Singular Deformations in Mechanics

Southern Illinois University At Carbondale, Carbondale IL

Investigators

Abstract

Proposal: DMS-0405646 PI: Scott J Spector Institution: Southern Illinois University Title: Singular Deformations in Mechanics ABSTRACT The Principal Investigator will analyze singular solutions to certain systems of partial differential equations as well as singular minimizers to the corresponding energy. The problems involve: existence of singular solutions to parabolic systems; existence of minimizers with singularities for problems in the calculus of variations; regularity, fine properties, and the asymptotic behavior of singular minimizers; 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 motivation for these problems lies in the mechanics of solids. Mathematical models for elastomers, glass, glassy polymers, thin films, and ductile metals yield singular deformations that are used to predict the failure of such materials under the application of external forces. The failures of interest are those due to the formation and growth of microscopic holes as well as the subsequent coalescence of such holes to form cracks in a material. When external forces are applied to an elastomer these holes become visible, grow in size, and then combine to break the material. Particular problems of interest are: the initiation and dynamic growth of a solitary hole; the manner in which holes combine to form cracks; the optimal location for hole formation; and, the manner in which a change in temperature can cause the formation of such holes. The research area of this grant is the mathematical analysis of equations that arise in Materials Science. The focus of the investigation is certain significant material failures in solids. The goal is the prediction of the formation and growth of voids and holes in materials that are crucial to the automotive and defense industries. If one wants to understand the reasons that materials fail 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.