Vigdergauz, S. A planar grained structure with a multiphase nested inclusion in a periodic cell: elastostatic solution and the equi-stressness. (English) Zbl 1370.74132 Math. Mech. Solids 21, No. 6, 709-724 (2016). Summary: 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. Cited in 2 Documents MSC: 74Q15 Effective constitutive equations in solid mechanics 74K99 Thin bodies, structures 74S70 Complex-variable methods applied to problems in solid mechanics Keywords:plane elasticity problem; multiphase lattices; shape optimization; Kolosov-Muskhelishvili potentials; hoop stresses; extremal elastic structures; Hashin-Shtrikman bounds PDFBibTeX XMLCite \textit{S. Vigdergauz}, Math. Mech. Solids 21, No. 6, 709--724 (2016; Zbl 1370.74132) Full Text: DOI