Analysis of Stability and Instability for Elastic Materials

弹性体，软胶粘剂和玻璃状聚合物的数学模型分析产生奇异变形，可用于预测这些材料在外力作用下的失效。我们感兴趣的失败是由于表面上的折痕的形成，以及表面和内部微观孔的产生和生长。当施加外力时，这些孔变得可见，尺寸变大，然后结合起来破坏材料。在这个项目中解决的特殊问题是：在软胶粘剂薄膜中产生孔；单孔的形成和生长；以及在弹性体中受到严重压缩的表面折痕的形成。这类现象的数学分析涉及：变分法问题中有奇点和无奇点的极小值的存在性和唯一性；奇异极小化器的正则性和精细性质欧拉-拉格朗日方程的多重解的物理相关性；在文献中许多地方构造的径向最小值是否确实是能量的全局最小值，特别是当考虑非径向摄动时；并且，确定拟线性椭圆系统的已知奇异解是否确实是变分学中相应问题的极小解。这个项目的重点是对材料科学中出现的方程进行数学分析。在现代技术先进的材料上进行的实验揭示了材料失效的另一种机制，即，这些材料的断裂方式可能不符合工程师为经典材料开发的通常预期。例如，虽然用于制造传统飞机的金属会慢慢疲劳，但用于制造现代飞机的复合材料偶尔会表现出更快的退化。该项目的目标是更好地理解两种特定的破坏机制：材料表面缺陷（折痕）的形成以及材料表面和内部孔的产生和生长。被分析的特殊材料是胶粘剂薄膜和某些被称为弹性体的橡胶聚合物。

SpectorDMS-1107899 The analysis of mathematical models for elastomers, soft adhesives, and glassy polymers yields singular deformations that can be used to predict the failure of such materials under the application of external forces. The failures of interest are those due to the formation of creases on the surface and also the creation and growth of microscopic holes, both on the surface and in the interior. When external forces are applied these holes become visible, grow in size, and then combine to break the material. Particular problems that are addressed in this project are: the creation of holes in thin films of soft adhesives; the initiation and growth of a solitary hole; and the formation of surface creases in elastomers that are subject to severe compressions. The mathematical analysis of such phenomena involves: the existence and uniqueness of minimizers with and without singularities for problems in the calculus of variations; regularity and fine properties of singular minimizers; the physical relevance of multiple solutions to the corresponding Euler-Lagrange equations; whether or not radial minimizers, which have been constructed in numerous places in the literature, are indeed the global minimizers of the energy, especially when nonradial perturbations are considered; and, the determination of whether known singular solutions to quasilinear elliptic systems are indeed minimizers of the corresponding problem in the calculus of variations. The focus of this project is the mathematical analysis of equations that arise in Materials Science. Experiments on modern technologically sophisticated materials have uncovered alternative mechanisms of material failure, i.e., the way such materials break may not follow the usual expectations that engineers have developed for classical materials. For example, while the metals used to make a traditional aircraft slowly fatigue, the composites used to construct a modern plane occasionally exhibit a more rapid degradation. The goal of this project is to obtain a better understanding of two particular destructive mechanisms: the formation of defects (creases) on the surface of a material and the creation and growth of holes both on the surface and in the interior of a material. The particular materials that are analyzed are thin films of adhesives and certain rubbery polymers that are known as elastomers.