Completing Eringen’s nonlocal elasticity theory and its connection with surface elasticity

Eringen’s nonlocal elasticity theory is known to suffer from boundary-related inconsistencies, which arise from the presence of additional Boundary Conditions, commonly referred to as Constitutive (CBCs), which are embedded in the Green’s function-type attenuation functions and supplement the problem BCs. To avoid over-determination, the method of kernel modification has been recently proposed, that enforces consistency between the CBCs and the prescribed BCs. In so doing, the kernel can no longer be of the difference type. Still, we prove that the influence of the system boundaries extends beyond the issue of over-determination. More specifically, we show that the differential and the integral formulation of nonlocal elasticity are equivalent provided that a boundary term is set to zero, that amounts to requiring that the motion equations are satisfied on the boundary. Indeed, this condition is necessary for the problem closure, because, once the BCs are incorporated into the kernels, they are automatically satisfied by any general solution of the associated differential formulation. Moreover, we show that Eringen’s single-integral formulation fails to accommodate inhomogeneous BCs. Therefore, an extended integral formulation is introduced that matches the differential formulation in the presence of surface loads. This extended formulation admits a natural reinterpretation in terms of surface elasticity, thereby clarifying the role of boundary effects in nonlocal continua. As an application, the generalized Rayleigh problem is examined for a half-plane with an elastically constrained surface, which reveals the existence of localized surface waves that have no counterpart in classical elasticity.