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--- doc:user:elements:volumes:hyper_functionbased [2025/11/14 15:53] – [Generalized Neo-Hookean Material with Thermal Expansion] vanhulle
+++ doc:user:elements:volumes:hyper_functionbased [2026/01/15 13:15] (current) – [Holzapfel-Gasser-Ogden Anisotropic Material] vanhulle
@@ Line 8: / Line 8: @@
 (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**.
@@ Line 26: / Line 29: @@
 | Density                                                 |  ''MASS_DENSITY''  |  ''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''   |
 | Orthotropic axis                                    |    ''ORTHO_AX1_X''       |  -   |
@@ Line 60: / Line 63: @@
 | Density                                                 |  ''MASS_DENSITY''  |  ''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''   |
 | Orthotropic axis                                    |    ''ORTHO_AX1_X''       |  -   |
@@ Line 76: / Line 79: @@
 | Heat Capacity $C_p$                |  ''HEAT_CAPACITY''   |  ''TO/TM''   |
 | 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 ======
-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 ====
 Here we create the same material as ''NeoHookeanHyperMaterial'' (see [[doc:user:elements:volumes:hyper_materials]])
+$$
+W = C_1\left(\bar{I_1} - 3\right) + \frac{k_0}{2}\left[ \left(J-1\right)^2 + \ln^2 J\right]
+$$
   ## Neo-Hookean Material
   materialset.define(1, FunctionBasedHyperMaterial)
   materialset(1).put(MASS_DENSITY, rho)
-  materialset(1).put(RUBBER_PENAL, k0)
+  materialset(1).put(RUBBER_PENAL,  k0)
-  materialset(1).put(HYPER_ELAST_POTENTIAL_NO, 1)
+  materialset(1).put(HYPER_ELAST_POTENTIAL_NUMS,   [1])
-  materialset(1).put(HYPER_VOL_POTENTIAL_NO, 2)
+  materialset(1).put(HYPER_VOL_POTENTIAL_NUMS,     [2])
-  materialset(1).put(HYPER_INELAST_POTENTIAL_NO, 3)
+  materialset(1).put(HYPER_INELAST_POTENTIAL_NUMS, [3])
 The elastic deviatoric potential is the ''NeoHookeanHyperPotential'':
   ## Elastic (deviatoric) potential
-  materlawset = domain.getMaterialLawSet()
   materlawset.define(1, NeoHookeanHyperPotential)
   materlawset(1).put(HYPER_C1, C1)
@@ Line 102: / Line 138: @@
   materlawset.define(2, QuadLogVolumicPotential)
-Isotropic thermal expansion is added as an ''InelasticPotential'':
+Isotropic thermal expansion is added using a ''InelasticPotential'':
   ## Inelastic potential
   materlawset.define(3, ThermalIsotropicExpansion)
   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 ====
+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''.
+In this case, the material represents arterial wall tissues with two fiber families.
+$$
+W = C_1\left(\bar{I_1} - 3\right) + \frac{k_1}{2k_2}\sum_{i=1}^2\left[ e^{k_2\left< d\left(\bar{I}_1-3\right) + \left(1-3d\right)\left(\bar{I}_4^{(i)}-1\right)\right>^2}-1 \right] + \frac{k_0}{2}\text{ln}^2 J
+$$
+  ## HGO Material
+  materialset.define(1, FunctionBasedHyperMaterial)
+  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_VOL_POTENTIAL_NUMS,      [3])
+The elastic deviatoric potential is the sum of the ''NeoHookeanHyperPotential'' (isotropic matrix part)
+  materlawset.define(1, NeoHookeanHyperPotential)
+  materlawset(1).put(HYPER_C1, C1)
+and ''HolzapfelGasserOgdenHyperPotential'' with two separate fiber directions (+-beta in the xy-plane).
