Mathematical Sciences: Asymptotic Estimates for Boundary- Value Problems in Linear and Nonlinear Continuum Mechanics

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9622748 Horgan The asymptotic behavior of solutions in continuum mechanics is a broad topic of considerable mathematical and technological interest. For the specific geometries of a three-dimensional cylinder or a two-dimensional rectangle, the spatial decay of solutions of elliptic partial differential equations (or systems of equations) arises in consideration of the classic Saint-Venant principle in elasticity theory. Such problems also arise in entry flows for viscous incompressible fluids. Generalizations to include principles of Phragmen- Lindelof type are also of interest. In this proposal, it is planned to analytically investigate a sequence of boundary-value problems for second-order and fourth-order elliptic partial differential equations. Both linear and nonlinear, isotropic and anisotropic problems are considered. The results of such investigations have widespread technological impact. In particular, rigorously obtained asymptotic estimates for the rate of decay of solutions of continuum mechanics problems are immediately applicable in engineering analysis and design. The problems to be investigated involve mathematical models for advanced composite materials and structures. %%% Structures of technological interest such as automotive and aircraft parts, rocket casings, helicopter blades, hollow shafts and containment vessels, are often constructed of layers of anisotropic, filament or fiber-reinforced materials which must be designed to remain elastic. The proper analysis of such structures in current high-performance technology requires careful mathematical analysis of basic issues. It is proposed to study the safe and efficient performance of these structures using results from the theory of partial differential equations. In particular, simplified, cost-effective asymptotic methods will be used. The scope of the investigation will include both linear and nonlinear problems, and one of the underlying goals is that of efficient use of materials. It is known, for example, that considerations of nonlinear effects in composites often lead to striking differences from predictions of linearized theories. In view of the rapid utilization of advanced composite materials in current technology, mathematical studies on large deformations of such materials promise to have widespread impact on the mechanics and materials knowledge and technology base. Applications to specific composite architectures of interest, for example, to the Boeing Company and NASA will be made. ***