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 | Dependency |
|---|---|---|
| Density | MASS_DENSITY | TO/TM |
| NeoHookean coefficient ($C_1$) | RUBBER_C1 | TO/TM |
| Initial bulk modulus ($k_0$) | RUBBER_PENAL | TO/TM |
| Material temperature evolution law | TEMP | TM |
| Thermal expansion coefficient ($\alpha$) | THERM_EXPANSION | TO/TM |
Neo-Hookean hyperelastic law, using a Cauchy stress tensor $\boldsymbol{\sigma}$, stress in the current configuration.
Here, the TEMP parameter is not relevant anymore.
| Name | Metafor Code | Dependency |
|---|---|---|
| Density | MASS_DENSITY | TO/TM |
| NeoHookean coefficient ($C_1$) | RUBBER_C1 | TO/TM |
| Initial bulk modulus ($k_0$) | RUBBER_PENAL | TO/TM |
| Thermal expansion coefficient ($\alpha$) | THERM_EXPANSION | TO/TM |
| Conductivity | CONDUCTIVITY | TO/TM |
| Heat capacity | HEAT_CAPACITY | TO/TM |
| Dissipated thermoelastic power fraction | DISSIP_TE | - |
| Dissipated (visco)plastic power fraction (Taylor-Quinney factor) | DISSIP_TQ | - |
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 | Dependency |
|---|---|---|
| Density | MASS_DENSITY | TO/TM |
| Mooney-Rivlin coefficient ($C_1$) | RUBBER_C1 | TO/TM |
| Mooney-Rivlin coefficient ($C_2$) | RUBBER_C2 | TO/TM |
| Initial bulk modulus ($k_0$) | RUBBER_PENAL | TO/TM |
| Material temperature evolution law | TEMP | TM |
| Thermal expansion coefficient ($\alpha$) | THERM_EXPANSION | TO/TM |
Version < 3554
This material has no analytical material tangent stiffness. The latter should be computed by pertubation (global or material).
See STIFFMETHOD in the element properties of Volume elements.
Mooney-Rivlin hyperelastic law, using a Cauchy stress tensor $\boldsymbol{\sigma}$, stress in the current configuration.
Here, the TEMP parameter is not relevant anymore.
| Name | Metafor Code | Dependency |
|---|---|---|
| Density | MASS_DENSITY | TO/TM |
| Mooney-Rivlin coefficient ($C_1$) | RUBBER_C1 | TO/TM |
| Mooney-Rivlin coefficient ($C_2$) | RUBBER_C2 | TO/TM |
| Initial bulk modulus ($k_0$) | RUBBER_PENAL | TO/TM |
| Thermal expansion coefficient ($\alpha$) | THERM_EXPANSION | TO/TM |
| Conductivity | CONDUCTIVITY | TO/TM |
| Heat capacity | HEAT_CAPACITY | TO/TM |
| Dissipated thermoelastic power fraction | DISSIP_TE | - |
| Dissipated (visco)plastic power fraction (Taylor-Quinney factor) | DISSIP_TQ | - |
Holzapfel-Gasser-Ogden (invariant-based) anisotropic hyperelastic law, using a Cauchy stress tensor $\boldsymbol{\sigma}$, stress in the current configuration. This model is particularly suited to predict the response of biological tissues.
(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 $.
