Atomistic-to-continuum convergence for quasi-static crack growth in brittle materials

We study the atomistic-to-continuum limit for a model of quasi-static crack evolution driven by time-dependent boundary conditions. We consider a two-dimensional atomic mass spring system whose interactions are modeled by classical interaction potentials, supplemented by a suitable irreversibility condition accounting for the breaking of atomic bonding. In a simultaneous limit of vanishing interatomic distance and discretized time step, we identify a continuum model of quasistatic crack growth in brittle fracture [G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46 (1998) 1319–1342] featuring an irreversibility condition, a global stability, and an energy balance. The proof of global stability relies on a careful adaptation of the jump-transfer argument in [G. A. Francfort and C. J. Larsen, Existence and convergence for quasi-static evolution in brittle fracture, Commun. Pure Appl. Math. 56 (2003) 1465–1500] to the atomistic setting.