Mathematical Sciences: Cavitation in Solids

PI：Scott J. Spector，南伊利诺伊大学卡本代尔分校 职务名称：空穴在固体ID# DMS-9403862摘要：主要研究者将分析某些问题的变分法和某些偏微分方程，以确定条件下，这些问题有奇异的解决方案。基本方程是那些在弹性和粘弹性中出现的方程，所需的奇异性是不连续的。将被考虑的问题包括：奇异解的存在性拟线性椭圆系统的存在性，和容许的标准，奇异解的双曲系统，存在的最小值与奇异性的问题，在变分法;并且，在本发明中，确定一个拟线性椭圆方程组的已知间断解是否实际上是微积分中相应问题的极小化变化. 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。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。感兴趣的故障是由于微观孔的形成。当外力作用于某些塑料时，这些孔的尺寸会增大，然后联合收割机在材料中形成裂纹。在用于构造光纤电缆的玻璃中，已经发现当激光束沿着电缆的长度被引导时形成这样的孔。将解决的具体问题是：孤立孔的起始和动态增长;孤立孔表面上二次裂纹的形成;以及温度变化可能导致形成这种孔的方式。 ______________________________________________________________________________

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. ______________________________________________________________________________