Lateral Bending Vibration of Nanoscale Ultra-Thin Beams Using a Semi-Continuum Model

Abstract

A semi-continuum dynamical model involving the relaxation effects is constructed to investigate the mechanical properties of the lateral bending vibration of an ultra-thin beam with a nanoscale thickness. The governing equation of vibration is derived based on the Hamilton principle. Three different kinds of end supporting conditions including fully clamped, simply supported and cantilevered nanoscale ultra-thin beams are considered and various numerical results are analyzed and discussed in detail. Some comparisons between the semi-continuum and classical continuum models are presented. It is concluded that the variational tends of natural frequencies depend on the values of relaxation coefficient. For different relaxation coefficients, the natural frequencies may decrease or increase with an increase in the number of atomic layers. Hence an explanation is proposed for some contradictions that the bending stiffness of the nanoscale beams should be enhanced or weakened in current non-local elasticity theory. Moreover, the relations between the Young's modulus and the number of atomic layers are developed and it is clearly seen that the small scale has a significant effect on the Young's modulus. When the number of atomic layers, or the thickness of the beam is sufficiently large, the small scale effects disappear and the value of the Young's modulus is just recovered to the corresponding classical bulk material, which also proves the validity of the semi-continuum dynamical model proposed in this study.