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--- doc:user:elements:volumes:hyper_dev_potential [2026/07/02 14:55] – vanhulle
+++ doc:user:elements:volumes:hyper_dev_potential [2026/07/02 15:47] (current) – [EightChainHyperPotential] vanhulle
@@ Line 76: / Line 76: @@
 === 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 ^
-| Neo-Hookean coefficient ($C_1$)  |  ''HYPER_C1''  |  ''TO/TM''  |
+| Shear modulus ($\mu$)  |  ''HYPER_MU''  |  ''TO/TM''  |
+| Locking stretch ($\lambda_{lock}$)  |  ''HYPER_LOCK_STRETCH''  |  ''TO/TM''  |
@@ Line 177: / 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 192: / Line 215: @@
 ^   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''  |  -  |