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--- doc:user:elements:volumes:hyper_dev_potential [2026/01/15 14:12] – [GeneralizedMaxwellHyperPotential] vanhulle
+++ doc:user:elements:volumes:hyper_dev_potential [2026/07/02 15:47] (current) – [EightChainHyperPotential] vanhulle
@@ Line 71: / Line 71: @@
 | Yeoh coefficient ($C_2$)  |  ''HYPER_C2''  |  ''TO/TM''  |
 | Yeoh coefficient ($C_3$)  |  ''HYPER_C3''  |  ''TO/TM''  |
+===== EightChainHyperPotential =====
+=== Description ===
+The deviatoric part of the isotropic Eight-Chain (or Arruda-Boyce) hyperelastic law writes
+$$
+            W^e_{EC}\left( \bar{\lambda^*}, \theta_0\right) = N k_B \theta_0 n \left[\frac{\bar{\lambda^*}}{\sqrt{n}}\mathcal{L}^{-1}\left[\frac{\bar{\lambda^*}}{\sqrt{n}}\right] + \text{ln}\left(\frac{\mathcal{L}^{-1}\left[\frac{\bar{\lambda^*}}{\sqrt{n}}\right]}{\text{sinh }\mathcal{L}^{-1}\left[\frac{\bar{\lambda^*}}{\sqrt{n}}\right]}\right)\right],
+$$ where $N$ is the number of chains per unit volume, $k_B$ is Boltzmann constant, $\theta_0$ is the reference temperature, $n$ is the number of chains and $\mathcal{L}^{-1}$ is the inverse Langevin function.
+$\bar{\lambda^*}$ is the effective distortional stretch and writes
+$$
+    \bar{\lambda}^*=\sqrt{\frac{\bar{I}_1}{3}}.
+$$
+The Cauchy stress writes
+$$
+    \boldsymbol{\sigma}=\frac{\mu}{J \bar{\lambda}^*}\frac{\mathcal{L}\left(\frac{\bar{\lambda}^*}{\lambda^{lock}}\right)}{\mathcal{L}\left(\frac{1}{\lambda^{lock}}\right)}\text{dev}\left(\bar{\mathbf{B}}\right),
+$$
+where the locking stretch $\lambda_{lock}=\sqrt{n}$ and shear modulus
+$$
+\mu = \frac{N k_B\theta_0}{3}\lambda^{lock} \mathcal{L}^{-1}\left(\frac{1}{\lambda^{lock}}\right)
+$$
+have been introduced to simplify the formulation.
+[[https://www.sciencedirect.com/science/article/abs/pii/0022509693900136 | Ellen M. Arruda, Mary C. Boyce,
+A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials, Journal of the Mechanics and Physics of Solids, Volume 41, Issue 2, 1993, Pages 389-412.]]
+=== Parameters ===
+^   Name                                                  ^  Metafor Code  ^ Dependency ^
+| Shear modulus ($\mu$)  |  ''HYPER_MU''  |  ''TO/TM''  |
+| Locking stretch ($\lambda_{lock}$)  |  ''HYPER_LOCK_STRETCH''  |  ''TO/TM''  |
@@ Line 168: / Line 200: @@
 === 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**.
+In the rheological analogy, the generalized Maxwell visco-elastic model consists in a main [[doc:user:elements:volumes:hyper_dev_potential|elastic deviatoric 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 [[doc:user:elements:volumes:hyper_dev_branchl|Maxwell branch material law]].
 {{ :doc:user:references:materials:maxwellgrid.png?400 |}}
@@ Line 178: / Line 210: @@
 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) ===
+=== Parameters ===
 ^   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''  |  -  |
+| Array of numbers defining the [[doc:user:elements:volumes:hyper_dev_branchl|Maxwell branch law]] [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''  |  -  |
-==== MaxwellBranch ====
-=== Description ===