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

A weak form of the statement conservation of linear momemtum for the quasi-static case is obtained by using Eqs.(2.5.1-2) and (2.5.1-4): $$\begin{equation} \delta W=\int_{b}\left[\delta\mathbf{v}^{s}\cdot\left(\divg\boldsymbol{\sigma}+\rho\mathbf{b}\right)+\delta p\,\divg\left(\mathbf{v}^{s}+\mathbf{w}\right)\right]dv=0\,,\label{eq193} \end{equation}$$ where $b$ is the domain of interest defined on the solid matrix, $\delta\mathbf{v}^{s}$ is a virtual velocity of the solid and $\delta p$ is a virtual pressure of the fluid [77]. $dv$ is an elemental volume of $b$ . Using the divergence theorem, this expression may be rearranged as $$\begin{equation} \begin{aligned}\delta W & =\int_{\partial b}\delta\mathbf{v}^{s}\cdot\mathbf{t}\,da+\int_{\partial b}\delta p\,w_{n}\,da+\int_{b}\delta\mathbf{v}^{s}\cdot\rho\mathbf{b}\,dv\\ & -\int_{b}\boldsymbol{\sigma}:\grad\delta\mathbf{v}^{s}\,dv-\int_{b}\left(\mathbf{w}\cdot\grad\delta p-\delta p\,\divg\mathbf{v}^{s}\right)\,dv \end{aligned} \,,\label{eq194} \end{equation}$$ where $\delta\mathbf{d}^{s}=\left(\grad\delta\mathbf{v}^{s}+\grad^{T}\delta\mathbf{v}^{s}\right)/2$ is the virtual rate of deformation tensor, $\mathbf{t}=\boldsymbol{\sigma}\cdot\mathbf{n}$ is the total traction on the surface $\partial b$ , and $w_{n}=\mathbf{w}\cdot\mathbf{n}$ is the component of the fluid flux normal to $\partial b$ , with $\mathbf{n}$ representing the unit outward normal to $\partial b$ . $da$ represents an elemental area of $\partial b$ . In this type of problem, essential boundary conditions are prescribed for $\mathbf{u}$ and $p$ , and natural boundary conditions are prescribed for $\mathbf{t}$ and $w_{n}$ . In the expression of Eq.(3.2-2), $\delta W\left(\boldsymbol{\chi}^{s},p,\delta\mathbf{v}^{s},\delta p\right)$ represents the virtual work.