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--- doc:user:elements:volumes:hyper_dev_potential [2026/01/15 13:17] – [BonetBurtonHyperPotential] vanhulle
+++ doc:user:elements:volumes:hyper_dev_potential [2026/01/15 14:12] (current) – [MaxwellBranch] vanhulle
@@ Line 162: / Line 162: @@
-====== Rheological Laws ======
+====== Visco-elastic Potentials ======
-{{:doc:user:ico-worker.png?25|Under construction}} Other laws will follow with the addition of visco-elasticity.
+===== GeneralizedMaxwellHyperPotential =====
-===== CombinedElasticPotential =====
 === Description ===
-The ''CombinedElasticPotential'' material law allows to combine two deviatoric hyperelastic potentials together as
+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
 $$
-\boldsymbol{\sigma}^e = \boldsymbol{\sigma}^e_1 + \boldsymbol{\sigma}^e_2
+\boldsymbol{\sigma}^{n+1} = \boldsymbol{\sigma}^{n+1}_0+ \sum_{j=1}^N \mathbf{h}_j^{n+1},
 $$
-This can be illustrated using the following analogous rheological element
+where $\boldsymbol{\sigma}_0$ is the stress in the main elastic branch and $\mathbf{h}_j$ is the non-equilibrium stress from Maxwell branch $j$.
-{{ :doc:user:references:materials:rheoelast.png?300 |}}
+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''  |  -  |
-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.