Mathematical Sciences: Cavitation in Solids

题目：固体中的空化问题摘要：首席研究员将分析变分学和偏微分方程中的某些问题，以确定这些问题具有奇异解的条件。基础方程是那些出现在弹性和粘弹性和期望的奇点是一个不连续。将考虑的问题包括：拟线性椭圆系统奇异解的存在性；双曲型系统奇异解的存在性及可容性准则变分学问题的奇异极小值的存在性在变分学中，确定拟线性椭圆系统的已知不连续解是否为相应问题的最小解。首席研究员将分析的问题出现在许多塑料和玻璃的数学模型中。本研究的目的是预测这些材料在外力和/或高温作用下的失效。我们感兴趣的失败是由于微观孔洞的形成。当外力作用于某些塑料时，这些孔的尺寸会增大，然后结合在一起形成材料中的裂缝。在用于构建光纤电缆的玻璃中，人们发现当激光束沿着电缆的长度向下照射时，就会形成这样的孔。将要解决的具体问题是：孤立洞的产生和动态增长；裂缝：在一个单独的洞的表面上形成的次级裂缝；以及温度变化导致这些洞形成的方式。______________________________________________________________________________

PI: Scott J. Spector, Southern Illinois University at Carbondale Title: Cavitation in Solids ID# DMS-9403862 Abstract: The principal investigator will analyze certain problems in the calculus of variations and certain partial differential equations in order to determine conditions under which these problems have singular solutions. The underlying equations are those that arise in elasticity and viscoelasticity and the desired singularity is a discontinuity. Problems that will be considered include: the existence of singular solutions to quasilinear elliptic systems; the existence of, and admissibility criteria for, singular solutions to hyperbolic systems; the existence of minimizers with singularities for problems in the calculus of variations; and, the determination of whether known discontinuous solutions to a quasilinear elliptic system are in fact minimizers of the corresponding problem in the calculus of variations. The problems that the principal investigator will analyze arise in mathematical models for a number of plastics and glasses. The purpose of this investigation is to predict the failure of such materials under the application of external forces and/or high temperatures. The failures of interest are those due to the formation of microscopic holes. When external forces are applied to certain plastics these holes grow in size and then combine to form cracks in the material. In glasses that are used to construct fiber optic cables such holes have been found to form when a laser beam is directed down the length of the cable. Particular issues that will be addressed are: the initiation and dynamic growth of a solitary hole; the formation of secondary cracks upon the surface of a solitary hole; and, the manner in which a change in temperature can cause the formation of such holes. ______________________________________________________________________________