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.. _sect-theory_bs:

Basic sums and the effective conductivity of composites
=======================================================

Consider the effective conductivity of polydispersed fiber inclusions of
conductivity :math:`\lambda_f` of different sizes randomly embedded in a
matrix of conductivity \ :math:`1` (see :ref:`Fig. 1 <fib-figure>`). 

.. _fib-figure:

.. figure:: fibCompPoly.png
   :alt: There is no picture!
   :width: 60%

   Fig. 1  Model of a polydispersed fibrous composite.

.. _ref-cell_periods:

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

.. _cell-figure:

.. figure:: cell2.svg
   :alt: There is no picture!
   :width: 60%

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

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

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

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

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

.. math::
   :label: nuk_const

   \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 [#BM2001]_ obtained the closed form
representation of the effective conductivity (see also [#GMN2018]_)

.. math::
   :label: lambda

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

where

.. math::

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

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

.. _theorem_Bq:

**Theorem** [#Naw2016]_ The algorithm for generating the symbolic representations of
the coefficients :math:`B_q` takes the form of a recurrence relation of
the first order:

.. math::
   :label: coeff_Bq

   \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}

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

.. math::
   :label: tr_rule

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

.. math::
   :label: eSum

   %\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}

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

.. math::

   \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}

The sum :eq:`eSum`  is called the *basic sum of the multi-order*
:math:`{\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 :math:`t_n=1`. From the above definitions we see that
every coefficient :math:`B_q` (:math:`q=1,2,3, \ldots`) forms the linear
combination of basic sums. All basic sums included in the coefficient
:math:`B_q` are called *basic sums of order* :math:`q`.

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

.. math::

   \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}


.. rubric:: References

.. [#Naw2016] W. Nawalaniec, *Algorithms for computing symbolic representations of basic e–sums and their application to composites*, Journal of Symbolic Computation, Vol. 74, 328345,2016.

.. [#BM2001] L. Berlyand, V. Mityushev, *Generalized Clausius–Mossotti formula for random composite with circular fibers*. J. Stat. Phys. 102 (1/2), 115–145, 2001

.. [#GMN2018] S. Gluzman, V. Mityushev, W. Nawalaniec, *Computational Analysis of Structured Media*. Academic Press (Elsevier), 2018