Bifurcation Analysis and Computation in Elliptic and Multiphase Problems of Nonlinear Elasticity

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T. Healey and P. Rosakis plan to study various problems from solid continuummechanics governed by nonlinear partial differential equations. A majorthrust of the proposed work will be on the static morphology and dynamicevolution of microstructure due to phase transitions in two-dimensionalmodels of shape-memory alloys. In contrast to well-known absolute,minimum-energy approaches to the statics problem, we plan to employ moderntechniques of symmetry-breaking, bifurcation theory. The latter enablessystematic determination of metastable states, which we believe our ensuingdynamical studies will show to be associated with hysteresis. Anotheraspect of the work will involve applications of a generalized topological degree(developed previously by Healey & Simpson) to global bifurcation problemsof nonlinear elasticity - these will be the first such results inthree-dimensional elasticity. The overall goals of the work are to detectnew nonlinear phenomena, obtain constitutive restrictions that are bothphysically and mathematically reasonable for general classes of materials,and to develop new tools/approaches to such nonlinear problems - bothanalytical and computational. The proposed approach comprises a blend ofsolid continuum mechanics, materials science, nonlinear analysis, symmetryand 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 engineeringalloys as well as more traditional engineering materials/structures. Theproposed investigation is strongly interdisciplinary, combining models andtechniques from mechanics, materials science and modern mathematics.Broadly speaking the work will provide new mathematical approaches todifficult nonlinear problems arising in structural/mechanical engineeringand materials sciences. This has the potentia of leading to a better understanding ofmaterial behavior and design - including the design of "smart" structures.