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.