The analysis of mathematical models for elastomers, soft adhesives, and glassy polymers yields singular deformations that can be used to predict the failure of such materials under the application of external forces. The failures of interest are those due to the formation of creases on the surface and also the creation and growth of microscopic holes, both on the surface and in the interior. When external forces are applied these holes become visible, grow in size, and then combine to break the material. Particular problems that are addressed in this project are: the creation of holes in thin films of soft adhesives; the initiation and growth of a solitary hole; and the formation of surface creases in elastomers that are subject to severe compressions. The mathematical analysis of such phenomena involves: the existence and uniqueness of minimizers with and without singularities for problems in the calculus of variations; regularity and fine properties of singular minimizers; the physical relevance of multiple solutions to the corresponding Euler-Lagrange equations; whether or not radial minimizers, which have been constructed in numerous places in the literature, are indeed the global minimizers of the energy, especially when nonradial perturbations are considered; and, the determination of whether known singular solutions to quasilinear elliptic systems are indeed minimizers of the corresponding problem in the calculus of variations.
The focus of this project is the mathematical analysis of equations that arise in Materials Science. Experiments on modern technologically sophisticated materials have uncovered alternative mechanisms of material failure, i.e., the way such materials break may not follow the usual expectations that engineers have developed for classical materials. For example, while the metals used to make a traditional aircraft slowly fatigue, the composites used to construct a modern plane occasionally exhibit a more rapid degradation. The goal of this project is to obtain a better understanding of two particular destructive mechanisms: the formation of defects (creases) on the surface of a material and the creation and growth of holes both on the surface and in the interior of a material. The particular materials that are analyzed are thin films of adhesives and certain rubbery polymers that are known as elastomers.