The strain-energy density function $W$ is expressed as the sum of an isotropic (=matrix) and anisotropic (=fibers) contribution: $$ W\left(\bar{I}_1,\bar{I}_4,J \right) = W_{iso}\left(\bar{I}_1,J \right) + W_{ani}\left(\bar{I}_1,\bar{I}_4\right) $$
The isotropic contribution takes the form of a generalized Neo-Hookean model: $$ W_{iso}\left(\bar{I}_1,J \right) = C_1\left(\bar{I}_1 -3\right) +K f\left(J\right) = C_1\left(\bar{I}_1 -3\right) +\frac{k_0}{2}\text{ln}^2 J $$
The anisotropic contribution to the strain energy density function writes: $$ W_{ani}\left(\bar{I}_1,\bar{I}_4\right) = \frac{k_1}{2k_2} \sum_{\alpha=1}^n \left[ e^{k_2\left<E_\alpha \right>^2} - 1 \right] = \frac{k_1}{2k_2} \sum_{\alpha=1}^n \left[ e^{k_2\left<d(\bar{I}_1-3)+(1-3d)(\bar{I}_4^\alpha - 1)\right>^2} - 1 \right], $$ where $k_1$[MPa] and $k_2$[-] are material parameters characterizing all fiber families in the material. $d\in[0,~\frac{1}{3}]$ is a parameter accounting for fiber dispersion, with $d=0$ corresponding to perfectly aligned fibers whilst $d=\frac{1}{3}$ corresponds to randomly oriented fibers (isotropic response). The model adds up to three different families of fibers ($\alpha \leq 3$), with their initial orientation given by $\mathbf{a}^\alpha = \left[a_{\alpha x},~a_{\alpha y},~a_{\alpha z} \right]$. Fiber directions do not have to be orthogonal.
More information and mathematical derivations, such as the analytical tangent stiffness matrix, can be found in metafor_hgo.pdf.
| Name | Metafor Code |
|---|---|
| Density | MASS_DENSITY |
| Mooney-Rivlin coefficient ($C_1$) | RUBBER_C1 |
| Initial bulk modulus ($k_0$) | RUBBER_PENAL |
| HGO parameter $k_1$ | HGO_K1 |
| HGO parameter $k_2$ | HGO_K2 |
| Fiber dissipation $d$ (optional, default=0) | HGO_DISP |
| Direction of $1^{st}$ fiber family $\mathbf{a}^1$ | HGO_FIB1_X, HGO_FIB1_Y, HGO_FIB1_Z |
| Direction of $2^{nd}$ fiber family $\mathbf{a}^2$ | HGO_FIB2_X, HGO_FIB2_Y, HGO_FIB2_Z |
| Direction of $3^{rd}$ fiber family $\mathbf{a}^3$ | HGO_FIB3_X, HGO_FIB3_Y, HGO_FIB3_Z |
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 | Dependency |
|---|---|---|
| Density | MASS_DENSITY | TO/TM |
| Initial bulk modulus ($k_0$) | HYPER_K0 | TO/TM |
| Initial shear modulus ($g_0$) | HYPER_G0 | TO/TM |
| Material temperature evolution law | TEMP | TM |
| Thermal expansion coefficient ($\alpha$) | THERM_EXPANSION | TO/TM |
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:
$$ U^{dev}= \dfrac{g_0}{4} \ln \left(\hat{\mathbf{C}}\right):\ln \left(\hat{\mathbf{C}}\right) $$
| Name | Metafor Code | Dependency |
|---|---|---|
| Density | MASS_DENSITY | TO/TM |
| Initial bulk modulus ($k_0$) | HYPER_K0 | TO/TM |
| Initial shear modulus ($g_0$) | HYPER_G0 | TO/TM |
| Material temperature evolution law | TEMP | TM |
| Thermal expansion coefficient ($\alpha$) | THERM_EXPANSION | TO/TM |
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 | TO/TM |
| Initial bulk modulus ($k_0$) | HYPER_K0 | TO/TM |
| Initial shear modulus ($g_0$) | HYPER_G0 | TO/TM |
| Number of the material law which defines the yield stress $\sigma_{yield}$ | YIELD_NUM | - |
| Material temperature evolution law | TEMP | TM |
| Thermal expansion coefficient ($\alpha$) | THERM_EXPANSION | TO/TM |
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 | TO/TM |
| Initial bulk modulus ($k_0$) | HYPER_K0 | TO/TM |
| Number of the hyperelastic law | HYPER_FUNCTION_NO | - |
| Material temperature evolution law | TEMP | TM |
| Thermal expansion coefficient ($\alpha$) | THERM_EXPANSION | TO/TM |
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 | TO/TM |
| Initial bulk modulus ($k_0$) | HYPER_K0 | TO/TM |
| 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 | - |
| Material temperature evolution law | TEMP | TM |
| Thermal expansion coefficient ($\alpha$) | THERM_EXPANSION | TO/TM |