Nonlinear Analysis:
Hyperelastic Material Analysis
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Section 3 – Nonlinear Analysis
Objectives
Module 3 – Hyperelastic Materials
Page 2

The objectives of this module are to:

Provide an introduction to the theory and methods used to
perform analyses using hyperelastic materials.

Discuss the characteristics and limitations of hyperelastic material
models.

Develop the finite deformation quantities used to define the strain
energy density functions of hyperelastic materials.

Learn from an example: The deformation in an o-ring subjected to
a pressure shows how the theory relates to the input data required
by Autodesk Simulation Multiphysics.
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Hyperelastic Materials:
Applications

Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 3
Hyperelastic material models
are used with materials that
respond elastically when
subjected to large deformations.
Elastomers

Some of the most common
applications to model are:
(i) the rubbery behavior of a
polymeric material
 (ii) polymeric foams that can be
subjected to large reversible shape
changes (e.g. a sponge)
 (iii) biological materials

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Biological
Materials
Foams
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Hyperelastic Models:
Definition
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 4
A hyperelastic material model
derives its stress-strain relationship
from a strain-energy density
function.
Hyperelastic material models are
non-linearly elastic, isotropic, and
strain-rate independent.
Many polymers are nearly
incompressible over small to
moderate stretch values.
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Nonlinear response of a
typical polymer
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Hyperelastic Models:
Variety of Models
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 5
• Each material model contains
constants that must be determined
experimentally.
Autodesk Simulation
Hyperelastic Material Models
• Which material model to use
depends on which one best
matches the behavior of the
material in the stretch range of
interest.
• A good discussion of material tests
needed to define hyperelastic
material parameters may be found
at www.axelproducts.com
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•Neo-Hookian
•Mooney-Rivlin
•Ogden
•Yeoh
•Arruda-Boyce
•Vander Waals
•Blatz - Ko
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Hyperelastic Models:
Typical Applications
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 6
 Deciding which hyperelastic material
model to use is not easy.
 Each model contains coefficients which
must be determined by fitting the model
to experimental data.
 The “best” model is the one that best
matches the experimental data over the
stretch range of interest.
 Multiaxial tests are generally required to
obtain a good match between a
particular material model and the
experimental data.
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Strain Invariant Based Models
Neo-Hookean
Mooney-Rivlin
Yeoh
Arruda-Boyce
polymers,
moderate
stretch levels
Mooney-Rivlin is a commonly
used model for polymers.
Blatz-Ko
polyurethane
foams
Stretch Based Models
Ogden
Vander Waals
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High stretch
levels
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Hyperelastic Models:
Limitations
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 7
• Hyperelastic models are
reversible meaning that there is
no difference between load and
unload response.
• Hyperelastic models assume
stable behavior (i.e. there is no
difference in response between
the first and any other load
event).
• They are perfectly elastic and do
not develop a residual strain.
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Unload
Load
From: Dorfmann A., Ogden R.W., “A Constitutive
Model for the Mullins Effect with Permanent Set in
Particle-Filled Rubber”, Int. J. Solids Structures,
41, 1855-1878, 3004.
The Mullins Effect is a type of
response not covered by
hyperelastic material models.
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Hyperelastic Models:
No Viscous Effects
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 8
 Hyperelastic Models are strain-rate
independent (i.e. it doesn’t matter
how fast or slow the load is applied).
0.009
0.008
0.007
0.006
Stress
 In addition to the Mullins Effect,
creep, relaxation, and losses due to a
sinusoidal input cannot be modeled
using a hyperelastic material model.
Relaxation Curves for a Linear
Viscoelastic Material
0.005
0.004
2 times
0.002
 Viscoelastic material models covered
in Module 4 of this Section can be
used to capture some of these
phenomena.
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2 times
0.003
2 times
2 times
0.001
0
0
1
2
3
4
5
6
Tim e, sec.
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Hyperelastic Models:
Strain Energy Density Functions
• The stress-strain relationship
for a hyperelastic material is
derived from a strain-energy
density function, W.
• The strain-energy density
functions for hyperelastic
materials are defined in terms
of finite deformation quantities
(i.e. Green’s strain, invariants of
the Cauchy-Green deformation
tensor, or principal stretch
ratios).
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Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 9
Stresses are determined
from the derivatives of the
strain-energy density
functions
W
W
Sij 
2
Eij
Cij
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Section 3 – Nonlinear Analysis
Finite Deformation Theory:
Material Configurations
Module 3 – Hyperelastic Materials
Page 10
 Consider an arbitrary line
element defined by points P &
Q in the undeformed
configuration.
Q*
P*
y,y*
The same points are defined
by P* and Q* in the deformed
configuration.
f,g & h are functions that
define the relationship
between coordinates in the
deformed and undeformed
configurations.
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Deformed
Configuration
Q
Undeformed
Configuration
P
x,x*
z,z*
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x *  f ( x, y , z )
y *  g ( x, y , z )
z *  h ( x, y , z )
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Section 3 – Nonlinear Analysis
Finite Deformation Theory:
Deformation Gradient
Module 3 – Hyperelastic Materials
Page 11
The differential changes in the
coordinates of the deformed and
undeformed configurations are:
f
f
f
dx* 
dx 
dy 
dz
x
y
z
g
g
g
dy* 
dx 
dy 
dz
x
y
z
h
h
h
dz* 
dx 
dy 
dz
x
y
z
The deformation gradient
is defined as
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 f
 x
dx * 

