Refined boundary conditions on the free surface of an elastic half-space taking into account non-local effects

Abstract

The dynamic response of a homogeneous halfspace, with a traction-free surface, is considered within the framework of non-local elasticity. The focus is on the dominant effect of the boundary layer on overall behaviour. A typical wavelength is assumed to considerably exceed the associated internal lengthscale. The leading-order long-wave approximation is shown to coincide formally with the 'local' problem for a half-space with a vertical inhomogeneity localized near the surface. Subsequent asymptotic analysis of the inhomogeneity results in an explicit correction to the classical boundary conditions on the surface. The order of the correction is greater than the order of the better-known correction to the governing differential equations. The refined boundary conditions enable us to evaluate the interior solution outside a narrow boundary layer localized near the surface. As an illustration, the effect of nonlocal elastic phenomena on the Rayleigh wave speed is investigated.

Free vibrations of nonlocally elastic rods

“…Thus, in the general case, the DNM is inconsistent since it does not allow rigorous treatment of boundary conditions in terms of stresses, and does not provide an adequate description of deformations of a nanostructure in the near-boundary domain. Similar conclusions were presented in another recent work [22] for the conventional Gaussian kernel, comparing corrections arising from the boundary layer and the nonlocal effect within the media. It is concluded in the above-mentioned paper that the nonlocal effects within the media are of higher asymptotic order than those arising from rigorous asymptotic treatment of boundary conditions.…”

“…Obviously, an error of results obtained on the basis of DNM may be enourmous if a nanorod is sufficiently short. This conclusion on inconsistency of DNM is also confirmed by results of [22], studying the dynamic response of a non-locally elastic half-space with a traction-free surface. As for PNIM, solutions (65) and (66) satisfy the boundary conditions of clamped ends with accuracy of O(ε), and modes (66) and (67) obtained for the rods with one and two free edges, respectively, do not satisfy the additional conditions (31), (32).…”

Nonlocal axial vibration of the multiple Bishop nanorod system

“…there is no influence of nonlocal parameter; same as in macro-scale modeling), we get back to the classical stress–strain relation of the elastic body. It should be noted that more rigorous methods for analyzing the nonlocal mechanical behavior of different structural elements such as beams and plates are given in various works [48–51].…”

Exact solutions for the bending of Timoshenko beams using Eringen’s two-phase nonlocal model

“…Recently, the size-dependent behavior of plates was analyzed in [19], which demonstrated for the first time the importance of nonlocal effect in the thickness direction. In [20], through asymptotic analysis, the effect of the nonlocal integral model for a half space was shown to take the form of boundary layers localized near the surface. Taking into consideration the nonlocal properties across the plate thickness, the long-wave low-frequency approximations of 3D dynamic equations in nonlocal integral elasticity were derived for plate bending and extension in [21].…”