The mechanically-based approach to 3D non-local linear elasticity theory: Long-range central interactions

This paper presents the generalization to a three-dimensional (3D) case of a mechanically-based
approach to non-local elasticity theory, recently proposed by the authors in a one-dimensional (1D) case.
The proposed model assumes that the equilibrium of a volume element is attained by contact forces
between adjacent elements and by long-range forces exerted by non-adjacent elements. Specifically,
the long-range forces are modelled 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 proposed 3D formulation is developed both in a mechanical and in a variational context. For this
the elastic energy functionals of the solid with long-range interactions are introduced, based on the principle
of virtual work to set the proper correspondence between the mechanical and the kinematic variables
of the model. Numerical applications are reported for 2D solids under plane stress conditions