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HGO unconstrained

Module: solid

Category: material

Type string: "HGO unconstrained"

Parameters

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

Description

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

The strain-energy function is given by:

\[ \begin{aligned}\Psi_{r} & =\frac{c}{2}\left(I_{1}-3\right)-c\ln J+\frac{k_{1}}{2k_{2}}\sum_{\alpha}\left(\exp\left(k_{2}\left\langle E_{\alpha}\right\rangle ^{2}\right)-1\right)\\ & +\frac{K_{0}}{2}\left(\frac{J^{2}-1}{2}-\ln J\right) \end{aligned} \]

The fiber strain is

\[ E_{\alpha}=\kappa\left(I_{1}-3\right)+\left(1-3\kappa\right)\left(I_{4\alpha}-1\right) \]

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

Unlike the uncoupled Holzapfel-Gasser-Ogden material presented in Section Holzapfel-Gasser-Ogden, this unconstrained version does not enforce isochoric deformation. This unconstrained model may be used to describe the porous solid matrix of a biphasic or multiphasic tissue model, where pore volume may change in response to influx or efflux of interstitial fluid.

Example: 

<material id="2" type="HGO unconstrained">
  <c>7.64</c>
  <k1>996.6</k1>
  <k2>524.6</k2>
  <gamma>49.98</gamma>
  <kappa>0.226</kappa>
  <k>7.64e3</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. ↩↩