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

The linearization of the various terms in $\delta W_{\text{int}}$ along $\Delta\tilde{p}$ yields $$\begin{equation} D\left(\mathbf{S}:\delta\mathbf{\dot{E}}\right)\left[\Delta\tilde{p}\right]\,dV=-\Delta\tilde{p}\,\divg\delta\mathbf{v}\,dv\,,\label{eq307} \end{equation}$$ $$\begin{equation} D\left(\mathbf{W}\cdot\Grad\delta\tilde{p}-\delta\tilde{p}\frac{\partial J}{\partial t}\right)\left[\Delta\tilde{p}\right]\,dV=-\grad\delta\tilde{p}\cdot\tilde{\mathbf{k}}\cdot\grad\Delta\tilde{p}\,dv\,,\label{eq308} \end{equation}$$ $$\begin{equation} D\left(\mathbf{J}^{\alpha}\cdot\Grad\delta\tilde{c}^{\alpha}-\delta\tilde{c}^{\alpha}\frac{\partial\left(J\varphi^{w}\tilde{\kappa}^{\alpha}\tilde{c}^{\alpha}\right)}{\partial t}\right)\left[\Delta\tilde{p}\right]\,dV=-\frac{\tilde{\kappa}^{\alpha}\tilde{c}^{\alpha}}{d_{0}^{\alpha}}\grad\delta\tilde{c}^{\alpha}\cdot\mathbf{d}^{\alpha}\cdot\tilde{\mathbf{k}}\cdot\grad\Delta\tilde{p}\,dv\,.\label{eq309} \end{equation}$$