Analysis of Cavitation in Solids

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The focus of the research supported by this award is themathematical analysis of certain significant material failures insolids. The goal of this endeavor is the qualitative prediction ofthe formation and growth of voids. Toward this end the principalinvestigator will continue his studies of the relevant nonlinearpartial differential equations, which constitute the mathematicalmodel, in order to determine conditions under which these problemshave singular solutions. The underlying equations are those thatarise in elasticity and viscoelasticity and the desiredsingularities are point discontinuities. Problems that will beconsidered include: the existence of singular solutions toquasilinear elliptic systems; the existence of, and admissibilitycriteria for, singular solutions to hyperbolic systems; theexistence of singular solutions to certain parabolic systems; theexistence of minimizers with singularities for problems in thecalculus of variations; regularity, fine properties, and theasymptotic behavior of singular minimizers; the optimal locationfor an isolated singularity; and, the determination of whetherknown singular solutions to a quasilinear elliptic system areindeed minimizers of the corresponding problem in the calculus ofvariations.The research area of this grant is the mathematical analysis ofequations that arise in Materials Science. The most common way todetermine when a material will fail under the influence of externalforces is to subject a piece of the actual material to tensileloads until failure occurs, i.e., "pull on it until it breaks".This is fine if one is interested only in the gross properties ofthe material. However, if one wants to understand the reasons formaterial failure then one must have recourse to mathematical modelsof the material. Experiments on certain rubbery polymers, calledelastomers, have shown that when one pulls on an elastomer smallholes appear in the material. These holes then grow in size andcombine to form cracks. A similar phenomenon has been observed inoptical fibers. Catastrophic failure, due to a series of holesthat cascade down the core of the fiber, can occur when excessivepower is applied. These holes seriously degrade the ability of thefiber to transmit information. In this grant, the principalinvestigator will uncover the mechanisms that cause the creationand growth of holes in polymers and glasses by examining systems ofpartial differential equations from the theory of elasticity andviscoelasticity.