Calculus of Variations and Geometric Measure Theory

G. Dal Maso - R. Toader

A model for the quasi-static growth of brittle fractures: existence and approximation results

created on 30 Mar 2001
modified on 20 Jun 2002


Published Paper

Inserted: 30 mar 2001
Last Updated: 20 jun 2002

Journal: Arch. Rational Mech. Anal.
Volume: 162
Pages: 101-135
Year: 2002


We give a precise mathematical formulation of a variational model for the irreversible quasi-static evolution of brittle fractures proposed by G.A. Francfort and J.-J. Marigo, and based on Griffith's theory of crack growth. In the two-dimensional case we prove an existence result for the quasi-static evolution and show that the total energy is an absolutely continuous function of time, although we can not exclude that the bulk energy and the surface energy may present some jump discontinuities. This existence result is proved by a time discretization process, where at each step a global energy minimization is performed, with the constraint that the new crack contains all cracks formed at the previous time steps. This procedure provides an effective way to approximate the continuous time evolution