## 5 Step 1: Computation of Structural Sums  
The dependence of effective constants on shapes and locations was established in [21]. So, we are able to write an approximate analytical formula for the effective conductivity tensor (23) for 2D two-phase composites with circular inclusions. Let a composite be represented by a periodicity cell \(Q\) formed by the two fundamental translations defined by the vectors denoted as \(\omega_{1}\) and \(\omega_{2}\) defined on the complex plane \(\mathbb{C}\):  
\[Q=\bigg{\{}z=t_{1}\omega_{1}+t_{2}\omega_{2}\in\mathbb{C}:-\frac{1}{2}<t_{1},t _{2}<\frac{1}{2}\bigg{\}}. \tag{24}\]  
Without loss of generality, the periods \(\omega_{1}\) and \(\omega_{2}\) satisfy the conditions \(\omega_{1}>0\) and \(\mathrm{Im}\ \omega_{2}>0\). Let us normalize the area of \(Q\) to unity, so that \(\omega_{1}\mathrm{Im}\ \omega_{2}=1\). Let \(\mathrm{i}=\sqrt{-1}\) be the imaginary unit. We will consider below the particular cases of lattices, such as the hexagonal lattice with  
\[\omega_{1}=\frac{\sqrt{2}}{\sqrt{3}},\quad\omega_{2}=\frac{\sqrt{2}}{\sqrt{3 }}\frac{1+\mathrm{i}\sqrt{3}}{2}, \tag{25}\]  
and the square lattice with  
\[\omega_{1}=1,\quad\omega_{2}=\mathrm{i}. \tag{26}\]  
Figure 3: Double periodic set of inclusions generated by the inclusions in the unit cell shown in Figure 2.  
The considered doubly periodic structure can be represented by a plane torus denoted by \(\mathcal{Q}=\{Q+m_{1}\omega_{1}+m_{2}\omega_{2}:\ m_{1},m_{2}\in\mathbb{Z}\}\), where \(\mathbb{Z}\) denotes the set of integers. Introduce the lattice sums [47]:  
\[S_{p}=\sum_{(m_{1},m_{2})\in\mathbb{Z}^{2}\setminus(m_{1},m_{2})\
eq(0,0)} \frac{1}{(m_{1}\omega_{1}+m_{2}\omega_{2})^{p}},\quad p=2,3,\ldots \tag{27}\]  
The conditionally convergent sum \(S_{2}\) is defined by the Eisenstein summation method [47], and \(S_{2}=\pi\) for the square and hexagonal arrays. The exact and recurrence formulas for the lattice sums can be found both in Appendix A. Consider \(N\) mutually disjoint disks \(D_{k}=\{z\in\mathbb{C}:\ |z-a_{k}|<r\}\), where \(a_{k}\) denotes the center of \(D_{k}\) in \(Q\) and \(r_{k}\) its radius. Let \(D=Q\setminus\cup_{k=1}^{N}(D_{k}\cup L_{k})\) denote the complement of all the closures of \(D_{k}\) to \(Q\). An example of disks \(D_{k}\) is shown in Figure 2b. We will use below the Eisenstein functions \(E_{p}(z)\) (\(p=2,3,\ldots\)) related to the elliptic Weierstrass functions in the following way [34]:  
\[E_{2}(z)=\wp(z)+S_{2},\quad E_{p}(z)=-\frac{(-1)^{p}}{(p-1)!}\frac{\mathrm{d}^ {p-2}\wp(z)}{\mathrm{d}z^{p-2}},\quad p=3,4,\ldots \tag{28}\]  
Following [34] (Chapter 4), let us introduce the structural sums  
\[e_{m_{1},\ldots,m_{q}}=\frac{1}{N^{\frac{2}{2}+1}}\sum_{k_{0},k_{1},\ldots,k_ {n}}E_{m_{1}}(a_{k_{0}}-a_{k_{1}})\mathbf{C}E_{m_{2}}(a_{k_{1}}-a_{k_{2}}) \ldots\mathbf{C}^{q+1}E_{m_{q}}(a_{k_{q-1}}-a_{k_{q}}), \tag{29}\]  
where \(\alpha=m_{1}+\cdots+m_{q}\). It is assumed for shortness that  
\[E_{p}(a_{k}-\ a_{m})=:S_{p}\ \text{and}\ \wp(a_{k}-a_{m}):=0,\ \text{for}\ a_{m}=a_{k}\ (m,k=1,2,\ldots,N). \tag{30}\]  
For instance, \(e_{2}\) takes the following form:  
\[e_{2}=\frac{1}{N^{2}}\sum_{k,m=1}^{N}E_{2}(a_{k}-a_{m})\equiv\pi+\frac{1}{N^{ 2}}\sum_{k,m=1}^{N}\wp(a_{k}-a_{m}). \tag{31}\]  
The structural sums, including \(e_{2}\) and their relations to macroscopic anisotropy, were discussed in [37]. A set of relations among the structural sums for macroscopically isotropic composites was established. In particular, the isotropy is maintained up to \(O(f^{3})\) when \(e_{2}=\pi\). Using the complex structural sums, we form the matrix structural sums:  
\[\mathbf{e}_{m_{1},\ldots,m_{q}}=\left( \begin{array}{cc}\text{Re}\ e_{m_{1}, \ldots,m_{q}}&-\text{Im}\ e_{m_{1},\ldots,m_{q}}\\ -\text{Im}\ e_{m_{1},\ldots,m_{q}}&\text{Re}\ e_{m_{1},\ldots,m_{q}}^{*}\\ \end{array} \right), \tag{32}\]  
where \(e_{m_{1},\ldots,m_{q}}^{*}\) has the same form except at the terms \(E_{2}(a_{k}-a_{m})\), which have to be replaced by the expressions \(2\pi-E_{2}(a_{k}-a_{m})\).