Mathematical Sciences: Cavitation in Solids

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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. ______________________________________________________________________________