MAXWELL’S APPROACH TO EFFECTIVE CONDUCTIVITYAND ITS LIMITATIONS by V. MITYUSHEV † (Dept. Computer Sciences and Computer Methods) and N. RYLKO (Dept. Technology, Pedagogical University, ul. Podchorazych 2, Krakow 30-084, Poland) [Received 26 June 2012. Revise 6 December 2012. Accepted 28 January 2013] Summary Maxwell’s approach to calculate the effective constants of composites is discussed in many recent works. There are also various modifications such as effective medium approximations and differential schemes which are referred to self-consistent methods (SCMs). The following key question was stated in the literature: Are the SCMs valid only for dilute composites when the interactions among inclusions do not matter? In the present article, the rigorous answer is given for non-overlapping inclusions arbitrarily distributed in the plane. Most attention is paid to circular disks where the final formulae can be explicitly written. A series in the contrast parameter for the effective conductivity is truncated and the second- and third-order terms are analysed. It is shown that for macroscopically isotropic composites the second-order term does not depend on the location of inclusions while the third-order term does. This implies that any SCM is valid up to the third-order term. Moreover, it is shown that SCMs can be applied to macroscopically anisotropic composites only within the first-order approximation. 1. Introduction Analytical formulae for the transport properties of unidirectional cylinders are of considerable interest in electrical or thermal conductivity, dielectric permittivity and other fields governed by Laplace’s equation. Many approximations for the effective moduli of composites are based on the solution of the single inclusion problems and of its applications to dilute composites (effective medium approximation, differential scheme and so forth (1)). Such an approach in the theory of composites was first proposed by Mossotti in 1847 and is called ‘Maxwell’s approach’, its extensions are called ‘self-consistent methods’ (SCMs). There are also other terms, we use the abbreviation SCMs to be concise. There are also exact and approximate formulae for various types of composites. Consider, for instance, identical circular inclusions of conductivity λ embedded in a host of the normalised unit conductivity. The famous Clausius–Mossotti (Maxwell Garnett) approximation for the effective conductivity is valid for dilute composites (2, 3) λ e ≈ 1 + ρν 1 - ρν , (1) † <mityu@up.krakow.pl> Q. Jl Mech. Appl. Math, Vol. 66. No. 2 © TheAuthor, 2013. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oup.com Advance Access publication 26 February 2013. doi:10.1093/qjmam/hbt003 Downloaded from https://academic.oup.com/qjmam/article/66/2/241/1866250 by guest on 06 June 2022