Optimal Multimaterial Composites and Exotic Structures

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This award supports the research program of the Principal Investigator in the area of optimal design of composite materials, which are important structural materials in manufacturing (for example). Modern technological capabilities such as 3d-printing, micro- and nano-fabrication, allow a huge variety of material structures to be manufactured, and one wants to know the "best" structure, or how composite microstructures can be optimized. This project develops methods of structural optimization of multimaterial composites. Optimal composites lead to metamaterials that utilize the extreme properties. Optimal composites are crucial to structural optimization because optimal designs are made from specially tailored composites. Numerous applications call for optimal design of multimaterial composites, but so far the vast majority of related results deals only with two-material optimal composites because of limitations of existing theory. This project fills this gap. Optimal microstructures of multiphase composites drastically differ from the structures of optimal two-phase composites. The latter have an intuitively expected topology: a strong material always surrounds weak inclusions. In contrast, optimal three-material structures show a large variety of patterns: they may contain a strong envelope, "hubs" of intermediate material connected by anisotropic "pathways", and other configurations that reveal a geometrical essence of optimality. The topologies of optimal structures cannot be easily guessed, and the proposed research develops a regular way to construct them.Optimal structures and bounds for effective properties of composites are studied by methods of convex analysis. The problem of optimal multimaterial composite structure is formulated as a problem of constructing a quasiconvex envelope of a multi-well Lagrangian. Two methods will be developed: a novel method of bounds derivation that accounts for inequalities due to ordering of gradient fields in optimal structures (weaker fields corresponding to stronger materials) and a complementary method for building minimizing sequences that realize the bounds to determine the structure of an optimal composite. It is assumed that fields in the materials are known from optimality requirements derived from the bounds, and the geometry is chosen so that these fields are compatible. Special attention is paid to optimal 3d structures of multiphase composites; the developed methods for structures can be used for constructing metamaterials.