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doc:user:elements:volumes:hyper_functionbased [2026/01/15 10:46] – [Generalized Neo-Hookean Material with Thermal Expansion] 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 ======
-Some example materials from the literature using ''FunctionBasedHyperMaterial''.+Some example materials from the literature using ''FunctionBasedHyperMaterial'' and ''VeFunctionBasedHyperMaterial''.
  
 ==== Generalized Neo-Hookean Material with Thermal Expansion ==== ==== Generalized Neo-Hookean Material with Thermal Expansion ====
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 The elastic deviatoric potential is the ''NeoHookeanHyperPotential'': The elastic deviatoric potential is the ''NeoHookeanHyperPotential'':
   ## Elastic (deviatoric) potential   ## Elastic (deviatoric) potential
-  materlawset = domain.getMaterialLawSet() 
   materlawset.define(1, NeoHookeanHyperPotential)   materlawset.define(1, NeoHookeanHyperPotential)
   materlawset(1).put(HYPER_C1, C1)   materlawset(1).put(HYPER_C1, C1)
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   materlawset.define(3, ThermalIsotropicExpansion)   materlawset.define(3, ThermalIsotropicExpansion)
   materlawset(3).put(HYPER_THERM_EXPANSION, alpha)   materlawset(3).put(HYPER_THERM_EXPANSION, alpha)
 +  
 +==== Visco-hyperelastic Neo-Hookean Material (Generalized Maxwell) ====
 +
 +  ## Visco-elastic Neo-Hookean with n Maxwell branches
 +  materialset.define(1, VeFunctionBasedHyperMaterial)
 +  materialset(1).put(MASS_DENSITY, rho)
 +  materialset(1).put(RUBBER_PENAL,  k0)
 +  materialset(1).put(HYPER_VE_POTENTIAL_NUMS,   [1])
 +  materialset(1).put(HYPER_VOL_POTENTIAL_NUMS,  [2])
 +
 +We create the ''GeneralizedMaxwellHyperPotential'' grid base:
 +  ## Generalized Maxwell Potential
 +  materlawset.define(1, GeneralizedMaxwellHyperPotential)
 +
 +The main spring of the grid is the ''NeoHookeanHyperPotential'' (purely elastic):
 +  ### Main spring is Neo-Hookean
 +  materlawset.define(101, NeoHookeanHyperPotential)
 +  materlawset(101).put(HYPER_C1, C1)
 +  materlawset(1).put(HYPER_MAIN_POTENTIAL_NO, 101) #define law 101 as main spring
 +  
 +We add n parallel ''MaxwellBranch'' to the grid:
 +  ### n Maxwell Branches
 +  for i in range(0, n):
 +      materlawset.define(102+i, MaxwellBranch)
 +      materlawset(102+i).put(HYPER_MAXWELL_GAMMA,   gamma[i])
 +      materlawset(102+i).put(HYPER_VE_TAU,            tau[i])
 +      materlawset(1).append(HYPER_MAXWELL_BRANCH_NUMS, 102+i) #add law 102+i to the set of parallel Maxwell branches
 +      
 +The volumetric potential is the ''QuadLogVolumicPotential'':
 +  ## Volumetric potential
 +  materlawset.define(2, QuadLogVolumicPotential)
  
 ==== Holzapfel-Gasser-Ogden Anisotropic Material ==== ==== Holzapfel-Gasser-Ogden Anisotropic Material ====
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   materialset.define(1, FunctionBasedHyperMaterial)   materialset.define(1, FunctionBasedHyperMaterial)
   materialset(1).put(MASS_DENSITY, rho)   materialset(1).put(MASS_DENSITY, rho)
-  materialset(1).put(RUBBER_PENAL, k0)+  materialset(1).put(RUBBER_PENAL,  k0)
   materialset(1).put(HYPER_ELAST_POTENTIAL_NUMS, [1, 2])   materialset(1).put(HYPER_ELAST_POTENTIAL_NUMS, [1, 2])
-  materialset(1).put(HYPER_VOL_POTENTIAL_NUMS, [3])+  materialset(1).put(HYPER_VOL_POTENTIAL_NUMS,      [3])
      