  g
dy *  
 dz *  x

 h

 x
 f
 x

g

F   
x
 h

 x
f 
z  dx
g   
dy

z  
h   dz

z 
f
y
g
y
h
y
f
y
g
y
h
y
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dx*  F dx
f 
z 
g 
z 
h 

z 
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Finite Deformation Theory:
Mapping Functions
The displacements u,v and w in the
x, y and z directions can be used to
determine the mapping functions
f, g and h.
x*  x  u ( x, y, z )
y*  y  v( x, y, z )
z*  z  w( x, y, z )
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f  x  u ( x, y , z )
 g  y  v ( x, y , z )
h  z  w( x, y, z )
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Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 12
Using these functions,
the deformation
gradient becomes
Deformation Gradient
 u
u
u 
1  x


y

z


v
v
v 

F   
1
x
y
z 
 w
w
w 

1 
y
z 
 x
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Section 3 – Nonlinear Analysis
Finite Deformation Theory:
Stretch Tensor
The Deformation Gradient can be
broken down into a product of
two matrices.
Module 3 – Hyperelastic Materials
Page 13
U 
F   RU   V R
The matrix [R] is an orthogonal
rotation matrix, and [U] and [V]
are symmetric matrices that are
called the right and left stretch
tensors.
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V 
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The Right Stretch Tensor
because it appears on the
right of the rotation
matrix.
The Left Stretch Tensor
because it appears on the
left of the rotation matrix.
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Finite Deformation Theory:
Right Cauchy-Green Deformation Tensor
The change in length squared of
the line element PQ in the
deformed configuration is
  dx  dx F  F dx
* T
dS  dx
2
T
*
T
dS  dx Cdx
2
T
Where [C] is the right CauchyGreen deformation tensor given
by
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 14
As shown below, the right
Cauchy-Green deformation
tensor is equal to the
square of the right stretch
tensor.
C   U  R RU 
T
T
C   U  U   U 
T
2
C  F  F 
T
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Finite Deformation Theory:
Principal Stretch Ratios
The principal stretch ratios could be
found by extracting the eigenvalues
of [U] or [V].
This is typically not done since it
would require that the rotation
matrix [R] be found.
Instead it is more customary to find
the square of the principal stretch
ratios by extracting the eigenvalues
of the Cauchy-Green deformation
tensor.
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Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 15
Principal Stretch Ratios
1 , 2 , 3
Equation used to define
[U] and [V]
F   RU   V R
Equation used to find the
square of the principal
stretch ratios.
C  U  U   U 
T
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Section 3 – Nonlinear Analysis
Finite Deformation Theory:
Stretch Ratio Invariants
Module 3 – Hyperelastic Materials
Page 16
The square of the principal stretch
ratios can be found from the
equation

 C11  2

det  C12
 C13


C
C12

2
22

C23


C23   0
C33  2 
C13

The coefficients of the
characteristic equation are
invariants of [C] and can be
written in terms of the
principal stretch ratios as

which results in the characteristic
equation
   I  
2 3
2 2
1
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 
 I 2   I3  0
2
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Principal Stretch Invariants
I1      
2
1
2
2
2
3
I2         
2 2
1 2
2 2
2 3
2 2
3 1
I3    
2 2 2
1 2 3
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Section 3 – Nonlinear Analysis
Finite Deformation Theory:
Volumetric Strain
Module 3 – Hyperelastic Materials
Page 17
Elastomers are nearly incompressible while
undergoing moderate stretch (i.e. there is
no volume change).
Deformed Volume
V  dx*  dy*  dz*
Original Volume
Vo  dx  dy  dz
Cube of material
in the deformed
configuration
dy
dz
Volume Ratio
dx
dV
dx  dy  dz
 *
 x  y z  123
*
*
dVo dx  dy  dz
Incompressible material constraint
1  1  2  3       
2
Incompressible
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dV
 1  1  2  3 
dVo
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2
1
2
2
2
3
 I 3  det( F )
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Section 3 – Nonlinear Analysis
Finite Deformation Theory:
Multiplicative Decomposition of F
Module 3 – Hyperelastic Materials
Page 18
The principal stretch invariants can be used to
describe the strain energy functions of
materials that are incompressible.
~
F  Fvol  F
1
3
Fvol  J I
For materials that are nearly incompressible,
it is necessary to define volumetric and
deviatoric portions of the deformation
gradient.

1
3
~
FJ F
Volumetric
Contribution
Deviatoric
Contribution
The determinant of F is equal to the ratio of
the deformed and undeformed
configurations.
R.J. Flory, Thermodynamic Relations for High Elastic
Materials, Trans. Faraday Soc., 57, (1961) 829-838.
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Section 3 – Nonlinear Analysis
Finite Deformation Theory:
Incompressibility Constraint
The determinant of F is equal to the
ratio of the deformed and
undeformed configurations.
The determinant of the volumetric
contribution is equal to one.
The determinant of the deviatoric
contribution is equal determinant of
the deformation gradient, J.
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Module 3 – Hyperelastic Materials
Page 19
V
~
 det F  det Fvol  det F
V0
det Fvol
 J 13

 0

 0
0
J
1
0
3
0 
3

1
   J 3   1


1 
J 3

~
det F  det F  J
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Section 3 – Nonlinear Analysis
Finite Deformation Theory:
Relationships
Module 3 – Hyperelastic Materials
Page 20
~
~ ~
The principal invariants I1 , I 2 and I 3of the deviatoric Cauchy~ ~T ~
Green deformation tensor C  F F are related to the principal
invariants I1, I2 and I3 of the Cauchy-Green deformation tensor C
by the following relationships.
~
I1  J 1 3 I1
~
I 2  J 4 3 I 2
~
I3  1
~ ~
~
Similarly, the deviatoric principal stretches 1 , 2 and 3 are related
to the principal stretches 1, 2, and 3 by the equations:
~
1  J 1 31
~
2  J 1 32
~
3  J
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1 3
3
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I. Doghri, Mechanics of Deformable Solids:
Linear, Nonlinear, Analytical and
Computational Aspects, Springer-Verlag,
2000, p. 374.
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Hyperelastic Material Models:
Rivlin Polynomial Model
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 21
A general and widely used form of strain energy density function,
W, was proposed by Rivlin

m
i ~
j
~
2m
W   Cij I1  3 I 2  3   Dm  J  1

i , j 0


M 1
where Cij and Dm are material constants.
The left hand term controls the distortional response of the
material while the right hand term controls the volumetric
response.
Note the left hand term is written in terms of the principal
invariants of the deviatoric portion of the Cauchy-Green
deformation tensor.
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Hyperelastic Material Models:
Rivlin Polynomial Model
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 22
Several hyperelastic material models in Autodesk Simulation
Multiphysics are obtained from the Rivlin polynomial model by
selecting different values for i, j and M. Examples are shown below.
Neo-Hookian
Mooney-Rivlin
i=1, j=0 and M=1

