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--- doc:user:elements:volumes:hyper_dev_potential [2025/11/14 12:17] – created vanhulle
+++ doc:user:elements:volumes:hyper_dev_potential [2025/11/14 15:33] (current) – [Anisotropic Elastic Potentials] vanhulle
@@ Line 71: / Line 71: @@
 | Yeoh coefficient ($C_2$)  |  ''HYPER_C2''  |  ''TO/TM''  |
 | Yeoh coefficient ($C_3$)  |  ''HYPER_C3''  |  ''TO/TM''  |
+====== Anisotropic Elastic Potentials ======
+The ''AnisoElasticPotential'' material law regroups elastic anisotropic contibutions to the deviatoric part of the strain-energy density function in a set of $n$ principal directions as
+$$
+W_{dev} = \sum^n_{i=1} W_{dev}^{(i)} \left(\bar{I}_1, \bar{I}_2, \bar{I}_3, \bar{I}_4^{(i)}, \bar{I}_5^{(i)} \right) = \sum^n_{i=1} W_{dev}^{(i)} \left(\bar{I}_1, \bar{I}_2, J, \bar{I}_4^{(i)}, \bar{I}_5^{(i)} \right)
+$$
+At the moment, a maximum of 3 separate principal directions can be given to the material law.
+=== Reminders ===
+$$
+\mathbf{M}^{(i)} = \mathbf{a}_0^{(i)} \otimes \mathbf{a}_0^{(i)}
+$$
+$\mathbf{a}_0^{(i)} = \left[a_x^{(i)}, a_y^{(i)}, a_z^{(i)}\right]_0$ is the $i^{th}$ principal direction in the reference ($t_0$) material frame (:!:).
+$$
+\bar{\mathbf{N}}^{(i)} = \bar{\mathbf{F}}\mathbf{M}^{(i)}\bar{\mathbf{F}}^T
+$$
+$$
+\bar{I}_4^{(i)} = \text{tr}\left(\bar{\mathbf{C}}\mathbf{M}^{(i)}\right) = \text{tr}\left(\bar{\mathbf{B}}\bar{\mathbf{N}}^{(i)}\right) = \left(\bar{\mathbf{F}}\mathbf{a}_0^{(i)}\right):\left(\bar{\mathbf{F}}\mathbf{a}_0^{(i)}\right) = J^{-\frac{2}{3}}I_4^{(i)}
+$$
+$$
+\bar{I}_5^{(i)} = \text{tr}\left(\bar{\mathbf{C}}^2\mathbf{M}^{(i)}\right) = \text{tr}\left(\bar{\mathbf{B}}^2\bar{\mathbf{N}}^{(i)}\right) = J^{-\frac{4}{3}}I_5^{(i)}
+$$
+===== HolzapfelGasserOgdenHyperPotential =====
+=== Description ===
+The deviatoric part of the anisotropic Holzapfel-Gasser-Ogden hyperelastic law for the $i^{th}$ direction writes
+$$
+W_{\text{HGO},~dev}^{(i)}\left(\bar{I}_1, \bar{I}_4^{(i)} \right) = \frac{k_1}{2k_2}\left[ e^{k_2\left< d\left(\bar{I}_1-3\right) + \left(1-3d\right)\left(\bar{I}_4^{(i)}-1\right)\right>^2}-1 \right] = \frac{k_1}{2k_2}\left[ e^{k_2\left<E^{(i)}\right>^2}-1 \right]
+$$
+where $k_1$ and $k_2$ are material parameters characterizing the fibers and $d\in\left[0, \frac{1}{3}\right]$ is a parameter accounting for fiber dispersion.
+=== Remarks ===
+  * $d=0$ corresponds to perfectly aligned fibers whilst $d=\frac{1}{3}$ corresponds to randomly aligned fibers (isotropic response)
+  * $W_{dev}$ only affects the traction behavior of the material as $W_{dev}=0$ when $E^{(i)}=0$ (Macauley brackets)
+$$
+\begin{split} \left<E^{(i)} \right> =
+\left\{\begin{array}{ll}
+E^{(i)} & \text{if } E^{(i)} \geq 0 \\
+& \text{if } E_\alpha < 0
+\end{array} \right.
+\end{split}
+$$
+Mathematical derivations, such as the analytical tangent stiffness matrix, can be found in {{ :doc:user:references:materials:vanhulle_250107_slides_hgo_metafor_final.pdf |this presentation}}.