+  materlawset.define(2, HolzapfelGasserOgdenHyperPotential)
+  materlawset(2).put(HYPER_HGO_K1,  k1)
+  materlawset(2).put(HYPER_HGO_K2,  k2)
+  materlawset(2).put(HYPER_HGO_DISP, d)
+  # 2 fiber orientations in the xy-plane (+-beta)
+  materlawset(2).put(HYPER_FIBS_THETA, [beta, -beta])
+  materlawset(2).put(HYPER_FIBS_DELTA, [  0.,    0.]) # NB: facultative if 0
+The volumetric potential is the ''LogarithmicVolumicPotential''
+  ## Volumetric potential
+  materlawset.define(3, LogarithmicVolumicPotential)
 ==== Bonet-Burton Transversely-Isotropic Material ====
+Here we create a Bonet-Burton transversely isotropic hyperelastic material as presented in [[https://www.sciencedirect.com/science/article/pii/S0045782597003393 | Bonet J., Burton A. J., 1998, A simple orthotropic, transversely isotropic hyperelastic constitutive equation for large strain computations,
+Computer Methods in Applied Mechanics and Engineering, 162, 151-164.]]
+For this material, the strain-energy density function writes
+$$
+W = \frac{\mu}{2}\left(\bar{I}_1 - 3\right) + \left[\alpha + \beta \left( \bar{I}_1-3 \right) + \gamma \left( \bar{I}^{(i)}_4 -1\right)\right]\left(\bar{I}^{(i)}_4 - 1\right) - \frac{1}{2}\alpha \left(\bar{I}^{(i)}_5 -1\right) + \frac{\lambda}{2}(J-1)^2
+$$
+The different constants from $W$ are related to the engineering material constants in the pricipal (1) direction and in the transverse (2, 3) directions [[https://pubs.aip.org/aip/acp/article/909/1/142/624705/Numerical-Modeling-of-Transversely-Isotropic | Abd Latif M., Perić D., Dettmer W. (2007). Numerical Modeling of Transversely Isotropic Elastic Material at Small and Finite Strains. 909. 142-146.]]
+$$
+\mu = \frac{E_2}{2\left(1+\nu_{23}\right)} ~~~~~|~~~~~ \lambda = \frac{-E_2\left(E_2\nu_{12}^2 + E_1\nu_{23}\right)}{\left(2E_2\nu_{12}^2+E_1\left[\nu_{23}-1\right]\right)\left(1+\nu_{23}\right)}
+$$
+$$
+    \alpha = \frac{1}{4}\left(\frac{E_2}{1+\nu_{23}}-2G_{12}\right) ~~~~~|~~~~~ \beta = \frac{E_2}{4}\left(\frac{E_1-2E_1\nu_{12}}{2E_2\nu_{12}^2+E_1\left(\nu_{23}-1\right)}+\frac{1}{1+\nu_{23}}\right)
+$$
+$$
+   \gamma = \frac{1}{16}\left(2E_1 - 4G_{12} - \frac{E_1 E_2\left(1-2\nu_{12}\right)^2}{2E_2\nu_{12}^2+E_1\left(\nu_{23}-1\right)}-\frac{E_2}{1+\nu_{23}}\right)
+$$
+  ## Bonet-Burton Material
+  materialset.define(1, FunctionBasedHyperMaterial)
+  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_VOL_POTENTIAL_NUMS,      [3])
+The elastic deviatoric potential is the sum of the ''NeoHookeanHyperPotential'' (isotropic matrix part)
+  materlawset.define(1, NeoHookeanHyperPotential)
+  materlawset(1).put(HYPER_C1, mu/2.) #mu=2C1
+and ''BonetBurtonHyperPotential'' with one fiber directions (+beta in xy-plane).
+  materlawset.define(2, BonetBurtonHyperPotential)
+  materlawset(2).put(HYPER_BB_ALPHA,   alpha)
+  materlawset(2).put(HYPER_BB_BETA,     beta)
+  materlawset(2).put(HYPER_BB_GAMMA,   gamma)
+  materlawset(2).put(HYPER_BB_USE_LNJ, false)
+  # 1 fiber orientation in the xy-plane
+  materlawset(2).put(HYPER_FIBS_THETA, [beta])
+  materlawset(2).put(HYPER_FIBS_DELTA,   [0.])
+The volumetric potential is the ''QuadraticVolumicPotential''
+  ## Volumetric potential
+  materlawset.define(3, QuadraticVolumicPotential)