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doc:user:elements:volumes:hyper_functionbased [2026/01/15 11:05] – [Generalized Neo-Hookean Material with Thermal Expansion] vanhulledoc:user:elements:volumes:hyper_functionbased [2026/07/02 14:15] (current) – 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)}
 $$ $$
  
-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 $\psi_{vol}^{(i)}$ 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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 ^   Name                                                  ^  Metafor Code  ^ Dependency ^ ^   Name                                                  ^  Metafor Code  ^ Dependency ^
 | Density                                                 |  ''MASS_DENSITY''  |  ''TO/TM''  | | 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|hyperelastic potential laws]]  [1, 2,...]             |  ''HYPER_ELAST_POTENTIAL_NUMS''   |  -  | 
-| 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_vol_potential|volumic potential laws]] [1, 2,...] \\ (default = QuadraticVolumicPotential)               |  ''HYPER_VOL_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_inel_potential|inelastic potential laws]] [1, 2,...] \\ (default = None)             |  ''HYPER_INELAST_POTENTIAL_NUMS''  |  -  |
-| Number of the [[doc:user:elements:volumes:hyper_inel_potential|inelastic potential law]]  \\ (default = None)              |  ''HYPER_INELAST_POTENTIAL_NO''  |  -  |+
 | 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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 ^   Name                                                  ^  Metafor Code  ^ Dependency ^ ^   Name                                                  ^  Metafor Code  ^ Dependency ^
 | Density                                                 |  ''MASS_DENSITY''  |  ''TO/TM''  | | 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|hyperelastic potential laws]]  [1, 2,...]             |  ''HYPER_ELAST_POTENTIAL_NUMS''   |  -  | 
-| 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_vol_potential|volumic potential laws]] [1, 2,...] \\ (default = QuadraticVolumicPotential)               |  ''HYPER_VOL_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_inel_potential|inelastic potential laws]] [1, 2,...] \\ (default = None)             |  ''HYPER_INELAST_POTENTIAL_NUMS''  |  -  |
-| Number of the [[doc:user:elements:volumes:hyper_inel_potential|inelastic potential law]]  \\ (default = None)              |  ''HYPER_INELAST_POTENTIAL_NO''  |  -  |+
 | 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 
 +$$ 
 +\psi = \sum_{i=1}^{N_{ve}}\psi_{ve}^{(i)} + \sum_{i=1}^{N_{e}}\psi_{e}^{(i)} + \sum_{i=1}^{N_{vol}}\psi_{vol}^{(i)} 
 +$$ 
 + 
 +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''  | 
 +| 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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   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(HYPER_ELAST_POTENTIAL_NUMS,   [1])   materialset(1).put(HYPER_ELAST_POTENTIAL_NUMS,   [1])
   materialset(1).put(HYPER_VOL_POTENTIAL_NUMS,     [2])   materialset(1).put(HYPER_VOL_POTENTIAL_NUMS,     [2])
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   ## Volumetric potential   ## Volumetric potential
   materlawset.define(2, QuadLogVolumicPotential)   materlawset.define(2, QuadLogVolumicPotential)
 +  materlawset(2).put(HYPER_COMPR_MODULUS, k0)
      
 Isotropic thermal expansion is added using a ''InelasticPotential'': Isotropic thermal expansion is added using a ''InelasticPotential'':
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   materialset.define(1, VeFunctionBasedHyperMaterial)   materialset.define(1, VeFunctionBasedHyperMaterial)
   materialset(1).put(MASS_DENSITY, rho)   materialset(1).put(MASS_DENSITY, rho)
-  materialset(1).put(RUBBER_PENAL,  k0) 
   materialset(1).put(HYPER_VE_POTENTIAL_NUMS,   [1])   materialset(1).put(HYPER_VE_POTENTIAL_NUMS,   [1])
   materialset(1).put(HYPER_VOL_POTENTIAL_NUMS,  [2])   materialset(1).put(HYPER_VOL_POTENTIAL_NUMS,  [2])
  
