====== Hyperelastic materials ======

===== NeoHookeanHyperMaterial =====

=== Description ===

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 $.

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



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

=== Parameters ===
^   Name                                                  ^  Metafor Code  ^
| Density                                                 |''MASS_DENSITY''|
| NeoHookean coefficient ($C_1$)                          | ''RUBBER_C1''  | 
| Initial bulk modulus ($k_0$)                            |''RUBBER_PENAL''| 

===== MooneyRivlinHyperMaterial =====

=== Description ===

Mooney-Rivlin 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 $.

$$
 W\left(I_1,I_2,J\right)  =  \bar{W}\left(\bar{I_1},\bar{I_2}\right) + K f\left(J\right) = C_1\left(\bar{I_1} - 3\right) + C_2\left(\bar{I_2} - 3\right)+ \frac{k_0}{2}\left[ \left(J-1\right)^2 + \ln^2 J\right]
$$



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

=== Parameters ===
^   Name                                                  ^  Metafor Code  ^
| Density                                                 |''MASS_DENSITY''|
| Mooney-Rivlin coefficient ($C_1$)                          | ''RUBBER_C1''  | 
| Mooney-Rivlin coefficient ($C_2$)                          | ''RUBBER_C2''  | 
| Initial bulk modulus ($k_0$)                            |''RUBBER_PENAL''| 


===== NeoHookeanHyperPk2Material =====

=== Description ===

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:

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

=== Parameters ===

^   Name                                                  ^     Metafor Code   ^
| Density                                                 |  ''MASS_DENSITY''  |
| Initial bulk modulus ($k_0$)                            |    ''HYPER_K0''    |
| Initial shear modulus ($g_0$)                           |    ''HYPER_G0''    |

===== LogarihtmicHyperPk2Material =====

=== Description ===

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:

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

=== Parameters ===

^   Name                                             ^     Metafor Code     ^
| Density                                                 |  ''MASS_DENSITY''  |
| Initial bulk modulus ($k_0$)  |    ''HYPER_K0''    |
| Initial shear modulus ($g_0$)     |    ''HYPER_G0''    | 

===== EvpIsoHLogarithmicHyperPk2Material =====

=== Description ===
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:

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

=== Parameters ===

^   Name                                                                     ^     Metafor Code     ^  Dependency ^
| Density                                                                    |  ''MASS_DENSITY''  |    -     |
| Initial bulk modulus ($k_0$)                                               |    ''HYPER_K0''    |    -     |
| Initial shear modulus ($g_0$)                                              |    ''HYPER_G0''    |    -     | 
| Number of the material law which defines the yield stress $\sigma_{yield}$ |    ''YIELD_NUM''   |    -     |

===== FunctionBasedHyperPk2Material =====

=== Description ===

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 [[doc:user:elements:volumes:hyper_viscoelastic]].

=== Parameters ===

^   Name                                                  ^     Metafor Code     ^  Dependency ^
| Density                                                 |  ''MASS_DENSITY''           |    -     |
| Initial bulk modulus ($k_0$)                            |    ''HYPER_K0''             |    -     |
| Number of the hyperelastic law                          |    ''HYPER_FUNCTION_NO''    |    -     | 


===== VeIsoHyperPk2Material =====

=== Description ===

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 [[doc:user:elements:volumes:hyper_viscoelastic]].

=== Parameters ===

^   Name                                             ^     Metafor Code     ^  Dependency ^
| Density                                                 |    ''MASS_DENSITY''         |    -     |
| Initial bulk modulus ($k_0$)  |    ''HYPER_K0''             |    -     |
| Number of the main viscoelastic law              |    ''MAIN_FUNCTION_NO''     |    -     | 
| Number of the first Maxwell viscoelastic law      |    ''MAXWELL_FUNCTION_NO1'' |    -     |
| Number of the second Maxwell viscoelastic law (optional)    |    ''MAXWELL_FUNCTION_NO2'' |    -     |
| Number of the third Maxwell viscoelastic law (optional)    |    ''MAXWELL_FUNCTION_NOI'' |    -     |