Conductivity of two‐dimensional composites with randomly distributed elliptical inclusions

Abstract

Analytical approximate formulae for the effective conductivity tensor of the two-dimensional composites with elliptical inclusions are derived in the form of polynomial approximations in concentration. New formulae explain the seeming contradiction between various formulae derived in the framework of self consistent methods. Random composites with high conducting inclusions of two different shapes (elliptical and circle) of the same area are compared. It is established that greater relative concentration of ellipses increases the effective conductivity.

Effective conductivity of a random suspension of highly conducting spherical particles

“…Different approxmate formulae/solutions for the mathemat-ical model hold under restrictions usually not discussed by authors. A serious methodological mistake may follow when intermediate manipulations are valid only within the precision O(f ), while the final formula is claimed to work with a higher precision, see an explicit example in [22]. In particular, it follows from our formulae (5.21)- (5.22) that it is impossible to write a universal higher order formula independent on locations of inclusions.…”

“…We can say that it is certainly methodological, not a computational error, related to intuitive physical investigation of the conditionally convergent integral discussed in [7]. The critical review of the limit transition from finite to infinite number of inclusions in applications of self-consistent and cluster methods is presented in [8,20,21,22] for 2D conductivity problems. In particular, it was demonstrated that self-consistent and cluster methods with various intuitive corrections are frequently applied within the accuracy f , but the obtained results are applied to high concentrations without any justification.…”