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

数学科学：线性和非线性连续介质力学中边值问题的渐近估计

• 批准号: 9622748
• 负责人: Cornelius Horgan
• 金额: $ 7万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622748&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Asymptotic; Estimates; Boundary

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. ***

9622748 Horgan 连续介质力学中解的渐近行为是一个广泛的话题，具有相当大的数学和技术兴趣。 对于三维圆柱体或二维矩形的特定几何形状，椭圆偏微分方程（或方程组）解的空间衰减是考虑到弹性理论中经典的圣维南原理而产生的。 此类问题也出现在粘性不可压缩流体的入口流中。 包括 Phragmen-Lindelof 类型原理的概括也很有趣。 在该提案中，计划分析研究二阶和四阶椭圆偏微分方程的一系列边值问题。 线性和非线性、各向同性和各向异性问题都被考虑。 此类调查的结果具有广泛的技术影响。 特别是，严格获得的连续介质力学问题解的衰减率的渐近估计可立即应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。 %%% 具有技术意义的结构，例如汽车和飞机零件、火箭外壳、直升机叶片、空心轴和安全壳，通常由各向异性、长丝或纤维增强材料层构成，这些材料必须设计成保持弹性。 在当前高性能技术中对此类结构进行正确分析需要对基本问题进行仔细的数学分析。 建议利用偏微分方程理论的结果来研究这些结构的安全有效的性能。 特别是，将使用简化的、具有成本效益的渐近方法。 研究范围将包括线性和非线性问题，基本目标之一是有效利用材料。 例如，众所周知，对复合材料中非线性效应的考虑通常会导致与线性理论的预测存在显着差异。 鉴于先进复合材料在当前技术中的快速应用，对此类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。 ***