## 4 Deterministic Composite  
Consider the infinite double periodic unidirectional array of cylinder fibers depicted in Figure 2, represented by the square \(Q=\{(x_{1},x_{2})\in\mathbb{R}^{2}:-\frac{1}{2}<x_{j}<\frac{1}{2}\}\). The square contains \(N\) equal circular inclusions of radius \(r\) distributed without overlap. Let the coordinates of circular inclusions and their radius \(r\) be known and fixed. This is an example of a deterministic composite material since its geometry is fixed and precisely defined. Let the square cell \(Q\) periodically extend to the plane \(\mathbb{R}^{2}\). Therefore, the homogenization principles are satisfied, and one can state the proper periodicity problem to determine the effective constants. The following conditions on the potential \(u(x_{1},x_{2})\) in the host can be considered:  
\[u(x_{1}+1,x_{2})-u(x_{1},x_{2})=1,\quad u(x_{1},x_{2}+1)-u(x_{1},x_{2})=0. \tag{15}\]  
The condition (15) means that the external flux parallel to the axis \(x_{1}\) is applied to the square array:  
\[\mathbf{q}_{ext}=-(1,0). \tag{16}\]  
The temperature jump equals \(1\) along the axis \(x_{1}\) and \(0\) along the axis \(x_{2}\). The conditions of the perfect contact have the form  
\[u(\mathbf{x})=u_{1}(\mathbf{x}),\quad\frac{\partial u}{\partial\mathbf{n}}( \mathbf{x})=\lambda_{1}\frac{\partial u_{1}}{\partial\mathbf{n}}(\mathbf{x}) \text{ on }\partial D^{+}. \tag{17}\]  
Figure 2: (**a**) Unidirectional fibrous composite. (**b**) Section perpendicular to fibers. In the considered case, the quasi-periodicity conditions (15) can be replaced with the boundary conditions for the domain \(Q\):  
\[u\left(\pm\frac{1}{2},x_{2}\right)=\pm\frac{1}{2},\quad\frac{\partial u}{ \partial x_{2}}\left(x_{1},\pm\frac{1}{2}\right)=0. \tag{18}\]  
The effective conductivity tensor \(\mathbf{\lambda}_{e}\) can be determined by the averaged value over \(Q\):  
\[\langle f(\mathbf{x})\rangle=\frac{1}{|\mathcal{Q}|}\int_{Q}f(\mathbf{x})\ d \mathbf{x}. \tag{19}\]  
The value \(\langle\mathbf{q}\rangle\) stands for the averaged flux and \(\widehat{\
abla u}\) is the macroscopic temperature gradient over the considered sample. In the case of Equation (16), \(\widehat{\
abla u}=(1,0)\) and \(\langle\mathbf{q}\rangle=\mathbf{q}_{ext}\). The effective conductivity tensor \(\mathbf{\lambda}_{e}\) in the case of the perfect contact between inclusions and host is defined as the proportionality coefficient in equation [34]:  
\[\langle\mathbf{q}\rangle=-\mathbf{\lambda}_{e}\widehat{\
abla u}. \tag{20}\]  
This approach is similar to the black box operator discussed in [23], meaning that the input is given by \(\widehat{\
abla u}\), and the output \(\langle\mathbf{q}\rangle\) ought to be measured or calculated. The tensor \(\mathbf{\lambda}_{e}\) determines the conductive properties of the sample \(Q\). Equation (20) is linear. It relates two constant vectors \(\langle\mathbf{q}\rangle\) and \(\widehat{\
abla u}\) in \(\mathbb{R}^{3}\). It follows from linear algebra that a general linear relation between two vectors is determined by a matrix called the effective conductivity tensor:  
\[\mathbf{\lambda}_{e}=\left( \begin{array}{c}\lambda_{11}\ \lambda_{12}\ \lambda_{13}\\ \lambda_{21}\ \lambda_{22}\ \lambda_{23}\\ \lambda_{31}\ \lambda_{32}\ \lambda_{33}\end{array} \right). \tag{21}\]  
The tensor (21) is symmetric, \(\lambda_{ij}=\lambda_{ji}\). Hence, it can be reduced to the diagonal form. The tensor \(\mathbf{\lambda}_{e}\) expresses the anisotropic effective properties of the homogenized material, i.e., different conductivity in different directions. Let the matrix Equation (21) be represented in the form \(\mathbf{\lambda}_{e}=\lambda_{e}\mathbf{I}\), where \(\mathbf{I}\) denotes the unit matrix. In this case, the material is isotropic macroscopically, and the scalar \(\lambda_{e}\) is called the effective conductivity of the composite material. The same statements hold in the case of heat conduction perpendicular to the unidirectional fibrous materials, depicted in Figures 2 and 3.