 +We create the ''GeneralizedMaxwellHyperPotential'' grid base:
   ## Generalized Maxwell Potential   ## Generalized Maxwell Potential
   materlawset.define(1, GeneralizedMaxwellHyperPotential)   materlawset.define(1, GeneralizedMaxwellHyperPotential)
  
 +The main spring of the grid is the ''NeoHookeanHyperPotential'' (purely elastic):
   ### Main spring is Neo-Hookean   ### Main spring is Neo-Hookean
   materlawset.define(101, NeoHookeanHyperPotential)   materlawset.define(101, NeoHookeanHyperPotential)
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   materlawset(1).put(HYPER_MAIN_POTENTIAL_NO, 101) #define law 101 as main spring   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   ### n Maxwell Branches
   for i in range(0, n):   for i in range(0, n):
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   ## Volumetric potential   ## Volumetric potential
   materlawset.define(2, QuadLogVolumicPotential)   materlawset.define(2, QuadLogVolumicPotential)
 +  materlawset(2).put(HYPER_COMPR_MODULUS, k0)
 +  
 +==== Visco-hyperelastic Three-Network Material (Nonlinear Generalized Maxwell) ====
 +Here we create a Three-Network material as presented in [[https://www.sciencedirect.com/science/chapter/monograph/pii/B978032331150200008X | Jörgen Bergström, 2015, 8 - Viscoplasticity Models, Mechanics of Solid Polymers, William Andrew Publishing, 371-436.]]. This material model can be used to model the "plastic" behavior of thermoplastic polymers.
  
