Gianni Dal Maso, SISSA, Classe di matematica, Trieste
"Quasi-static growth of brittle fractures: a variational model based on local minimization"
The talk will deal with a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo. The main feature of this new model is that, in the discrete-time formulation, in each step we do not consider absolute minimizers of the energy, but, in a sense, we look for local minimizers which are sufficiently close to the approximate solution obtained in the previous step. This is done by introducing in the variational problem an additional term which penalizes the distance between the approximate solutions at two consecutive times. The continuous-time version of this model is obtained by passing to the limit as the time step tends to zero. It satisfies (for almost every time) some minimality conditions which are slightly different from those considered in by Francfort and Marigo and by Toader and myself in our previous paper, but are still enough to prove (under suitable regularity assumptions on the crack path) that the classical Griffith's criterion holds at the crack tips. If no initial crack is present and if the data of the problem are sufficiently smooth, no crack will develop in this new model, provided the penalization term is large enough.
TUESDAY, October 8, 2002