Module elastic_hyperviscous 

Elastic-hyperviscous solid constitutive models.

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Elastic-hyperviscous solid constitutive models are defined by an elastic stress tensor function and a viscous dissipation function.

\mathbf{P}:\dot{\mathbf{F}} - \mathbf{P}^e(\mathbf{F}):\dot{\mathbf{F}} \geq 0

The second law of thermodynamics is satisfied by ensuring that the viscous dissipation function is convex and vanishes at zero rate,

\frac{\partial\psi}{\partial\dot{\mathbf{F}}}:\dot{\mathbf{F}} - \psi \geq 0 \quad\text{and}\quad \psi(\mathbf{F},\mathbf{0}) = 0

and by minimizing the dissipation potential less the stress power with respect to the deformation gradient rate, yielding the stress.

\mathbf{P} = \frac{\partial\phi}{\partial\dot{\mathbf{F}}}

Consequently, the rate tangent stiffness associated with the first Piola-Kirchhoff stress is symmetric for these constitutive models.

\mathcal{U}_{iJkL} = \mathcal{U}_{kLiJ}

Traits

ElasticHyperviscous

Required methods for elastic-hyperviscous solid constitutive models.

FirstOrderMinimize

First-order optimization methods for elastic-hyperviscous solid constitutive models.

SecondOrderMinimize

Second-order optimization methods for elastic-hyperviscous solid constitutive models.