# Homogenized Balance Equations for Nonlinear Poroelastic Composites
## 3 The Asymptotic Homogenization Method  
Here, we summarise the fluid-structure interaction problem in the reference configuration that we introduced in the previous section. We will then be ready to perform a multiscale analysis.  
\[\
abla_{\mathbf{X}}\cdot\mathbf{\sigma}_{\alpha} =0\] in \[\Omega_{\alpha}^{0}, \tag{32}\] \[\
abla_{\mathbf{X}}\cdot\mathbf{\sigma}_{\mathrm{II}} =0\] in \[\Omega_{\mu}^{0},\] (33) \[\
abla_{\mathbf{X}}\cdot(\mathbf{\sigma}_{\mathrm{f}}\mathsf{G}_{ \mathrm{f}}^{\mathrm{T}}) =0\] in \[\Omega_{\mathrm{f}}^{0},\] (34) \[\
abla_{\mathbf{X}}\cdot(\mathsf{G}_{\mathrm{f}}\mathbf{V}) =0\] in \[\Omega_{\mathrm{f}}^{0},\] (35) \[\frac{\partial\mathbf{u}_{\alpha}}{\partial t} =\mathbf{V}\] on \[\Gamma_{\alpha}^{0},\] (36) \[\frac{\partial\mathbf{u}_{\mathrm{II}}}{\partial t} =\mathbf{V}\] on \[\Gamma_{\mu}^{0},\] (37) \[\mathbf{\sigma}_{\mathrm{f}}\mathsf{G}_{\mathrm{f}}^{\mathrm{T}} \mathbf{n}_{\alpha}^{0} =\sigma_{\mathrm{a}}\mathbf{n}_{\alpha}^{0}\] on \[\Gamma_{\alpha}^{0},\] (38) \[\mathbf{\sigma}_{\mathrm{f}}\mathsf{G}_{\mathrm{f}}^{\mathrm{T}} \mathbf{n}_{\mathrm{II}}^{0} =\sigma_{\mathrm{II}}\mathbf{n}_{\mathrm{II}}^{0}\] on \[\Gamma_{\mu}^{0},\] (39) \[\mathbf{\sigma}_{\mathrm{a}}\mathbf{n}_{\alpha}^{0} =\sigma_{\mathrm{I}}\mathbf{n}_{\mathrm{a}\mathrm{a}\mathrm{a} \mathrm{a}}^{0}\] on \[\Gamma_{\alpha\mathrm{II}}^{0}.\] (40) \[\mathbf{u}_{\alpha} =\mathbf{u}_{\mathrm{II}}\] on \[\Gamma_{\alpha\mathrm{II}}^{0}, \tag{41}\]  
We have that the constitutive relationships for the fluid and the multiple solid phases are given as:  
\[\mathbf{\sigma}_{\mathrm{f}}\mathsf{G}_{\mathrm{f}}^{\mathrm{T}} =-p\mathsf{G}_{\mathrm{f}}^{\mathrm{T}}+\mu((\
abla_{\mathbf{X}} \mathbf{V})\mathsf{F}_{\mathrm{f}}^{-1}\mathsf{G}_{\mathrm{f}}^{\mathrm{T}}+ \mathsf{F}_{\mathrm{f}}^{-\mathrm{T}}(\
abla_{\mathbf{X}}\mathbf{V})^{\mathrm{ T}}\mathsf{G}_{\mathrm{f}}^{\mathrm{T}})\] in \[\Omega_{\mathrm{f}}^{0}, \tag{42}\] \[\mathbf{\sigma}_{\mathrm{a}} =\frac{\partial\psi_{\mathrm{a}}}{\partial\mathbf{F}_{\mathrm{a}}}\] in \[\Omega_{\mathrm{a}}^{0},\] (43) \[\mathbf{\sigma}_{\mathrm{II}} =\frac{\partial\psi_{\mathrm{II}}}{\partial\mathbf{F}_{\mathrm{ II}}}\] in \[\Omega_{\mathrm{II}}^{0}. \tag{44}\]  
We then use the constitutive relationship (42) with the incompressibility constraint (35) in the balance Equation (34) to obtain:  
\[0 =-\mathsf{G}_{\mathrm{f}}^{\mathrm{T}}\
abla_{\mathbf{X}}p+\mu \
abla_{\mathbf{X}}\cdot((\
abla_{\mathbf{X}}\mathbf{V})\mathsf{G}_{ \mathrm{f}}\mathsf{F}_{\mathrm{f}}^{-\mathrm{T}}+\mathsf{G}_{\mathrm{f}}^{ \mathrm{T}}(\
abla_{\mathbf{X}}\mathbf{V})^{\mathrm{T}}\mathsf{F}_{\mathrm{f} }^{-\mathrm{T}})\] in \[\Omega_{\mathrm{f}}^{0}. \tag{45}\]  
We also have the deformation gradients for each of the solid phases:  
\[\mathbf{F}_{\mathrm{a}} =\mathbf{I}+\
abla_{\mathbf{X}}\mathbf{u}_{\alpha}\] in \[\Omega_{\mathrm{a}}^{0}, \tag{46}\] \[\mathbf{F}_{\mathrm{II}} =\mathbf{I}+\
abla_{\mathbf{X}}\mathbf{u}_{\mathrm{II}}\] in \[\Omega_{\mathrm{II}}^{0}, \tag{47}\]  
\(\forall\alpha=1,\dots,K\). We assume that the system possesses two distinct length scales. The whole domain has a length scale, which we denote by \(L\), and this is referred to as the macroscale. There also exists a second length scale, which we denote by \(d\), and this describes the pores-scale. The length \(d\) is comparable to the intersubphase distance and the size of the pores. Therefore, to capture the true difference between the two scales, it is useful to perform a nondimensional analysis of the system of PDEs (32)-(47). This nondimensionalisation is carried out in the next section.  
We should note that this fluid-structure interaction is the nonlinear counterpart to the one found in [27] for linear poroelastic composites. We also highlight that this FSI problem is formulated with the intention that all elastic phases are in contact with each other and the fluid.