T. Healey and P. Rosakis plan to study various problems from solid continuum mechanics governed by nonlinear partial differential equations. A major thrust of the proposed work will be on the static morphology and dynamic evolution of microstructure due to phase transitions in two-dimensional models of shape-memory alloys. In contrast to well-known absolute, minimum-energy approaches to the statics problem, we plan to employ modern techniques of symmetry-breaking, bifurcation theory. The latter enables systematic determination of metastable states, which we believe our ensuing dynamical studies will show to be associated with hysteresis. Another aspect of the work will involve applications of a generalized topological degree (developed previously by Healey & Simpson) to global bifurcation problems of nonlinear elasticity - these will be the first such results in three-dimensional elasticity. The overall goals of the work are to detect new nonlinear phenomena, obtain constitutive restrictions that are both physically and mathematically reasonable for general classes of materials, and to develop new tools/approaches to such nonlinear problems - both analytical and computational. The proposed approach comprises a blend of solid continuum mechanics, materials science, nonlinear analysis, symmetry and group-theoretic methods and computation.
The analysis of models of nonlinear materials at a very general level is fundamental to the understanding of shape-memory effects in certain advanced engineering alloys as well as more traditional engineering materials/structures. The proposed investigation is strongly interdisciplinary, combining models and techniques from mechanics, materials science and modern mathematics. Broadly speaking the work will provide new mathematical approaches to difficult nonlinear problems arising in structural/mechanical engineering and materials sciences. This has the potentia of leading to a better understanding of material behavior and design - including the design of "smart" structures.