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

The linearization of the first term in $\delta W_{\text{int}}$ along $\Delta\tilde{c}^{\gamma}$ yields $$\begin{equation} D\left(\mathbf{S}:\delta\mathbf{\dot{E}}\right)\left[\Delta\tilde{c}^{\gamma}\right]dV=\Delta\tilde{c}^{\gamma}\left(\boldsymbol{\sigma}_{\gamma}^{\prime}:\delta\mathbf{d}-R\theta\frac{\partial\left(\Phi\tilde{\kappa}^{\alpha}\tilde{c}^{\alpha}\right)}{\partial\tilde{c}^{\gamma}}\divg\delta\mathbf{v}\right)\,dv\,,\label{eq310} \end{equation}$$ where $$\begin{equation} \boldsymbol{\sigma}_{\gamma}^{\prime}=J^{-1}\mathbf{F}\cdot\frac{\partial\mathbf{S}^{e}}{\partial\tilde{c}^{\gamma}}\cdot\mathbf{F}^{T}\label{eq311} \end{equation}$$ represents the spatial tangent of the stress with respect to the effective concentration. The next term is $$\begin{equation} D\left(\mathbf{W}\cdot\Grad\delta\tilde{p}\right)\left[\Delta\tilde{c}^{\gamma}\right]\,dV=\grad\delta\tilde{p}\cdot\mathbf{w}_{\gamma}^{\prime}\,dv\,,\label{eq312} \end{equation}$$ where $$\begin{equation} \begin{aligned}\mathbf{w}_{\gamma}^{\prime} & \equiv J^{-1}\mathbf{F}\cdot D\mathbf{W}\left[\Delta\tilde{c}^{\gamma}\right]=-\tilde{\mathbf{k}}_{\gamma}^{\prime}\cdot\left(\grad\tilde{p}+R\theta\sum\limits _{\beta}\frac{\tilde{\kappa}^{\beta}}{d_{0}^{\beta}}\mathbf{d}^{\beta}\cdot\grad\tilde{c}^{\beta}\right)\Delta\tilde{c}^{\gamma}\\ & -R\theta\tilde{\mathbf{k}}\cdot\frac{\tilde{\kappa}^{\gamma}}{d_{0}^{\gamma}}\mathbf{d}^{\gamma}\cdot\grad\Delta\tilde{c}^{\gamma}-\Delta\tilde{c}^{\gamma}R\theta\tilde{\mathbf{k}}\cdot\sum\limits _{\beta}\left(\frac{\partial}{\partial\tilde{c}^{\gamma}}\left(\frac{\tilde{\kappa}^{\beta}}{d_{0}^{\beta}}\right)\mathbf{d}^{\beta}+\frac{\tilde{\kappa}^{\beta}}{d_{0}^{\beta}}\mathbf{d}_{\gamma}^{\beta\prime}\right)\cdot\grad\tilde{c}^{\beta} \end{aligned} \label{eq313} \end{equation}$$ and $$\begin{equation} \tilde{\mathbf{k}}_{\gamma}^{\prime}=J^{-1}\mathbf{F}\cdot\frac{\partial\tilde{\mathbf{K}}}{\partial\tilde{c}^{\gamma}}\cdot\mathbf{F}^{T},\quad\mathbf{d}_{\gamma}^{\beta\prime}=J^{-1}\mathbf{F}\cdot\frac{\partial\mathbf{d}^{\beta}}{\partial\tilde{c}^{\gamma}}\cdot\mathbf{F}^{T}\label{eq314} \end{equation}$$ are the spatial tangents of the effective hydraulic permeability and solute diffusivity with respect to the effective concentration.