9704730 Healey We plan to carry out research in global nonlinear analysis of partial differential equations of nonlinear elastostatics. A major thrust of the proposed work will be focused on applications of a new general existence tool developed recented by Simpson & myself (from past partial NSF support), viz., a generalization of the Leray-Schauder degree for global continuation and bifurcation. In the context a concrete problems, we seek a-prior bounds, symmetry/positivity properties, etc., all with a view toward obtaining meaningful constitutive restrictions for entire classes of materials. Our study will include both traditional (strongly elliptic) problems and those involving phase change (loss of ellipticity). For the latter, a new approach based upon higher-gradient regularization, global continuation and singular limits is being proposed. The analysis of such models at a very general level is fundamental to the understanding of traditional engineering materials/structures and martensitic transformations and shape-memory effects, which are observed in many advanced engineering alloys. The work has two major goals: (i) To obtain new qualitative results and detect new phenomena - of both mathematical and physical significance; (ii) To obtain new global-continuation (existence) results in problems of 2 and 3-dimensional elasticity - including problems involving phase transformations. Broadly speaking,the proposed work will provide important mathematical underpinnings to difficult nonlinear problems arising in traditional engineering fields like structural & mechanical engineering and also in more modern areas like materials science. The work has the potential to: (i) deliver new mathematical tools for the analysis of difficult problems of engineering practice,leading ultimately to safer and more optimal design of structures; (ii) lead to a better understanding of the nonlinear material behavior of certain engineering alloys, with potential application s to manufacturing engineering and the design of non-passive or "smart" structures.
