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--- doc:user:elements:volumes:hyper_dev_potential [2025/11/14 15:31] – [HolzapfelGasserOgdenHyperPotential] 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 78: / Line 110: @@
 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 160: @@
 | 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''  |
-| Direction of the first principal (fiber) direction ($a^1_x$) |  ''HYPER_FIB1_X''  |  -  |
+| Array of $\theta$ angles defining the principal directions [$\theta_1$,...,$\theta_n$] |  ''HYPER_FIBS_THETA''  |  -  |
-| Direction of the first principal (fiber) direction ($a^1_y$) |  ''HYPER_FIB1_Y''  |  -  |
+| Array of $\theta$ angles defining the principal directions [$\delta_1$,...,$\delta_n$] |  ''HYPER_FIBS_DELTA''  |  -  |
-| Direction of the first principal (fiber) direction ($a^1_z$) |  ''HYPER_FIB1_Z''  |  -  |
-| Direction of the second principal (fiber) direction ($a^2_x$) |  ''HYPER_FIB2_X''  |  -  |
-| Direction of the second principal (fiber) direction ($a^2_y$) |  ''HYPER_FIB2_Y''  |  -  |
-| Direction of the second principal (fiber) direction ($a^2_z$) |  ''HYPER_FIB2_Z''  |  -  |
-| Direction of the third principal (fiber) direction ($a^3_x$) |  ''HYPER_FIB3_X''  |  -  |
-| Direction of the third principal (fiber) direction ($a^3_y$) |  ''HYPER_FIB3_Y''  |  -  |
-| Direction of the third principal (fiber) direction ($a^3_z$) |  ''HYPER_FIB3_Z''  |  -  |
@@ Line 150: / Line 182: @@
 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 158: / Line 190: @@
 | 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''  |  -  |
-====== 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 [[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 |}}
+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 |}}
+=== Parameters ===
+^   Name                                                  ^  Metafor Code  ^ Dependency ^
+| Number of the main elastic potential $\sigma_0$  |  ''HYPER_MAIN_POTENTIAL_NO''  |  -  |
+| Array of numbers defining the [[doc:user:elements:volumes:hyper_dev_branchl|Maxwell branch law]] [1, 2, ...]  |  ''HYPER_MAXWELL_BRANCH_NUMS''  |  -  |
-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.