A planar grained structure with a multiphase nested inclusion in a periodic cell: Elastostatic solution and the equi-stressness

Abstract

By using the specially constructed Kolosov–Muskhelishvili potentials a concise formulation of 2D elastostatic problems for general regular structures with multiphase nested inclusions is obtained in complex-variable terms. Analytical averaging of the stress/strain fields over the representative cell of the structure gives its effective moduli in the perturbation-like form that was known thus far only for less complicated phase arrangements. These derivations are further extended to prove the existence of the equi-stress nested inclusions under the square symmetry of the structure. In sharp contrast to the one-phase (homogeneous) inclusion, they no longer saturate the attainable Hashin–Shtrikman bounds on the effective bulk modulus, but continue to be a subject by themselves.