doc:user:elements:volumes:hyper_materials
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doc:user:elements:volumes:hyper_materials [2013/07/11 14:25] – created joris | doc:user:elements:volumes:hyper_materials [2024/04/12 14:55] (current) – radermecker | ||
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+ | ====== Hyperelastic materials ====== | ||
+ | |||
+ | ===== NeoHookeanHyperMaterial ===== | ||
+ | |||
+ | === Description === | ||
+ | |||
+ | Neo-Hookean hyperelastic law, using a '' | ||
+ | |||
+ | (Quasi-)incompressibility is treated by a volumetric/ | ||
+ | |||
+ | $$ | ||
+ | | ||
+ | $$ | ||
+ | |||
+ | |||
+ | |||
+ | $$ | ||
+ | U^{dev}=\dfrac{g_0}{2} \left[\text{tr}\right(\hat{\mathbf{C}}\left)-3\right] | ||
+ | $$ | ||
+ | |||
+ | === Parameters === | ||
+ | ^ | ||
+ | | Density | ||
+ | | NeoHookean coefficient ($C_1$) | ||
+ | | Initial bulk modulus ($k_0$) | ||
+ | |||
+ | ===== MooneyRivlinHyperMaterial ===== | ||
+ | |||
+ | === Description === | ||
+ | |||
+ | Mooney-Rivlin hyperelastic law, using a '' | ||
+ | |||
+ | (Quasi-)incompressibility is treated by a volumetric/ | ||
+ | |||
+ | $$ | ||
+ | | ||
+ | $$ | ||
+ | |||
+ | |||
+ | |||
+ | $$ | ||
+ | U^{dev}=\dfrac{g_0}{2} \left[\text{tr}\right(\hat{\mathbf{C}}\left)-3\right] | ||
+ | $$ | ||
+ | |||
+ | === Parameters === | ||
+ | ^ | ||
+ | | Density | ||
+ | | Mooney-Rivlin coefficient ($C_1$) | ||
+ | | Mooney-Rivlin coefficient ($C_2$) | ||
+ | | Initial bulk modulus ($k_0$) | ||
+ | |||
+ | |||
+ | ===== NeoHookeanHyperPk2Material ===== | ||
+ | |||
+ | === Description === | ||
+ | |||
+ | Neo-Hookean hyperelastic law, using a '' | ||
+ | |||
+ | 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 === | ||
+ | |||
+ | ^ | ||
+ | | Density | ||
+ | | Initial bulk modulus ($k_0$) | ||
+ | | Initial shear modulus ($g_0$) | ||
+ | |||
+ | ===== LogarihtmicHyperPk2Material ===== | ||
+ | |||
+ | === Description === | ||
+ | |||
+ | Logarithmic hyperelastic law, using a '' | ||
+ | |||
+ | 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: | ||
+ | |||
+ | $$ | ||
+ | U^{dev}= \dfrac{g_0}{4} \ln \left(\hat{\mathbf{C}}\right): | ||
+ | $$ | ||
+ | |||
+ | === Parameters === | ||
+ | |||
+ | ^ | ||
+ | | Density | ||
+ | | Initial bulk modulus ($k_0$) | ||
+ | | Initial shear modulus ($g_0$) | ||
+ | |||
+ | ===== EvpIsoHLogarithmicHyperPk2Material ===== | ||
+ | |||
+ | === Description === | ||
+ | Logarithmic hyperelastic law, using a '' | ||
+ | |||
+ | 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: | ||
+ | |||
+ | $$ | ||
+ | U^{dev}= \dfrac{g_0}{4} \ln \left(\hat{\mathbf{C}}^{el}\right): | ||
+ | $$ | ||
+ | |||
+ | === Parameters === | ||
+ | |||
+ | ^ | ||
+ | | Density | ||
+ | | Initial bulk modulus ($k_0$) | ||
+ | | Initial shear modulus ($g_0$) | ||
+ | | Number of the material law which defines the yield stress $\sigma_{yield}$ | '' | ||
+ | |||
+ | ===== FunctionBasedHyperPk2Material ===== | ||
+ | |||
+ | === Description === | ||
+ | |||
+ | Hyperelastic law, using a '' | ||
+ | |||
+ | 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: | ||
+ | |||
+ | === Parameters === | ||
+ | |||
+ | ^ | ||
+ | | Density | ||
+ | | Initial bulk modulus ($k_0$) | ||
+ | | Number of the hyperelastic law | '' | ||
+ | |||
+ | |||
+ | ===== VeIsoHyperPk2Material ===== | ||
+ | |||
+ | === Description === | ||
+ | |||
+ | Viscoelastic hyperelastic law, using a '' | ||
+ | |||
+ | 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 : | ||
+ | |||
+ | $$ | ||
+ | U^{dev}= U^{dev}_{\text{main, | ||
+ | $$ | ||
+ | |||
+ | The dissipation potential is written as: | ||
+ | |||
+ | $$ | ||
+ | \Delta t \phi^{dev}= \Delta t \phi^{dev}_{\text{main, | ||
+ | $$ | ||
+ | |||
+ | 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, | ||
+ | |||
+ | === Parameters === | ||
+ | |||
+ | ^ | ||
+ | | Density | ||
+ | | Initial bulk modulus ($k_0$) | ||
+ | | Number of the main viscoelastic law | '' | ||
+ | | Number of the first Maxwell viscoelastic law | '' | ||
+ | | Number of the second Maxwell viscoelastic law (optional) | ||
+ | | Number of the third Maxwell viscoelastic law (optional) | ||