The shape of Maxwell's equivalent inhomogeneity and ‘strange’ properties of regular polygons and other symmetric domains

Abstract

The Maxwell concept of equivalent inhomogeneity is re-examined in the context of elastic composite, porous or micro-cracked materials of periodic structure. It is demonstrated that accurate estimates for the effective elastic properties can be obtained by the modified Maxwell approach that employs the proper shape of the equivalent inhomogeneity and explicitly accounts for the geometry of the cluster and for the interactions between its constituents. In addition, it is shown that regular polygonal and some other symmetric inhomogeneities possess ‘strange’and remarkable properties. Under the action of uniform far-field loads, the averages of certain combinations of stresses inside these inhomogeneities are constant and coincide with those for a circular inhomogeneity.