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doc:user:elements:volumes:hyper_functionbased [2026/01/15 11:08] – [Visco-hyperelastic Neo-Hookean Material (Generalized Maxwell)] vanhulledoc:user:elements:volumes:hyper_functionbased [2026/01/15 13:15] (current) – [Holzapfel-Gasser-Ogden Anisotropic Material] vanhulle
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 (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 $. (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 a **deviatoric** $W_{dev}$ and **volumetric** $W_{vol}$ contribution:+The strain-energy density function $\psi$ is expressed as the sum of **deviatoric (elastic)** $\psi_{e}$ and **volumetric** $\psi_{vol}$ contributions:
 $$ $$
-W\left(\bar{I}_1,\bar{I}_2, J, \bar{I}_4, \bar{I}_5 \right) = W_{dev}\left(\bar{I}_1,\bar{I}_2, J, \bar{I}_4, \bar{I}_5 \right) + W_{vol}\left( J \right) = W_{dev}\left(\bar{I}_1,\bar{I}_2, J, \bar{I}_4, \bar{I}_5 \right) + k_0 \mathcal{f}\left( J \right)+\psi = \sum_{i=1}^{N_{e}}\psi_{e}^{(i)} + \sum_{i=1}^{N_{vol}}\psi_{vol}^{(i)} = \sum_{i=1}^{N_{e}}\psi_{e}^{(i)} + k_0 \sum_{i=1}^{N_{vol}}\mathcal{f}^{(i)}\left( J \right)
 $$ $$
  
-The deviatoric potential $W_{dev}$ is defined using a hyperelastic potential law defined in [[doc:user:elements:volumes:hyper_dev_potential]] whilst the volumetric potential $\mathcal{f}(J)$ is defined using a volumic potential law in [[doc:user:elements:volumes:hyper_vol_potential]].+The deviatoric (elastic) potentials $\psi_{e}^{(i)}$ are defined using hyperelastic potential laws defined in [[doc:user:elements:volumes:hyper_dev_potential]] whilst volumetric potentials $\mathcal{f}^{(i)}(J)$ are defined using volumic potential laws in [[doc:user:elements:volumes:hyper_vol_potential]].
  
-It is also possible to add inelastic deformations $\mathbf{F}^{in}$ (//e.g.// thermal expansion) by using an inelastic potential law in [[doc:user:elements:volumes:hyper_inel_potential]]. The elastic part of the total deformation gradient $\mathbf{F}^e$ writes+It is also possible to add **inelastic** deformations $\mathbf{F}_{in}$ (//e.g.// thermal expansion) by using inelastic potential laws in [[doc:user:elements:volumes:hyper_inel_potential]]. The total deformation gradient $\mathbf{F}$ writes
 $$ $$
-\mathbf{F}^e = \mathbf{F}\left(\mathbf{F}^{in}\right)^{-1}+\mathbf{F} = \mathbf{F}_e\prod_{i=1}^{N_{in}}\mathbf{F}_{in}^{(i)}
 $$ $$
 +
 +This material can be summarized into the following analogous rheological model:
 +{{ :doc:user:references:materials:fbgrid.png?900 |}}
  
