Analysis of Cavitation in Solids

Southern Illinois University At Carbondale, Carbondale IL

Investigators

Abstract

The focus of the research supported by this award is the mathematical analysis of certain significant material failures in solids. The goal of this endeavor is the qualitative prediction of the formation and growth of voids. Toward this end the principal investigator will continue his studies of the relevant nonlinear partial differential equations, which constitute the mathematical model, 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 singularities are point discontinuities. 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 singular solutions to certain parabolic systems; the existence of minimizers with singularities for problems in the calculus of variations; regularity, fine properties, and the asymptotic behavior of singular minimizers; the optimal location for an isolated singularity; and, the determination of whether known singular solutions to a quasilinear elliptic system are indeed minimizers of the corresponding problem in the calculus of variations. The research area of this grant is the mathematical analysis of equations that arise in Materials Science. The most common way to determine when a material will fail under the influence of external forces is to subject a piece of the actual material to tensile loads until failure occurs, i.e., "pull on it until it breaks". This is fine if one is interested only in the gross properties of the material. However, if one wants to understand the reasons for material failure then one must have recourse to mathematical models of the material. Experiments on certain rubbery polymers, called elastomers, have shown that when one pulls on an elastomer small holes appear in the material. These holes then grow in size and combine to form cracks. A similar phenomenon has been observed in optical fibers. Catastrophic failure, due to a series of holes that cascade down the core of the fiber, can occur when excessive power is applied. These holes seriously degrade the ability of the fiber to transmit information. In this grant, the principal investigator will uncover the mechanisms that cause the creation and growth of holes in polymers and glasses by examining systems of partial differential equations from the theory of elasticity and viscoelasticity.