Mathematical Sciences Department
PhD Dissertation Defense
Pooya Yousefi
Thursday, April 20, 2023
11:00 am - 1:00 pm
Stratton Hall 306
and via Zoom: 942 4665 0321
https://wpi.zoom.us/j/94246650321
Title: Quasi-Static Griffith Fracture Evolution with Boundary Loads
Abstract: In this dissertation, we study the well-posedness of a variational formulation for modeling quasi-static evolution of cracks in elastic materials under boundary loads. Quasi-static evolution of fracture for displacement loads, i.e., Dirichlet boundary conditions, has been studied extensively in the past couple of decades, using models based on global and local minimization. However, boundary loads, i.e., Neumann boundary conditions, had been seen as problematic with the usual variational formulation, due to a straightforward non-existence argument.
Recently, a variational formulation, namely alternate minimization, was proposed as a method for finding solutions for fracture problem with boundary loads. Adopting this method, we study existence of quasi-static fracture evolutions under time-varying boundary loads.
Global minimizers of the quasi-static Dirichlet problem have always balanced the sum of stored elastic plus crack dissipated surface energies. Nonetheless, even though our formulation for the quasi-static Neumann problem is based on global minimization, we show that evolutions here do not necessarily satisfy this energy balance, and describe how there can be decreases in the energy. Note that decrease in the sum of stored and dissipated energies in time might be expected since the effect of kinetic energy caused by the jumps in the evolution of cracks is not considered in the quasi-static energy equation. We also give estimates on how big energy drops can be.
Dissertation Committee:
Professor Christopher J. Larsen, WPI (Dissertation Advisor)
Professor Gilles A. Francfort, Université Paris-Nord
Professor Marcus Sarkis-Martins, WPI
Professor Nima Rahbar, WPI
Professor Guanying Peng, WPI