This section contains all material laws which allow to define the deviatoric part of the strain-energy density function $W_{dev}$

The ElasticPotential material law regroups elastic isotropic deviatoric strain-energy density functions as $$ W_{dev} = W^e_{dev}\left(\bar{I}_1, \bar{I}_2, \bar{I}_3\right) = W^e_{dev}\left(\bar{I}_1, \bar{I}_2, J\right) $$

$$ \bar{I}_1 = \text{tr}\bar{\mathbf{B}} = \text{tr}\bar{\mathbf{C}} = \bar{\mathbf{F}}:\bar{\mathbf{F}} = J^{-\frac{2}{3}}I_1 $$ $$ \bar{I}_2 = \frac{1}{2}\left[ \left(\text{tr}\bar{\mathbf{B}}\right)^2 - \text{tr}\bar{\mathbf{B}}^2 \right] = \frac{1}{2}\left[ \left(\text{tr}\bar{\mathbf{C}}\right)^2 - \text{tr}\bar{\mathbf{C}}^2 \right] = J^{-\frac{4}{3}}I_2 $$ $$ \bar{I}_3 = \text{det}\bar{\mathbf{B}} = \text{det}\bar{\mathbf{C}} = 1 $$

The deviatoric part of the isotropic Neo-Hookean hyperelastic law writes $$ W^e_{\text{NH},~dev} \left(\bar{I}_1\right) = \frac{\mu}{2}\left(\bar{I}_1 - 3\right) = \frac{G}{2}\left(\bar{I}_1 - 3\right) = C_1\left(\bar{I}_1 - 3\right) $$ where $\mu$ (or $G$) is the shear modulus and $C_1$ is the equivalent Neo-Hookean parameter.

The deviatoric part of the isotropic Mooney-Rivlin hyperelastic law writes $$ W^e_{\text{MR},~dev} \left(\bar{I}_1, \bar{I}_2\right) = \frac{\mu_1}{2}\left(\bar{I}_1 - 3\right) + \frac{\mu_2}{2}\left(\bar{I}_2 - 3\right) = C_1\left(\bar{I}_1 - 3\right) + C_2\left(\bar{I}_2 - 3\right) $$ where $C_1$ and $C_2$ are Mooney-Rivlin coefficients.

The equivalent shear modulus $G$ writes $$ G = \mu_1 + \mu_2 = 2\left(C_1+C_2\right) $$

The deviatoric part of the isotropic Yeoh hyperelastic law writes $$ W^e_{\text{MR},~dev} \left(\bar{I}_1\right) = C_1\left(\bar{I}_1 - 3\right) + C_2\left(\bar{I}_1 - 3\right)^2 + C_3\left(\bar{I}_1 - 3\right)^3 $$ where $C_1$, $C_2$ and $C_3$ are Yeoh coefficients.

The equivalent shear modulus $G$ writes $$ G = 2 \left[ C_1+2C_2\left(\bar{I}_1-3\right)+3C_3\left(\bar{I}_1-3\right)^2\right] $$