial data onl ma not ield realistic behavior

in other modes.

Ogden noted that a minimum of three terms should be used.*

1.0 < 1 < 2.0 and 1 > 0 The second term represents stiffening at larger strains:

2 . 2 2 1 The third term represents behavior in compression:

3 < -0.5 and 3 < 0 with 3


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Customer Training Material... Ogden Guidelines

Yeoh added some additional insights on the Ogden constants by

examining the Ogden equation:*

In shear, when |i| > 2.0, material stiffens with increasing strain. Conversely,i . , .insensitive to sign of i)

When i is negative, it has a large contribution to compressive behavior butsmall contribution to tensile behavior. For ositive small values of thecompression behavior is insensitive to i and behaves like neo-Hookeanmaterial (1=2).

-

1.2 < 1 < 1.6 and 1 > 0 (small-strain behavior)2 ~ 6.0 and 2 > 0 with 2


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Customer Training Material... Ogden Foam Model Guidelines

The Ogden compressible foam model behaves in a similar fashion to

the regular Ogden model for incompressible rubber:

i drastically.

Conversely, larger positive values of i affects tensile behavior(significant hardening)



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Customer Training Material... Blatz-Ko Model

The Blatz-Ko modelis specifically for compressible polyurethane

foam rubber with the following form:

where is the shear modulus.

+= 52

2 33

2 II

W

The bulk modulus is defined as 5/3.This implies

= 0.25.

( )

( )

213

12

+=

Note that I2 and I3 are regular (not deviatoric)

second and third strain invariants.

( )( )

25.0213

25.012

+=

a 47% volume percent polyurethane foam-type

rubber.

3=



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Customer Training Material... Blatz-Ko Model

The Blatz-Ko model can be thought of as a subset of Ogden

compressible foam model, with N=1, 1=-, 1=-2, 1=0.5.

( ) ( )

+

++

=

+

++=

15.02

32

5.022

3

2

2

2

13

2

321

JJW

( ) ( )

+ =

+++=

223

222

32

2

2

3

2

2

2

1

IIW

JW

+= 52

2 3

3

2

3

II

IW

As will be shown later, the effective Poissons ratio can also be

determined from , which again leads to the assumption of =0.25 for



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- .


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Customer Training Material... Incompressibility Considerations

The Ogden Compressible Foam and Blatz-Ko models are for

compressible foam-type rubbers. The deviatoric and volumetric

terms of strain energy are tightly coupled.

For the nearly incompressible rubber models,the volumetric term is often presented as one

( )

=

==

J

d

W

i

i

i

b

1

1

22

1

21

,

Recall that the term J is ratio of current to

original volume. Undeformed state is J=1.

= J

-J

dW

d

b ln2

11 2

3

For cases of Wb1, only d1 is usually considered (= Wb

2).

The selections of Wb and the bulk modulus value (=2/d) do not tend toa ect results much unless the model is signi icantly stretched leading tofinite volume change) or highly confined.

For the full incom ressible case with d=0 this volumetric term W is



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ignored (J=1, volume preserved).


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Customer Training Material... Poissons Ratio

For nearly- or fully-incompressible materials, the material

compressibility parameter dcan be estimated as follows:

The initial bulk modulus can be estimated

and written in terms of the initial shear

modulus

The material compressibility parameter is( )E

12

=

+=

proportional to the inverse of the initial bulk

modulus

The initial bulk modulus is provided in the

( )

( )

213

12

+=

previous slides for each of the hyperelastic

materials

The material compressibility parameter can

( )

2

0.5for21

therefore be written in terms of the initialbulk modulus as shown on the right,

assuming nearly-or fully-incompressible ( )

o

212

=



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e av or. o


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Customer Training Material... Poissons Ratio

The material compressibility parameter dis not present in

compressible models since the volumetric term is coupled:

,

assumed to be =0.25 For the compressible Ogden model, Poissons ratio can be calculated as

follows, assuming i is constant ():= E

N

ii ( ) 1212 +=+

( )

213 =

E

+=

=

=

1

2

1

N

iiio

io

( ) ( )( )

362

31211

=++

+=+

( )

213

+=

+=

=

3

12

1

o

o

i

21

=

+=



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Customer Training MaterialReferences for Material Data Input

ANSYS Online Help References:

1. ANSYS Elements Reference, Section 2.5.2 Hyperelastic Material

2. ANSYS Structural Analysis Guide, Section 8.3.1.3 Hyperelasticity

3. ANSYS Inc. Theor Reference Section 4.7 H erelasticit



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Mechanical-Nonlin 13.0 App7A Hyper

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