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This material uses a strain-dependent permeability tensor that is transversely isotropic in the reference configuration, and accommodates strain-induced anisotropy [7]: $$\begin{equation} \begin{aligned}\mathbf{k} & =k_{0r}\left(\frac{J-\varphi_{r}^{s}}{1-\varphi_{r}^{s}}\right)^{\alpha_{0}}e^{M_{^{0}}\left(J^{2}-1\right)/2}\mathbf{I}\\ & +\left(\frac{k_{1r}^{T}}{J^{2}}\left(\mathbf{b}-\mathbf{m}\right)+\frac{k_{2r}^{T}}{2J^{4}}\left[2\mathbf{b}^{2}-\left(\mathbf{m}\cdot\mathbf{b}+\mathbf{b}\cdot\mathbf{m}\right)\right]\right)\left(\frac{J-\varphi_{r}^{s}}{1-\varphi_{r}^{s}}\right)^{\alpha_{T}}e^{M_{T}\left(J^{2}-1\right)/2}\\ & +\left(\frac{1}{J^{2}}k_{1r}^{A}\mathbf{m}+\frac{1}{2J^{4}}k_{2r}^{A}\left(\mathbf{m}\cdot\mathbf{b}+\mathbf{b}\cdot\mathbf{m}\right)\right)\left(\frac{J-\varphi_{r}^{s}}{1-\varphi_{r}^{s}}\right)^{\alpha_{A}}e^{M_{A}\left(J^{2}-1\right)/2}\,, \end{aligned} \label{eq508} \end{equation}$$ where $\mathbf{m}$ is a second order tensor representing the spatial structural tensor describing the axial direction, given by $$\begin{equation} \mathbf{m}=\mathbf{F}\cdot\left(\mathbf{V}\otimes\mathbf{V}\right)\cdot\mathbf{F}^{T}\,,\label{eq509} \end{equation}$$ and $\mathbf{V}$ is a unit vector along the axial direction. Note that the permeability in the reference state ($\mathbf{F}=\mathbf{I}$ ) is given by $\mathbf{k}=\left(k_{0r}+k_{1r}^{T}+k_{2r}^{T}\right)\mathbf{I}+\left(k_{1r}^{A}-k_{1r}^{T}+k_{2r}^{A}-k_{2r}^{T}\right)\left(\mathbf{V}\otimes\mathbf{V}\right)$ .