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

The AnisoElasticPotential material law regroups elastic anisotropic contibutions to the deviatoric part of the strain-energy density function in a set of $n$ principal directions as $$ W_{dev} = \sum^n_{i=1} W_{dev}^{(i)} \left(\bar{I}_1, \bar{I}_2, \bar{I}_3, \bar{I}_4^{(i)}, \bar{I}_5^{(i)} \right) = \sum^n_{i=1} W_{dev}^{(i)} \left(\bar{I}_1, \bar{I}_2, J, \bar{I}_4^{(i)}, \bar{I}_5^{(i)} \right) $$

$$ \mathbf{M}^{(i)} = \mathbf{a}_0^{(i)} \otimes \mathbf{a}_0^{(i)} $$ $\mathbf{a}_0^{(i)} = \left[a_x^{(i)}, a_y^{(i)}, a_z^{(i)}\right]_0$ is the $i^{th}$ principal direction in the reference ($t_0$) material frame (). $$ \bar{\mathbf{N}}^{(i)} = \bar{\mathbf{F}}\mathbf{M}^{(i)}\bar{\mathbf{F}}^T $$ $$ \bar{I}_4^{(i)} = \text{tr}\left(\bar{\mathbf{C}}\mathbf{M}^{(i)}\right) = \text{tr}\left(\bar{\mathbf{B}}\bar{\mathbf{N}}^{(i)}\right) = \left(\bar{\mathbf{F}}\mathbf{a}_0^{(i)}\right):\left(\bar{\mathbf{F}}\mathbf{a}_0^{(i)}\right) = J^{-\frac{2}{3}}I_4^{(i)} $$ $$ \bar{I}_5^{(i)} = \text{tr}\left(\bar{\mathbf{C}}^2\mathbf{M}^{(i)}\right) = \text{tr}\left(\bar{\mathbf{B}}^2\bar{\mathbf{N}}^{(i)}\right) = J^{-\frac{4}{3}}I_5^{(i)} $$

The deviatoric part of the anisotropic Holzapfel-Gasser-Ogden hyperelastic law for the $i^{th}$ direction writes $$ W_{\text{HGO},~dev}^{(i)}\left(\bar{I}_1, \bar{I}_4^{(i)} \right) = \frac{k_1}{2k_2}\left[ e^{k_2\left< d\left(\bar{I}_1-3\right) + \left(1-3d\right)\left(\bar{I}_4^{(i)}-1\right)\right>^2}-1 \right] = \frac{k_1}{2k_2}\left[ e^{k_2\left<E^{(i)}\right>^2}-1 \right] $$ where $k_1$ and $k_2$ are material parameters characterizing the fibers and $d\in\left[0, \frac{1}{3}\right]$ is a parameter accounting for fiber dispersion.

• $d=0$ corresponds to perfectly aligned fibers whilst $d=\frac{1}{3}$ corresponds to randomly aligned fibers (isotropic response)
• $W_{dev}$ only affects the traction behavior of the material as $W_{dev}=0$ when $E^{(i)}=0$ (Macauley brackets)

$$ \begin{split} \left<E^{(i)} \right> = \left\{\begin{array}{ll} E^{(i)} & \text{if } E^{(i)} \geq 0 \\ 0 & \text{if } E_\alpha < 0 \end{array} \right. \end{split} $$

Mathematical derivations, such as the analytical tangent stiffness matrix, can be found in this presentation.

The deviatoric part of the anisotropic Bonet-Burton hyperelastic law for the $i^{th}$ direction writes $$ W_{\text{BB},~dev}^{(i)}\left(\bar{I}_1, \bar{I}_4^{(i)}, \bar{I}_5^{(i)} \right) = \left[\alpha + \beta \left( \bar{I}_1-3 \right) + \gamma \left( \bar{I}^{(i)}_4 -1\right)\right]\left(\bar{I}^{(i)}_4 - 1\right) - \frac{1}{2}\alpha \left(\bar{I}^{(i)}_5 -1\right) $$ or alternatively, $$ W_{\text{BB},~dev}^{(i)}\left(J, \bar{I}_4^{(i)}, \bar{I}_5^{(i)} \right) = \left[\alpha + \beta \text{ln}J + \gamma \left( \bar{I}^{(i)}_4 -1\right)\right]\left(\bar{I}^{(i)}_4 - 1\right) - \frac{1}{2}\alpha \left(\bar{I}^{(i)}_5 -1\right) $$ where $\alpha$, $\beta$ and $\gamma$ are material parameters which are related to the engineering material constants from the fibers and matrix (see Bonet-Burton material example).

Mathematical derivations, such as the analytical tangent stiffness matrix, can be found in this presentation.