Speaker: G.A. Francfort (Universit\'e Paris 13)
Title: A variational view of brittle fracture evolution
Abstract:
The theory of fracture usually referred to as that of Griffith shows many
drawbacks: it does not initiate cracks; it is powerless when trying to predict
the crack path; it does not know how to handle sudden crack jumps, .....
Jean-Jacques Marigo and I have proposed a model based on energy minimization
which does away with many of those obstacles, while departing as little as
feasible from Grifffith's theory.
I will first describe the details of the proposed model, show how it does away
with the above mentioned drawbacks and evoke its specific shortcomings.
From a mathematical stanpoint, the model resembles a kind of evolutionary image
segmentation problem in the sense of Mumford \& Shah. Chris Larsen and I have
shown the existence of a solution to the evolution for the weak -- \`a la De
Giorgi -- formulation of the problem. I will describe the result and briefly
evoke the method that was used. I will also mention the non-trivial extensions,
obtained in collaboration with Gianni Dal Maso and Rodica Toader, to the case of
a non-convex bulk energy.
The model is readily amenable to numerics through various regularization of the
energy which "Gamma-converge" to the original energy. This is the work of
Blaise Bourdin (partly in collaboration with Antonin Chambolle). For lack of
time, I will not discuss numerical issues, but merely illustrate the talk with
Bourdin's computations in one or two cases that are well beyond the scope of
classical brittle fracture