Energy-minimizing inclusions in a planar elastic structure with macroisotropy

We consider a thin elastic plate in which identical foreign inclusions are perfectly embedded on the nodes of a doubly periodic lattice. Both materials are taken to be homogeneous and isotropic, differing in their bulk and shear moduli. At a given proportion of the above phases, a macroanisotropic material with extremal bulk modulus may be constructed by the equistress concept (Cherepanov 1974; Vigdergauz 1994; Grabovsky and Kohn 1995a,b) which is used to identify the optimal inclusion shape. Here, this approach is further employed to find the extremal macroisotropic materials when the design domain is a hexagonal periodic cell. Within these settings we consider two different inclusion arrays in which the single inclusion is centered in the cell, or six inclusions are equally spaced on the circle. Actually, the second geometry generates the triangle lattice. Both cases are believed to have never been considered in this context. We find the optimal shapes using an effective numerical procedure in which a stable series expansion technique is employed to arrive ata linear algebraic system of a small computational size.