Effective Conductivity of an Isotropic Heterogeneous Medium of Lognormal Conductivity Distribution

Abstract

The study aims at deriving the effective conductivity Kef of a three-dimensional heterogeneous medium whose local conductivity K(x) is a stationary and isotropic random space function of lognormal distribution and finite integral scale IY. We adopt a model of spherical inclusions of different K, of lognormal pdf, that we coin as a multi-indicator structure. The inclusions are inserted at random in an unbounded matrix of conductivity K 0 within a sphere $\Omega $, of radius R 0, and they occupy a volume fraction n. Uniform flow of flux $% U_{\infty }$ prevails at infinity. The effective conductivity is defined as the equivalent one of the sphere $\Omega ,$ under the limits $n\rightarrow 1$ and $R_{0}/I_{Y}\rightarrow \infty .$ Following a qualitative argument, we derive an exact expression of Kef by computing it at the dilute limit $% n\rightarrow 0.$ It turns out that Kef is given by the well-known self-consistent or effective medium argument. The above result is validated by accurate numerical simulations for $\sigma _{Y}^{2}$ $\leq $ 10 and for spheres of uniform radii. By using a faced-centered cubic lattice arrangement, the values of the volume fraction are in the interval 0<n < 0.7. The simulations are carried out by the means of an analytic element procedure. To exchange space and ensemble averages, a large number N=10000 of inclusions is used for most simulations. We surmise that the self-consistent model is an exact one for this type of medium that is different from the multi-Gaussian one.