2002/06/29 by Gianni Dal Maso, Maso, Gianni Dal, Rodica Toader +1
Computer Science · Engineering · Mathematics · #35A35 #35B30 #35J25 #35R35 #49Q10 #74R10 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in engineering #math.AP #msc:35A35 #msc:35B30 #msc:35J25 #msc:35R35 #msc:49Q10 #msc:74R10
paper · pdf · doi:10.48550/arxiv.math/0207002
20 pages
arxiv created 2002/06/29 · openalex publication_date 2002/06/29 · arxiv updated 2009/11/30 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
We study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo. 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 Francfort and Marigo and 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. We prove also 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.