Calculus of variations on rigid elastic structures

帕克扎德DMS-0907844这项工作是研究人员和他的合作者通过变分方法研究材料弹性性质的长期项目的一部分。在这里，重点是当弹性体表现出某种刚性行为时的情况。通过这种方式，研究结构的几何性质，例如二维曲面的各种等距空间和无穷小等距空间的性质就显得尤为重要。这与经典微分几何的不同之处在于，给定曲面或变形的正则性较弱。第一个目标是通过Gamma收敛方法从三维非线性弹性理论中识别和严格推导薄壳的变分理论。所导出的理论自然依赖于弹性能或体力相对于壳层厚度的标度范围。在这一背景下，研究者研究了涉及各种弱可微等距空间的问题。同时，研究了所得理论的性质，如解的多解性和正则性，以及它们对壳的几何形状的依赖性。另一项研究是对观察到的非无应力构型的研究，例如在叶子生长过程中观察到的构型。研究了切向非均匀三维非线性弹性模型，并用Gamma极限方法对模型进行了降维和分析。定量精确度估计也作为处理这些变分问题的强有力的分析工具而被研究。研究者在非线性弹性理论的统一保护伞下研究不同的力学现象。弹性材料对不同的运动边界条件或体力表现出不同的响应。这一事实在弹性数学理论中引起了许多有趣的问题。这一理论的主要目的是用一些共同的数学原理来解释各种明显不同的现象。非线性理论的变分方法在处理这些问题时是非常有效的。它还有助于严格推导与体力的不同标度区有关的弹性壳或板的模型。在后一种情况下，这种方法的优点在于它可以预测适当的模型和弹性板在给定力的比例或运动边界条件下的响应，而不需要任何先验假设，而不需要除三维非线性弹性一般原理之外的任何先验假设。这种方法的另一个特点是，它可以产生以前没有考虑过的新模型。

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.