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 the nonlocal interaction.The pairwise interactions between points are mediated by a bond potential of multi-well type while multi-point interactions are associated with the 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 the material degradation and cavi-tation.We first show the well-posedness of the peridynamic formulation and that peridy-namic evolutions exist in the Sobolev space H2.We show that the finite element approxi-mations converge to the H2 solutions uniformly as measured in the mean square norm.For linear continuous finite elements,the convergence rate is shown to be Ct Δt+Cs h2∕ε 2,where ε is the size of the horizon,h is the mesh size,and Δt is the size of the time step.The constants Ct and Cs are independent of Δt and h and may depend on ε 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 the dynamic crack propagation that support the the-oretical convergence rate.
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