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.