Nonlocal continuum mechanics based on distributions

This paper presents a derivation of the balance laws of nonlocal continuum mechanics, based on the concept of microelements (very small collections of atoms or molecules). Mechanical fields (mass density, stress tensor, internal energy density, …) are assumed to be distributions (generalized functions), over a microelement surface and volume, influenced by distant microelements. Collection of these distributions in the limit when the surface and volume elements approach zero, defines the nonlocal mechanical fields. By this process, singularities and discontinuities that may be present in microelements, are smoothed out. The nonlocal balance laws are then established, having no nonlocal residuals.