perm-ref-ortho¶

Module: biphasic

Category: materialprop

Type string: "perm-ref-ortho"

Parameters¶

Name	Description	Default	Units
`perm0`	perm0	1	[L^4/F.t]
`M0`	M0	0	[]
`alpha0`	alpha0	0	[]
`perm1`	perm1		[L^4/F.t]
`perm2`	perm2		[L^4/F.t]
`M`	M		[]
`alpha`	alpha		[]

Description¶

The material type for a poroelastic material with strain-dependent permeability which is orthotropic in the reference configuration is perm-ref-ortho.

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

\[ \mathbf{k}=k_{0}\mathbf{I}+\sum\limits_{a=1}^{3}k_{1}^{a}\mathbf{m}_{a}+k_{2}^{a}\left(\mathbf{m}_{a}\cdot\mathbf{b}+\mathbf{b}\cdot\mathbf{m}_{a}\right)\,, \]

\[ \begin{aligned}k_{0} & =k_{0r}\left(\frac{J-\varphi_{r}^{s}}{1-\varphi_{0}}\right)^{\alpha_{^{0}}}e^{M_{^{0}}\left(J^{2}-1\right)/2}\\ k_{1}^{a} & =\frac{k_{1r}^{a}}{J^{2}}\left(\frac{J-\varphi_{r}^{s}}{1-\varphi_{0}}\right)^{\alpha_{^{a}}}e^{M_{^{a}}\left(J^{2}-1\right)/2}\\ k_{2}^{a} & =\frac{k_{2r}^{a}}{2J^{4}}\left(\frac{J-\varphi_{r}^{s}}{1-\varphi_{0}}\right)^{\alpha_{^{a}}}e^{M_{a}\left(J^{2}-1\right)/2} \end{aligned} ,\quad a=1,2,3\,. \]

\(J\) is the Jacobian of the deformation, i.e. \(J=\det\mathbf{F}\) where \(\mathbf{F}\) is the deformation gradient. \(\mathbf{m}_{a}\) are second order tensors representing the spatial structural tensors describing the orthogonal planes of symmetry, given by

\[ \mathbf{m}_{a}=\mathbf{F}\cdot\left(\mathbf{V}_{a}\otimes\mathbf{V}_{a}\right)\cdot\mathbf{F}^{T},\quad a=1-3, \]

where \(\mathbf{V}_{a}\) are orthonormal vectors normal to the planes of symmetry. 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 given by \(\mathbf{k}=k_{0r}\mathbf{I}+\sum\limits_{a=1}^{3}\left(k_{1r}^{a}+k_{2r}^{a}\right)\mathbf{V}_{a}\otimes\mathbf{V}_{a}\).

Example:

<permeability name="Permeability" type="perm-ref-ortho">
  <perm0>0.001</perm0>
  <perm1>0.01, 0.02, 0.03</perm1>
  <perm2>0.001, 0.002, 0.003</perm2>
  <M0>0.5</M0>
  <M>1.5, 2.0, 2.5</M>
  <alpha0>1.5</alpha0>
  <alpha>2, 2.5, 3</alpha>
</permeability>