Finite element convergence for state-based peridynamic fracture models

We establish the a-priori convergence rate for finite element approximations of a class of nonlocal nonlinear fracture models. We consider state based peridynamic models where the force at a material point is due to both the strain between two points and the change in volume inside the domain of nonlocal interaction. The pairwise interactions between points are mediated by a bond potential of multi-well type while multi point interactions are associated with volume change mediated by a hydrostatic strain potential. The hydrostatic potential can either be a quadratic function, delivering a linear force-strain relation, or a multi-well type that can be associated with material degradation and cavitation. We first show the well-posedness of the peridynamic formulation and that peridynamic evolutions exist in the Sobolev space $H^2$. We show that the finite element approximations converge to the $H^2$ solutions uniformly as measured in the mean square norm. For linear continuous finite elements the convergence rate is shown to be $C_t \Delta t + C_s h^2/\epsilon^2$, where $\epsilon$ is the size of horizon, $h$ is the mesh size, and $\Delta t$ is the size of time step. The constants $C_t$ and $C_s$ are independent of $\Delta t$ and $h$ and may depend on $\epsilon$ through the norm of the exact solution. We demonstrate the stability of the semi-discrete approximation. The stability of the fully discrete approximation is shown for the linearized peridynamic force. We present numerical simulations with dynamic crack propagation that support the theoretical convergence rate.