Variational Fracture with Loads

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For a large range of applications, from civil infrastructure to national defense, understanding the failure of materials is critical. The ability to predict failure depends on modeling and on the mathematics available to formulate and analyze models, and to justify numerical methods. Over the last twenty years, there have been significant mathematical advances in this area, particularly in fracture mechanics, but these results are largely limited to models without applied forces. However, the inclusion of applied forces is relevant for applications. This project aims to improve the current mathematical formulation for fracture in materials in equilibrium with applied forces, to extend this formulation to models for material evolution, and to study certain features of these evolutions related to the presence of applied forces. The project offers training research opportunities for doctoral students. The ability to accurately predict failure depends on the quality of the underlying mathematical modeling of defects, as well as on understanding fundamental properties of solutions. Until recently, variational models for static and quasi-static fracture have been limited to Dirichlet boundary conditions, since there do not exist solutions to the seemingly most natural formulation that includes Neumann boundary conditions, i.e., boundary loads. The aim of the project is to improve on a recently introduced static formulation for variational fracture with boundary loads which can have solutions, to extend this static model to the quasi-static case and show existence of solutions, and to study the possibility of existence of energy drops in quasi-static models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.