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

Overview 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 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

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

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/