The investigator uses variational methods to study material damage, with an emphasis on properties of damaged regions and their time evolution. Examples of damage include brittle cracks, cracking due to fatigue, and debonding of thin films. Some discrete-time models for damage evolution exist in the engineering community, but these are generally ad hoc and not known to correspond to limiting continuous-time models. The investigator studies the existence of the limiting models, basic properties of the damage regions, and the justification for extending numerical methods for static problems to methods for the quasi-static, continuous-time problems. Damage to materials is of great technological importance not only because it results in material failure, but also because it plays an important role in building certain nanostructures, for example by selectively debonding thin films. The project seeks to greatly improve our understanding of these materials by improving the mathematical models and studying basic properties of their solutions, as well as by developing and justifying numerical methods. Mathematics Ph.D. students are included in the project, and are trained in the mathematical modeling and analysis of important interdisciplinary problems.