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

The first term in the internal work becomes $$\begin{equation} \begin{aligned}\int_{b}\boldsymbol{\sigma}:\grad\delta\mathbf{v}\,dv & =\int_{B}\mathbf{F}\cdot\mathbf{S}:\Grad\delta\mathbf{v}\,dV\end{aligned} \,,\label{eq:work-material-1-1} \end{equation}$$ where $\mathbf{S}=J\cdot\mathbf{F}^{-1}\cdot\boldsymbol{\sigma}\cdot\mathbf{F}^{-T}$ is the second Piola-Kirchhoff stress for the solid material. In general, $\boldsymbol{\sigma}$ (and thus, $\mathbf{S}$ ) is only a function of the solid strain, such as the right Cauchy-Green tensor $\mathbf{C}=\mathbf{F}^{T}\cdot\mathbf{F}$ or the Green-Lagrange strain $\mathbf{E}=\left(\mathbf{C}-\mathbf{I}\right)/2$ . $$\begin{equation} D\mathbf{E}\left[\Delta\mathbf{u}\right]=\frac{\alpha_{f}}{2}\left(\Grad^{T}\Delta\mathbf{u}\cdot\mathbf{F}+\mathbf{F}^{T}\cdot\Grad\Delta\mathbf{u}\right)\,.\label{eq:edy-E-linearization} \end{equation}$$ Therefore, following the standard approach in solid mechanics, the linearization of $\mathbf{S}$ is $$\begin{equation} \begin{aligned}D\mathbf{S}\left[\Delta\mathbf{u}\right] & =\frac{\partial\mathbf{S}}{\partial\mathbf{E}}:D\mathbf{E}\left[\Delta\mathbf{u}\right]\\ & =\alpha_{f}\boldsymbol{\mathbb{C}}:\frac{1}{2}\left(\Grad^{T}\Delta\mathbf{u}\cdot\mathbf{F}+\mathbf{F}^{T}\cdot\Grad\Delta\mathbf{u}\right)\\ & =\alpha_{f}\boldsymbol{\mathbb{C}}:\left(\mathbf{F}^{T}\,\underline{\otimes}\,\mathbf{F}^{T}\right):\Delta\boldsymbol{\varepsilon} \end{aligned} \,,\label{eq:edy-solid-stress-linearization} \end{equation}$$ where $\boldsymbol{\mathbb{C}}$ is the material elasticity tensor. Now, the linearization of the first term in $\delta W_{int}$ is $$\begin{equation} \boxed{\begin{aligned} & D\left(\int_{B}\mathbf{F}\cdot\mathbf{S}:\Grad\delta\mathbf{v}\,dV\right)\left[\Delta\mathbf{u}\right]\\ & =\int_{v}\alpha_{f}\left(\grad\delta\mathbf{v}:\grad\Delta\mathbf{u}\cdot\boldsymbol{\sigma}+\grad\delta\mathbf{v}:\boldsymbol{\mathcal{C}}:\grad\Delta\mathbf{u}\right)\,dv \end{aligned} }\,.\label{eq:edy-Dint-1} \end{equation}$$ where $\boldsymbol{\mathcal{C}}$ is the spatial elasticity tensor. Similarly, the second term in $\delta W_{int}$ produces $$\begin{equation} \boxed{D\left(\int_{B}\delta\mathbf{v}\cdot\rho_{r}\mathbf{a}\,dV\right)\left[\Delta\mathbf{u}\right]=\int_{b}\delta\mathbf{v}\cdot\frac{\alpha_{m}}{\beta\Delta t^{2}}\rho\Delta\mathbf{u}\,dv}\,.\label{eq:edy-Dint-2} \end{equation}$$