Effective conductivity of a random suspension of highly conducting spherical particles

Abstract

Randomly distributed non-overlapping perfectly conducting spheres are embedded in a conducting matrix with the concentration of inclusions f . Jeffrey (1973) suggested an analytical formula valid up to O(f 3 ) for macroscopically isotropic random composites. A conditionally convergent sum arose in the spatial averaging. In the present paper, we apply a method of functional equations to random composites and correct Jeffrey's formula. The main revision concerns the proper investigation of the conditionally convergent sum and correction the f 2 -term. A new model of symbolic computations is developed in order to compute the effective conductivity tensor. The corresponding algorithm is realized up to O(f 10 3 ). The obtained formulae explicitly demonstrate the dependence of the effective conductivity tensor on the deterministic and probabilistic distributions of inclusions in the f 2 -term, and in the f 3 -term. This leads to the conclusion that some previous formulae presented as universal, i.e., valid for all random composites, may be actually applied only to dilute or to special composites when interaction between inclusions do not matter.

Refinement of the Maxwell formula for composite reinforced by circular cross-section fibers. Part I: using the Schwarz alternating method

“…We will use Schwarz alternating method (SAM) [9,[22][23][24][25][26] to solve the problem, as it yields a solution of the problem within the (01) approximation of SAM, which refers to the problem formulated for circular inclusions…”

“…3. Refinement of Maxwell's formula can be based on some physically justified hypotheses [18], and by constructing the higher-order approximations of some asymptotic processes [7,9,[12][13][14]17,22,23].…”