 Note that all computations are done with respect to the **orthotropic axes**. Note that all computations are done with respect to the **orthotropic axes**.
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 | Density                                                 |  ''MASS_DENSITY''  |  ''TO/TM''  | | Density                                                 |  ''MASS_DENSITY''  |  ''TO/TM''  |
 | Initial bulk modulus ($k_0$)                            |  ''RUBBER_PENAL''  |  ''TO/TM''  | | Initial bulk modulus ($k_0$)                            |  ''RUBBER_PENAL''  |  ''TO/TM''  |
-| Number of the [[doc:user:elements:volumes:hyper_dev_potential|hyperelastic potential law]]               |  ''HYPER_ELAST_POTENTIAL_NO''  |  -  | +| Array of numbers defining the [[doc:user:elements:volumes:hyper_dev_potential|hyperelastic potential laws]]  [1, 2,...]             |  ''HYPER_ELAST_POTENTIAL_NUMS''   |  -  | 
-| Number of the [[doc:user:elements:volumes:hyper_vol_potential|volumic potential law]] \\ (default = QuadraticVolumicPotential)               |  ''HYPER_VOL_POTENTIAL_NO''  |  -  | +| Array of numbers defining the [[doc:user:elements:volumes:hyper_vol_potential|volumic potential laws]] [1, 2,...] \\ (default = QuadraticVolumicPotential)               |  ''HYPER_VOL_POTENTIAL_NUMS''  |  -  | 
-| Number of the [[doc:user:elements:volumes:hyper_inel_potential|inelastic potential law]]  \\ (default = None)              |  ''HYPER_INELAST_POTENTIAL_NO''  |  -  |+| Array of numbers defining the [[doc:user:elements:volumes:hyper_inel_potential|inelastic potential laws]] [1, 2,...] \\ (default = None)             |  ''HYPER_INELAST_POTENTIAL_NUMS''  |  -  |
 | Material temperature evolution law                      |  ''TEMP''  |    ''TM''   | | Material temperature evolution law                      |  ''TEMP''  |    ''TM''   |
 | Orthotropic axis                                    |    ''ORTHO_AX1_X''       |  -   | | Orthotropic axis                                    |    ''ORTHO_AX1_X''       |  -   |
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 | Density                                                 |  ''MASS_DENSITY''  |  ''TO/TM''  | | Density                                                 |  ''MASS_DENSITY''  |  ''TO/TM''  |
 | Initial bulk modulus ($k_0$)                            |  ''RUBBER_PENAL''  |  ''TO/TM''  | | Initial bulk modulus ($k_0$)                            |  ''RUBBER_PENAL''  |  ''TO/TM''  |
-| Number of the [[doc:user:elements:volumes:hyper_dev_potential|hyperelastic potential law]]               |  ''HYPER_ELAST_POTENTIAL_NO''  |  -  | +| Array of numbers defining the [[doc:user:elements:volumes:hyper_dev_potential|hyperelastic potential laws]]  [1, 2,...]             |  ''HYPER_ELAST_POTENTIAL_NUMS''   |  -  | 
-| Number of the [[doc:user:elements:volumes:hyper_vol_potential|volumic potential law]] \\ (default = QuadraticVolumicPotential)               |  ''HYPER_VOL_POTENTIAL_NO''  |  -  | +| Array of numbers defining the [[doc:user:elements:volumes:hyper_vol_potential|volumic potential laws]] [1, 2,...] \\ (default = QuadraticVolumicPotential)               |  ''HYPER_VOL_POTENTIAL_NUMS''  |  -  | 
-| Number of the [[doc:user:elements:volumes:hyper_inel_potential|inelastic potential law]]  \\ (default = None)              |  ''HYPER_INELAST_POTENTIAL_NO''  |  -  |+| Array of numbers defining the [[doc:user:elements:volumes:hyper_inel_potential|inelastic potential laws]] [1, 2,...] \\ (default = None)             |  ''HYPER_INELAST_POTENTIAL_NUMS''  |  -  |
 | Material temperature evolution law                      |  ''TEMP''  |    ''TM''   | | Material temperature evolution law                      |  ''TEMP''  |    ''TM''   |
 | Orthotropic axis                                    |    ''ORTHO_AX1_X''       |  -   | | Orthotropic axis                                    |    ''ORTHO_AX1_X''       |  -   |
