Dynamic Brittle Fracture as a Small Horizon Limit of Peridynamics

We consider the nonlocal formulation of continuum mechanics described by peridynamics. We provide a link between peridynamic evolution and brittle fracture evolution for a broad class of peridynamic potentials associated with unstable peridynamic constitutive laws. Distinguished limits of peridynamic evolutions are identified that correspond to vanishing peridynamic horizon. The limit evolution is associated with dynamic brittle fracture and satisfies a dynamic energy inequality expressed in terms of the kinetic energy of the motion together with a bulk elastic energy and a Griffith surface energy. It corresponds to the simultaneous evolution of elastic displacement and brittle fracture with displacement fields satisfying the wave equation inside the cracking domain. The wave equation provides the dynamic coupling between elastic waves and the evolving fracture path inside the media. The elastic moduli, wave speed and energy release rate for the evolution are explicitly determined by moments of the peridynamic influence function and the peridynamic potential energy.