Hyperelastic law, using a Cauchy stress tensor $\boldsymbol{\sigma}$, stress in the current configuration.

(Quasi-)incompressibility is treated by a volumetric/deviatoric multiplicative split of the deformation gradient, i.e. $\bar{\mathbf{F}} = J^{-1/3}\mathbf{F}$. Hence the deviatoric potential is based on reduced invariants of $\bar{\mathbf{b}} =\bar{\mathbf{F}}\bar{\mathbf{F}}^T $.

The strain-energy density function $W$ is expressed as the sum of a deviatoric $W_{dev}$ and volumetric $W_{vol}$ contribution: $$ W\left(\bar{I}_1,\bar{I}_2, J, \bar{I}_4, \bar{I}_5 \right) = W_{dev}\left(\bar{I}_1,\bar{I}_2, J, \bar{I}_4, \bar{I}_5 \right) + W_{vol}\left( J \right) = W_{dev}\left(\bar{I}_1,\bar{I}_2, J, \bar{I}_4, \bar{I}_5 \right) + k_0 \mathcal{f}\left( J \right) $$

The deviatoric potential $W_{dev}$ is defined using a hyperelastic potential law defined in Deviatoric Potentials whilst the volumetric potential $\mathcal{f}(J)$ is defined using a volumetric potential law in Volumic Potentials.

It is also possible to add inelastic deformations $\mathbf{F}^{in}$ (e.g. thermal expansion) by using an inelastic potential law in Inelastic Potentials. The elastic part of the total deformation gradient $\mathbf{F}^e$ writes $$ \mathbf{F}^e = \mathbf{F}\left(\mathbf{F}^{in}\right)^{-1} $$

Note that all computations are done with respect to the orthotropic axes.

Thermo-mechanical hyperelastic law with orthotropic thermal conduction law, using a Cauchy stress tensor $\boldsymbol{\sigma}$, stress in the current configuration.

Thermal conduction writes in the orthotropic frame $$ \boldsymbol{K}~\nabla T = \left[ \begin{array}{c c c} K_1 & 0 & 0 \\ 0 & K_2 & 0 \\ 0 & 0 & K_3 \end{array} \right] \nabla T, $$ where $\boldsymbol{K}$ is the orthotropic conduction matrix (in material axes) and $\nabla T$ is the temperature gradient.

Some example materials from the literature using FunctionBasedHyperMaterial.