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--- doc:user:elements:volumes:hyper_dev_potential [2025/11/14 15:13] – [BonetBurtonHyperPotential] vanhulle
+++ doc:user:elements:volumes:hyper_dev_potential [2026/01/15 14:12] (current) – [MaxwellBranch] vanhulle
@@ Line 78: / Line 78: @@
 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 ===
@@ Line 121: / Line 128: @@
 | 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''  |
+| 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''  |  -  |
@@ Line 141: / Line 150: @@
 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}}.
+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 ===
@@ Line 149: / Line 158: @@
 | 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''  |  -  |