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). Fig. 1 Model of a polydispersed fibrous composite.

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). Fig. 2 Polydispersed two–dimensional composite modelled as a two-periodic cell $$Q_{(0,0)}$$.

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  obtained the closed form representation of the effective conductivity (see also )

(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  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

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