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Holzapfel-Gasser-Ogden

Module: solid

Category: material

Type string: "Holzapfel-Gasser-Ogden"

Parameters

Name Description Default Units
density density 1 [M/L^3]
k bulk modulus 0 [P]
pressure_model pressure_model 3 []
c c 0 [P]
k1 k1 0 [P]
k2 k2 0 []
kappa kappa 0 []
gamma gamma 0 [d]
mat_axis []

Description

The material type for the uncoupled Holzapfel-Gasser-Ogden material 1 is Holzapfel-Gasser-Ogden.

This material model uncouples deviatoric and volumetric behaviors. The deviatoric strain-energy function is given by:

\[ \Psi_{r}=\tilde{\Psi}_{r}\left(\tilde{\mathbf{C}}\right)+U\left(J\right) \]

where

\[ \tilde{\Psi}_{r}=\frac{c}{2}\left(\tilde{I}_{1}-3\right)+\frac{k_{1}}{2k_{2}}\sum_{\alpha=1}^{2}\left(\exp\left(k_{2}\left\langle \tilde{E}_{\alpha}\right\rangle ^{2}\right)-1\right) \]

and the default volumetric strain energy function is

\[ U\left(J\right)=\frac{k}{2}\left(\frac{J^{2}-1}{2}-\ln J\right) \]

The fiber strain is

\[ \tilde{E}_{\alpha}=\kappa\left(\tilde{I}_{1}-3\right)+\left(1-3\kappa\right)\left(\tilde{I}_{4\alpha}-1\right) \]

where \(\tilde{I}_{1}=tr\tilde{\mathbf{C}}\) and \(\tilde{I}_{4\alpha}=\mathbf{a}_{\alpha r}\cdot\tilde{\mathbf{C}}\cdot\mathbf{a}_{\alpha r}\). The Macaulay brackets around \(\left\langle \tilde{E}_{\alpha}\right\rangle\) indicate that this term is zero when \(\tilde{E}_{\alpha}<0\) and equal to \(\tilde{E}_{\alpha}\) when this strain is positive.

There are two fiber families along the vectors \(\mathbf{a}_{\alpha r} (\alpha=1,2)\), lying in the \(\left\{ \mathbf{e}_{1},\mathbf{e}_{2}\right\}\) plane of the local material axes \(\left\{ \mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\right\}\), making an angle \(\pm\gamma\) with respect to \(\mathbf{e}_{1}\). Each fiber family has a dispersion \(\kappa\), where \(0\le\kappa\le\frac{1}{3}\). When \(\kappa=0\) there is no fiber dispersion, implying that all the fibers in that family act along the angle \(\pm\gamma\); the value \(\kappa=\frac{1}{3}\) represents an isotropic fiber dispersion. All other intermediate values of \(\kappa\) produce a \(\pi\)-periodic von Mises fiber distribution, as described in 1. \(c\) is the shear modulus of the ground matrix; \(k_{1}\) is the fiber modulus and \(k_{2}\) is the exponential coefficient.

Example: 

<material id="2" type="Holzapfel-Gasser-Ogden">
  <c>7.64</c>
  <k1>996.6</k1>
  <k2>524.6</k2>
  <gamma>49.98</gamma>
  <kappa>0.226</kappa>
  <k>1e5</k>
</material>

  1. Gasser, T Christian, Ogden, Ray W, and Holzapfel, Gerhard A, "Hyperelastic modelling of arterial layers with distributed collagen fibre orientations", J R Soc Interface 3, 6 (2006), pp. 15-35. ↩↩