New Mathematical Methods for Dynamic Fracture Evolution

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For a large range of applications, from civil infrastructure to national defense, understanding the failure of materials is critical. Yet, our ability to predict this failure is limited by both modeling, which is somewhat ad hoc, and the mathematics available to formulate and analyze models, as well as to justify numerical methods. These issues are most severe in dynamic problems, such as impacts, when loading changes quickly. The main goal of this project is the development of new mathematical methods for dynamic fracture evolution. In particular, the principal investigator (PI) will extend methods for regular crack paths to more realistic paths, with kinking and branching. A second goal is to address fundamental mathematical issues that are necessary for further progress in completely general settings. Finally, the PI will study phase-field approximations of fracture, which have become very popular tools in the engineering community but remain poorly understood.The ability to accurately predict failure depends on the quality of the underlying mathematical models of defects as well as on understanding fundamental properties of solutions. When crack paths are regular, mathematical methods are available to study these evolutions. However, when they are not, the only methods so far involve considering the paths to be limits of more regular paths. The main technical issue here is that strong convergence of the corresponding elastodynamics is necessary for energy balance, as well as for other properties of solutions, but this convergence remains open in many situations. Another fundamental issue is uniqueness of elastodynamic solutions for a given crack path. The investigator will show uniqueness in certain settings, and explore general consequences, such as bounds on crack speed. The final goal of the project is to analyze phase-field models for fracture. While very popular in the engineering community, a number of properties, including whether they approximate the correct surface energy, or satisfy a maximal dissipation condition, remain open questions.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.