Clamped Neo-Hookean Material
Constitutive Law
This MPM Material is an isotropic, elastic-plastic material in large strains using a hyperelastic formulation. The elastic part is a neo-Hookean material. Plasticity occurs when the elongation in either tensile of compressive elongation reaches a critical value. This material is based on similar material using in a paper to animate snow mechanics[1]. Although the model was based on engineering analysis of snow, it was simplified for efficiency in animation and for ease it creating a variety of responses.
The elastic regime of the material using a neo-Hookean material:
[math]\displaystyle{ W = \Phi\bigl(\mathbf{F}_E,G(J_P),\lambda(J_P)\bigr) }[/math]
where \Phi() is a neo-Hookean potential energy function that depends on the current elastic deformation gradient (\mathbf{F}_E) and shear and Lamé moduli G(J_P) and \lambda(J_P). The implementation here allows two different neo-Hookean laws. The first uses the law defined for the standard neo-Hookean material. The second uses the law proposed in Stomakhin et al.[1]:
\Phi\bigl(\mathbf{F}_E,G(J_P),\lambda(J_P)\bigr) = G(J_P)\sum_k (\lambda_k-1)^2 + {\lambda(J_P)\over 2)(J_E-1)^2
where \lambda_k are the principal elongations and J_E is the determinant of \mathbf{F}_E. The Cauchy stress by this law is
\sigma =
The plasticity is implement as follows:
Material Properties
The material properties are given in the following table.
| Property | Description | Units | Default |
|---|---|---|---|
| CritComp | Critical compression extension | none | 0.025 |
| CritComp | Critical tensile extension | none | 0.0075 |
| xihard | Hardening coefficient | none | 10 |
| Elastic | Enter 0 to basic elastic stresses on the model in Ref. [1]. Enter 1 to base elastic stresses on the neo-Hookean material. | none | 0 |
| (other) | Properties to define underlying neo-Hookean material (note that UJOption is always 1 when Elastic is 0, but can be any option when Elastic is 1) | varies | varies |
Examples
These commands model snow: