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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 [2026/01/15 14:12] (current) – [MaxwellBranch] 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
+$$
+W_{dev} = \sum^n_{i=1} W_{dev}^{(i)} \left(\bar{I}_1, \bar{I}_2, \bar{I}_3, \bar{I}_4^{(i)}, \bar{I}_5^{(i)} \right) = \sum^n_{i=1} W_{dev}^{(i)} \left(\bar{I}_1, \bar{I}_2, J, \bar{I}_4^{(i)}, \bar{I}_5^{(i)} \right)
+$$
+The principal directions are defined using spherical coordinates and the (radius-)**longitude-lattitude convention**, allowing to reduce the set of parameters to $\theta$ and $\delta$ for each direction. These angles must be given in **degrees** and with respect to the **material reference frame** as shown in the figure below.
+{{ :doc:user:references:materials:coordCylA0.png?300   }}
+Note that if only one of $\theta$ or $\delta$ is specified, the other one is considered 0$^\circ$.
+=== 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 parameter ($d$)  |  ''HYPER_HGO_DISP''  |  ''TO/TM''  |
+| Array of $\theta$ angles defining the principal directions [$\theta_1$,...,$\theta_n$] |  ''HYPER_FIBS_THETA''  |  -  |
+| Array of $\theta$ angles defining the principal directions [$\delta_1$,...,$\delta_n$] |  ''HYPER_FIBS_DELTA''  |  -  |
+===== 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_v2.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''  |
+| Array of $\theta$ angles defining the principal directions [$\theta_1$,...,$\theta_n$] |  ''HYPER_FIBS_THETA''  |  -  |
+| Array of $\theta$ angles defining the principal directions [$\delta_1$,...,$\delta_n$] |  ''HYPER_FIBS_DELTA''  |  -  |
+====== Visco-elastic Potentials ======
+===== GeneralizedMaxwellHyperPotential =====
+=== Description ===
+In the rheological analogy, the generalized Maxwell visco-elastic model consists in a **main elastic potential** (main spring) put in parallel with several **Maxwell branches**, which are made of a spring and a damper in series. Each Maxwell branch must be defined using the **MaxwellBranch material law**.
+{{ :doc:user:references:materials:maxwellgrid.png?400 |}}
+The Cauchy stress in the current configuration writes
+$$
+\boldsymbol{\sigma}^{n+1} = \boldsymbol{\sigma}^{n+1}_0+ \sum_{j=1}^N \mathbf{h}_j^{n+1},
+$$
+where $\boldsymbol{\sigma}_0$ is the stress in the main elastic branch and $\mathbf{h}_j$ is the non-equilibrium stress from Maxwell branch $j$.
+The non-equilibrium stress in the current configuration in a Maxwell branch writes (trapezoidal integration)
+$$
+    \begin{align*}
+    \mathbf{h}_j^{n+1}
+    \approx e^{-\frac{\Delta t}{\tau_j}} \frac{1}{\Delta J} \Delta F ~\mathbf{h}_j^{n}(\Delta F)^T + \Gamma_j \frac{1 - e^{-\frac{\Delta t}{\tau_j}}}{\frac{\Delta t}{\tau_j}}\left[ \boldsymbol{\sigma}^{n+1}_0 - \frac{1}{\Delta J} \Delta F ~~\boldsymbol{\sigma}^{n}_0(\Delta F)^T\right]
+    \end{align*}
+$$
+where $\Delta \mathbf{F} = \mathbf{F}^{n+1}\left(\mathbf{F}^{n}\right)^{-1}$ and $\Delta J = \text{det}\left(\Delta \mathbf{F}\right)$.
+=== Parameters (GeneralizedMaxwellHyperPotential) ===
+^   Name                                                  ^  Metafor Code  ^ Dependency ^
+| Number of the main elastic potential $\sigma_0$  |  ''HYPER_MAIN_POTENTIAL_NO''  |  -  |
+| Array of numbers defining the Maxwell branches [1, 2, ...]  |  ''HYPER_MAXWELL_BRANCH_NUMS''  |  -  |
+=== Parameters (MaxwellBranch) ===
+^   Name                                                  ^  Metafor Code  ^ Dependency ^
+| Normalized Maxwell stiffness $\Gamma$  |  ''HYPER_MAXWELL_GAMMA''  |  ''TO/TM''  |
+| Relaxation time $\tau$  |  ''HYPER_VE_TAU''  |  ''TO/TM''  |
+| Boolean parameter, use trapezoidal integration (=False, default) or mid-point rule (=True)  |  ''HYPER_MAXWELL_USE_MPR''  |  -  |