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

For an elastic solid, the internal virtual work is $$\begin{equation} \begin{aligned}\delta W_{int} & =\int_{v}\boldsymbol{\sigma}:\grad\delta\mathbf{u}\,dv\\ & =\sum_{a}\left[\begin{array}{cc} \delta\mathbf{u}_{a} & \delta\mathbf{w}_{a}\end{array}\right]\left[\begin{array}{c} \mathbf{f}_{a}^{u}\\ \mathbf{f}_{a}^{w} \end{array}\right] \end{aligned} \,,\label{eq:virtual-work-internal} \end{equation}$$ where $$\begin{equation} \begin{aligned}\mathbf{f}_{a}^{u} & =\int_{v}\boldsymbol{\sigma}\cdot\grad\left(\frac{1+\xi_{3}}{2}M_{a}\right)\,dv\\ \mathbf{f}_{a}^{w} & =\int_{v}\boldsymbol{\sigma}\cdot\grad\left(\frac{1-\xi_{3}}{2}M_{a}\right)\,dv \end{aligned} \,.\label{eq:shell-internal-force} \end{equation}$$ For the external work of body forces, $$\begin{equation} \begin{aligned}\delta W_{ext} & =\int_{v}\delta\mathbf{u}\cdot\rho\mathbf{b}\,dv\\ & =\sum_{a=1}^{m}\delta\mathbf{u}_{a}\cdot\mathbf{f}_{a}^{u}+\delta\mathbf{w}_{a}\cdot\mathbf{f}_{a}^{w} \end{aligned} \,,\label{eq:virtual-work-external} \end{equation}$$ where $$\begin{equation} \begin{aligned}\mathbf{f}_{a}^{u} & =\int_{v}\frac{1+\xi_{3}}{2}M_{a}\rho\mathbf{b}\,dv\\ \mathbf{f}_{a}^{w} & =\int_{v}\frac{1-\xi_{3}}{2}M_{a}\rho\mathbf{b}\,dv \end{aligned} \,.\label{eq:shell-external-force} \end{equation}$$