The proposal studies optimal structured materials. These problems require non-standard methods of calculus of variations that are developed. The first project deals with materials or structures with inner instabilities. Under ballistic impact, these materials dissipate and radiate the energy in an abnormal high rate; the effect can be used in various protective devices. The second project aims to find the optimal structures for multi-component composites. This problem is not solved even for the simplest cases; the optimal structures are significantly different from optimal two-material structures. The third project deals with specific problem of ``inverse optimization". It aims on understanding of optimality of natural structures, as trunks of trees or human bones. Here we want to determine the goal functional of the structural optimization from the known structure and external conditions.
The three chapters of the proposal treat novel problem in theoretical material science that require development of new mathematical methods. All problems originate in science and engineering. The first projects is concerned with new structures with a very high ability to resist shocks. Such materials can be used in various protective devices or containers. The second project addresses optimal structures of composites that are made of three or more components. The study of such structures is demanded by many engineering applications because the majority of natural or artificial composites are made of more than two materials. The last project aims to develop mathematical methods for evaluating the optimality of biological structures. The goal here is to understand in what sense biology is optimizing its structures and to use biological "solutions" to evolutionary challenge in engineering applications.
