# Basic sums and the effective conductivity of composites¶

Consider the effective conductivity of polydispersed fiber inclusions of conductivity $$\lambda_f$$ of different sizes randomly embedded in a matrix of conductivity $$1$$ (see Fig. 1).

A cross–section of such composite is considered to be the periodic two–dimensional lattice $$\mathcal{Q}$$, defined by complex numbers $$\omega_{1}$$ and $$\omega _{2}$$ on the complex plane $$\mathbb{C}$$. The $$(0,0)$$-cell is introduced as the parallelogram $$Q_{(0,0)}:=\{z=t_{1}\omega_{1}+t_{2}\omega_{2}:-1/2<t_{j}<1/2\;(j=1,2)\}$$ (see Fig. 2).

The lattice $$\mathcal{Q}$$ consists of the cells $$Q_{(m_{1},m_{2})}:=\{z\in \mathbb{C}:z-m_{1}\omega_{1}-m_{2}\omega _{2}\in Q_{(0,0)}\}$$, where $$m_{1}$$ and $$m_{2}$$ run over integer numbers.

Inclusions are modelled by $$N$$ non-overlapping disks of different radii $$r_j$$ ($$j=1,2,3,\ldots,N$$). Thus, the total concentration of inclusions equals

$\nu=\pi\sum_{j=1}^N r_j^2.$

Let $$r=\displaystyle\max_{1 \leq j \leq N}r_j$$ and introduce the constants

(1)$\label{eq:nuConst} % \nu_j =\frac{r_j^2}{r^2}, \quad j=1,2,3,\ldots,N, \nu_j =\left(\frac{r_j}{r}\right)^2, \quad j=1,2,3,\ldots,N$

describing polydispersity of inclusions. V.Mityushev [2] obtained the closed form representation of the effective conductivity (see also [3])

(2)$\label{eq:eff1} \lambda=1 +2\rho \nu \sum_{q=0}^{\infty }B _{q}\nu^{q},$

where

$\label{eq:rho} \rho=\frac{\lambda_f-1}{\lambda_f+1}$

is the Bergman’s contrast parameter  and the constants $$B_q$$ are given by the following theorem, as linear combinations of basic sums (defined later in this section).

Theorem [1] The algorithm for generating the symbolic representations of the coefficients $$B_q$$ takes the form of a recurrence relation of the first order:

(3)$\begin{split}\label{eq:2.18} \begin{array}{lll} B_0=1&\\ B_1=\pi^{-1}\rho e_{2}, & \\ B_2=\pi^{-2}\rho^2 e_{2,2}, & \\ B_q=\pi^{-1}\beta B_{q-1}, & q=3,4,5..., \end{array}\end{split}$

where $$\beta$$ is the substitution operator modifying every basic sum in $$B_{q-1}$$ according the transformation rule:

(4)$\label{mainRule} e_{p_1,p_2,\ldots,p_n}\longmapsto\rho e_{2,p_1,p_2,\ldots,p_n}-\frac{p_2}{p_1-1}e_{p_1+1,p_2+1,p_3,\ldots,p_n}.$

Consider a set of points $$a_k$$ $$(k= 1, 2,3,\ldots, N)$$ in the cell $$Q_{(0,0)}$$. Let $$n$$ be a natural number; $$k_0, k_1\ldots, k_n$$ be integers from 1 to $$N$$; $$k_j\geq 2$$. Let $$\mathbf{C}$$ be the operator of complex conjugation. The following sums, encountered in (3)  and (4), were introduced by V.Mityushev:

(5)$\begin{split}%\label{eq:eSum} \begin{array}{c} e^{\nu_0,\nu_1,\nu_2,\ldots,\nu_n}_{p_1,p_2,p_3,\ldots,p_n}= \displaystyle{\frac{1}{\eta^{\delta+1}} {\sum_{k_0,k_1,\ldots,k_n}}} \nu^{t_0}_{k_0}\nu^{t_1}_{k_1}\nu^{t_2}_{k_2}\cdots\nu^{t_n}_{k_n} E_{p_1}(a_{k_0}-a_{k_1})\overline{E_{p_2}(a_{k_1}-a_{k_2})}\\ \times E_{p_3}(a_{k_2}-a_{k_3})\cdots \mathbf{C}^{n+1} E_{p_n}(a_{k_{n-1}}-a_{k_n}), \end{array}\end{split}$

where $$\eta=\sum_{j=1}^N\nu_j$$ and $$\delta ={\frac{1}{2}\sum^n_{j=1}p_j}$$. Functions $$E_k$$ ($$k=2,3\ldots$$) are Eisenstein functions corresponding to the two-periodic cell $$Q_{(0,0)}$$ (see section on Eisenstein functions), and the superscripts $$t_j$$ $$(j=0,1,2\ldots,n)$$ are given by the recurrence relations

$\begin{split}\label{eq:ep2nuk} \begin{array}{ll} t_0=1, \\ t_j=p_j-t_{j-1},\quad & j=1,2,3,\dots,n. \end{array}\end{split}$

The sum (5)  is called the basic sum of the multi-order $${\mathbf p}=(p_{1},\ldots,p_{n})$$. Hereafter, the superscripts for basic sum are omitted for the purpose of conciseness. In addition, following  we have $$t_n=1$$. From the above definitions we see that every coefficient $$B_q$$ ($$q=1,2,3, \ldots$$) forms the linear combination of basic sums. All basic sums included in the coefficient $$B_q$$ are called basic sums of order $$q$$.

In case of the composite modelled by $$N$$ identical disks, where $$\nu_j=1$$ ($$j=1,2,3,\ldots,N$$), basic sums $$e_2$$ and $$e_{2,2}$$ take the following forms:

$\begin{split}\begin{array}{lll} e_{2}&= & \displaystyle{\frac{1}{N^2}} \displaystyle{\sum_{k_0=1}^{N}}\;\displaystyle{\sum_{k_1=1}^{N}}E_{2}(a_{k_0}-a_{k_1}),\\ e_{2,2}&= & \displaystyle{\frac{1}{N^3}} \displaystyle{\sum_{k_0=1}^{N}}\;\displaystyle{\sum_{k_1=1}^{N}}\;\displaystyle{\sum_{k_2=1}^{N}}E_{2}(a_{k_0}-a_{k_1})\overline{E_{2}(a_{k_1}-a_{k_2})}. \end{array}\end{split}$

References

 [1] Nawalaniec, Algorithms for computing symbolic representations of basic e–sums and their application to composites, Journal of Symbolic Computation, Vol. 74, 328345,2016.
 [2] Berlyand, V. Mityushev, Generalized Clausius–Mossotti formula for random composite with circular fibers. J. Stat. Phys. 102 (1/2), 115–145, 2001
 [3] Gluzman, V. Mityushev, W. Nawalaniec, Computational Analysis of Structured Media. Academic Press (Elsevier), 2018