+=== Parameters ===
+^   Name                                                  ^  Metafor Code  ^ Dependency ^
+| Holzapfel-Gasser-Ogden coefficient ($k_1$)  |  ''HYPER_HGO_K1''  |  ''TO/TM''  |
+| Holzapfel-Gasser-Ogden coefficient ($k_2$)  |  ''HYPER_HGO_K2''  |  ''TO/TM''  |
+| Fiber dispersion fraction ($d$)  |  ''HYPER_HGO_DISP''  |  ''TO/TM''  |
+| Direction of the first principal (fiber) direction ($a^1_x$) |  ''HYPER_FIB1_X''  |  -  |
+| Direction of the first principal (fiber) direction ($a^1_y$) |  ''HYPER_FIB1_Y''  |  -  |
+| Direction of the first principal (fiber) direction ($a^1_z$) |  ''HYPER_FIB1_Z''  |  -  |
+| Direction of the second principal (fiber) direction ($a^2_x$) |  ''HYPER_FIB2_X''  |  -  |
+| Direction of the second principal (fiber) direction ($a^2_y$) |  ''HYPER_FIB2_Y''  |  -  |
+| Direction of the second principal (fiber) direction ($a^2_z$) |  ''HYPER_FIB2_Z''  |  -  |
+| Direction of the third principal (fiber) direction ($a^3_x$) |  ''HYPER_FIB3_X''  |  -  |
+| Direction of the third principal (fiber) direction ($a^3_y$) |  ''HYPER_FIB3_Y''  |  -  |
+| Direction of the third principal (fiber) direction ($a^3_z$) |  ''HYPER_FIB3_Z''  |  -  |
+===== BonetBurtonHyperPotential =====
+=== Description ===
+The deviatoric part of the anisotropic Bonet-Burton hyperelastic law for the $i^{th}$ direction writes
+$$
+W_{\text{BB},~dev}^{(i)}\left(\bar{I}_1, \bar{I}_4^{(i)}, \bar{I}_5^{(i)} \right) = \left[\alpha + \beta \left( \bar{I}_1-3 \right) + \gamma \left( \bar{I}^{(i)}_4 -1\right)\right]\left(\bar{I}^{(i)}_4 - 1\right) - \frac{1}{2}\alpha \left(\bar{I}^{(i)}_5 -1\right)
+$$
+where $\alpha$, $\beta$ and $\gamma$ are material parameters which are related to the engineering material constants from the fibers and matrix (see [[doc:user:elements:volumes:hyper_functionbased|Bonet-Burton material example]]). This model is actually directly derived from small-strain orthotropic (transversely isotropic) elasticity.
+=== Remarks ===
+Alternatively, another implementation of this material law is available where the hyperlastic law writes
+$$
+W_{\text{BB}}^{(i)}\left(J, \bar{I}_4^{(i)}, \bar{I}_5^{(i)} \right) = \left[\alpha + \beta~\text{ln}J + \gamma \left( \bar{I}^{(i)}_4 -1\right)\right]\left(\bar{I}^{(i)}_4 - 1\right) - \frac{1}{2}\alpha \left(\bar{I}^{(i)}_5 -1\right)
+$$
+by using the parameter ''HYPER_BB_USE_LNJ=true''.
+Note that in this case, $W_{\text{BB}}^{(i)}$ is not purely deviatoric since there is a coupling between $J$ and $\bar{I}_4^{(i)}$. Therefore, this formulation also contributes to the volumetric part of the deformation gradient.
+Mathematical derivations, such as the analytical tangent stiffness matrix, and information regarding the second form of the hyperelastic law can be found in {{ :doc:user:references:materials:vanhulle_251106_slides_function_based_hyper_v1.pdf |this presentation}}.
+=== Parameters ===
+^   Name                                                  ^  Metafor Code  ^ Dependency ^
+| Bonet-Burton coefficient ($\alpha$)  |  ''HYPER_BB_ALPHA''  |  ''TO/TM''  |
+| Bonet-Burton coefficient ($\beta$)  |  ''HYPER_BB_BETA''  |  ''TO/TM''  |
+| Bonet-Burton coefficient ($\gamma$)  |  ''HYPER_BB_GAMMA''  |  ''TO/TM''  |
+| Use the alternative Bonet-Burton law with $\beta~\text{ln}J$ \\ boolean: ''true'' (default) |  ''HYPER_BB_USE_LNJ''  |  ''TO/TM''  |
+| Direction of the first principal (fiber) direction ($a^1_x$) |  ''HYPER_FIB1_X''  |  -  |
+| Direction of the first principal (fiber) direction ($a^1_y$) |  ''HYPER_FIB1_Y''  |  -  |
+| Direction of the first principal (fiber) direction ($a^1_z$) |  ''HYPER_FIB1_Z''  |  -  |
+| Direction of the second principal (fiber) direction ($a^2_x$) |  ''HYPER_FIB2_X''  |  -  |
+| Direction of the second principal (fiber) direction ($a^2_y$) |  ''HYPER_FIB2_Y''  |  -  |
+| Direction of the second principal (fiber) direction ($a^2_z$) |  ''HYPER_FIB2_Z''  |  -  |
+| Direction of the third principal (fiber) direction ($a^3_x$) |  ''HYPER_FIB3_X''  |  -  |
+| Direction of the third principal (fiber) direction ($a^3_y$) |  ''HYPER_FIB3_Y''  |  -  |
+| Direction of the third principal (fiber) direction ($a^3_z$) |  ''HYPER_FIB3_Z''  |  -  |
+====== Rheological Laws ======
+{{:doc:user:ico-worker.png?25|Under construction}} Other laws will follow with the addition of visco-elasticity.
+===== CombinedElasticPotential =====
+=== Description ===
+The ''CombinedElasticPotential'' material law allows to combine two deviatoric hyperelastic potentials together as
+$$
+\boldsymbol{\sigma}^e = \boldsymbol{\sigma}^e_1 + \boldsymbol{\sigma}^e_2
+$$
+This can be illustrated using the following analogous rheological element
+{{ :doc:user:references:materials:rheoelast.png?300 |}}
+The main purpose of this element is to create anisotropic hyperelastic materials, as they are often composed of an isotropic (generally a Neo-Hookean) matrix component and an anisotropic fibrous component (see [[doc:user:elements:volumes:hyper_functionbased|anisotropic material examples]]). Nonetheless, this material law can also be used to add two or more deviatoric potentials, since ''CombinedElasticPotential'' can combine with itself.