Microstructures of Composites of Extremal Rigidity and Exact Bounds on the Associated Energy Density

We study the elastic composites of extremal stiffness, i.e., periodic structures that provide minimal energy density in a given stress field (we call them composites of the minimal compliance or of the maximal stiffness). The composites are assembled from two isotropic materials taken in a prescribed proportion, and may be of an arbitrary microstructure.Two-sided bounds for the energy density are obtained for composites of maximal stiffness and composites of minimal stiffness. One bound is an inequality valid for all structures independently of their geometry, and the second one corresponds to the energy stored in a composite with a specially chosen microstructure.Porous media are studied in detail: microstructures of porous composites of maximal stiffness are explicitly determined for all possible external stress fields. Also, the structures of composites that possess minimal stiffness in any given external strains are explicitly found in the case when one of the constituents is absolutely rigid.We discuss the link of obtained optimal microstructures with the composites that are optimal for the plane problem of elasticity; they have been obtained by the authors earlier.1 The application of obtained results to the optimal design problems is discussed.To solve the problem we develop and use the technique of construction of the quasiconvex envelopes for nonconvex integrands of the corresponding multidimensional variational problems.