Variational Aspects of the Physically-Based Approach to 3D Non-Local Continuum Mechanics

This paper is the generalization to a three dimensional case of a physically-based approach to non-local mechanics, recently proposed by the authors in one-dimensional case. The proposed model assumes that the equilibrium of a volume element is attained by contact forces between adjacent elements and by long-range central forces exerted by non-adjacent elements. Specifically, the long-range forces are modeled as central body forces depending on the relative displacement between the centroids of the volume elements, measured along the line connecting the centroids. Further, the long-range forces are assumed to be proportional to a proper, material-dependent, distance-decaying function and to the products of the interacting volumes. Consistently with the modelling of the long-range forces as central body forces, the static boundary conditions enforced on the free surface of the solid involve only local stress due to contact forces. The model coalesces with the well-known Eringen integral model of non-local elasticity for unbounded domains but it remains substantially different in case of bounded domain.