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: 1338869
• 负责人: Marta Lewicka
• 金额: $ 45.1万
• 依托单位: University of Pittsburgh
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1338869&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.

LewickaDMS-0846996 该项目的一个统一主题是当域接近极限中的“简并”区域时给定方程的渐近行为。 所考虑的问题是非线性变分或偏微分方程，所讨论的简并性可以采取多种形式：维数损失、正则性损失或无界性。 第一组问题涉及非线性弹性数学理论，该理论研究三维弹性体的大机械变形。 研究者进行了一项长期的研究计划，其范围包括：通过伽玛收敛的方法推导低维壳理论并理解它们与中面几何的联系，分析曲面的（无穷小）等距以及刚度对曲面的影响。导出的理论，研究非线性现象，例如给定壳体欠压下的屈曲和起泡（其中一个应用与植物生长有关）。 该项目中的第二组问题与流体动力学有关。 当边界行为退化时，各种结构和尺度的边界不规则性都被视为极限。 其他问题涉及无界通道中燃烧的行进前沿，以及纳维边界条件下三维薄壳中具有大量初始数据的解的动力学。 由于各种几何形状的弹性薄（或“简并”）物体在物理世界中无处不在，因此精确理解控制其平衡的定律具有许多潜在的应用。 例如，许多生长组织（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构造，人们希望通过人造手段来复制它们。 弹性数学理论中一个长期存在的相关问题是从全三维物体的非线性理论出发严格预测此类低维物体的理论。 对于板材，最近的一项努力已经对此类理论的层次结构进行了严格的论证，具体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（当中表面弯曲时），尽管在文献和工程应用中广泛使用了它们的临时概括，但从数学角度来看，人们所知甚少。 研究人员识别了固体和流体动力学连续介质力学中的几个非线性问题，这些问题自然出现在特定的简并域中，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有趣的联系。 他们有潜力在以下领域提供有用的观察结果：结构力学，同时整合向不同教育水平的更广泛社区展示科学成果的目标。