On the variational limit of a class of nonlocal functionals related to peridynamics

In this paper we study variational problems on a bounded domain for a nonlocal elastic energy of peridynamic-type which result in nonlinear systems of nonlocal equations. The well-posedness of variational problems is established via a careful study of the associated energy spaces. In the event of vanishing nonlocality we establish the convergence of the nonlocal energy to a corresponding local energy using the method of $\Gamma$-convergence. Building upon existing techniques, we prove an L^p-compactness result (on bounded domains) based on near-boundary estimates that is used to study the variational limit of minimization problems subject to various volumetric constraints. For energy functionals in suitable forms, we find the corresponding limiting energy explicitly. As a special case, the classical Navier-Lamé potential energy is realized as a limit of linearized peridynamic energy offering a rigorous connection between the nonlocal peridynamic model to classical mechanics for small uniform strain.