Bounds and asymptotic results for the effective electromagnetic properties of a locally periodic distribution of conducting inclusions in a conducting matrix

The medium considered in this paper is made of two conducting materials distributed as locally periodic inclusions in a matrix. Whatever the concentration of each phase is, the homogenization method gives the variations of the electric and magnetic fields at the level of the microstructure, and furnishes a rigorous deductive procedure for obtaining the governing equations of the homogenized medium. The expressions of the effective conductivity and magnetic permeability coefficients are analogous, and it is shown that the effective conductivity and dielectric permittivity tensors are related. Moreover, when the medium is isotropic on the macroscopic scale, upper and lower bounds are derived for its effective electromagnetic constants. These bounds depend on the concentration of each phase as well as on the geometry of the medium on the microscopic scale. First order asymptotic expansions are also derived for the electric constants (i) for large concentrations of cubic inclusions, close to packing conditions (ii) for low concentrations of spherical inclusions. In this last case, we recover the classical expression of the effective conductivity of a dilute suspension of spheres which was derived by Maxwell and Rayleigh on the basis of random and periodic distributions, respectively. For cubic or spherical inclusions, our bounds are compared with Hashin and Shtrikmanʹs ones, established on the basis of a random distribution.