Vibration of nonlocal strain gradient beams incorporating Poisson's ratio and thickness effects

Poisson's ratio is an extremely important physical parameter of materials, which can be defined as the ratio of transverse strain to longitudinal strain. Due to the fact that the Poisson's ratio for common structural materials is within a very narrow domain (i.e., from 0.25 to 0.35), the Poisson's ratio effect on mechanical behaviors of nanobeams is few investigated. However, the Poisson's ratio for isotropic solids is from 1 to 0.5, and the structures with special Poisson's ratio, especially the negative Poisson's ratio, have attracted wide attention. Thus, it is necessary to study the Poisson's ratio effect on mechanical behaviors of nanobeams. Free vibration of nanobeams was performed by many works based on nonlocal strain gradient theory (NSGT), but most of them did not consider the size-dependent effect in thickness direction. In this study, the effects of Poisson's ratio and thickness are examined for the vibration behaviors of nanobeams via NSGT. Based on NSGT with the thickness effect, the analytical solution for the bending frequencies of hinged-hinged nanobeams are derived. When neglecting the thickness effect in NSGT, the stiffness-hardening effect is underestimated, especially for thin beams where the size-dependent effect of loss of neighboring atoms on the beam's surface is significant. By numerical results and the molecular dynamics simulation, it is found that the stiffness-hardening effect in [110] copper nanowire attenuates with an increased vibrational mode number. Moreover, the Poisson's ratio effect in vibration frequency of reflexyne and flexyne nanobeams is different. The Poisson's ratio effect in flexyne and reflexyne nanobeams with the thickness effect is extraordinarily remarkable.