Mathematical Sciences: Symmetry Methods and Nonlinear Analysis in Elastomechanics

9407738 Healey拟议工作的主要重点将是分析具有对称性的偏微分方程，并将其应用于弹性力学。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。所提出的WRK有三个基本目标：(I)获得新的定性结果和发现新的现象；(Ii)获得二维和三维弹性问题的整体延拓结果；(Iii)进一步探索和利用椭圆型方程整体分支问题中对称性和节点性质之间的联系。这项工作对“真实世界”的工程问题有几个潜在的应用。首先，对称结构在工程实践中无处不在--部分是由于美学，部分是由于最佳承载能力(通常基于线性分析)。然而，这种结构也容易受到对称性破坏的不稳定性的影响。这项研究的成果将有助于提供一种系统的方法来分析这种不稳定性(必然是非线性的)。其次，对一类具体问题的定性非线性分析，由一般本构定律参数化，通常说明材料行为在达到失稳门槛时的具体作用。通过这种方式，我们能够着眼于结构的工程设计来对材料行为进行分类。

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.