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--- doc:user:elements:volumes:hyper_materials [2025/05/20 15:50] – C = Right Cauchy-Green tensor lacroix
+++ doc:user:elements:volumes:hyper_materials [2025/11/13 16:11] (current) – vanhulle
@@ Line 87: / Line 87: @@
 | Dissipated (visco)plastic power fraction (Taylor-Quinney factor)  |     ''DISSIP_TQ''     |      -      |
-===== HolzapfelGasserOgdenHyperMaterial =====
-=== Description ===
-Holzapfel-Gasser-Ogden (invariant-based) anisotropic hyperelastic law, using a ''Cauchy'' stress tensor $\boldsymbol{\sigma}$, stress in the current configuration. This model is particularly suited to predict the response of **biological tissues**.
-(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 an **isotropic** (=**matrix**) and **anisotropic** (=**fibers**) contribution:
-$$
-W\left(\bar{I}_1,\bar{I}_4,J \right) = W_{iso}\left(\bar{I}_1,J \right) + W_{ani}\left(\bar{I}_1,\bar{I}_4\right)
-$$
-The **isotropic** contribution takes the form of a **generalized Neo-Hookean** model:
-$$
-W_{iso}\left(\bar{I}_1,J \right) = C_1\left(\bar{I}_1 -3\right) +K f\left(J\right) = C_1\left(\bar{I}_1 -3\right) +\frac{k_0}{2}\text{ln}^2 J
-$$
-The **anisotropic** contribution to the strain energy density function writes:
-$$
-W_{ani}\left(\bar{I}_1,\bar{I}_4\right) = \frac{k_1}{2k_2} \sum_{\alpha=1}^n \left[ e^{k_2\left<E_\alpha \right>^2} - 1 \right] = \frac{k_1}{2k_2} \sum_{\alpha=1}^n \left[ e^{k_2\left<d(\bar{I}_1-3)+(1-3d)(\bar{I}_4^\alpha - 1)\right>^2} - 1 \right],
-$$
-where $k_1$[MPa] and $k_2$[-] are material parameters characterizing all fiber families in the material. $d\in[0,~\frac{1}{3}]$ is a parameter accounting for **fiber dispersion**, with $d=0$ corresponding to **perfectly aligned** fibers whilst $d=\frac{1}{3}$ corresponds to **randomly oriented** fibers (isotropic response). The model adds up to three different families of fibers ($\alpha \leq 3$), with their initial orientation given by $\mathbf{a}^\alpha = \left[a_{\alpha x},~a_{\alpha y},~a_{\alpha z} \right]$. Fiber directions do not have to be orthogonal.
-More information and mathematical derivations, such as the analytical tangent stiffness matrix, can be found in {{ :doc:user:elements:volumes:metafor_hgo.pdf |}}.
-=== Parameters ===
-^   Name                             ^  Metafor Code           ^
-| Density                            |''MASS_DENSITY''         |
-| Mooney-Rivlin coefficient ($C_1$)  | ''RUBBER_C1''           |
-| Initial bulk modulus ($k_0$)       |''RUBBER_PENAL''         |
-| HGO parameter $k_1$   |''HGO_K1''  |
-| HGO parameter $k_2$   |''HGO_K2''  |
-| Fiber dissipation $d$ (optional, default=0)  |''HGO_DISP''  |
-| Direction of $1^{st}$ fiber family $\mathbf{a}^1$  | ''HGO_FIB1_X'', ''HGO_FIB1_Y'', ''HGO_FIB1_Z''  |
-| Direction of $2^{nd}$ fiber family $\mathbf{a}^2$  | ''HGO_FIB2_X'', ''HGO_FIB2_Y'', ''HGO_FIB2_Z''  |
-| Direction of $3^{rd}$ fiber family $\mathbf{a}^3$  | ''HGO_FIB3_X'', ''HGO_FIB3_Y'', ''HGO_FIB3_Z''  |
 ===== NeoHookeanHyperPk2Material =====