Nonlocal Theories for Free Crack Propagation in Brittle Materials - Lecture 2

Explore the mathematics of upscaling for nonlocal methods in brittle material fracture in this second lecture of a three-part series. Delve into appropriate notions of weak convergence, compactness, and Gamma-convergence as they relate to nonlocal theories for free crack propagation. Learn how to connect nonlocal short-range forces acting over small length scales with dynamic free crack evolution observed at the macroscopic scale in brittle media. Examine the peridynamic formulation, which allows for both continuous and discontinuous deformations associated with cracks, and understand how it can be used to represent dynamics at mesoscopic length scales. Discover how analysis shows that the resulting mesoscopic dynamics is well-posed and how the macroscopic evolution corresponds to the simultaneous evolution of the fracture surface and linear elastic displacement away from the crack set. Gain insights into how elastic moduli, wave speed, and energy release rate for the macroscopic evolution are explicitly determined by moments of the nonlocal potential energy.