We discuss mathematical and computational issues related to some nonlocal balance laws and in particular the peridynamic theory for materials models. A vector calculus for nonlocal operators is presented which provides a rigorous fundation to pose abstract nonlocal balance laws with reduced regularity requirements. We address some basic mathematical and computational issues and explore the connections with local models. We also discuss questions concerning finite dimensional approximations of such nonlocal models, such as convergence, conditioning, a priori and a posteriori error analysis and adaptive methods.
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