Effective conductivity of a random suspension of highly conducting spherical particles

Abstract

• A constructive approach for random 3D composites is developed. • The question of conditionally divergent series for random composites is resolved. • The famous Jeffreys formula for the effective conductivity of suspensions is revised and extended. • A theory of 3D Eisenstein functions is developed. 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. An algorithm for symbolic computations of the effective conductivity tensor is developed and performed 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.