Neo-Hookean 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 $\bar{\mathbf{b}} =\bar{\mathbf{F}}\bar{\mathbf{F}}^T$.

$$W\left(I_1,I_2,J\right) = \bar{W}\left(\bar{I_1},\bar{I_2}\right) + K f\left(J\right) = C_1\left(\bar{I_1} - 3\right) + \frac{k_0}{2}\left[ \left(J-1\right)^2 + \ln^2 J\right]$$

$$U^{dev}=\dfrac{g_0}{2} \left[\text{tr}\right(\hat{\mathbf{C}}\left)-3\right]$$

Neo-Hookean hyperelastic law, using a PK2 tensor.

The potential per unit volume is computed based on the average compressibility over the element, ($\theta$):

$$U^{vol}=\dfrac{k_0}{2} \left[\ln\right(\theta\left)\right]^2$$

The deviatoric potential is computed based on a Cauchy tensor with a unit determinant:

$$U^{dev}=\dfrac{g_0}{2} \left[\text{tr}\right(\hat{\mathbf{C}}\left)-3\right]$$

Logarithmic hyperelastic law, using a PK2 tensor.

The potential per unit volume is computed based on the average compressibility of the element, ($q$):

$$U^{vol}=\dfrac{k_0}{2} \left[\ln\right(\theta\left)\right]^2$$

The deviatoric potential is computed based on a Cauchy tensor with a unit determinant:

$$U^{dev}= \dfrac{g_0}{4} \ln \left(\hat{\mathbf{C}}\right):\ln \left(\hat{\mathbf{C}}\right)$$

Logarithmic hyperelastic law, using a PK2 tensor.

The potential per unit volume is computed based on the average compressibility of the element, ($\theta$):

$$U^{vol}=\dfrac{k_0}{2} \left[\ln\right(\theta\left)\right]^2$$

The deviatoric potential is computed based on a Cauchy tensor with a unit determinant:

$$U^{dev}= \dfrac{g_0}{4} \ln \left(\hat{\mathbf{C}}^{el}\right):\ln \left(\hat{\mathbf{C}}^{el}\right)$$

Hyperelastic law, using a PK2 tensor. Its function applied on the strain spectral decomposition is a user law.

The potential per unit volume is computed based on the average compressibility of the element, ($\theta$):

$$U^{vol}=\dfrac{k_0}{2} \left[\ln\right(\theta\left)\right]^2$$

The deviatoric potential is computed based on a hyperelastic user function defined in Viscoelastic laws.

Viscoelastic hyperelastic law, using a PK2 tensor. The law includes a main branch (spring and dashpot in parallel) and one or several Maxwell branches (spring and dashpot in series).

Each branch has its behavior corresponding to a viscoelastic law, supplied by the user.

The potential per unit volume is computed based on the average compressibility of the element, ($\theta$):

$$U^{vol}=\dfrac{k_0}{2} \left[\ln\right(\theta\left)\right]^2$$

The deviatoric potential is computed based on the viscoelastic laws :

$$U^{dev}= U^{dev}_{\text{main,elastic}}\left(\hat{C}\right) + \sum_{Maxwell} U^{dev}_{\text{Maxwell,elastic}}\left(\hat{C}^{\text{el}}\right)$$

The dissipation potential is written as:

$$\Delta t \phi^{dev}= \Delta t \phi^{dev}_{\text{main,viscous}}\left( \exp{\frac{\ln{\Delta\hat{C}}}{\Delta t}} \right) + \sum_{Maxwell} \Delta t \phi^{dev}_{\text{Maxwell,viscous}}\left(\exp{\frac{\ln{\Delta C^{\text{vis}}}}{\Delta t}} \right)$$

where $$\Delta\hat{C} = {\hat{F}^n}^{-T} \hat{C}^{n+1} {\hat{F}^n}^{-1}$$

$$\Delta C^{\text{vis}} = {{F^{\text{vis}}}^n}^{-T} {C^{\text{vis}}}^{n+1} {{F^{\text{vis}}}^n}^{-1}$$

The potentials $U^{dev}_{\text{main,elastic}},~~U^{dev}_{\text{Maxwell,elastic}},~~\phi^{dev}_{\text{main,viscous}},~~\phi^{dev}_{\text{Maxwell,viscous}}$ are hyperelastic functions defined in Viscoelastic laws.