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doc:user:elements:volumes:hyper_dev_potential [2025/11/14 15:13] – [BonetBurtonHyperPotential] vanhulledoc:user:elements:volumes:hyper_dev_potential [2026/07/02 15:47] (current) – [EightChainHyperPotential] vanhulle
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 | Yeoh coefficient ($C_2$)  |  ''HYPER_C2''  |  ''TO/TM''  | | Yeoh coefficient ($C_2$)  |  ''HYPER_C2''  |  ''TO/TM''  |
 | Yeoh coefficient ($C_3$)  |  ''HYPER_C3''  |  ''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''  |
  
  
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 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) 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 === === Reminders ===
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 | Holzapfel-Gasser-Ogden coefficient ($k_1$)  |  ''HYPER_HGO_K1''  |  ''TO/TM''  | | Holzapfel-Gasser-Ogden coefficient ($k_1$)  |  ''HYPER_HGO_K1''  |  ''TO/TM''  |
 | Holzapfel-Gasser-Ogden coefficient ($k_2$)  |  ''HYPER_HGO_K2''  |  ''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''  |  -  |
  
  
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 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. 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 === === Parameters ===
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 | Bonet-Burton coefficient ($\gamma$)  |  ''HYPER_BB_GAMMA''  |  ''TO/TM''  | | 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''  | | 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 [[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
 +$$
 +\boldsymbol{\sigma}^{n+1} = \boldsymbol{\sigma}^{n+1}_0+ \sum_{j=1}^N \mathbf{h}_j^{n+1},
 +$$
 +where $\boldsymbol{\sigma}_0$ is the stress in the main elastic branch and $\mathbf{h}_j$ is the non-equilibrium stress from Maxwell branch $j$.
 +
 +
 +
 +=== 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''  |  -  |
 +
 +
 +
  
  
doc/user/elements/volumes/hyper_dev_potential.1763129582.txt.gz · Last modified: by vanhulle
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