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

连续介质力学中解的渐近行为是一个具有相当大的数学和技术兴趣的广泛话题。对于三维圆柱体或二维矩形的特定几何形状，考虑到弹性理论中的经典圣维南原理，产生了椭圆型偏微分方程（或方程组）解的空间衰减。这类问题也出现在粘性不可压缩流体的入口流动中。概括包括Phragmen- Lindelof类型的原则也是有趣的。本文拟分析研究一类二阶和四阶椭圆型偏微分方程的边值问题。考虑了线性和非线性、各向同性和各向异性问题。这些调查的结果具有广泛的技术影响。特别是，连续介质力学问题的解的衰减速率的严格的渐近估计立即适用于工程分析和设计。要研究的问题涉及先进复合材料和结构的数学模型。具有技术价值的结构，如汽车和飞机部件、火箭外壳、直升机叶片、空心轴和密封容器，通常由各向异性、长丝或纤维增强材料层构成，其设计必须保持弹性。在当前的高性能技术中，对这种结构进行适当的分析需要对基本问题进行仔细的数学分析。提出利用偏微分方程理论的结果来研究这些结构的安全有效性能。特别是，将使用简化的、具有成本效益的渐近方法。调查的范围将包括线性和非线性问题，其中一个潜在的目标是有效利用材料。例如，众所周知，考虑复合材料的非线性效应往往会导致与线性化理论预测的显著差异。鉴于先进复合材料在当前技术中的快速应用，对这种材料大变形的数学研究有望对力学和材料知识和技术基础产生广泛的影响。将应用于感兴趣的特定复合结构，例如波音公司和美国宇航局。＊＊＊

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. ***