Nonlinear Problems of Elasticity for Multiphase Solids and Shells

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Proposal: DMS-0406161PI: Timothy J HealeyInstitution: Cornell UniversityTitle: Nonlinear Problems of Elasticity for Multiphase Solids and ShellsABSTRACTT. Healey plans to study various problems from nonlinear elasticity, with applications to multi-phase, shape-memory solids and nonlinearly elastic shells - especially bio-membranes or bio-shells. For the first class of problems, coming from a field dominated by global-energy minimization techniques, the main goal of the work is provide new tools and results concerning meta-stable equilibria in the presence of loading and interfacial energy. For shell problems the main goal is to understand non-axisymmetric equilibria of nominally spherical shells, which has recently become an important question for lipid-bilayer membranes or "bio-shells", which are ubiquitous in molecular-biological structures. The nonlinear partial differential equations governing these models are essentially untouched - from the point of view of analysis. The proposed work is interdisciplinary and challenging, requiring tools and perspectives from several different fields, e.g., nonlinear analysis and partial differential equations, bifurcation theory, nonlinear continuum physics, computational methods, symmetry ideas, materials science and bio-physics.The proposed work will provide new mathematical models and predictive analysis for two seemingly different physical phenomena - phase transitions in shape-memory solids and in bio-membranes. Of course the detailed mathematical models for the two phenomena are distinct, but the overall approach and philosophy of the project is the same. As verified in experiments, there is a "perfect" or homogeneous state, which loses stability, after which more exotic patterns emerge. The goal of the work is to provide new mathematical tools to predict and understand the emergence and persistence of such phenomena from a quantitative point of view. The work has direct connections to technology: Shape-memory alloys are important in applications as "smart materials" and can potentially provide actuation capabilities on the micro scale; the deformability and mechanical behavior of bio-membranes is closely related to cell function, which is of central importance in biology.