This talk is on the mathematical analysis of a certain class of linear nonlocal systems that are derived from a nonlocal model in continuum mechanics, called peridynamics. The system is made up of coupled integral equations. I will briefly discuss their derivation and then show well poshness of the problem via basic variational analysis. Along the way, we will study associated energy spaces and establish connections with classical function spaces. By taking limit of certain parameters, we establish the convergence of a sequence of nonlocal energies to a limiting local (gradient-based) energy via Gamma convergence. As a special case the classical Navier-Lame potential energy will be realized as a limit of these nonlocal energy offering a rigorous connection between the nonlocal peridynamic model to classical mechanics