On the Convergence of the Mesh-Free Discretization of a Non-Linear Bond-Based Peridynamic Model with a Bond-Breakage Criterion

We show the convergence of a commonly used mesh-free discretization of a dynamic non-linear bond-based peridynamic model with irreversible damage implemented via a bond-breakage criterion for a fixed horizon $\delta >0$. The model chosen here is a mathematically well-posed regularization of a discontinuous peridynamic operator often found in the literature, which assumes that fracture results from strain in tension only. We explicitly include the one-point quadrature used in this scheme in the analysis. The convergence is with respect to the natural norm of the solution space given by $C^1([0,T],L^{\mathrm{\infty }}(\mathrm{\Omega }{)}^d)$, and therefore $L^{\mathrm{\infty }}(\mathrm{\Omega }{)}^d$ in the spatial coordinates. This result heavily constrains the spatial regularity of the continuous solution which we prove afterwards. More specifically, for locally $L$-Lipschitz continuous input data, local Lipschitz bounds are derived for the displacement, regardless of damage. This is consistent with interpreting the peridynamic model used here as an approximation to a truly discontinuous fracture model with a length-scale parameter given by the horizon $\delta >0$. We note that we do not deal with the limit of the solutions as $\delta \searrow 0$.