# Homogenized Balance Equations for Nonlinear Poroelastic Composites
## 2 Formulation of the Fluid-Structure Interaction Problem
### Fluid-Structure Interaction in Lagrangian Coordinates
The equations governing the fluid in the reference configuration are therefore given by:  
\[\
abla_{\mathbf{x}}\cdot(\boldsymbol{\sigma}_{\text{f}}\text{G}_{\text{f}}^{ \text{T}})=0\quad\text{in}\quad\Omega_{\text{f}}^{0}, \tag{24}\]  
with the fluid stress tensor \(\boldsymbol{\sigma}_{\text{f}}\) transformed as:  
\[\boldsymbol{\sigma}_{\text{f}}\text{G}_{\text{f}}^{\text{T}}=-p\text{G}_{ \text{f}}^{\text{T}}+\mu((\
abla_{\mathbf{x}}\mathbf{V})\mathbf{F}_{\text{f}}^ {-1}\text{G}_{\text{f}}^{\text{T}}+\mathbf{F}_{\text{f}}^{-\text{T}}(\
abla_ {\mathbf{x}}\mathbf{V})^{\text{T}}\text{G}_{\text{f}}^{\text{T}})\quad\text{ in}\quad\Omega_{\text{f}}^{0}, \tag{25}\]  
to correspond to the fluid balance equation, where \(\mathbf{V}\) is the fluid velocity in the reference configuration. The incompressibility condition also transforms as follows:  
\[\
abla_{\mathbf{x}}\cdot(\text{G}_{\text{f}}\mathbf{V})=0\quad\text{in}\quad \Omega_{\text{f}}^{0}. \tag{26}\]  
The Stokes' problem in the reference configuration is obtained by substituting (25) into (24) and using (26), that is,  
\[0=-\text{G}_{\text{f}}^{\text{T}}\
abla_{\mathbf{x}}p+\mu\
abla_{\mathbf{x}} \cdot((\
abla_{\mathbf{x}}\mathbf{V})\text{G}_{\text{f}}\mathbf{F}_{\text{f}}^ {-\text{T}}+\text{G}_{\text{f}}^{\text{T}}(\
abla_{\mathbf{x}}\mathbf{V})^{ \text{T}}\mathbf{F}_{\text{f}}^{-\text{T}})\quad\text{in}\quad\Omega_{\text{ f}}^{0}. \tag{27}\]  
Now that all our governing equations are described in the Lagrangian framework, we are able to describe the interface conditions between the fluid and solid phases that we require to set up an appropriate FSI problem. We note that these interface conditions take place on the interfaces between the fluid and the elastic subphases in the reference configuration. Defining the interfaces between the fluid phase and each of the \(\alpha\) subphases in the reference configuration as \(\Gamma_{\alpha}^{0}:=\partial\Omega_{\alpha}^{0}\cap\partial\Omega_{\text{f}}^ {0}\) and defining the interface between the matrix and the fluid phase, again in the reference configuration, as \(\Gamma_{\text{ii}}^{0}:=\partial\Omega_{\text{ii}}^{0}\cap\partial\Omega_{ \text{f}}^{0}\), we can impose the continuity of the velocities and stresses on the various interfaces. We therefore have:  
\[\frac{\partial\mathbf{u}_{\alpha}}{\partial t}=\mathbf{V} \qquad\text{on}\qquad\Gamma_{\alpha}^{0}, \tag{28}\] \[\boldsymbol{\sigma}_{\text{f}}\text{G}_{\text{f}}^{\text{T}} \mathbf{n}_{\alpha}^{0}=\sigma_{\text{a}}\mathbf{n}_{\alpha}^{0} \qquad\text{on}\qquad\Gamma_{\alpha}^{0},\] (29) \[\mathbf{n}\mathbf{d}\mathbf{a}=J\mathbf{F}^{-\mathsf{T}}\mathbf{N}\mathrm{d} \mathbf{A}, \tag{21}\]0 (30) \[\mathbf{n}\mathbf{d}\mathbf{a}=J\mathbf{F}^{-\mathsf{T}}\mathbf{N}\mathrm{d} \mathbf{A}, \tag{21}\]1  
\(\forall\alpha=1,\dots,K\). We have that \(\partial\mathbf{u}_{\alpha}/\partial t\) and \(\partial\mathbf{u}_{\text{ii}}/\partial t\) are the solid velocities in each of the subphases and the matrix, respectively. We have that \(\mathbf{n}_{\alpha}^{0}\) and \(\mathbf{n}_{\text{ii}}^{0}\) are the unit outward normals to the interfaces \(\Gamma_{\alpha}^{0}\) and \(\Gamma_{\text{ii}}^{0}\), respectively. Our complete FSI problem in the reference configuration is therefore given by (2)-(5), (10), (11), (14) and (24)-(31). Within the next section, we carry out our analysis by first nondimensionalising the system of partial differential equations (PDEs) that we formulate in the reference configuration within this section. We introduce two well-separated length scales that allow us to apply the two-scale asymptotic homogenization technique to the nondimensionalised PDEs. This allows the derivation of the macroscale governing equations.