# Homogenized Balance Equations for Nonlinear Poroelastic Composites
## 2 Formulation of the Fluid-Structure Interaction Problem
The balance equation is given as:  
\[\
abla_{\mathbf{x}}\cdot\mathbf{\sigma}_{\rm f}=0\quad\text{in}\quad\Omega^{t}_{ \rm f}, \tag{6}\]  
where we denoted the fluid stress tensor by \(\sigma_{\rm f}\). The fluid is assumed to be an incompressible Newtonian fluid and, so, has the constitutive law:  
\[\mathbf{\sigma}_{\rm f}=-p\mathbf{I}+\mu((\
abla_{\mathbf{x}}\mathbf{v})+(\
abla _{\mathbf{x}}\mathbf{v})^{\rm T})\quad\text{in}\quad\Omega^{t}_{\rm f}, \tag{7}\]  
where \(\mathbf{v}\) is the fluid velocity, \(p\) is the fluid pressure and \(\mu\) is the fluid viscosity. As we consider an incompressible fluid, the incompressibility condition reads:  
\[\
abla_{\mathbf{x}}\cdot\mathbf{v}=0\quad\text{in}\quad\Omega^{t}_{\rm f}. \tag{8}\]  
We can substitute the constitutive law (7) into the balance Equation (6), and then, by using (8), we obtain the Stokes problem:  
\[0=-\
abla_{\mathbf{x}}p+\mu\
abla_{\mathbf{x}}\cdot((\
abla_{\mathbf{x}} \mathbf{v})+(\
abla_{\mathbf{x}}\mathbf{v})^{\rm T})\quad\text{in}\quad\Omega ^{t}_{\rm f}. \tag{9}\]In order to set up an appropriate fluid-structure interaction problem, we require interface conditions between the fluid and the various solid phases and also interface conditions between the various solid subphases and the matrix. The conditions we impose are the continuity of velocities, the continuity of stresses and the continuity of displacements. This presents an issue since the fluid and solid equations are described in different coordinate systems. The fluid equations are currently presented in Eulerian coordinates and the solid equations in Lagrangian coordinates. It is not possible to properly express continuity on the interface between the fluid and solid whilst the governing equations are in different coordinate systems. For this reason, we only describe the interface conditions between the elastic subphases and the matrix here and wait until Section 2.1 to describe the interface conditions between the fluid and solids once we have formulated all equations in Lagrangian coordinates. We define the interface between each elastic subphase and the matrix as \(\Gamma^{0}_{\alpha\Pi}:=\partial\Omega^{0}_{\alpha}\cap\partial\Omega^{0}_{ \Pi}\) and impose continuity of stresses and displacements, namely:  
Figure 1.0 Figure 1.1  
\(\forall\alpha=1,\dots,K\), where we define the unit vector normal to the interface \(\Gamma^{0}_{\alpha\Pi}\) as \(\mathbf{n}^{0}_{\alpha\Pi}\), and it is pointing into the subphase \(\Omega^{0}_{\alpha}\) and where \(\mathbf{u}_{\alpha}\) and \(\mathbf{u}_{\Pi}\) are the elastic displacements in each subphase and the matrix, respectively. We describe the deformation from \(\mathcal{B}_{0}\) to \(\mathcal{B}_{t}\) by the deformation gradient \(\mathbf{F}=\text{Grad}\chi\). For convenience, we use the notation:  
\[\mathbf{F}= \begin{cases}\mathbf{F}_{\alpha}&\text{in}\quad\Omega^{0}_{\alpha },\\ \mathbf{F}_{\Pi}&\text{in}\quad\Omega^{0}_{\Pi},\\ \mathbf{F}_{\text{f}}&\text{in}\quad\Omega^{0}_{\text{f}}.\end{cases}  \tag{12}\]  
We note that our deformation gradient is in general discontinuous. It is however useful here to consider the alternative definition of the deformation gradient:  
\[\mathbf{F}=\mathbf{I}+\text{Grad}\mathbf{u}, \tag{13}\]  
where \(\mathbf{I}\) is the identity tensor and \(\text{Grad}\mathbf{u}\) is the gradient operator of the elastic displacement. We use the notation Grad with a capital G to denote the gradient in Lagrangian coordinates and grad for Eulerian coordinates. We can specialise relationship (13) in each of the reference elastic subphases as follows:  
Figure 1: A 2D sketch that shows the poroelastic composite microstructure of the body in the reference configuration \(\mathcal{B}_{0}\). It also shows the deformation \(\mathbf{F}\) and gives the resulting microstructure of the deformed/current configuration \(\mathcal{B}_{t}\). In both configurations, the porous matrix is shown in red, the subphases in green and the fluid in blue. \[\mathbf{F}=