The investigator determines optimal microstructures and new bounds for effective conductivity or stiffness of multimaterial composites. Mathematically, the problem is equivalent to the relaxation of a multivariable variational problem with a piece-wise quadratic multiwell Lagrangian. The investigator studies new sufficient conditions (bounds) and special minimizing sequences (microstructures). A new technique called localized polyconvexification is explored that refines the known polyconvexification procedure by deriving and accounting for new pointwise constraints on minimizers. A complementary technique is considered for determining complicated minimizing laminate-based sequences. Several examples of optimal conducting and elastic multimaterial structures are worked out; there could be used for optimal design of multimaterial structures.
The investigator studies how to place several materials of different conductivity or stiffness in a periodic composite structure in order to maximize its overall conductivity or stiffness. In a sense, the problem remains a jigsaw puzzle that asks to build an optimal mosaic from pieces of different physical properties. The mosaic can be two- or three-dimensional. Work on this problem was started in 1963 by the famous paper by Hashin and Shtrikman. The problem is solved for two mixing materials, and the suggested novel approach allows to extend the results to more than two materials. The obtained optimal microstructures are diverse and unpredictable. The best possible composites from given components can be used in manufacturing of artificial materials and structural design. The underlying mathematics, when developed, should also help to better understand and control smart materials and phase transitions in solids and similar phenomena of self-organization in heterogeneous materials.
