Table of Contents
Hyperelastic materials
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 Viscoelastic laws.
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 Viscoelastic laws.
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 | - |