The thermal effect on vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory

Abstract

This paper is concerned with the use of the nonlocal Timoshenko beam model for free vibration analysis of single-walled carbon nanotubes (CNTs) including the thermal effect. Unlike the Euler beam model, the Timoshenko beam model allows for the effects of transverse shear deformation and rotary inertia. These effects become significant for CNTs with small length-to-diameter ratios that are normally encountered in applications such as nanoprobes. The elastic Timoshenko beam model is reformulated using the nonlocal differential constitutive relations of Eringen (1972 Int. J. Eng. Sci. 10 1–16). The study focuses on the wave dispersion caused not only by the rotary inertia and the shear deformation in the traditional Timoshenko beam model but also by the nonlocal elasticity characterizing the microstructure of CNTs in a wide frequency range up to terahertz. Numerical results are presented using the nonlocal beam theory to bring out the effect of both the nonlocal parameter and the temperature change on the properties of transverse vibrations of CNTs. The exact nonlocal Timoshenko beam solution presented here should be useful to engineers who are designing microelectromechanical and nanoelectromechanical devices.

Terahertz wave propagation in a fluid-conveying single-walled carbon nanotube with initial stress subjected to temperature and magnetic fields

“…It can be found that the critical stresses increase with the increase in temperature change at the same fluid velocity. This means that the nanobeam will be stiffer when it works under the low or room field [31,44]. It can be found in Eqs.…”

“…Murmu and McCarthy [29,30] studied the influence of longitudinal magnetic fields on the vibration of DWCNTs based on a nonlocal approach. Benzair et al [31] discussed the thermal effect on wave characteristics of SWCNTs, in which the exact nonlocal Timoshenko beam solution was presented.…”

Nonlocal effect on the free vibration of short nanotubes embedded in an elastic medium

“…However, in most of these works, only the transverse vibrations of these structural elements have been investigated using different one-dimensional beam theories; see for example Refs. [1][2][3][4][5][6][7][8][9][10][11][12]. In some other works, the longitudinal vibration of these structural elements has been studied [13,14].…”

The dispersion characteristics of the waves propagating in a spinning single-walled carbon nanotube

“…Owing to the lattice feature of the CNTs, traditional continuum mechanics is not applicable straightforwardly but the need for a continuum model is obvious especially for a substitution of the more expensive and less-flexible molecular dynamics (MD) model for CNT vibration simulation. There are already some vibration studies in this regard using the so-called Timoshenko beam theory [3][4][5] and the Eringen's nonlocal elasticity theory [6] where special constitutive relationship has been employed to replace the traditional strain and stress relation, hereafter referred to as the nonlocal Timoshenko beam theory. Hamilton's principle [7,8] and Flügge's shell theory [8,9] are two other approaches, which have resulted in some energy formalisms [7] for our reference.…”

“…The purposes of these investigations were to understand the effects of tube size, wavenumber, nanotube-lattice length, temperature-induced axial strain, and beam support nature, etc. on the CNT wave and/or vibration characteristics including phase speed, natural frequencies and/or mode shapes [3][4][5][8][9][10]. The primary objective was to find data to underlie future design and development of nanoelectromechanical systems (NEMS) devices [10].…”