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--- doc:user:elements:volumes:hyper_materials [2015/01/26 12:12] – joris
+++ doc:user:elements:volumes:hyper_materials [2024/04/12 14:55] – radermecker
@@ Line 1: / Line 1: @@
 ====== 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 =====
@@ Line 24: / Line 74: @@
 | Density                                                 |  ''MASS_DENSITY''  |
 | Initial bulk modulus ($k_0$)                            |    ''HYPER_K0''    |
 | Initial shear modulus ($g_0$)                           |    ''HYPER_G0''    |
 ===== LogarihtmicHyperPk2Material =====