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

The virtual work for the domain $b$ is given by $$\begin{equation} \delta W=\int_{b}\delta\mathbf{v}\cdot\left(\divg\boldsymbol{\sigma}+\rho\left(\mathbf{b}-\mathbf{a}\right)\right)\,dv\,,\label{eq:edy-virtual-work} \end{equation}$$ where $\delta\mathbf{v}$ is the virtual velocity. Using the divergence theorem, this virtual work may be expressed as the difference $\delta W=\delta W_{ext}-\delta W_{int}$ between external virtual work $\delta W_{ext}$ and internal virtual work $\delta W_{int}$ , where $$\begin{equation} \begin{aligned}\delta W_{int} & =\int_{b}\boldsymbol{\sigma}:\grad\delta\mathbf{v}\,dv+\int_{b}\delta\mathbf{v}\cdot\rho\mathbf{a}\,dv\,,\\ \delta W_{ext} & =\int_{\partial b}\delta\mathbf{v}\cdot\mathbf{t}\,da+\int_{b}\delta\mathbf{v}\cdot\rho\mathbf{b}\,dv\,, \end{aligned} \label{eq:edy-int-ext-virtual-work} \end{equation}$$ where $\mathbf{t}=\boldsymbol{\sigma}\cdot\mathbf{n}$ is the traction on the boundary $\partial b$ .