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.

Conducting nonoverlapping identical disks are embedded in a two-dimensional background. The set of disks is infinite. The disks are distributed in such a way that the obtained composite is macroscopically isotropic. Let the conductivity of inclusions be higher than the conductivity of the matrix. It is proved that the hexagonal (triangular) lattice of disks possess the minimal effective conductivity when the concentration is not high.

We study transport properties of equal infinite unidirectional circular cylinders that are arbitrarily distributed in a uniform host. The problem is reduced to a conjugation problem for the two-dimensional Laplace equation. The flux is written exactly in the form of a power series in Bergman's contrast parameter. Asymptotic formulae for the flux are deduced for densely packed inclusions. These formulae involve two small parameters: the ratio of the distance between the close discs to their radius and a dimensionless parameter that characterizes the degree of perfect conductor/isolator. Validity of the structural approximation is discussed.