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 the material used in a paper to animate snow mechanics. Although that 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 uses an isotropic, neo-Hookean material potential energy:

      [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.:

      [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 [math]\displaystyle{ \lambda_k }[/math] are the principal elongations and J_E is the determinant of [math]\displaystyle{ \mathbf{F}_E }[/math]. 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]

where [math]\displaystyle{ \vec b_k }[/math] are the eigenvectors of the elastic left Cauchy-Green strain tensor ([math]\displaystyle{ \mathbf{B} }[/math]).

The neo-Hookean mechanical properties are assumed to depend on the amount of plastic volume change using

      [math]\displaystyle{ G(J_P)= G(1)e^{\xi(1-J_P)} \qquad {\rm and} \qquad \lambda(J_P)= \lambda(1)e^{\xi(1-J_P)} }[/math]

where [math]\displaystyle{ \xi }[/math] is a hardening parameter and [math]\displaystyle{ G(1) }[/math] and [math]\displaystyle{ \lambda(1) }[/math] are the initial shear and Lamé moduli (i.e., before any plasticity). Note that in tension, the material softens, which can emulate fracture. In compression, the material hardens.

The plasticity is implemented as follows:

1. On each time step, a trial, updated deformation gradient is calculated and decomposed into its principle stretches.
2. If any stretch is greater then (1+θ_t), it is set to (1+θ_t), where θ_t is the critical tensile strain before tensile failure. If any stretch is less then (1-θ_c), it is set to (1-θ_c), where θ_c is the critical compressive strain before compressive failure. In other words, all elongations are "clamped" to be with the interval [1-θ_c,1+θ_t]
3. Once the elongations are clamped, the final elastic deformation is found. The total J is divided into elastic J_E and plastic J_P = J/J_E. The new mechanical properties are found from J_P and the stress is found from these properties and the elastic deformation.

Dissipated Energy

Although this plastic material will dissipate energy whenever the elongations are clamped, the currently implemented model does not calculate that energy. The only consequence of this fact is that you can not use this material to simulate heating caused by plastic deformation. Except for this simulation option, the dissipated energy plays no role in MPM results.

Residual Stress

In the presence of temperature or concentration changes, this material accounts for residual stresses by the process described for the Mooney-Rivlin material. This change only requires that the stress calculation replace [math]\displaystyle{ J_E }[/math] by [math]\displaystyle{ J_{eff} = J_E/J_{res} }[/math] and replace [math]\displaystyle{ \lambda_k }[/math] by [math]\displaystyle{ \lambda_k/\lambda_{res} }[/math]

Material Properties

The material properties are given in the following table.

Property	Description	Units	Default
CritComp	Critical compression extension θc	none	0.025
CritComp	Critical tensile extension θt	none	0.0075
xihard	Hardening coefficient ξ	none	10
Elastic	Enter 0 to basic elastic stresses on the model in Stomakhin et al.. 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

Note that any simulation that includes compression may get significantly stiffer as the analysis proceeds. to avoid instability due to lack of convergence, this material should always be use the the AdjustTimeStep custom tasks to change the time step due to stiffening.

History Data

This material uses history #1 to store the volumetric strain (i.e., the determinant of the deformation gradient or J) and history #2 to store the plastic volumetric strain (i.e., J_P). The total strain, which is elastic, is stored in the elastic strain variable, while the plastic strain stores the left Cauchy Green strain tensor.

Examples

These commands are suggested by Stomakin et. al. as reference material properties for animating snow:

Material "Snow","Snow Animation Material","ClampedNeohookean"
  E 0.14
  nu 0.2
  rho 0.4
  alpha 0
  CritComp .025
  CritTens .0074
  xihard 10
  Elastic 0
  Color 0.84,0.91,1
Done