Modeling of Neo-Hookean
Materials using FEM
By: Robert Carson
Overview
•
•
•
•
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Introduction
Background Information
Nonlinear Finite Element Implementation
Results
Conclusion
Introduction
• Neo-Hookean materials fall
under a classification of
materials known as
hyperelastic materials
– Elastomer often fall under this
category
• Hyperelastic materials have
evolving material properties
– Nonlinear material properties
– Often used in large
displacement applications so
also can suffer from nonlinear
geometries
Elastomer mold [1]
Solid Mechanics Brief Overview
Deformation Gradient:
Green Strain:
2nd Piola-Kirchoff Stress Tensor:
Stiffness Tensor for Hyperelastic Materials:
Neo-Hookean Material Properties
Neo-Hookean Free energy relationship:
Note: Neo-Hookean materials only depends on the shear modulus and the bulk
modulus constants as material properties
The Cauchy stress tensor can be simply found by using a push forward operation
to bring it back to the material frame
Material tangent stiffness matrix can be found in a similar manner as the
Cauchy stress tensor
Derivation of Weak Form
The weak form in the material frame is the same as we have derived in
class for the 3D elastic case.
Referential Weak Form
Isoparametric Deformation Gradients
Referential Gradient Matrix
The referential frame the gradient matrix is a full matrix. However, the
shape functions do not change as displacement changes.
While, the material frame the gradient matrix remains a sparse matrix.
However, the shape functions change as the displacement.
Total Lagrangian Form
Total Lagrangian form takes all the kinematic and static variables are referred back to the
initial configuration at t=0.
• By linearizing the nonlinear equations and taking appropriate substeps one can
approximate the nonlinear solution
Another formulation used called the Updated Lagrangian form refers all the kinematic and
static variables to the last updated configuration at t=t-1.
Newton-Rhapson Method
The residual vector shows us how far off the linearized version of the nonlinear
model is off from the correct solution.
We use the Newton-Rhapson method to approach a solution that is
“acceptable.”
We define [A] as the Jacobian matrix and will use it to find an appropriate
change in the displacements.
Jacobian Matrix
A common method to compute the Jacobian matrix is by taking the time
derivative of the internal forces.
The Jacobian matrix for each element is computed and then combine it into a
global matrix to find the change in displacements.
Kgeom Properties
ANSYS Compression Results
Material Properties: E=30MPa, ν=0.3, G=11.5 MPa, K=25MPa
Displacement in Y direction: -0.2m
ANSYS Tension Comparison
Material Properties: E=30MPa, ν=0.3, G=11.5 MPa, K=25MPa
Displacement in Y direction: 0.2m
ANSYS Shear Comparison
Material Properties: E=30MPa, ν=0.3, G=11.5 MPa, K=25MPa
Displacement in X direction: 0.5m
Material Response Comparison
0.8
0.7
0.7
Equivalent Strain (m/m)
Equivalent Strain (m/m)
0.6
0.5
0.4
0.3
0.2
0.1
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.5
-0.4
-0.3
-0.2
-0.1
Linear
Neo-Hookean
Simple Shear Response of Neo-Hookean and Linear Material
0.1
0.2
0.3
0.4
0.5
Uy (m)
Ux (m)
Linear
0
Neo-Hookean
Axial Loading Response of Neo-Hookean and Linear Material
Material Properties: E=30MPa, ν=0.3, G=11.5 MPa, K=25MPa
50.00
50.00
45.00
45.00
40.00
40.00
35.00
35.00
Error (%)
Error(%)
Error Comparison
30.00
25.00
20.00
30.00
25.00
20.00
15.00
15.00
10.00
10.00
5.00
5.00
0.00
0.00
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.5
-0.4
-0.3
-0.2
-0.1
Strain Error
Error of Simple Shear Response of Neo-Hookean and Linear
Material
0.1
0.2
0.3
0.4
Uy (m)
Ux (m)
Stress Error
0
Strain Error
Stress Error
Error of Axial Loading Response of Neo-Hookean and Linear
Material
0.5
Conclusion
• Hyperelastic materials are important to model
using nonlinear methods
– Even at small strains error can be noticeable
• Nonlinear materials can exhibit non symmetric
stress responses when loaded in the opposite
direction.
– Their response can be hard to predict without
modeling especially under complex loading
conditions
Thank You
• Any Questions?
References
• [1] http://www.polytek.com/