perm-ref-iso

Module: biphasic

Category: materialprop

Type string: "perm-ref-iso"

Parameters

Name	Description	Default	Units
perm0	perm0	1	[L^4/F.t]
perm1	perm1	0	[L^4/F.t]
perm2	perm2	0	[L^4/F.t]
M	M	0	[]
alpha	alpha	0	[]

Description

The material type for a biphasic material with strain-dependent permeability which is isotropic in the reference configuration is perm-ref-iso.

This material uses a strain-dependent permeability tensor that accommodates strain-induced anisotropy:

\[ \mathbf{k}=\left(k_{0r}\mathbf{I}+\frac{k_{1r}}{J^{2}}\mathbf{b}+\frac{k_{2r}}{J^{4}}\mathbf{b}^{2}\right)\left(\frac{J-\varphi_{r}^{s}}{1-\varphi_{0}}\right)^{\alpha}e^{M\left(J^{2}-1\right)/2}, \]

where \(J\) is the Jacobian of the deformation, i.e. \(J=\det\mathbf{F}\) where \(\mathbf{F}\) is the deformation gradient, and \(\mathbf{b}=\mathbf{F}\cdot\mathbf{F}^{T}\) is the left Cauchy-Green tensor. Here, \(\varphi_{r}^{s}\) is the referential solid volume fraction in the current configuration and \(\varphi_{0}=\varphi_{r}^{s}\left(0\right)\) is its value in the reference configuration. Note that the permeability in the reference state (\(\mathbf{F}=\mathbf{I}\)) is isotropic and given by \(\mathbf{k}=\left(k_{0r}+k_{1r}+k_{2r}\right)\mathbf{I}\).

Example:

<permeability name="Permeability" type="perm-ref-iso">
  <perm0>0.001</perm0>
  <perm1>0.005</perm1>
  <perm2>0.002</perm2>
  <M>1.5</M>
  <alpha>2</alpha>
</permeability>