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

The linearization of the various terms in $\delta W_{\text{int}}$ along $\Delta\tilde{p}$ yields $$\begin{equation} D\left(\mathbf{S}:\delta\mathbf{\dot{E}}\right)\left[\Delta\tilde{p}\right]\,dV=-\Delta\tilde{p}\,\divg\delta\mathbf{v}\,dv\,,\label{eq247} \end{equation}$$ $$\begin{equation} D\left(\mathbf{W}\cdot\Grad\delta\tilde{p}-\delta\tilde{p}\frac{\partial J}{\partial t}\right)\left[\Delta\tilde{p}\right]\,dV=-\grad\delta\tilde{p}\cdot\tilde{\mathbf{k}}\cdot\grad\Delta\tilde{p}\,dv\,,\label{eq248} \end{equation}$$ $$\begin{equation} D\left(\mathbf{J}\cdot\Grad\delta\tilde{c}-\delta\tilde{c}\frac{\partial\left(J\varphi^{w}\tilde{\kappa}\tilde{c}\right)}{\partial t}\right)\left[\Delta\tilde{p}\right]\,dV=-\frac{\tilde{\kappa}\tilde{c}}{d_{0}}\grad\delta\tilde{c}\cdot\mathbf{d}\cdot\tilde{\mathbf{k}}\cdot\grad\Delta\tilde{p}\,dv\,.\label{eq249} \end{equation}$$