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 | Heat Capacity $C_p$                |  ''HEAT_CAPACITY''   |  ''TO/TM''   | | Heat Capacity $C_p$                |  ''HEAT_CAPACITY''   |  ''TO/TM''   |
 | Dissipated thermoelastic power fraction $\eta_e$               |  ''DISSIP_TE''   |  -   | | Dissipated thermoelastic power fraction $\eta_e$               |  ''DISSIP_TE''   |  -   |
-| Dissipated (visco)plastic power fraction (Taylor-Quinney factor)                |  ''DISSIP_TQ''   |  -   |+| Dissipated (visco)plastic power fraction (Taylor-Quinney factor)                |  ''DISSIP_TQ''   |  -   |\ 
 + 
 +===== VeFunctionBasedHyperMaterial ===== 
 + 
 +=== Description === 
 +Visco-hyperelastic law, using a ''Cauchy'' stress tensor $\boldsymbol{\sigma}$, stress in the current configuration. 
 + 
 +This material is similar to ''FunctionBasedHyperMaterial'' with the addition of **visco-elastic** $\psi_{ve}$ contributions to the strain-energy density function $\psi$ as 
 +$$ 
 +\psi = \sum_{i=1}^{N_{ve}}\psi_{ve}^{(i)} + \sum_{i=1}^{N_{e}}\psi_{e}^{(i)} + k_0 \sum_{i=1}^{N_{vol}}\mathcal{f}^{(i)}\left( J \right) 
 +$$ 
 + 
 +The deviatoric visco-elastic potentials $\psi_{ve}^{(i)}$ are defined using visco-hyperelastic potential laws defined in [[doc:user:elements:volumes:hyper_dev_potential]]. 
 + 
 +Note that with this material, it is not mandatory to define any $\psi_{e}^{(i)}$. 
 + 
 +=== Parameters === 
 +^   Name                                                  ^  Metafor Code  ^ Dependency ^ 
 +| Density                                                 |  ''MASS_DENSITY''  |  ''TO/TM''  | 
 +| Initial bulk modulus ($k_0$)                            |  ''RUBBER_PENAL''  |  ''TO/TM''  | 
 +| Array of numbers defining the [[doc:user:elements:volumes:hyper_dev_potential|visco-hyperelastic potential laws]]  [1, 2,...]             |  ''HYPER_VE_POTENTIAL_NUMS''   |  -  | 
 +| Array of numbers defining the [[doc:user:elements:volumes:hyper_dev_potential|hyperelastic potential laws]]  [1, 2,...]  \\ (default = None)            |  ''HYPER_ELAST_POTENTIAL_NUMS''   |  -  | 
 +| Array of numbers defining the [[doc:user:elements:volumes:hyper_vol_potential|volumic potential laws]] [1, 2,...] \\ (default = QuadraticVolumicPotential)               |  ''HYPER_VOL_POTENTIAL_NUMS''  |  -  | 
 +| Array of numbers defining the [[doc:user:elements:volumes:hyper_inel_potential|inelastic potential laws]] [1, 2,...] \\ (default = None)             |  ''HYPER_INELAST_POTENTIAL_NUMS''  |  -  | 
 +| Material temperature evolution law                      |  ''TEMP''  |    ''TM''   | 
 +| Orthotropic axis                                    |    ''ORTHO_AX1_X''       |  -   | 
 +| Orthotropic axis                                    |    ''ORTHO_AX1_Y''       |  -   | 
 +| Orthotropic axis                                    |    ''ORTHO_AX1_Z''       |  -   | 
 +| Orthotropic axis                                    |    ''ORTHO_AX2_X''       |  -   | 
 +| Orthotropic axis                                    |    ''ORTHO_AX2_Y''       |  -   | 
 +| Orthotropic axis                                    |    ''ORTHO_AX2_Z''       |  -   | 
 +| Orthotropic axis initialized by mesh construction \\ boolean : True - False (default) \\ override OrthoAxis definition  |  ''ORTHO_INIT_AS_JACO''  | 
  
 ===== Example Materials ====== ===== Example Materials ======
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   materlawset(2).put(HYPER_HGO_DISP, d)   materlawset(2).put(HYPER_HGO_DISP, d)
   # 2 fiber orientations in the xy-plane (+-beta)   # 2 fiber orientations in the xy-plane (+-beta)
-  materlawset(2).put(HYPER_FIBS_THETAS, [beta, -beta]) +  materlawset(2).put(HYPER_FIBS_THETA, [beta, -beta]) 
-  materlawset(2).put(HYPER_FIBS_DELTA,  [  0.,    0.]) # NB: facultative if 0+  materlawset(2).put(HYPER_FIBS_DELTA, [  0.,    0.]) # NB: facultative if 0
    
      
doc/user/elements/volumes/hyper_functionbased.1768471713.txt.gz · Last modified: by vanhulle
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