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 $.
$$ W\left(I_1,I_2,J\right) = \bar{W}\left(\bar{I_1},\bar{I_2}\right) + K f\left(J\right) = C_1\left(\bar{I_1} - 3\right) + \frac{k_0}{2}\left[ \left(J-1\right)^2 + \ln^2 J\right] $$
| Name | Metafor Code |
|---|---|
| Density | MASS_DENSITY |
| NeoHookean coefficient ($C_1$) | RUBBER_C1 |
| Initial bulk modulus ($k_0$) | RUBBER_PENAL |
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 $.
$$ W\left(I_1,I_2,J\right) = \bar{W}\left(\bar{I_1},\bar{I_2}\right) + K f\left(J\right) = C_1\left(\bar{I_1} - 3\right) + C_2\left(\bar{I_2} - 3\right)+ \frac{k_0}{2}\left[ \left(J-1\right)^2 + \ln^2 J\right] $$
| 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 |
This material has no analytical material tangent stiffness. The latter should be computed by pertubation (global or material).
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:
$$ U^{dev}=\dfrac{g_0}{2} \left[\text{tr}\right(\hat{\mathbf{C}}\left)-3\right] $$
| Name | Metafor Code |
|---|---|
| Density | MASS_DENSITY |
| Initial bulk modulus ($k_0$) | HYPER_K0 |
| Initial shear modulus ($g_0$) | HYPER_G0 |
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:
$$ U^{dev}= \dfrac{g_0}{4} \ln \left(\hat{\mathbf{C}}\right):\ln \left(\hat{\mathbf{C}}\right) $$
| Name | Metafor Code |
|---|---|
| Density | MASS_DENSITY |
| Initial bulk modulus ($k_0$) | HYPER_K0 |
| Initial shear modulus ($g_0$) | HYPER_G0 |
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:
$$ U^{dev}= \dfrac{g_0}{4} \ln \left(\hat{\mathbf{C}}^{el}\right):\ln \left(\hat{\mathbf{C}}^{el}\right) $$
| 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 | - |
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.
| Name | Metafor Code | Dependency |
|---|---|---|
| Density | MASS_DENSITY | - |
| Initial bulk modulus ($k_0$) | HYPER_K0 | - |
| Number of the hyperelastic law | HYPER_FUNCTION_NO | - |
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.
| 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 | - |