Asymptotic derivation of nonlocal beam models from two-dimensional nonlocal elasticity

Abstract

The paper is focused on the possible justification of nonlocal beam models (at the macroscopic scale) from an asymptotic derivation based on nonlocal two-dimensional elasticity (at the material scale). The governing partial differential equations are expanded in Taylor series, through the dimensionless depth ratio of the beam. It is shown that nonlocal Bernoulli–Euler beam models can be asymptotically obtained from nonlocal two-dimensional elasticity, with a nonlocal length scale at the beam scale (macroscopic length scale) that may differ from the nonlocal length scale at the material scale. Only when the nonlocality is restricted to the axial direction are the two length scales coincident. In this specific nonlocal case, the nonlocal Bernoulli–Euler model emerged at the zeroth order of the asymptotic expansion, and the nonlocal truncated Bresse–Timoshenko model at the second order. However, in the general case, some new asymptotically-based nonlocal beam models are built which may differ from existing references nonlocal structural models. The natural frequencies for simply supported nonlocal beams are determined for each nonlocal model. The comparison shows that the models provide close results for low orders of frequencies and the difference increases with the order.

On governing equations for a nanoplate derived from the 3D gradient theory of elasticity

“…where σ (0) , σ (3) and σ (2) are polynomials of the third and fifth orders in z, respectively, with coefficients depending on ξ k , t, which are calculated by formulas ( 22), ( 36) and (42).…”

“…It was detected that capturing the nonlocal effects in the thickness direction leads to the first-order corrections to the bending and extensional stiffness. Finally, in a recently published paper Hache et al [42] used an asymptotic approach to derive governing equations for a nanobeam proceeding from the Eringen gradient theory of 2D elasticity accounting for nonlocality in both the axial and vertical directions. They showed that the built models accounting for nonlocallity in the vertical direction differ from the nonlocal Bernoulli-Euler model capturing the internal length scale effects only in the axial direction.…”

“…The general drawback of the 1D and 2D theories for nanosize bodies, obeying kinematic hypotheses, is that they ignore the nonlocal effects in the transverse direction. Apparently, the first and so far the only researches considering these effects were performed by Sajadi et al [19], Hache et al [42] and Chebakov et al [43]. In the first two papers, relying on Eringen’s purely nonlocal theory (PNLI model) with the 3D Gaussian kernel in the constitutive equations, the authors, in contrast to the aforementioned theories, neglected the size dependence in the plate plane and studied the effect of the internal length scale in the transverse direction.…”

On non-locally elastic Rayleigh wave

“…Most of the numerous publications on the subject use differential formulations (e.g. see [8][9][10]), which appear to be better suited for analytical treatment than the initial integral formulations. Moreover, due to the nature of non-local models governed by integral equations, the issue of their solvability naturally arises.…”

On integral and differential formulations in nonlocal elasticity

“…In the paper [10] the original integral nonlocal theory has been reduced to a differential form, which is much easier for implementation. The latter has been adapted for numerous scenarios, in particular for thin nano-structures, see [17,14,2] to name a few. In addition, the differential formulation in nonlocal elasticity formally has a lot of in common with a popular model of gradient elasticity, e.g.…”