Result: Effective Conductivity of an Isotropic Heterogeneous Medium of Lognormal Conductivity Distribution: Effective conductivity of an isotropic heterogeneous medium of lognormal conductivity distribution

Summary: The study aims at deriving the effective conductivity \(K_{\text{ef}}\) 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 \(I_{Y}\). 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 \(n\to 1\) and \(R_{0}/I_{Y}\to \infty\). Following a qualitative argument, we derive an exact expression of \(K_{\text{ef}}\) by computing it at the dilute limit \(n\to 0\). It turns out that \(K_{ef}\) 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