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 $$ W_{dev} = \sum^n_{i=1} W_{dev}^{(i)} \left(\bar{I}_1, \bar{I}_2, \bar{I}_3, \bar{I}_4^{(i)}, \bar{I}_5^{(i)} \right) = \sum^n_{i=1} W_{dev}^{(i)} \left(\bar{I}_1, \bar{I}_2, J, \bar{I}_4^{(i)}, \bar{I}_5^{(i)} \right) $$

The principal directions are defined using spherical coordinates and the (radius-)longitude-lattitude convention, allowing to reduce the set of parameters to $\theta$ and $\delta$ for each direction. These angles must be given in degrees and with respect to the material reference frame as shown in the figure below.

Note that if only one of $\theta$ or $\delta$ is specified, the other one is considered 0$^\circ$.

$$ \mathbf{M}^{(i)} = \mathbf{a}_0^{(i)} \otimes \mathbf{a}_0^{(i)} $$ $\mathbf{a}_0^{(i)} = \left[a_x^{(i)}, a_y^{(i)}, a_z^{(i)}\right]_0$ 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 $$ W_{\text{BB},~dev}^{(i)}\left(\bar{I}_1, \bar{I}_4^{(i)}, \bar{I}_5^{(i)} \right) = \left[\alpha + \beta \left( \bar{I}_1-3 \right) + \gamma \left( \bar{I}^{(i)}_4 -1\right)\right]\left(\bar{I}^{(i)}_4 - 1\right) - \frac{1}{2}\alpha \left(\bar{I}^{(i)}_5 -1\right) $$ 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). This model is actually directly derived from small-strain orthotropic (transversely isotropic) elasticity.

Alternatively, another implementation of this material law is available where the hyperlastic law writes $$ W_{\text{BB}}^{(i)}\left(J, \bar{I}_4^{(i)}, \bar{I}_5^{(i)} \right) = \left[\alpha + \beta~\text{ln}J + \gamma \left( \bar{I}^{(i)}_4 -1\right)\right]\left(\bar{I}^{(i)}_4 - 1\right) - \frac{1}{2}\alpha \left(\bar{I}^{(i)}_5 -1\right) $$ by using the parameter HYPER_BB_USE_LNJ=true.

Note that in this case, $W_{\text{BB}}^{(i)}$ is not purely deviatoric since there is a coupling between $J$ and $\bar{I}_4^{(i)}$. Therefore, this formulation also contributes to the volumetric part of the deformation gradient.

Mathematical derivations, such as the analytical tangent stiffness matrix, and information regarding the second form of the hyperelastic law can be found in this presentation.

In the rheological analogy, the generalized Maxwell visco-elastic model consists in a main elastic potential (main spring) put in parallel with several Maxwell branches, which are made of a spring and a damper in series. Each Maxwell branch must be defined using the MaxwellBranch material law.

The Cauchy stress in the current configuration writes $$ \boldsymbol{\sigma}^{n+1} = \boldsymbol{\sigma}^{n+1}_0+ \sum_{j=1}^N \mathbf{h}_j^{n+1}, $$ where $\boldsymbol{\sigma}_0$ is the stress in the main elastic branch and $\mathbf{h}_j$ is the non-equilibrium stress from Maxwell branch $j$.

The non-equilibrium stress in the current configuration in a Maxwell branch writes (trapezoidal integration) $$ \begin{align*} \mathbf{h}_j^{n+1} \approx e^{-\frac{\Delta t}{\tau_j}} \frac{1}{\Delta J} \Delta F ~\mathbf{h}_j^{n}(\Delta F)^T + \Gamma_j \frac{1 - e^{-\frac{\Delta t}{\tau_j}}}{\frac{\Delta t}{\tau_j}}\left[ \boldsymbol{\sigma}^{n+1}_0 - \frac{1}{\Delta J} \Delta F ~~\boldsymbol{\sigma}^{n}_0(\Delta F)^T\right] \end{align*} $$ where $\Delta \mathbf{F} = \mathbf{F}^{n+1}\left(\mathbf{F}^{n}\right)^{-1}$ and $\Delta J = \text{det}\left(\Delta \mathbf{F}\right)$.