TY - JOUR
T1 - A model for the quasi-static growth of a brittle fracture: existence and approximation results
JF - Math. Models Methods Appl. Sci., 12 (2002), no. 12, 1773
Y1 - 2002
A1 - Gianni Dal Maso
A1 - Rodica Toader
AB - We study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo.9 The main feature of our 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 L2-distance between the approximate solutions at two consecutive times. We study the continuous-time version of this model, obtained by passing to the limit as the time step tends to zero, and show that it satisfies (for almost every time) some minimality conditions which are slightly different from those considered in Refs. 9 and 8, 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. We also prove that, if no initial crack is present and if the data of the problem are sufficiently smooth, no crack will develop in this model, provided the penalization term is large enough.
PB - SISSA Library
UR - http://hdl.handle.net/1963/1571
U1 - 2547
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -