Mathematical Sciences: Analysis of Cavitation in Solids

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9703986 Spector The principal investigator will analyze certain partial differential equations as well as certain problems in the calculus of variations 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 minimizers with singularities for problems in the calculus of variations; and, the determination of whether known discontinuous solutions to quasilinear elliptic systems are in fact 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 loading 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 occurs in optical fibers. If one tries to send too much power through an optical fiber then a series of holes can form in the core of the fiber. 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. ***