Mathematical Sciences: Symmetry Methods and Nonlinear Analysis in Elastomechanics

小行星9407738 拟议的工作的主要重点将是分析偏微分方程的对称性与应用程序的机械。后者将涉及非线性弹性和粘弹性固体和结构的几何精确公式。建议wrk有三个基本目标：（i）获得新的定性结果和检测新的现象;（ii）获得全球连续的结果，为2和3维弹性问题;（iii）进一步追求和利用对称性和节点属性之间的联系，在全球分歧问题的椭圆型方程。 这项工作有几个潜在的应用“现实世界”的工程问题。首先，对称结构在工程实践中的出现是普遍存在的-部分原因是美学，部分原因是最佳承载能力（通常基于线性分析）。然而，这样的结构也容易受到破坏稳定性的影响。这项研究的成果将有助于提供一个系统的方法（必然非线性）分析这种不稳定性。第二，定性的非线性分析的具体类别的问题，参数化的一般本构关系，往往阐明了具体的作用，材料的行为inachieving阈值的不稳定性。通过这种方式，我们能够对材料行为进行分类，并着眼于结构的工程设计。

9407738 Healey The main focus of the proposed work will be the analysis of partial differential equations with symmetry in connection with applications to elastomechanics. The latter will involve geometrically exact formulations of nonlinearly elastic and visco-elastic solids and structures. The proposed wrk has three essential goals: (i) to obtain new qualitative results and detect new phenomena; (ii) to obtain global continuation results for problems of 2 and 3-dimensional elasticity; (iii) to further pursue and exploit the connection between symmetry and nodal properties in global bifurcation problems for elliptic equations. This work has several potential applications to "real-world" engineering problems. First, the occurence of symmetric structures in engineering practice is ubiquitous - due in part to esthetics, and in part to optimal load-carrying capacity (typically based upon linear analyses). However, such structures are also susceptible to symmetry-breaking instabilities. The fruits of this research will help to provide a systematic approach to the (necessarily nonlinear) analysis of such instabilities. Second, a qualitative nonlinear analysis of a concrete class of problems, parameterized by general constitutive laws, often illuminates the specific roles of material behavior inattaining thresholds of instability. In this way we are able to classify material behavior with an eye toward engineering design of structures.