Electromagnetic properties of random materials*

Scale dependence bounds on the electromagnetic properties are studied in the setting of spatially random linear materials with statistically homogeneous and spatially ergodic random microstructures. First, from the Hill-Mandel homogenization conditions adapted to electric and magnetic fields, uniform boundary conditions are formulated for a statistical volume element (SVE). From these conditions, rigorous bounds are obtained on the macroscale (effective) electrical permittivity and magnetic permeability. Using computational electromagnetism methods, these bounds are obtained through numerical simulations for composites of two types: (i) 2D random checkerboard (two-phase) microstructures and (ii) 2D Gaussian correlated microstructure. The simulation results demonstrate a convergence of these bounds to the effective properties with increasing length scales.