$\newcommand{\Dev}{\operatorname{Dev}}$ $\newcommand{\dev}{\operatorname{dev}}$ $\newcommand{\grad}{\operatorname{grad}}$ $\newcommand{\Grad}{\operatorname{Grad}}$ $\newcommand{\divg}{\operatorname{div}}$ $\newcommand{\Ei}{\operatorname{Ei}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\Divg}{\operatorname{Div}}$ $\newcommand{\cay}{\operatorname{cay}}$ $\newcommand{\rot}{\operatorname{rot}}$

The linearization of the first term in $\delta W_{\text{int}}$ along $\Delta\mathbf{u}$ yields $$\begin{equation} \left(\mathbf{S}:\delta\dot{\mathbf{E}}\right)\left[\Delta\mathbf{u}\right]\,dV=\left[\delta\mathbf{d}^{s}:\boldsymbol{\mathcal{C}}:\Delta\boldsymbol{\varepsilon}+\boldsymbol{\sigma}:\left(\grad^{T}\Delta\mathbf{u}\cdot\grad\delta\mathbf{v}\right)\right]\,dv\,,\label{eq233} \end{equation}$$ where $\boldsymbol{\mathcal{C}}$ is the spatial elasticity tensor of the mixture, $$\begin{equation} \boldsymbol{\mathcal{C}}=\boldsymbol{\mathcal{C}}^{e}-\left(\tilde{p}+R\theta\,\Phi\tilde{\kappa}\tilde{c}\right)\left(\mathbf{I}\otimes\mathbf{I}-2\mathbf{I}\,\overline{\underline{\otimes}}\,\mathbf{I}\right)-R\theta\tilde{c}\,J\frac{\partial\left(\Phi\tilde{\kappa}\right)}{\partial J}\mathbf{I}\otimes\mathbf{I}\,,\label{eq234} \end{equation}$$ and $\boldsymbol{\mathcal{C}}^{e}$ is the spatial elasticity tensor of the solid matrix, $$\begin{equation} \boldsymbol{\mathcal{C}}^{e}=J^{-1}\left(\mathbf{F}\,\underline{\otimes}\,\mathbf{F}\right):2\frac{\partial\mathbf{S}^{e}}{\partial\mathbf{C}}:\left(\mathbf{F}^{T}\,\underline{\otimes}\,\mathbf{F}^{T}\right)\,.\label{eq235} \end{equation}$$ The linearization of the second term is $$\begin{equation} D\left(\mathbf{W}\cdot\Grad\delta\tilde{p}\right)\left[\Delta\mathbf{u}\right]\,dV=\grad\delta\tilde{p}\cdot\mathbf{w}_{u}^{\prime}\,dv\,,\label{eq236} \end{equation}$$ where $$\begin{equation} \begin{aligned}\mathbf{w}_{u}^{\prime} & \equiv J^{-1}\mathbf{F}\cdot D\mathbf{W}\left[\Delta\mathbf{u}\right]=-\left(\tilde{\boldsymbol{\mathcal{K}}}:\Delta\boldsymbol{\varepsilon}\right)\cdot\left(\grad\tilde{p}+R\theta\frac{\tilde{\kappa}}{d_{0}}\mathbf{d}\cdot\grad\tilde{c}\right)\\ & -\frac{R\theta}{d_{0}}\tilde{\mathbf{k}}\cdot\left(J^{2}\frac{\partial\left(J^{-1}\tilde{\kappa}\right)}{\partial J}\left(\divg\Delta\mathbf{u}\right)\mathbf{I}+2\tilde{\kappa}\,\Delta\boldsymbol{\varepsilon}\right)\cdot\mathbf{d}\cdot\grad\tilde{c}-\tilde{\kappa}\frac{R\theta}{d_{0}}\tilde{\mathbf{k}}\cdot\left(\boldsymbol{\mathcal{D}}:\Delta\boldsymbol{\varepsilon}\right)\cdot\grad\tilde{c}\,, \end{aligned} \label{eq237} \end{equation}$$ with $$\begin{equation} \begin{aligned}\tilde{\boldsymbol{\mathcal{K}}} & =J^{-1}\left(\mathbf{F}\,\underline{\otimes}\,\mathbf{F}\right):2\frac{\partial\tilde{\mathbf{K}}}{\partial\mathbf{C}}:\left(\mathbf{F}^{T}\,\underline{\otimes}\,\mathbf{F}^{T}\right)\,,\\ \boldsymbol{\mathcal{D}} & =J^{-1}\left(\mathbf{F}\,\underline{\otimes}\,\mathbf{F}\right):2\frac{\partial\mathbf{D}}{\partial\mathbf{C}}:\left(\mathbf{F}^{T}\,\underline{\otimes}\,\mathbf{F}^{T}\right)\,, \end{aligned} \label{eq238} \end{equation}$$ representing the spatial tangents, with respect to the strain, of the effective permeability and solute diffusivity, respectively. These fourth-order tensors exhibit minor symmetries but not major symmetry, as described recently [7]. Since $\tilde{\mathbf{K}}$ is given by substituting (2.6.2-3)$_{3}$ into (3.3-8)$_{1}$ , the evaluation of $\tilde{\boldsymbol{\mathcal{K}}}$ is rather involved and it can be shown that $$\begin{equation} \tilde{\boldsymbol{\mathcal{K}}}=2\left(\tilde{\mathbf{k}}\otimes\mathbf{I}-2\tilde{\mathbf{k}}\,\underline{\otimes}\,\mathbf{I}\right)-\left(\tilde{\mathbf{k}}\,\overline{\underline{\otimes}}\,\tilde{\mathbf{k}}\right):\boldsymbol{\mathcal{G}}\,,\label{eq239} \end{equation}$$ where $$\begin{equation} \begin{aligned}\boldsymbol{\mathcal{G}} & =2\left(\mathbf{k}^{-1}\otimes\mathbf{I}-2\mathbf{k}^{-1}\,\overline{\underline{\otimes}}\,\mathbf{I}\right)-\left(\mathbf{k}^{-1}\,\underline{\otimes}\,\mathbf{k}^{-1}\right):\boldsymbol{\mathcal{K}}\\ & +\frac{R\theta\tilde{c}}{d_{0}}J\frac{\partial}{\partial J}\left(\frac{\tilde{\kappa}}{\varphi^{w}}\right)\left(\mathbf{I}-\frac{\mathbf{d}}{d_{0}}\right)\otimes\mathbf{I}\\ & +\frac{R\theta\tilde{c}}{d_{0}}\frac{\tilde{\kappa}}{\varphi^{w}}\left(\mathbf{I}\otimes\mathbf{I}-2\mathbf{I}\,\underline{\otimes}\,\mathbf{I}-\frac{1}{d_{0}}\boldsymbol{\mathcal{D}}\right) \end{aligned} \label{eq240} \end{equation}$$ and $$\begin{equation} \boldsymbol{\mathcal{K}}=J^{-1}\left(\mathbf{F}\,\underline{\otimes}\,\mathbf{F}\right):2\frac{\partial\mathbf{K}}{\partial\mathbf{C}}:\left(\mathbf{F}^{T}\,\underline{\otimes}\,\mathbf{F}^{T}\right)\,.\label{eq241} \end{equation}$$ The next term in $\delta W_{\text{int}}$ linearizes to $$\begin{equation} -D\left(\delta\tilde{p}\frac{\partial J}{\partial t}\right)\left[\Delta\mathbf{u}\right]\,dV=-\delta\tilde{p}\frac{1}{\Delta t}\divg\Delta\mathbf{u}\,dv\,,\label{eq242} \end{equation}$$ where we used a backward difference scheme to approximate the time derivative, $$\begin{equation} \frac{\partial J}{\partial t}\approx\frac{1}{\Delta t}\left(J-J^{-\Delta t}\right)\,,\label{eq243} \end{equation}$$ and $\Delta t$ represents the time increment relative to the previous time point. The next term is given by $$\begin{equation} D\left(\mathbf{J}\cdot\Grad\delta\tilde{c}\right)\left[\Delta\mathbf{u}\right]\,dV=\grad\delta\tilde{c}\cdot\mathbf{j}_{u}^{\prime}\,dv,\label{eq244} \end{equation}$$ where $$\begin{equation} \begin{aligned}\mathbf{j}_{u}^{\prime} & \equiv J^{-1}\mathbf{F}\cdot D\mathbf{J}\left[\Delta\mathbf{u}\right]=\left(J\frac{\partial\tilde{\kappa}}{\partial J}\left(\divg\Delta\mathbf{u}\right)\mathbf{d}+\tilde{\kappa}\boldsymbol{\mathcal{D}}:\Delta\boldsymbol{\varepsilon}\right)\cdot\left(-\varphi^{w}\grad\tilde{c}+\frac{\tilde{c}}{d_{0}}\mathbf{w}\right)\\ & +\tilde{\kappa}\mathbf{d}\cdot\left(-\varphi^{s}\left(\divg\Delta\mathbf{u}\right)\grad\tilde{c}+\frac{\tilde{c}}{d_{0}}\left(2\Delta\boldsymbol{\varepsilon}-\left(\divg\Delta\mathbf{u}\right)\mathbf{I}\right)\cdot\mathbf{w}\right)+\tilde{\kappa}\frac{\tilde{c}}{d_{0}}\mathbf{d}\cdot\mathbf{w}_{u}^{\prime} \end{aligned} \,.\label{eq245} \end{equation}$$ Using a backward difference scheme for the time derivative, the last term is $$\begin{equation} D\left(\delta\tilde{c}\frac{\partial\left(J\varphi^{w}\tilde{\kappa}\tilde{c}\right)}{\partial t}\right)\left[\Delta\mathbf{u}\right]\,dV=\frac{\delta\tilde{c}}{\Delta t}\frac{\partial\left(J\varphi^{w}\tilde{\kappa}\right)}{\partial J}\tilde{c}\divg\Delta\mathbf{u}\,dv\,.\label{eq246} \end{equation}$$