New variational methods for fracture evolution

Worcester Polytechnic Institute, Worcester MA

Investigators

Abstract

For a large range of applications, from civil infrastructure to national defense, predicting the failure of materials and structures is critical. Our ability to predict failure depends on mathematical and computational models, and we need both to be as rigorous and physically justified as possible. Over the last 25 years, there have been significant mathematical advances in this area, which have directly improved computational models, particularly for fracture. However, these advances still have significant limitations - they are mostly restricted to models without applied forces, and existing mathematical models can entangle fracture nucleation and propagation in an unphysical way. The investigator recently formulated a mathematical model for fracture that isolates propagation and is compatible with all applied forces. The first goal of this project is to show existence of evolutions satisfying this principle. Another goal is to formulate models and show existence for evolutions satisfying both this principle and physical criteria for nucleation. The investigator also develops and studies improved computational models based on these mathematical results. The project includes training Ph.D. students. The ability to accurately predict material failure depends on the quality of mathematical models of defects, as well as on understanding basic properties of solutions. While successful in many ways, variational models for static and quasi-static fracture have been based on energy minimization which can force unphysical nucleation and is incompatible with body loads. Recent progress includes formulating a new variational principle that overcomes these limitations. It remains to show that there exist equilibrium states satisfying this principle, and similarly for quasi-static evolutions. Both are goals of this project. The addition of completely independent strength criteria for nucleation is also studied. A second direction is to substantially improve the connection between phase-field fracture implementations and models for sharp fracture. This utilizes the newly introduced variational principle, which can be studied using Gamma-convergence, even though it is completely spatially local. 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.