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

The strong form of the virtual work statement is $\delta W=0$ where $$\begin{equation} \begin{aligned}\delta W & =\int_{b}\delta\mathbf{v}^{s}\cdot\divg\boldsymbol{\sigma}^{s}\,dv\\ & +\int_{b}\delta\mathbf{w}\cdot\left[\divg\boldsymbol{\sigma}^{f}+\rho^{f}\left(\mathbf{b}-\mathbf{a}^{f}\right)\right]\,dv\\ & -\int_{b}\delta J^{f}\left[\frac{1}{J^{f}}\left(\dot{J}^{f}+\grad J^{f}\cdot\mathbf{w}\right)-\divg\mathbf{w}-\frac{\dot{J}^{s}}{J^{s}}\right]\,dv \end{aligned} \,.\label{eq:fsi-virtual-work-strong} \end{equation}$$ Here, $b$ is the mixture domain (whose boundaries are defined on the solid), $\delta\mathbf{v}^{s}$ is the virtual solid velocity, $\delta\mathbf{w}$ is the virtual relative fluid velocity, and $\delta J^{f}$ is the virtual energy density. The weak form of this statement may be written 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 $$\begin{equation} \begin{aligned}\delta W_{int} & =\int_{b}\boldsymbol{\sigma}^{s}:\grad\delta\mathbf{v}^{s}\,dv\\ & +\int_{b}\boldsymbol{\tau}:\grad\delta\mathbf{w}\,dv+\int_{b}\delta\mathbf{w}\cdot\left(\grad p+\rho^{f}\mathbf{a}^{f}\right)\,dv\\ & +\int_{b}\delta J^{f}\left[\frac{1}{J^{f}}\left(\dot{J}^{f}+\grad J^{f}\cdot\mathbf{w}\right)-\frac{\dot{J}^{s}}{J^{s}}\right]\,dv\\ & +\int_{b}\mathbf{w}\cdot\grad\delta J^{f}\,dv \end{aligned} \,,\label{eq:fsi-internal-virtual-work} \end{equation}$$ and $$\begin{equation} \begin{aligned}\delta W_{ext} & =\int_{\partial b}\delta\mathbf{v}^{s}\cdot\mathbf{t}^{s}\,da\\ & +\int_{\partial b}\delta\mathbf{w}\cdot\mathbf{t}^{\tau}\,da+\int_{b}\delta\mathbf{w}\cdot\rho^{f}\mathbf{b}\,dv\\ & +\int_{\partial b}\delta J^{f}w_{n}\,da \end{aligned} \,.\label{eq:fsi-external-virtual-work} \end{equation}$$ Here, $\mathbf{t}^{s}=\boldsymbol{\sigma}^{s}\cdot\mathbf{n}$ is the solid traction, $\mathbf{t}^{\tau}=\boldsymbol{\tau}\cdot\mathbf{n}$ is the fluid viscous traction, and $w_{n}=\mathbf{w}_{n}\cdot\mathbf{n}$ is the normal component of the relative fluid velocity on the boundary $\partial b$ , whose outward unit normal is $\mathbf{n}$ . In practice, both $\boldsymbol{\sigma}^{s}$ and $\mathbf{t}^{s}$ should contribute negligibly to $\delta W$ , but they cannot be set exactly to zero since we need some small elasticity to regularize the deforming mixture mesh.