 +The model is represented by a generalized Maxwell with three branches, two of them are nonlinear:
 +{{ :doc:user:references:materials:tnmgrid.png?200 |}}
 +
 +The three springs are ''EightChainHyperPotential''
 +  #spring 'A'
 +  materlawset.define(1, EightChainHyperPotential)
 +  materlawset(1).put(HYPER_MU, muA)
 +  materlawset(1).put(HYPER_LOCK_STRETCH, lambLockA)
 +  #spring 'B'
 +  materlawset.define(2, EightChainHyperPotential)
 +  materlawset(2).put(HYPER_MU, muB)
 +  materlawset(2).put(HYPER_LOCK_STRETCH, lambLockB)
 +  #spring 'C'
 +  materlawset.define(3, EightChainHyperPotential)
 +  materlawset(3).put(HYPER_MU, muC)
 +  materlawset(3).put(HYPER_LOCK_STRETCH, lambLockC)
 +  
 +with the addition of a common ''QuadraticVolumicPotential''
 +  # Volumetric potential
 +  materlawset.define(4, QuadraticVolumicPotential)
 +  materlawset(4).put(HYPER_COMPR_MODULUS, k0)
 +
 +Both dashpots are ''BergstromBoyceDashpot''
 +  #dashpot 'A'
 +  materlawset.define(5, BergstromBoyceDashpot)
 +  materlawset(5).put(DASHPOT_BB_GAMMADOT0, gammaDot0)
 +  materlawset(5).put(DASHPOT_BB_TAUHAT   ,   tauHatA)
 +  materlawset(5).put(DASHPOT_BB_A        ,         a)
 +  materlawset(5).put(DASHPOT_BB_M        ,        mA)
 +  # dependence of spring 'B' wrt dashpot 'A'
 +  materlawset(5).put(DASHPOT_BB_BETA,   beta)
 +  materlawset(5).put(DASHPOT_BB_MU_I,   muBi)
 +  materlawset(5).put(DASHPOT_BB_MU_F,   muBf)
 +  #dashpot 'B'
 +  materlawset.define(6, BergstromBoyceDashpot)
 +  materlawset(6).put(DASHPOT_BB_GAMMADOT0, gammaDot0)
 +  materlawset(6).put(DASHPOT_BB_TAUHAT   ,   tauHatB)
 +  materlawset(6).put(DASHPOT_BB_A        ,         a)
 +  materlawset(6).put(DASHPOT_BB_M        ,        mB)
 +  
 +We assemble the nonlinear branches A and B using ''NonLinearMaxwellBranch''
 +  # Maxwell Branch 'A'
 +  materlawset.define(7, NonLinearMaxwellBranch)
 +  materlawset(7).put(HYPER_MAXWELL_SPRING_NUM, 1)
 +  materlawset(7).put(HYPER_MAXWELL_SPRING_VOL_NUM, 4)
 +  materlawset(7).put(HYPER_MAXWELL_DASHPOT_NUM, 5)
 +  # Maxwell Branch 'B'
 +  materlawset.define(8, NonLinearMaxwellBranch)
 +  materlawset(8).put(HYPER_MAXWELL_SPRING_NUM, 2)
 +  materlawset(8).put(HYPER_MAXWELL_SPRING_VOL_NUM, 4)
 +  materlawset(8).put(HYPER_MAXWELL_DASHPOT_NUM, 6)
 +  # branch 'B' depends on branch 'A'
 +  materlawset(8).put(HYPER_MAXWELL_DEPENDENCE_NUM, 7)
 +
 +We assemble the ''GeneralizedMaxwellHyperPotential''
 +  materlawset.define(9, GeneralizedMaxwellHyperPotential)
 +  # main spring 'C'
 +  materlawset(9).put(HYPER_MAIN_POTENTIAL_NUM, 3)
 +  # Maxwell branches 'A' and 'B'
 +  materlawset(9).append(HYPER_MAXWELL_BRANCH_NUMS, [7, 8])
 +  
 +We create the material ''VeFunctionBasedHyperMaterial''
 +  materialset.define(1, VeFunctionBasedHyperMaterial)
 +  materialset(1).put(MASS_DENSITY, rho)
 +  materialset(1).put(HYPER_VE_POTENTIAL_NUMS,   [9])
 +  materialset(1).put(HYPER_VOL_POTENTIAL_NUMS,  [4])
 +
 +  
 ==== Holzapfel-Gasser-Ogden Anisotropic Material ==== ==== Holzapfel-Gasser-Ogden Anisotropic Material ====
 Here we create a Holzapfel-Gasser-Ogden anisotropic hyperelastic material as presented in [[https://pubmed.ncbi.nlm.nih.gov/15179858/ | Holzapfel G., Gasser T., Ogden R., 2004, Comparison of a multi-layer structural model for arterial walls with a Fung-type model, and issues of material stability, Journal of biomechanical engineering, 126, 264-75.]] Note that before version 3570 of Metafor, this material was implemented as ''HolzapfelGasserOgdenHyperMaterial''. Here we create a Holzapfel-Gasser-Ogden anisotropic hyperelastic material as presented in [[https://pubmed.ncbi.nlm.nih.gov/15179858/ | Holzapfel G., Gasser T., Ogden R., 2004, Comparison of a multi-layer structural model for arterial walls with a Fung-type model, and issues of material stability, Journal of biomechanical engineering, 126, 264-75.]] Note that before version 3570 of Metafor, this material was implemented as ''HolzapfelGasserOgdenHyperMaterial''.
Line 150: Line 253:
   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(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])
Line 164: Line 266:
   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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   ## Volumetric potential   ## Volumetric potential
   materlawset.define(3, LogarithmicVolumicPotential)   materlawset.define(3, LogarithmicVolumicPotential)
 +  materlawset(3).put(HYPER_COMPR_MODULUS, k0)
  
  
Line 196: Line 299:
   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(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])
Line 217: Line 319:
   ## Volumetric potential   ## Volumetric potential
   materlawset.define(3, QuadraticVolumicPotential)   materlawset.define(3, QuadraticVolumicPotential)
 +  materlawset(3).put(HYPER_COMPR_MODULUS, lambda)
  
doc/user/elements/volumes/hyper_functionbased.1768471554.txt.gz · Last modified: by vanhulle
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