Energy balance and damage for dynamic brittle fracture from a nonlocal formulation

A nonlocal model of peridynamic type 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 in $\mathbb{R}^d$, $d=2,3$. The new nonlocal model delivers a two point strain evolution on a subset of $\mathbb{R}^d\times\mathbb{R}^d$. This evolution provides an energy that interpolates between volume energy corresponding to elastic behavior and surface energy corresponding to failure. In general the deformation energy resulting in material failure over a region $R$ is given by 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 for $d=3$ (or length for $d=2$). The nonlocal field theory is shown to recover a solution of Naiver's equation outside a propagating flat traction free crack in the limit of vanishing spatial nonlocality. Simulations illustrate fracture evolution through generation of an internal traction free boundary as a wake left behind a moving strain concentration. Crack paths are seen to follow a maximal strain energy density criterion.