Robert Lipton: Dynamic brittle fracture as a well posed nonlocal initial value problem

A nonlocal model for dynamic brittle damage is introduced consisting of two phases, one elastic and the other inelastic. Evolution from the elastic to the inelastic phase depends on material strength. Existence and uniqueness of the displacement-failure set pair follow from the initial value problem. The displacement-failure pair satisfies energy balance. The length of nonlocality $\epsilon$ is taken to be small relative to the domain. The evolution provides an energy that interpolates between volume energy corresponding to elastic behavior and surface energy corresponding to failure. The deformation energy resulting in material failure over a region is a $d-1$ dimensional integral that is uniformly bounded as $\epsilon\rightarrow 0$. For fixed $\epsilon$, the failure energy is nonzero for $d-1$ dimensional regions $R$ associated with flat crack surfaces. This failure energy is the Griffith fracture energy given by the energy release rate multiplied by area of the crack. For flat cracks the nonlocal field theory is shown to recover a solution of Naiver’s equation inside intact material adjacent to a propagating flat traction free crack in the limit of vanishing spatial nonlocality. For curved or more generally countably rectifiable cracks the failure energy is Griffith fracture energy but only in the $\epsilon=0$ limit. The limit deformation is found to be in SBD. A numerical scheme is introduced and shown to be asymptotically compatible to the $\epsilon=0$ limit.