# Homogenized Balance Equations for Nonlinear Poroelastic Composites
## 3 The Asymptotic Homogenization Method
### The Two-Scale Asymptotic Homogenization Method  
Here, we now introduce the asymptotic homogenization technique, which we will use to derive the effective macroscale governing equations for our structure. As described above, we have the pore-scale, denoted by \(d\), and the macroscale, which is the average size of the whole material, denoted by \(L\). We assume that these scales are well separated, i.e.,  
\[\epsilon=\frac{d}{L}\ll 1. \tag{67}\]  
We can describe \(\epsilon\) as the scale separation parameter. We require a local scale spatial variable, which will capture the pore-scale variations of each of the fields appearing in (50)-(65), that is,  
\[\tilde{\mathbf{Y}}=\frac{\mathbf{X}}{\epsilon}. \tag{68}\]  
We also have the macroscale variable:  
\[\tilde{\mathbf{X}}=\mathbf{X}. \tag{69}\]  
The newly introduced spatial variables \(\tilde{\mathbf{X}}\) and \(\tilde{\mathbf{Y}}\) represent the macroscale and the pore-scale respectively and are to be considered formally independent. The gradient operator also transforms, by the application of the chain rule, to become an operator of both scales, which we can write as:  
\[\
abla_{\mathbf{X}}\rightarrow\
abla_{\tilde{\mathbf{X}}}+\frac{1}{\epsilon} \
abla_{\tilde{\mathbf{Y}}}. \tag{70}\]  
We also make the assumption that all the fields in (50)-(65), \(\forall\alpha=1,...,K\), are functions of the two spatial variables \(\tilde{\mathbf{X}}\) and \(\tilde{\mathbf{Y}}\) and that every field can be written as a power series expansion in \(\epsilon\):  
\[\varphi^{\epsilon}(\tilde{\mathbf{X}},\tilde{\mathbf{Y}},t)=\sum_{l=0}^{ \infty}\varphi^{(l)}(\tilde{\mathbf{X}},\tilde{\mathbf{Y}},t)\epsilon^{l}, \tag{71}\]  
where \(\varphi\) is used to denote a general field in (50)-(65). **Remark 2** (Pore-scale periodicity). : _We assume that every field \(\varphi^{(l)}\) in our analysis is \(\tilde{\mathbf{Y}}\)-periodic, and this allows us to focus our attention on a single periodic cell in our structure. This is a technical assumption that is related solely to the microscale. The fields can still vary with respect to the macroscale. Making this assumption means that we formulate the pore-scale differential problems on this periodic cell, and it is these problems that are to be solved to determine the macroscale model coefficients. In [40], the authors assumed that all the fields are periodic in the pore-scale variable and are able to compute the coefficients of the model of standard poroelasticity obtained via asymptotic homogenization. In comparison, in [41], the macroscale model of poroelasticity was solved. As such, although the effective coefficients used are those computed by following the methodology described in [40], the illustrated solution solely depends on the macroscale and is in general not periodic. It is possible to relax this assumption that all fields are \(\tilde{\mathbf{Y}}\)-periodic and assume instead that all the fields are locally bounded. This assumption is less strict than local periodicity and means that all the fields are finite with respect to the pore-scale variable \(\tilde{\mathbf{Y}}\) when \(\epsilon\to 0\), but not necessarily periodic. This assumption however only permits the functional form of the macroscale model to be derived and does not in general allow for the model coefficients to be computed without further assumptions (see [21; 23] for further details) in multidimensional problems [23]._  
_We have that our periodic cell could potentially contain a variety of subphases and these subphases could possess various geometries and elastic properties. This assumption is particularly useful as it allows us to solve the differential problems obtained from the asymptotic homogenization technique on a single periodic cell instead of the whole material domain, therefore reducing the computational complexity. This periodic cell where we solve the differential problems is shown in Figure 2._  
**Remark 3** (Macroscopic uniformity).