Theory and Applications of Coupled Mathematical Models for Environmentally-friendly Multiscale Materials Systems

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Many current advanced technologies rely heavily on environmentally harmful components. One of such components is lead whose contents may exceed 60 weight percent in commercially available cutting-edge materials such as piezoelectrics. These materials and their composites demonstrate excellent performance in many applications where strong coupling between mechanical and electric fields is required. However, there is a strong impetus and increasingly strict environmental regulations to eliminate the environmentally harmful components without compromising the materials performance. This task is connected with a multitude of challenges, some of which require much more close attention of applied mathematicians. Building on our successful work in the development of coupled multiscale mathematical models, the proposed research program expands it to new areas, crucial for both the development of state-of-the-art mathematical theories and innovative applications. The program is aimed at further advancement in the development and applications of coupled mathematical models for the analysis of environmentally-friendly multiscale materials systems. The main goal is to account systematically for coupling and multiscale nature of the problem in this development. Application-wise, major focus will be given to several key classes of multiscale environmental-friendly materials systems such as piezoelectric composites. In the first approximation, elegantly simple mathematical concepts and formulations can be applied to some such systems, e.g., by viewing them as those consisting of two main components - a matrix, considered as a background material, and crystalline-particle-based inclusions. However, many mathematical challenges in this field still are on only scarcely explored horizons. In addressing its long-term goals, the program will, firstly, allow a systematic study of properties of such systems, accounting for nonlinear and nonlocal coupled effects, and for the polycrystalline structure of inclusions. Secondly, it will carry out mathematical and numerical analysis of the developed models with microstructures, as well as with performance-enhancing nano-additions. Thirdly, it will systematically study an important class of geometrically-tailored systems. Finally, given their ubiquitous nature, it is expected that the models and tools developed in this program will assist in addressing other challenging problems of mathematics and its applications. Indeed, the results may not be restricted to just the exemplifications given and can be useful in studying other important systems in science and engineering. The program's core includes fundamental and applied research in mathematical modelling that will lead to state-of-the-art scientific advances and will entail positive impacts on important global challenges. Training of highly qualified personnel and expanding international collaboration will be effectively integrated with the development of the program.