Optimal Distribution of the Nonoverlapping Conducting Disks

Abstract

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.

Conductivity of two‐dimensional composites with randomly distributed elliptical inclusions

“…where C is the operator of complex conjugation, the subscript in e 22...2 contains L of twos. It was proved in [31] that for macroscopically isotropic composites e 2 = π and e 22...2 ≥ π L , L > 1.…”

Basic sums and their random dynamic changes in description of microstructure of 2D composites

“…It is known that S n ¼ 0 for odd n. For even n, the sums (2.2) can be easily computed through the rapidly convergent infinite sums [27] …”

Mathematical Models of Elastic and Piezoelectric Fields in Two-Dimensional Composites

“…In particular, the so-called problem of the divergent integral [59,69,70] is easily solved by use of the Eisenstein summation. It is convenient to treat the considered problems as R-linear problems and Riemann-Hilbert problems for multiply connected domains [56,58,60,61,63,64]. The advantages of this approach are demonstrated in [12,71].…”