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 [math]\displaystyle{ \Phi() }[/math] is a neo-Hookean potential energy function that depends on the current elastic deformation gradient ([math]\displaystyle{ \mathbf{F}_E }[/math]) and shear and Lamé moduli [math]\displaystyle{ G(J_P) }[/math] and [math]\displaystyle{ \lambda(J_P) }[/math]. 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]:
[math]\displaystyle{ \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 }[/math]
where \lambda_k are the principal elongations and J_E is the determinant of \mathbf{F}_E. The Cauchy stress by this law is
[math]\displaystyle{ \mathbf{\sigma} = \sum_k \left( {2G(J_P)\over J_E} \lambda_k(\lambda_k-1)+ \lambda(J_p)(J_E-1)\right) \vec b_k\otimes\vec b_k }[/math]
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: