## 2 Principles of Homogenization  
In the present section, we outline the principles of homogenization theory based on differential equations with highly oscillated coefficients [7; 8; 9]. The first task of homogenization theory is to determine the conditions of homogenization, i.e., when the macroscopic constants exist for a composite. The second task is to state the boundary value problem for local fields to compute the effective constants. Consider a conductive material characterized locally by the symmetric tensor \(\lambda(\mathbf{x})\) in a sample \(Q^{\prime}\) represented by a measurable bounded function [9]. Here, \(\mathbf{x}=(x_{1},x_{2})\) denotes the spatial variable. The local Fourier law  
\[\mathbf{q}=-\boldsymbol{\lambda}(\mathbf{x})\
abla T,\quad\mathbf{x}\in Q^{ \prime}, \tag{1}\]  
relates the heat flux \(\mathbf{q}\) with the temperature gradient \(\
abla T\). The function \(T(\mathbf{x})\) belongs to a Sobolev space, while its gradient \(\
abla T\) belongs to the space \(L_{2}\)[9]. Let \(\boldsymbol{\lambda}(\mathbf{x})\) be a statistically stationary spatial process. Hence, \(\boldsymbol{\lambda}(\mathbf{x})\) is a typical realization of a stationary random field [9] (Chapter 7). In particular, a periodic function \(\lambda(\mathbf{x})\) realizes a stationary random field. We now proceed to discuss this case of periodic fields in detail. Two linearly independent vectors \(\omega_{j}\) (\(j=1,2\)) form a translation in \(\mathbb{R}^{2}\):  
\[\mathbf{x}\mapsto\mathbf{x}+\sum_{j=1,2}m_{j}\omega_{j},\]  
where \(m_{j}\) runs over the integers. A fundamental domain is a subset of the space that contains exactly one point from all its translation images, such as a parallelogram formed by the vectors \(\omega_{j}\) (\(j=1,2\)). The concept of homogenization for periodic composite materials is illustrated in Figure 1 for periodic, where the fundamental domain (small cell) \(Q\) is periodically continued up to the large sample \(Q^{\prime}\). The same conception of the fundamental domain \(Q\) holds in \(\mathbb{R}^{3}\). The local conductivity \(\lambda(\mathbf{x})\) satisfies the following condition:  
\[\lambda(\mathbf{x}+\omega_{j})=\lambda(\mathbf{x})\quad(j=1,2). \tag{2}\]  
Substitution of the Equation (1) into the divergence-free condition \(\
abla\cdot\mathbf{q}=0\) yields the second-order elliptic partial differential equation  
\[\
abla\cdot(\lambda(\mathbf{x})\
abla T)=0,\quad\mathbf{x}\in Q^{\prime}. \tag{3}\]  
The principles of homogenization are deduced from the multiple-scale analysis. It requires introducing the so-called slow and fast, independent spatial variables. A homogeneous medium with effective properties at the macroscale level gives the same response to a \"slow\" perturbations of the original heterogeneous medium. The \"fast\" variable determines the effective properties through a periodic boundary value problem. At the same time, the typical size of a \"slow\" variable is much larger than the size of periodically repeated cells, described by the \"fast\" variable. The \"slow\" variable corresponds to the homogenized composite. The following mathematical visualization represented in Figure 1 can precisely describe this approach. The sample \(Q^{\prime}\) is considered on the macroscale. Let this sample be periodic on a microscale. The fast variable \(\mathbf{y}=\varepsilon^{-1}\mathbf{x}\) is introduced, where \(\varepsilon\) denotes a small positive parameter. The variables \(\mathbf{x}\) and \(\mathbf{y}\) are assumed to be independent. Introduce the square periodicity cell on the microscale:  
\[Q=\{\mathbf{x}=(x_{1},x_{2})\in Q^{\prime}:\;-\frac{\varepsilon}{2}<x_{j}< \frac{\varepsilon}{2}\;(j=1,2)\}. \tag{4}\]  
The cell \(Q\) is defined by two fundamental vectors. The two vectors are denoted as \(\omega_{1}^{\prime}=\varepsilon\omega_{1}\) and \(\omega_{2}^{\prime}=\varepsilon\omega_{2}\), where \(\omega_{1}=(1,0)\) and \(\omega_{2}=(0,1)\).