Calculus of variations on rigid elastic structures

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弹性特性的长期项目的一部分。 这里，重点关注弹性体表现出某种刚性行为的情况。 以这种方式，结构的几何特性的研究，例如二维表面的等距和无穷小等距的各种空间的性质脱颖而出。 这与经典微分几何的不同之处在于给定表面或变形的规律性较弱。 第一个目标是通过伽玛收敛方法从非线性弹性的 3 维理论中识别并严格推导薄壳的变分理论。 导出的理论自然取决于弹性能或物体力在壳厚度方面的标度范围。 在这种情况下，研究者研究涉及弱可微等距的各种空间的问题。 同时，研究了衍生理论的性质，例如解的多重性和规律性，或者它们对壳几何形状的依赖性。 另一条研究方向是研究观察到的非无应力配置，例如在叶子的生长过程中。 研究了切向异质三维非线性弹性模型，采用Gamma极限方法对模型进行降维分析。 定量刚性估计也被作为解决这些变分问题的强大分析工具进行研究。 研究人员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边界条件或体积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。 该理论的主要目标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方面非常有效。 它还有助于严格推导与不同体力尺度范围相关的弹性壳或板的模型。 在后一种情况下，该方法的优势在于它可以针对给定的力比例或运动学边界条件预测适当的模型以及弹性板的响应，除了 3 维非线性弹性的一般原理之外，无需任何先验假设。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.