Finite element approximation of nonlocal dynamic fracture models

In this work we estimate the convergence rate for time stepping schemes applied to nonlocal dynamic fracture modeling. Here we use the nonlocal formulation given by the bond based peridynamic equation of motion. We begin by establishing the existence of $ H^2 $ peridynamic solutions over any finite time interval. For this model the gradients can become large and steep slopes appear and localize when the non-locality of the model tends to zero. In this treatment spatial approximation by finite elements are used. We consider the central-difference scheme for time discretization and linear finite elements for discretization in the spatial variable. The fully discrete scheme is shown to converge to the actual $ H^2 $ solution in the mean square norm at the rate $ C_t\Delta t +C_s h^2/\epsilon^2 $. Here $ h $ is the mesh size, $ \epsilon $ is the length scale of nonlocal interaction and $ \Delta t $ is the time step. The constants $ C_t $ and $ C_s $ are independent of $ \Delta t $, and $ h $. In the absence of nonlinearity a CFL like condition for the energy stability of the central difference time discretization scheme is developed. As an example we consider Plexiglass and compute constants in the a-priori error bound.