Mathematical Sciences: Cavitation in Solids

题：固体中的空化ID#dms-9403862摘要：主要研究人员将分析变分和某些偏微分方程中的某些问题，以确定这些问题有奇异解的条件。基本的方程是那些在弹性和粘弹性中出现的方程，所需的奇点是不连续的。将考虑的问题包括：拟线性椭圆组奇异解的存在性；双曲组奇异解的存在性和可容许性准则；变分中问题的奇异点极小元的存在性；以及确定拟线性椭圆组的已知间断解是否实际上是变分中相应问题的极小值。首席研究员将分析的问题出现在一些塑料和玻璃的数学模型中。这项研究的目的是预测这种材料在外力和/或高温作用下的失效。令人感兴趣的失败是那些由于形成微观孔造成的失败。当外力作用于某些塑料时，这些孔的大小会增大，然后结合在一起，在材料中形成裂缝。在用于建造光缆的玻璃中，人们发现当激光沿光缆的长度方向照射时，就会形成这样的孔。将讨论的具体问题包括：孤立孔的萌生和动态生长；在孤立孔表面形成二次裂纹；以及温度变化可能导致这种孔形成的方式。______________________________________________________________________________

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. ______________________________________________________________________________