CAREER: Thin shells - problems in nonlinear elasticity and fluid dynamics

职业：薄壳 - 非线性弹性和流体动力学问题

• 批准号: 0846996
• 负责人: Marta Lewicka
• 金额: $ 40.43万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0846996&HistoricalAwards=false
• 关键词: CAREER; Thin; shells; problems; nonlinear

LewickaDMS-0846996 A unifying theme in this project is the asymptotic behaviorof given equations as the domain approaches a "degenerate" regionin the limit. The considered problems are nonlinear variationalor partial differential equations, and the degeneracy in questioncan take various forms: loss of dimension, loss of regularity, orunboundedness. The first set of questions relates to themathematical theory of nonlinear elasticity, which studies largemechanical deformations of three-dimensional elastic bodies. Theinvestigator undertakes a long-term research program, with thescope of: deriving lower-dimensional shell theories through themethods of Gamma-convergence and understanding their connectionswith the geometry of the mid-surface, analyzing the(infinitesimal) isometries of surfaces and the effects ofrigidity on the derived theories, studying nonlinear phenomenasuch as buckling and blistering for a given shell undercompression (one application is related to plant growth). Thesecond set of questions in this project relates to fluiddynamics. Boundary irregularities of various structures andscales are considered in the limit when the boundary behaviorbecomes degenerate. Other problems concern traveling fronts incombustion in unbounded channels, and the dynamics of solutionswith large initial data under the Navier boundary conditions inthin three-dimensional shells. Because elastic thin (or otherwise "degenerate") objects ofvarious geometries are ubiquitous in the physical world, theprecise understanding of laws governing their equilibria has manypotential applications. For example, many growing tissues(leaves, flowers, or marine invertebrates) exhibit complicatedconfigurations during their free growth and one would like toreproduce them with man-made means. A related long-standingproblem in the mathematical theory of elasticity is to rigorouslypredict theories of such lower-dimensional objects starting fromthe nonlinear theory of full three-dimensional objects. Forplates, a very recent effort has lead to rigorous justificationof a hierarchy of such theories, depending on the magnitude ofthe applied forces and resulting in stretching, crumpling,bending, or a combination of these. For shells (when themid-surface is curved), despite extensive use of their ad hocgeneralizations in the literature and engineering applications,much less is known from the mathematical point of view. Theinvestigator identifies several nonlinear problems in continuummechanics of solids and fluid dynamics, naturally posed inspecific degenerate domains, with the intention of rigorouslyunderstanding the behavior of the system based on generalprinciples. At the heart of the program are interestingconnections between calculus of variations, differentialequations, geometry, material science, fluid dynamics, numericalanalysis, and even biology. They have a potential to deliveruseful observations in e.g. structural mechanics, whileintegrating the goal of exposing scientific results to a broadercommunity at various education levels.

在这个项目中，一个统一的主题是给定方程的渐近行为，当域接近极限的“退化”区域时。所考虑的问题是非线性变分或偏微分方程，所讨论的退化可以采取各种形式：维数损失，正则性损失，无界性。第一组问题涉及非线性弹性的数学理论，研究三维弹性体的大力学变形。研究者承担了一个长期的研究计划，其范围是：通过伽马收敛方法推导低维壳体理论，并理解它们与中表面几何形状的联系，分析（无穷小）表面等距和刚度对导出理论的影响，研究非线性现象，如给定壳体欠压缩的屈曲和起泡（一个应用与植物生长有关）。这个项目的第二组问题与流体动力学有关。当边界行为退化时，在极限中考虑了各种结构和尺度的边界不规则性。其他问题涉及在无界通道中燃烧的行进锋面，以及在三维壳体内的Navier边界条件下具有大初始数据的解的动力学。由于各种几何形状的弹性薄（或其他“退化”）物体在物理世界中无处不在，因此对控制其平衡的定律的精确理解具有许多潜在的应用。例如，许多生长中的组织（叶子、花或海洋无脊椎动物）在自由生长过程中表现出复杂的结构，人们希望用人工方法来繁殖它们。在弹性数学理论中，一个长期存在的问题是如何从全三维物体的非线性理论出发，严格地预测这类低维物体的理论。例如，最近的一项研究已经对这些理论的等级进行了严格的论证，这些理论的等级取决于所施加的力的大小，并导致拉伸、皱缩、弯曲或这些的组合。对于壳（当中表面是弯曲的），尽管在文献和工程应用中广泛使用了它们的广义化，但从数学的角度来看，我们所知道的要少得多。研究者在固体和流体动力学的连续力学中确定了几个非线性问题，自然地提出了非特定的退化域，目的是根据一般原理严格理解系统的行为。该计划的核心是变分法，微分方程，几何，材料科学，流体动力学，数值分析，甚至生物学之间的有趣联系。它们有可能在结构力学等领域提供有用的观察结果，同时整合了在不同教育水平的更广泛的社区中展示科学成果的目标。