The investigator studies the evolution of defects in materials (e.g., fracture, damage, dislocation plasticity) using a progression of models from global minimization-based quasi-statics to full dynamics. Global minimization models are now well understood, but very little is understood in the more physically realistic settings of local minimization-based and dynamic problems. Specifically, existence is open for both cohesive and sharp-interface (Griffith) dynamic fracture, and effective macro-scale models for local minimization-based dislocation plasticity are unknown. The investigator seeks to make significant progress on these and related problems.
The propagation of defects in materials is of obvious fundamental importance, yet in the most physically realistic settings mathematical foundations and analysis are lacking, leaving engineering models largely ad hoc. The investigator works on developing mathematical support in these areas, leading to improved models, better understanding of solutions, and improved (and justified) algorithms for computing simulations.
Infrastructure tends to be both over-engineered against failure, and subject to periodic catastrophic failure. This highlights the fact that a fundamental understanding of material failure is still lacking. In particular, we need to gain a mathematical understanding of the nucleation and evolution of defects in materials. This project addressed some fundamental questions about such defect evolution, as well as trained PhD, Masters, and undergraduate students in the analysis of these problems. A preliminary basic issue in the mathematics of dynamic fracture evolution is whether there exist solutions to the appropriate equations for the evolving material, even if the crack set evolution is specified. With a collaborator, G. Dal Maso, we answered this question in the affirmative. With another collaborator, V. Slastikov, we formulated new mathematical models for cohesive dynamic fracture -- a setting that is more physically realistic, but more mathematically challenging than the usual setting. Also in the setting of cohesive fracture, I formulated new mathematical models for quasi-static (slowly growing) fracture, with a finite stress threshold for nucleation (the creation of new cracks). Prior to this, it was generally presumed that one needs an infinite threshold for the mathematical problem to be solvable. With an undergraduate student, I showed that in models for dynamic Griffith fracture (non-cohesive), cracks must grow continuously (there can not instantly be increases in fracture). Finally, with another collaborator, I extended some prior work on elastic damage evolution to more physically realistic elastic settings. This project also supported ongoing work with a PhD student, addressing a long standing problem about quasi-static cohesive fracture evolution.
