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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)
DataGent ModelTwo - Term Model
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)
DataGent ModelTwo - Term Model
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)
DataGent ModelTwo - Term Model
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)
DataGent ModelTwo - Term Model
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)
DataGent ModelTwo - Term Model
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
DataGent ModelTwo - Term Model
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
Recommended