# Homogenized Balance Equations for Nonlinear Poroelastic Composites
## 2 Formulation of the Fluid-Structure Interaction Problem  
We assume that we have a continuum body that is not subject to any surface or body forces to have a reference configuration, which we denote by \(\mathcal{B}_{0}\in\mathbb{R}^{3}\). The body \(\mathcal{B}_{0}\) has a periodic microstructure that consists of the union of a porous hyperelastic matrix \(\Omega_{\text{f}}^{0}\), an interconnected fluid compartment \(\Omega_{\text{f}}^{0}\) and a set \(\Omega_{\text{i}}^{0}\) of \(K\) disjoint hyperelastic subphases \(\Omega_{\text{a}}^{0}\), where:  
\[\Omega_{\text{i}}^{0}=\bigcup_{\delta=1}^{K}\Omega_{\text{a}}^{0}, \tag{1}\]and \(\tilde{\Omega}^{0}=\tilde{\Omega}^{0}_{\rm I}\cup\tilde{\Omega}^{0}_{\rm II}\cup \tilde{\Omega}^{0}_{\rm I}\). We note that when any domain, interface or normal vector has the superscript \(0\) then it is in the reference configuration. We provide a sketch of this structure as shown in Figure 1. The body undergoes a deformation described by the deformation function \(\chi\), and the deformed body is denoted by \(\mathcal{B}_{\rm I}\). Each point \(\mathbf{x}\in\mathcal{B}_{\rm I}\) is such that \(\mathbf{x}=\chi(\mathbf{X},t)\) with \(\mathbf{X}\in\mathcal{B}_{0}\) being the point in the reference configuration. We assume that the periodicity of the body's microstructure is preserved during the deformation, and we now denote the deformed solid porous matrix as \(\Omega^{\rm I}_{\rm II}\), the deformed connected fluid compartment as \(\Omega^{\rm I}_{\rm I}\) and the set of \(K\) disjoint deformed subphases \(\Omega^{t}_{\alpha}\) as \(\Omega^{t}_{\rm I}\) in \(\mathcal{B}_{\rm I}\). When any domain, interface or normal vector has the superscript \(t\), this denotes the current configuration. The deformation from \(\mathcal{B}_{0}\) to \(\mathcal{B}_{\rm I}\) is described by the deformation gradient \(\mathbf{F}=\mathrm{Grad}\chi\). Figure 1 highlights the description of this structure pictorially. We now wish to describe the equations for the fluid and the solid compartments and the interface conditions in our structure. These equations however are not all in the same coordinate systems. We wish to work in the reference configuration so we have to perform a pull back to some of the equations that are described in the current configuration. We begin by describing all equations in their natural coordinate systems before discussing the coordinate transformations. The balance equations for the various hyperelastic domains \(\Omega_{\alpha}\) and \(\Omega_{\rm II}\) can be written as, \(\forall\alpha=1,\ldots,K\),  
\[\
abla_{\mathbf{X}}\cdot\mathbf{\sigma}_{\alpha}=0\quad\text{in}\quad\Omega^{0}_{ \alpha}, \tag{2}\]  
and:  
\[\
abla_{\mathbf{X}}\cdot\mathbf{\sigma}_{\rm II}=0\quad\text{in}\quad\Omega^{0}_{ \rm II}, \tag{3}\]  
where we neglect any volume forces and inertia. The tensors \(\mathbf{\sigma}_{\alpha}\) and \(\mathbf{\sigma}_{\rm II}\) are the solid stresses. Each subphase \(\Omega_{\alpha}\) has the stress tensor \(\mathbf{\sigma}_{\alpha}\), and the porous matrix \(\Omega_{\rm II}\) has solid stress tensor \(\mathbf{\sigma}_{\rm II}\). The matrix and the subphases are anisotropic, hyperelastic solids, and therefore, the constitutive laws for \(\mathbf{\sigma}_{\alpha}\) and \(\mathbf{\sigma}_{\rm II}\) are given in terms of strain energy functions,  
\[\mathbf{\sigma}_{\alpha} =\frac{\partial\psi_{\alpha}}{\partial\mathbf{F}_{\alpha}}\quad \text{in}\quad\Omega^{0}_{\alpha}, \tag{4}\] \[\mathbf{\sigma}_{\rm II} =\frac{\partial\psi_{\rm II}}{\partial\mathbf{F}_{\rm II}}\quad \text{in}\quad\Omega^{0}_{\rm II}, \tag{5}\]  
where \(\mathbf{F}_{\alpha}\) and \(\mathbf{F}_{\rm II}\) are the deformation gradients in the subphases and the matrix, respectively, and \(\psi_{\alpha}\) and \(\psi_{\rm II}\) are the strain energy functions in the subphases and matrix, respectively. We do not define a specific strain energy function at this stage and wait until Section 4.1 to specify it. We also require equations for the fluid phase.