A New I1-Based Hyperelastic Model for Rubber Elastic Materials

A New I1-Based Hyperelastic Model for Rubber Elastic Materials

Oscar Lopez-Pamies

October 12-14, 2011 · Evanston, ILSES 2011

2 2 21 1 2 3 1 2 3

3 3 12 2

+ otherwise

I ifW

m ml l l l l l

ìïï - = + + - =ï= íï ¥ïïî

( ) ( )

1. Mathematical simplicity (amenable to analytical solutions for fundamental BVP and implementation in FEM codes)

2. Physical interpretation (Gaussian statistics )

3. Good agreement with experiments (for some elastomers and up to reasonably large deformations)

• Ever since its derivation in the 1940’s, there have been numerous (more than 40!!!) “refinements” of the Neo-Hookean model (*). Many of them do not conform with features 1, 2, and/or 3

Vahapoglu & Karadeniz (2003)

(*)

Neo-Hookean model

• A more appropriate basic measure of strain

I I( ; ) ( ),a

a aj a aa

-

= - Î1

1 1

33

Note: i) it generalizes the linear NH measureI( )a-

13

I I( ; ) ( )j = -1 1

1 3

• Proposed modelr

r rM

rr r

W I I( ) ( )a

a ama

-

=

= -å1

1 11

33

2

r r,m a — real-valued material parameters

M — terms to be included in the summationLopez-Pamies (2010)

Proposed constitutive model

ii) it linearizes properly (unlike )iii) it neglects I2 for mathematical simplicity

• The resulting Cauchy stress is simply given by

• And the incremental tangent modulus by

r rM

Tr

r

Wp I pT I FF I

Fa am- -

=

æ ö¶ ÷ç= - = -÷ç ÷÷ç¶ è øå 1 1

11

3

r r r rM M

r r rr r

WI IL ( ) F F+

Fa a a aa m m- - - -

= =

æ ö æ ö¶ ÷ ÷ç ç= = - Ä÷ ÷ç ç÷ ÷÷ ÷ç ç¶ è ø è øå å

21 2 1 1

1 121 1

2 3 1 3

− These closed-form quantities are needed for the solution of BVPs, and for the implementation in numerical codes such as ABAQUS.

with principal stresses

r rM

i r ir

t I p i( , , )a am l- -

=

æ ö÷ç= - =÷ç ÷÷çè øå 1 1 2

11

3 1 2 3

1. Mathematical simplicity

• Non-Gaussian statistical mechanics model of Beatty (2003)TI pT ( )FF I= Y -

1

( )1

1

1

3

3 3AB

InkT

I

h

h

-

Y =/

/

r rAB r

r

nkTW I C I( ) ( )

¥

=

= -å1 11

33

Here, contains information about the statistical distribution of the underlying polymeric chains

I( )Y1

• Example: Arruda-Boyce model

with energy

Note: By choosing and rr r

rnkTCm -= 13rra =

rr r

M

rr r

W I I( ) ( )a

a ama

-

=

= -å1

1 11

33

2

reduces to the AB model

2. Physical significance of material parameters

• The proposed energy admits the polynomial representation:i iM

ir r

i r j

W I j Ii

( ) ( ) ( )a m- -¥

= = =

é ùæ ö÷çê ú= - -÷ç ÷ê úç ÷è øë ûå å

1 1

1 11 1 1

33

2 !

( )M M

r rr

r r

W I I I O I( ) ( ) ( ) ( )m a

m= =

-= - + - + -å å 2 3

1 1 1 11 1

13 3 3

2 12

M

rr

m m=

=å1

− Thus, in the limit of small deformations it reduces to

• In the small deformation regime: it linearizes properly and has no dependence on with

• In the moderate deformation regime: polynomial dependence on

ra

• In the large deformation regime: strong dependence on provides the means to capture the typical lock-up r

a

I( )-1

3

3. Predictive capabilities

• For demonstration purposes, consider the two-term model

W I I I( ) ( ) ( )a a

a a a am ma a

- -

= - + -1 2

1 1 2 2

1 1

1 1 1 2 11 2

3 33 3

2 2

Note: the material parameters will be determined from uniaxial data only by least-square fitting and by enforcing the proper linearization condition

, , ,m a m a1 1 2 2

• For comparison purposes, we also consider the response of the Gent model

m

Gm

J IW I

J( ) ln

m é ù-ê ú= - -ê úë û

11

31

2

where and Jm are material parametersm

m m m+ =1 2

Comparisons with 3 elastomers

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8l

S un (M

Pa)