 The elastic deviatoric potential is the sum of the ''NeoHookeanHyperPotential'' (isotropic matrix part)  The elastic deviatoric potential is the sum of the ''NeoHookeanHyperPotential'' (isotropic matrix part) 
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 and ''HolzapfelGasserOgdenHyperPotential'' with two separate fiber directions (+-beta in the xy-plane).  and ''HolzapfelGasserOgdenHyperPotential'' with two separate fiber directions (+-beta in the xy-plane). 
   materlawset.define(2, HolzapfelGasserOgdenHyperPotential)   materlawset.define(2, HolzapfelGasserOgdenHyperPotential)
-  materlawset(2).put(HYPER_HGO_K1, k1) +  materlawset(2).put(HYPER_HGO_K1,  k1) 
-  materlawset(2).put(HYPER_HGO_K2, k2)+  materlawset(2).put(HYPER_HGO_K2,  k2)
   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
    
      
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   ## Bonet-Burton Material   ## Bonet-Burton Material
   materialset.define(1, FunctionBasedHyperMaterial)   materialset.define(1, FunctionBasedHyperMaterial)
-  materialset(1).put(MASS_DENSITY, rho)+  materialset(1).put(MASS_DENSITY,    rho)
   materialset(1).put(RUBBER_PENAL, lambda)   materialset(1).put(RUBBER_PENAL, lambda)
-  materialset(1).put(HYPER_ELAST_POTENTIAL_NO, 1) +  materialset(1).put(HYPER_ELAST_POTENTIAL_NUMS, [1, 2]) 
-  materialset(1).put(HYPER_VOL_POTENTIAL_NO, 2)+  materialset(1).put(HYPER_VOL_POTENTIAL_NUMS,      [3])
      
-The elastic deviatoric potential is the sum of the ''NeoHookeanHyperPotential'' (isotropic matrix part) and ''BonetBurtonHyperPotential'' with one fiber directions. The addition between the two potentials is made using the CombinedElasticPotential.  +The elastic deviatoric potential is the sum of the ''NeoHookeanHyperPotential'' (isotropic matrix part)  
- +  materlawset.define(1, NeoHookeanHyperPotential) 
-  ## Elastic (deviatoric) potential +  materlawset(1).put(HYPER_C1, mu/2.) #mu=2C1
-  materlawset = domain.getMaterialLawSet() +
-  materlawset.define(1, CombinedElasticPotential) +
-  materlawset(1).put(HYPER_POTENTIAL1_NO, 3) #Neo-Hookean +
-  materlawset(1).put(HYPER_POTENTIAL2_NO, 4) #Bonet-Burton +
- +
-Isotropic Neo-Hookean potential +
-  materlawset.define(3, NeoHookeanHyperPotential) +
-  materlawset(3).put(HYPER_C1, mu/2.) #mu=2C1+
    
-Transversely isotropic Bonet-Burton potential with one fiber family +and ''BonetBurtonHyperPotential'' with one fiber directions (+beta in xy-plane).  
-  materlawset.define(4, BonetBurtonHyperPotential) +  materlawset.define(2, BonetBurtonHyperPotential) 
-  materlawset(4).put(HYPER_BB_ALPHA, alpha) +  materlawset(2).put(HYPER_BB_ALPHA,   alpha) 
-  materlawset(4).put(HYPER_BB_BETA, beta) +  materlawset(2).put(HYPER_BB_BETA,     beta) 
-  materlawset(4).put(HYPER_BB_GAMMA, gamma) +  materlawset(2).put(HYPER_BB_GAMMA,   gamma) 
-  materlawset(4).put(HYPER_BB_USE_LNJ, false) +  materlawset(2).put(HYPER_BB_USE_LNJ, false) 
-  # first fiber family with beta orientation +  # 1 fiber orientation in the xy-plane 
-  materlawset(4).put(HYPER_FIB1_X, np.cos(beta)) +  materlawset(2).put(HYPER_FIBS_THETA, [beta]) 
-  materlawset(4).put(HYPER_FIB1_Y, np.sin(beta))+  materlawset(2).put(HYPER_FIBS_DELTA,   [0.])
      
 The volumetric potential is the ''QuadraticVolumicPotential'' The volumetric potential is the ''QuadraticVolumicPotential''
   ## Volumetric potential   ## Volumetric potential
-  materlawset.define(2, QuadraticVolumicPotential)+  materlawset.define(3, QuadraticVolumicPotential)
  
doc/user/elements/volumes/hyper_functionbased.1768470360.txt.gz · Last modified: by vanhulle
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