Physically-based approach to the mechanics of strong non-local linear elasticity theory

In this paper the physically-based approach to non-local elasticity theory is introduced.
It is formulated by reverting the continuum to an ensemble of interacting volume elements.
Interactions between adjacent elements are classical contact forces while long-range
interactions between non-adjacent elements are modelled as distance-decaying central body
forces. The latter are proportional to the relative displacements rather than to the strain field
as in the Eringen model and subsequent developments. At the limit the displacement field is
found to be governed by an integro-differential equation, solved by a simple discretization
procedure suggested by the underlying mechanical model itself, with corresponding static
boundary conditions enforced in a quite simple form. It is then shown that the constitutive
law of the proposed model coalesces with the Eringen constitutive law for an unbounded domain
under suitable assumptions, whereas it remains substantially different for a bounded
domain. Thermodynamic consistency of the model also has been investigated in detail and
some numerical applications are presented for different parameters and different functional
forms for the decay of the long range forces. For simplicity, the problem is formulated for a
1D continuum while the general formulation for a 3D elastic solid has been reported in the
appendix.