1. Modeling of Neo-Hookean Materials using FEM
By: Robert Carson

2. Overview Introduction Background Information
Nonlinear Finite Element Implementation
Results
Conclusion

3. 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]

4. Solid Mechanics Brief Overview
Deformation Gradient:
Green Strain:
2nd Piola-Kirchoff Stress Tensor:
Stiffness Tensor for Hyperelastic Materials:

5. 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

6. 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.

7. Referential Weak Form

8. Isoparametric Deformation Gradients

9. 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.

10. 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.

11. 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.

12. 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.

13. Kgeom Properties

14. ANSYS Compression Results
Material Properties: E=30MPa, ν=0.3, G=11.5 MPa, K=25MPa
Displacement in Y direction: -0.2m

15. ANSYS Tension Comparison
Material Properties: E=30MPa, ν=0.3, G=11.5 MPa, K=25MPa
Displacement in Y direction: 0.2m

16. ANSYS Shear Comparison
Material Properties: E=30MPa, ν=0.3, G=11.5 MPa, K=25MPa
Displacement in X direction: 0.5m

17. Material Response Comparison
Simple Shear Response of Neo-Hookean and Linear Material
Axial Loading Response of Neo-Hookean and Linear Material
Material Properties: E=30MPa, ν=0.3, G=11.5 MPa, K=25MPa

18. Error Comparison
Error of Simple Shear Response of Neo-Hookean and Linear Material
Error of Axial Loading Response of Neo-Hookean and Linear Material

19. 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

20. Thank You
Any Questions?

21. References
[1]