i=1, j=0 and i=0, j=1 and M=1


~
W  C10 I1  3  D1  J  1



~
~
W  C10 I1  3  C01 I 2  3  D1  J  1
Yeoh
i=1,2 3, j=0 and M=1






2
3
~
~
~
W  C10 I1  3  C20 I1  3  C30 I1  3  D1  J  1
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Hyperelastic Material Models:
Common Mooney-Rivlin Constants
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 23
The constants C10 and C01 are
related to the instantaneous shear
modulus for a two parameter
Mooney-Rivlin model.
G0  2C10  C01 
Two Parameter MooneyRivlin Model
i=1, j=0 and i=0, j=1 and M=1




~
~
W  C10 I1  3  C01 I 2  3  D1  J  1
The constant D1 is related to the
instantaneous bulk modulus
0  2 D
1
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Hyperelastic Material Models:
Effects of Changing Constants
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 24
 For small strains, the shear
modulus and Young’s modulus are
related by the relationship
E
G
or
21   
E  2G 1  
 If the material is incompressible,
=0.5, and the above relationships
become
E
G
3
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E  3G
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 In terms of the MooneyRivlin constants,
G0  2C10  C01 
and
E0  6C10  C01 
These expressions show that
increasing either C10 or C01 will
increase the stiffness of the
material since G and E will
increase.
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Hyperelastic Material Models:
Mooney Plot
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 25
The relationship between the
stress and stretch for a uniaxial
test specimen can be written as
1 
C2 

  2   2  C1  
 
 

Plot of uniaxial stress-stretch data for a
two parameter incompressible MooneyRivlin Model

1

2   2 
 

This equation can be rewritten as

1

2   2 
 

 C1 
C2

o
o
o
o
o Slope  C2
C1
1

which is the equation for a straight
C10  C1
line.
C01  C2
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o
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Example Problem:
Problem Definition
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 26
The objective is to determine
the deformation in a rubber oring used to prevent leakage of
a 500 psi fluid between the
Rubber O-ring
housing and shaft.
Cut-away View
Close-up View
Shaft
Housing
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Example Problem:
Wedge Geometry
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 27
 Since nothing will vary around
the circumferential direction an
axisymmetric analysis will be
performed.
 This will reduce the size of the
problem without losing any of the
desired information.
 A 5-degree wedged shape
portion of the model is created in
Autodesk Inventor.
 This wedge will be used in
Autodesk Simulation
Multiphysics to create the
axisymmetric model.
© 2011 Autodesk
Y-Z Plane
An axisymmetric model requires that
the mesh be on the y-z plane (y is the
radial direction).
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Example Problem:
Analysis Type
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 28
 The analysis type is set to
Static Stress with Nonlinear
Material Models.
 This analysis type will allow the
selection of one of the
hyperelastic material models
when the element data is
defined.
 The analysis type can be
selected at startup or changed
by editing the Analysis Type in
the FEA Editor.
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Example Problem:
Axisymmetric Coordinate System
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 29
 When an axisymmetric analysis
is performed, the axis of
symmetry of the part must be
the z-axis as shown in the
figure.
Coordinate System Required
for an Axisymmetric Analysis
 The radial direction corresponds
to the +y direction.
 The radius is equal to zero when
y is equal to zero.
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Example Problem:
2D Axisymmetric Mesh
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 30
 There are several steps
involved in creating this
mesh and the details are
contained in the first of
the two videos for this
module.
 A 500 psi pressure load is
applied to one face of the
o-ring and the model is
constrained in the zdirection.
© 2011 Autodesk
Axisymmetric 2D Mesh
Pressure Load
Z-displacement constraint
The y-direction does not have to be
constrained because the circumferential
strain will limit the motion in this direction.
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Example Problem:
Element Type
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 31
 The 2D element type is
selected.
 2D element types can
be used to model plane
stress, plane strain, and
axisymmetric
problems.
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Example Problem:
Element Definition Data
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 32
 The Mooney-Rivlin
hyperelastic material
model is selected for the
O-ring material and
isotropic linear elastic
materials are selected for
the housing and shaft
parts.
 All parts use the
axisymmetric geometry
type.
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Example Problem:
Hyperelastic Material Definition
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 33
 A 2-constant
standard MooneyRivlin material is
selected. This
material is good for
moderate stretch
levels.
 The First Constant
(C10) and Second
Constant (C01)
material coefficients
are taken from the
Simulation library.
© 2011 Autodesk
Strain Energy Density Function




