Rigorous convergence results for the mesh-free discretization of peridynamic models are presented, beginning with the discretization of a linear state-based model without fracture, before moving on to a nonlinear bond-based model defined in [2] that includes irreversible damage. The analysis considers the error caused by the use of one-point quadrature that is part of the usual mesh-free discretization. The resulting model includes partial volume terms to improve the numerical behavior, as demonstrated in [3]. The convergence of the linear state-based numerical model is presented for a fixed choice of a singular weighting function commonly used in the literature while still allowing an arbitrary horizon. The assumptions on the input data are not significantly restrictive for practical purposes. In particular, they still allow discontinuities in the material parameters and external body forces. Results for the dynamic models using a central difference scheme then follow as well. While the linear models include material discontinuities, they do not include fracture-related phenomena. To consider damage, we extend our analysis to a nonlinear bond-based model using a bond-breakage criterion for an isotropic material. This model was introduced and shown to be well-posed in [2]. It is based on the prototype microelastic brittle (PMB) material first defined by Silling in [1]. The convergence of the solutions of the semi-discrete model using a mesh-free discretization and the fully discrete model are then shown. The results presented complement existing numerical convergence studies (see, e.g. [3]). To the best of our knowledge, they have not yet been rigorously proven for either model, especially not with the use of one-point quadrature. This research also serves as a starting point for further research concerning the convergence of mesh-free discretizations for other nonlocal models of peridynamic type compatible with discontinuities in the displacement field. REFERENCES [1] S. Silling and E. Askari, A meshfree method based on the peridynamic model of solid mechanics, Computers & Structures 83 (2005) 1526–1535. [2] Q. Du, Y. Tao and X. Tian, A peridynamic model of fracture mechanics with bond- breaking, Journal of Elasticity 132 (2018) 197–218. [3] P. Seleson and D. J. Littlewood, Numerical Tools for Improved Convergence of Mesh-free Peridynamic Discretizations (Springer International Publishing, Cham, 2018), pp. 1–27.