Maxwell's approach to effective conductivity and its limitations

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.