~
~
2
W  C10 I1  3  C01 I 2  3  2  J  1
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Example Problem:
Bulk Modulus (Incompressible)
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 34
The bulk modulus can
be related to the
shear modulus and
Poisson’s ratio
through the equation
2G 1  

31  2 
For an
incompressible
material =0.5, and
the bulk modulus is
infinite.
© 2011 Autodesk
Strain Energy Density Function




~
~
2
W  C10 I1  3  C01 I 2  3  2  J  1
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Example Problem:
Bulk Modulus (Nearly Incompressible)
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 35
The bulk modulus for a nearly incompressible MooneyRivlin material can be approximated using the following
procedure.
2G 1  
G
 is taken to be 0.5 in the numerator.


31  2  1  2
The shear modulus for a Mooney-Rivlin material is given by
G  2C10  C01 
2C10  C01 

1  2
© 2011 Autodesk
For C10 = 297 psi, C01=172 psi and =0.499, the bulk
modulus is computed to be 496,000 psi.
Small changes in  can cause large changes in 
i.e.   0.4988 400,000 psi.
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Example Problem:
Analysis Parameters
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 36
 The event duration is set to 1 second.
 Time is used only as an interpolation
parameter to determine the
percentage of load being applied.
 The pressure will be applied in 1000
time steps or load increments.
Hyperelastic materials are very
nonlinear and small time steps are
required to achieve converged
solutions.
 The load curve will increase the
pressure linearly from 0 to 1 seconds.
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Example Problem:
Analysis Results
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 37
 The von Mises stress is invariant to
hydrostatic stress states. This can be
seen in the figure.
Von Mises stress superimposed on the
deformed shape at 500 psi pressure
 The top half of the o-ring has
retained its circular shape due to the
contact constraints and the pressure
acting on the surface. In this
distortion free area the von Mises
stress is very low.
 The bottom half of the o-ring is not
constrained by the pressure and
distorts as it is squeezed into the
gland. This distorted area has larger
von Mises stresses.
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Example Problem:
Analysis Results
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 38
•
The shear stress in the upper half of
the o-ring is nearly zero. This is due
to the volumetric response in this
area. There is little contribution
from the deviatoric portion of the
constitutive equation.
•
The shear stress is greater in the
lower half where there is more
distortion. The deviatoric portion of
the constitutive equation is playing
a larger role in this area.
Shear Stress, yz , superimposed on the
deformed shape at 500 psi
Strain Energy Density Function




~
~
2
W  C10 I1  3  C01 I 2  3  2  J  1
Deviatoric
© 2011 Autodesk
Volumetric
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Example Problem:
Analysis Results
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 39
• The von Mises strain is also
invariant to hydrostatic
loading as seen in the figure.
Von Mises Strain superimposed on
deformed shape at 500 psi.
• The upper half of the o-ring
has very small strains.
• The strains in the lower half
where the distortion is taking
place has a max von Mises
strain of 40%.
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Summary
Section 3 – Nonlinear Analysis
Module 3 – Hyperelastic Materials
Page 40
 This module has provided an introduction to hyperelastic
materials and has demonstrated how to perform an analysis with
Autodesk Simulation Multiphysics using these material models.
 Hyperelastic materials are used to compute the large
deformation response of nonlinear elastic materials (i.e.
polymers, foams, and biological materials).
 The stress-strain response of hyperelastic materials are defined
by strain energy density functions expressed in terms of finite
deformation variables.
 Autodesk Simulation Multiphysics provides a library of
hyperelastic material models capable of modeling the behavior
of materials over a wide range of stretch.
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