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 这项工作是研究者和他的合作者通过变分法研究材料的弹性性质的长期项目的一部分。 这里，重点是当弹性体表现出某种刚性行为时的情况。 在这种方式下，结构的几何性质的研究，例如各种空间的等距和无穷小等距的二维表面的性质，来到了前列。 这不同于经典的微分几何中较弱的正则性的给定表面或变形。 第一个目标是从三维非线性弹性理论中通过Gamma收敛方法确定并严格推导出薄壳的变分理论。 导出的理论自然地依赖于壳厚度的弹性能或体积力的标度制度。 在这方面，调查工作的问题，涉及各种空间的weaklydifferentiable isometries。 与此同时，性质的衍生理论，如多重性和正则性的解决方案，或其依赖于壳的几何形状，进行了调查。 另一个研究方向是研究非无应力结构，例如在叶片生长中观察到的。 本文研究了切向非均匀三维非线性气动力模型，采用Gamma极限方法对模型进行降维分析。 本文还研究了作为处理这些变分问题的强有力的分析工具的定量可靠性估计。 研究者在非线性弹性理论的统一框架下研究不同的力学现象。弹性材料对不同的运动学边界条件或体积力表现出质的不同响应。 这一事实在弹性力学的数学理论中引起了许多有趣的问题。 这一理论的主要目标是用一些共有的数学原理来解释各种各样的、明显不同的现象。 非线性理论的变分方法在处理这些问题时是非常有效的。 它也有助于严格推导模型的弹性壳或板属于不同比例制度的体力。 在后一种情况下，这种方法的优势在于，它可以预测适当的模型，以及对给定的力或运动边界条件的比例的弹性板的响应，而没有任何先验假设以外的三维非线性弹性的一般原则。这种方法的另一个特点是，它可以导致以前没有考虑过的新模型。