Calculus of variations on rigid elastic structures

刚弹性结构的变分计算

• 批准号: 0907844
• 负责人: Mohammad Reza Pakzad
• 金额: $ 12.68万
• 依托单位: University of Pittsburgh
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0907844&HistoricalAwards=false
• 关键词: Calculus; variations; rigid; elastic; structures

PakzadDMS-0907844 This work is a part of a long-term project of theinvestigator and his collaborators to study the elasticityproperties of materials through methods of calculus ofvariations. Here, the focus is on situations when the elasticbodies exhibit some sort of rigid behavior. In this manner, thestudy of geometric properties of structures, e.g. the propertiesof various spaces of isometries and infinitesimal isometries of2-dimensional surfaces, comes to the fore. This differs fromclassical differential geometry in the weaker regularity of thegiven surface or of the deformations. A first objective is toidentify and rigorously derive variational theories for thinshells from the 3-dimensional theory of nonlinear elasticitythrough Gamma-convergence methods. The derived theoriesnaturally depend on the scaling regimes of the elastic energy orbody forces in terms of shell thickness. In this context, theinvestigator works on problems involving various spaces of weaklydifferentiable isometries. In parallel, properties of thederived theories, such as multiplicity and regularity ofsolutions, or their dependence on the geometry of the shell, areinvestigated. Another line of investigation is the study ofnon-stress-free configurations observed e.g. in the growth ofleaves. A tangentially heterogeneous 3-dimensional nonlinearelastic model is studied and the Gamma-limit approach is used toreduce the dimension and analyze the model. Quantitativerigidity estimates also are studied as a strong analytical toolin tackling these variational problems. The investigator studies different mechanical phenomenaunder the unifying umbrella of the nonlinear elasticity theory. Elastic materials exhibit qualitatively different responses todifferent kinematic boundary conditions or body forces. Thisfact has given rise to many interesting questions in themathematical theory of elasticity. The main goal of this theoryis to explain various, apparently different, phenomena in termsof some shared mathematical principles. The variational approachto the nonlinear theory has been very effective in dealing withthese questions. It has also been helpful in rigorously derivingmodels for elastic shells or plates pertaining to differentscaling regimes of the body forces. In the latter context, thestrength of this approach lies in the fact that it can predictthe appropriate model together with the response of the elasticplate for the given scaling of forces or kinematic boundaryconditions without any a priori assumptions other than thegeneral principles of the 3-dimensional nonlinear elasticity. Another feature of this approach is that it can lead to newmodels that were not previously considered.

Pakzaddms-0907844这项工作是The Investigator及其合作者长期项目的一部分，旨在通过瓦解的计算方法来研究材料的弹性。 在这里，重点是弹性界表现出某种刚性行为的情况。 以这种方式，结构的几何特性的特性，例如异构体和无穷小的异构体的各个属性的属性出现。 这不同于典型的差异几何形状在果膜表面或变形的较弱的规律性方面有所不同。 第一个目标是从非线性弹性通过伽马连接方法的三维理论中识别和严格地得出薄壳的变异理论。 衍生的理论汇总取决于弹性能量或体力的缩放状态，而壳厚度。 在这种情况下，Investigator致力于涉及各种弱分化异构体的各个空间的问题。 同时，被评估了，诸如多样性和溶液的依赖性或它们对壳的几何形状的属性。 另一个研究线是观察到的无压力构型的研究，例如在生长中。 研究了切向异质的3维非线弹性模型，并使用γ-限制方法进行锻炼，并分析模型。 还将定量估计值作为解决这些变异问题的强大分析工具研究。 研究者研究了不同的机械现象，即非线性弹性理论的统一伞。弹性材料在质量不同的运动学边界条件或身体力上表现出不同的反应。 这种事实在弹性理论中引起了许多有趣的问题。 该理论的主要目标是用一些共同的数学原理来解释各种，显然不同的现象。 非线性理论的变分方法在处理这些问题方面非常有效。 它也有助于对弹性壳或与身体力量不同的弹性壳或板的严格定型。 在后一种情况下，这种方法的特征在于，它可以预测适当的模型以及弹性板对给定力缩放或运动学边界条件的弹性板的响应，而没有任何先验假设，而没有任何先验假设。这种方法的另一个特征是它可以导致以前未考虑的新模特。