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

Abstract: We study the atomistic-to-continuum limit for a model of a 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 atmoic bonding. In a simultaneous limit of vanishing interatomic distance and discretized time step, we identify a continuum model of quasi-static crack growth in brittle fracture 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 by Francfort and Larsen to the atomistic setting.