In this talk, we present our new results on the numerical analysis of nonlocal fracture models. We begin by giving a brief introduction to the Peridynamic theory and the nonlocal potentials considered in our work. We consider a force interaction characterized by a double well potential. Here, one well, near zero strain, corresponds to the linear response of a material, and the other well, for large strain, corresponds to the softening of a material. We show the existence of a regularized model with evolving displacement field in either Hölder space or Sobolev space. Assuming exact solutions in Hölder space, we obtain apriori error estimates due to finite difference approximation. We show that the error converges to zero, uniformly in time, in the mean square norm. The rate depends on the nonlocal length scale and is proportional to O(Delta t + h^gamma / epsilon^2). We consider piecewise linear continuous elements. Theoretical claims are supported by numerical results. This is a joint work with Dr. Robert Lipton and is funded by the US Army Research Office under grant/award number W911NF1610456.