The investigator develops and studies models for the evolution of elastic damage (i.e., regions of weakened elastic properties) and fracture. Difficulties come from, among other things, the irreversibility of these phenomena, and the fact that existing models are essentially variational (static). The general strategy is to first develop quasi-static models that are consistent with these variational principles, and prove existence and approximation results. Next, the goal is to extend these results to the fully dynamic setting. The first step has been essentially completed for damage and brittle fracture, and the investigator works on developing corresponding methods for cohesive fracture. Dynamic models and their analysis are major open problems for all these phenomena, and are the main focus of this project.
Material damage and fracture prediction are of fundamental importance in the areas of materials and manufacturing, civil infrastructure, etc., yet in the most physically realistic setting (dynamics) there is no sound mathematical footing, making engineering models largely ad hoc. The investigator works on developing this mathematical support, as well as creating and rigorously justifying algorithms for computing approximate solutions. The project includes training mathematics Ph.D. students through work on important interdisciplinary problems.