DataGent ModelTwo - Term Model

0

0.5

1

1.5

2

2.5

3

3.5

4

1 2 3 4 5 6l

S (M

Pa)

Gent ModelTwo - Term Model

Biaxial Data

Shear Data

Treloar (1944)

Uniaxial TensionBiaxial Tension

Comparisons with a vulcanized rubber

Pure Shear“Tension” l =

31

-1

-0.5

0

0.5

1

0.5 1 1.5 2l

S un (M

Pa)


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 1.2 1.4 1.6 1.8 2 2.2l

S bi (M

Pa)


Meunier et al. (2008)

Uniaxial Tension & Compression

Biaxial Tension

Comparisons with a silicone rubber

-1.6

-1.2

-0.8

-0.4

0

0.5 0.6 0.7 0.8 0.9 1l

S ps (M

Pa)


0

0.2

0.4

0.6

0.8

1

1 1.2 1.4 1.6 1.8 2 2.2l

S ps (M

Pa)


Meunier et al. (2008)

Pure ShearCompression l =

31

Pure ShearTension l =

31

Comparisons with a silicone rubber

0

0.5

1

1.5

2

2.5

3

3.5

4

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8l

S un (M

Pa)


0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

S ss(M

Pa)

g


Lahellec et al. (2004)

Uniaxial Tension Simple Shear

Comparisons with a Michelin elastomer

Constitutive restriction on andr

mr

a• Necessary and sufficient conditions for strict polyconvexity

• Necessary and sufficient conditions for strong ellipticity

W I'( )>1

0 and W I IW I'( ) ''( )+ >1 1 1

2 0

− Sufficient conditions for strict polyconvexity (and therefore for strong ellipticity) in terms of the material parameters

rm > 0 and r

a >12

r M( , , ..., )= 1 2

W I'( )>1

0 and

( )i iW I I W I'( ) ''( )l l-+ - - >2 1

1 1 12 2 0 i( , , )= 1 2 3

Cavitation instabilitiesUndeformed FE model

Some specifics:

f p -

-

= ´

» ´

90

9

/6 10

0.5 10

Cavitation ensues whenever f f= ´5010

Nakamura & Lopez-Pamies (2011)

Mesh near cavity

l3

l1

l2

64,800 8-node brick elements

Initial volume fraction of cavity:

1

5x104

1x105

0 0.5 1 1.5 2 2.5 3

m/s m

ff0

l l l= =1 2 3

FEM Resultst1 = t2 = t3

f f 50/ 10under fixed f f = 5

0/ 10

t1 > t2 > t3

Near Cavity

0.0

0.5

1.0

1.5

2.0

2.5

-3 -2

-1 0

12

3

-3

-2

-1

0 1

2

Onset-of-cavitation surface: Michelin Rubber

m

t t ts

m+ +

= 1 2 3Hydrostatic stress:

Shear stresses:t t t t

t tm m- -

= =2 1 3 11 2,

/t m2

/t m1

msm

0

0.5

1

1.5

2

-2 -1 0 1 2

t t t= =1 2

msm

/t m

Axisymmetric Loading

Nakamura & Lopez-Pamies (2011)

− Although weakly, W is expected to depend on the second invariant I2. At the expense of sacrificing mathematical simplicity it may be of interest to consider

r sr r s s

M N

r sr sr s

W I I I I( , ) ( ) ( )a b

a a b bm ua b

- -

= =

= - + -å å1 1

1 2 1 21 1

3 33 3

2 2

− Compressibility effects may be readily added in a number of different ways, e.g.,

rr r

M M

r rr rr

W I J I J J'

( , ) ( ) ln ( )a

a a mm m

a

-

= =

= - - + -å å1

21 1

1 1

33 1

2 2

− Other effects that can be added include Mullins effect, hysteresis, as well as rate and thermal effects

• The proposed incompressible, isotropic model constitutes a practical platform from which to account for more levels of complexity to model elastomers

Generalizations

• We have proposed a new constitutive model for rubber like materials that: i) is mathematically simple, ii) contains parameters that may be given a physical significance, and iii) characterizes and predicts accurately the response of a variety of elastomers.

• In addition, the proposed I1-based model permits to readily check for conditions of polyconvexity and strong ellipticity, needed to understand the development of instabilities

• In view of its functional simplicity, it is straightforward to implement it in commercial finite element packages (e.g., ABAQUS) for the study of structural problems.

Final remarks

Documents

A New I1-Based Hyperelastic Model for Rubber Elastic Materials