Maxwell's equivalent inhomogeneity and remarkable properties of harmonic problems involving symmetric domains

This paper revisits the Maxwell concept of equivalent inhomogeneity in the context of two-dimensional harmonic problems involving composite or porous materials of periodic structure. As previously done for elasticity problems, here the scheme is modified to accommodate for the shape of the equivalent inhomogeneity and for the interactions between the constituents of the cluster. New numerical results for periodic materials with hexagonal arrangements of fibers (holes) demonstrate that, with these modifications, the scheme allows for accurate estimates of the effective material properties. It is also shown that, as for elasticity problems, some harmonic symmetric inhomogeneities possess remarkable properties. Under the action of uniform far-fields, the averages of the fields within these inhomogeneities preserve the structure of the applied far-fields.