While defects in materials play a fundamental role in material failure, their analysis remains a major challenge in applied mathematics. This is partly due to the difficulty of formulating precise mathematical models, and partly due to the difficulty of analyzing the free surfaces and singularities involved. The investigator extends recent successes in the analysis of globally minimizing and locally minimizing quasi-static evolutions to both locally stable quasi-static evolutions and dynamic evolutions. One goal is to develop and study new models for cohesive fracture and plasticity with softening, based on local stability rather than global minimality (which is mathematically problematic). The investigator also studies existence and analyzes fundamental properties of dynamic fracture solutions, based on models he formulated previously.
The failure of materials rests on the nucleation and evolution of defects such as cracks, plastic regions, and damage. The ability to accurately predict failure depends on the quality of the underlying mathematical models of these defects, as well as on understanding fundamental properties of solutions. Substantial challenges remain in these areas, both in formulating sound models and in the analysis of qualitative behavior of solutions. The investigator seeks to make fundamental progress on these fronts, by developing new models that are both mathematically well-posed and significantly more physically realistic than existing models, and performing the mathematical analysis necessary to assess their accuracy.
