Spatial and temporal limitations of atomic scale simulations necessitate the development of multi scale methodologies linking atomic and continuum scales. In this paper, we present formulations to evaluate continuum quantities of stress, strain and local elastic moduli of grain boundaries in a molecular dynamics setting. Energetics and deformation of symmetric tilt grain boundaries of Aluminum are studied using the stress - strain measures. The atomistic simulation studies of grain boundary sliding show a clear dependence of magnitude of sliding on grain boundary energy. Asymptotic expansion homogenization (AEH) is a mathematically rigorous approach for homogenization of periodic structures, which has been used extensively in composites and porous media. We propose a methodology and demonstrate the applicability of AEH to link atomic scale effects in much larger scale systems.