Speaker:
Alain Karma
Date of Talk:
11/12/10
Affiliation:
Department of Physics, Northeastern University
Title:
Mathematical Models of Fracture and Singular Perturbations
Abstract
Understanding crack propagation is important for many applications from the design of safe airplanes to earthquakes prediction. This talk will discuss two classes of mathematical models of fracture and their relationships. The first is the well-established continuum theory of fracture mechanics, which assumes that microscopic details of bond breaking and energy dissipation in a small region around the crack tip do not matter. All that matters in this theory is one experimentally measurable number, the fracture energy, which is the amount of energy required to create new fracture surfaces per unit area of crack advance. The second is the phase-field approach first developed to solve free-boundary problems that describe phase changes and subsequently extended to a wide range of interface dynamics problems including fracture. This approach attempts to describe both the short-scale physics of materials failure near the crack tip and macroscopic linear elasticity within a self-consistent set of partial differential equations that can be solved numerically and analyzed. I will discuss the results of a stability analysis of crack propagation under general loading condition which shows that the traditional continuum theory breaks down at short scale. I will then argue, based on computations and analysis of the phase-field model, that short scale cohesive forces analogous to surface tension in fluid mechanics act as a singular perturbation in the large scale crack evolution problem, revealing a deep analogy between fracture and interface pattern formation in a large class of systems.
