Module hyperelastic_viscoplastic 

Hyperelastic-viscoplastic solid constitutive models.

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Hyperelastic-viscoplastic solid constitutive models are defined by a Helmholtz free energy density function of the elastic deformation gradient, a plastic dissipation potential, and the Kröner-Lee decomposition of the deformation gradient.

\mathbf{P}:\dot{\mathbf{F}} - \dot{a}(\mathbf{F}_\mathrm{e}) \geq 0

The second law of thermodynamics is satisfied by ensuring that the dissipation potential is minimized and vanishes at zero rate,

\frac{\partial\phi}{\partial\mathbf{D}_\mathrm{p}}:\mathbf{D}_\mathrm{p} - \phi \geq 0 \quad\text{and}\quad \phi(\mathbf{0}) = 0

by requiring the inequality to hold for arbitrary deformation gradient rates, which yields the stress and a symmetric tangent,

\mathbf{P} = \frac{\partial a}{\partial\mathbf{F}} \quad\Longrightarrow\quad \mathcal{C}_{iJkL} = \mathcal{C}_{kLiJ}

and by minimizing the dissipation potential less the plastic power with respect to the plastic stretching rate, yielding the flow rule.

\mathbf{M}_\mathrm{e}' = \frac{\partial\phi}{\partial\mathbf{D}_\mathrm{p}} \quad\Longleftrightarrow\quad \mathbf{D}_\mathrm{p} = \frac{\partial\phi^*}{\partial\mathbf{M}_\mathrm{e}'}

The dissipation potential and its dual sum to the plastic power at the conjugate pair, which therefore must be non-negative.

\phi(\mathbf{D}_\mathrm{p}) + \phi^*(\mathbf{M}_\mathrm{e}') = \mathbf{M}_\mathrm{e}':\mathbf{D}_\mathrm{p} \geq 0

Traits

FirstOrderMinimize

First-order minimization methods for hyperelastic-viscoplastic solid constitutive models.

HyperelasticViscoplastic

Required methods for hyperelastic-viscoplastic solid constitutive models.

RootRkmkDaeMinimize

RKMK-DAE return-map methods for hyperelastic-viscoplastic solid constitutive models — the minimize-based sibling of crate::constitutive::solid::elastic_viscoplastic::RootRkmkDae, for models whose equilibrium is naturally posed as a potential minimization rather than a stress-residual root. F_p still advances on its group at the tableau’s own order; only how F is resolved within a stage differs. Blanket over any HyperelasticViscoplastic model, same as SecondOrderMinimize itself.

SecondOrderMinimize

Second-order minimization methods for hyperelastic-viscoplastic solid constitutive models.