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

The linearization of the first term in $\delta W_{\text{int}}$ along $\Delta\tilde{c}$ yields $$\begin{equation} D\left(\mathbf{S}:\delta\mathbf{\dot{E}}\right)\left[\Delta\tilde{c}\right]\,dV=\Delta\tilde{c}\left(\boldsymbol{\sigma}_{c}^{\prime}:\delta\mathbf{d}-R\theta\frac{\partial\left(\Phi\tilde{\kappa}\tilde{c}\right)}{\partial\tilde{c}}\divg\delta\mathbf{v}\right)\,dv\,,\label{eq250} \end{equation}$$ where $$\begin{equation} \boldsymbol{\sigma}_{c}^{\prime}=J^{-1}\mathbf{F}\cdot\frac{\partial\mathbf{S}^{e}}{\partial\tilde{c}}\cdot\mathbf{F}^{T}\label{eq251} \end{equation}$$ represents the spatial tangent of the stress with respect to the effective concentration. The next term is $$\begin{equation} D\left(\mathbf{W}\cdot\Grad\delta\tilde{p}\right)\left[\Delta\tilde{c}\right]\,dV=\grad\delta\tilde{p}\cdot\mathbf{w}_{c}^{\prime}\,dv\,,\label{eq252} \end{equation}$$ where $$\begin{equation} \begin{aligned}\mathbf{w}_{c}^{\prime} & \equiv J^{-1}\mathbf{F}\cdot D\mathbf{W}\left[\Delta\tilde{c}\right]=-\Delta\tilde{c}\,\tilde{\mathbf{k}}_{c}^{\prime}\cdot\left(\grad\tilde{p}+R\theta\frac{\tilde{\kappa}}{d_{0}}\mathbf{d}\cdot\grad\tilde{c}\right)\\ & -R\theta\tilde{\mathbf{k}}\cdot\left[\Delta\tilde{c}\left(\frac{\partial}{\partial\tilde{c}}\left(\frac{\tilde{\kappa}}{d_{0}}\right)\mathbf{d}+\frac{\tilde{\kappa}}{d_{0}}\mathbf{d}_{c}^{\prime}\right)\cdot\grad\tilde{c}+\frac{\tilde{\kappa}}{d_{0}}\mathbf{d}\cdot\grad\Delta\tilde{c}\right]\,, \end{aligned} \label{eq253} \end{equation}$$ and $$\begin{equation} \tilde{\mathbf{k}}_{c}^{\prime}=J^{-1}\mathbf{F}\cdot\frac{\partial\tilde{\mathbf{K}}}{\partial\tilde{c}}\cdot\mathbf{F}^{T}\label{eq254} \end{equation}$$ is the spatial tangent of the effective hydraulic permeability with respect to the effective concentration.