Planar grained structures with multiple inclusions in a periodic cell: Elastostatic solution and its potential applications

Abstract

The simple analytical expressions for the effective moduli and related quantities of interest in a planar doubly periodic matrix-inclusion structure previously obtained for only one inclusion in a cell are generalized on a finite number of non-intersecting inclusions each having its own local properties. As before, all phases are treated as linear and isotropic with perfectly bonding along smooth material interfaces. The derivations are performed by the complex variable technique applied to the quasi-periodic Weierstrassian zeta-function. Special attention is given to the equi-stress inclusion shapes (ESSs) where the analytical development can be fully completed. In particular, they are proved to saturate the multi-phase Hashin–Shtrikman bounds on the effective bulk modulus. The necessary condition of the ESSs existence is also found, although the question of whether they really exist is left aside. The results obtained form a basis for further numerical analysis of the attendant direct and optimization problems which are briefly discussed.