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doc:user:elements:volumes:hyper_materials

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doc:user:elements:volumes:hyper_materials [2014/10/01 17:42] – jorisdoc:user:elements:volumes:hyper_materials [2024/04/12 14:55] – radermecker
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 ====== Hyperelastic materials ====== ====== 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 ===== ===== NeoHookeanHyperPk2Material =====
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 Neo-Hookean hyperelastic law, using a ''PK2'' tensor. Neo-Hookean hyperelastic law, using a ''PK2'' tensor.
  
-The potential per unit volume is calculated based on the average compressibility over the element, ($\theta$): +The potential per unit volume is computed based on the average compressibility over the element, ($\theta$): 
  
 $$ $$
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 $$ $$
  
-The deviatoric potential is calculated based on a Cauchy tensor with a unit determinant:+The deviatoric potential is computed based on a Cauchy tensor with a unit determinant:
  
 $$ $$
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 | Density                                                 |  ''MASS_DENSITY''  | | Density                                                 |  ''MASS_DENSITY''  |
 | Initial bulk modulus ($k_0$)                            |    ''HYPER_K0''    | | Initial bulk modulus ($k_0$)                            |    ''HYPER_K0''    |
-| Initial shear modulus ($g_0$)                           |    ''HYPER_G0''    | +| Initial shear modulus ($g_0$)                           |    ''HYPER_G0''    |
  
 ===== LogarihtmicHyperPk2Material ===== ===== LogarihtmicHyperPk2Material =====
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 Logarithmic hyperelastic law, using a ''PK2'' tensor. Logarithmic hyperelastic law, using a ''PK2'' tensor.
  
-The potential per unit volume is calculated based on the average compressibility of the element, ($q$): +The potential per unit volume is computed based on the average compressibility of the element, ($q$): 
  
 $$ $$
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 $$ $$
  
-The deviatoric potential is calculated based on a Cauchy tensor with a unit determinant:+The deviatoric potential is computed based on a Cauchy tensor with a unit determinant:
  
 $$ $$
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 Logarithmic hyperelastic law, using a ''PK2'' tensor. Logarithmic hyperelastic law, using a ''PK2'' tensor.
  
-The potential per unit volume is calculated based on the average compressibility of the element, ($\theta$): +The potential per unit volume is computed based on the average compressibility of the element, ($\theta$): 
  
 $$ $$
Line 62: Line 112:
 $$ $$
  
-The deviatoric potential is calculated based on a Cauchy tensor with a unit determinant:+The deviatoric potential is computed based on a Cauchy tensor with a unit determinant:
  
 $$ $$
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 Hyperelastic law, using a ''PK2'' tensor. Its function applied on the strain spectral decomposition is a user law. Hyperelastic law, using a ''PK2'' tensor. Its function applied on the strain spectral decomposition is a user law.
  
-The potential per unit volume is calculated based on the average compressibility of the element, ($\theta$): +The potential per unit volume is computed based on the average compressibility of the element, ($\theta$): 
  
 $$ $$
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 $$ $$
  
-The deviatoric potential is calculated based on a hyperelastic user function defined in [[doc:user:elements:volumes:hyper_viscoelastic]].+The deviatoric potential is computed based on a hyperelastic user function defined in [[doc:user:elements:volumes:hyper_viscoelastic]].
  
 === Parameters === === Parameters ===
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 Each branch has its behavior corresponding to a viscoelastic law, supplied by the user. Each branch has its behavior corresponding to a viscoelastic law, supplied by the user.
  
-The potential per unit volume is calculated based on the average compressibility of the element, ($\theta$): +The potential per unit volume is computed based on the average compressibility of the element, ($\theta$): 
  
 $$ $$
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 $$ $$
  
-The deviatoric potential is calculated based on the viscoelastic laws :+The deviatoric potential is computed based on the viscoelastic laws :
  
 $$ $$
doc/user/elements/volumes/hyper_materials.txt · Last modified: 2024/05/02 11:15 by radermecker
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