Small-scale effects on transverse vibrational behavior of single-walled carbon nanotubes with arbitrary boundary conditions

Abstract

In the present paper, the differential transformation method is employed to develop a semianalytical solution for free transverse vibration of single-walled carbon nanotube (SWCNT) with arbitrary boundary conditions. The small scale effect is taken into consideration via Eringen's nonlocal elasticity theory while the transverse shear deformation effects and rotary inertia are taken into account in presented Timoshenko beam theory. Through variational formulation and the Hamilton's principle the governing differential equations and the boundary conditions are derived and then solved by a semi-analytical method called differential transformation method (DTM) for various frequency modes of beams and different edge conditions. Comparisons made between the present results and results reported by well-known references for special cases treated before, have confirmed accuracy and efficiency of the presented approach. The effects of several parameters such as transverse shear deformation effects, slenderness ratios, boundary conditions and small scale on vibration characteristics of SWCNT are examined. The present study illustrates that the vibration characteristics of an SWCNT are strongly dependent on the small scale parameters.

Flexural wave propagation in fluid-conveying carbon nanotubes with system uncertainties

“…Recently, the nonlocal continuum theory, which assumes that the stress tensor at any point is dependent on the whole strain field of the continuum, is widely used to characterize the small-scale effects of CNTs. This theory was firstly proposed by Eringen (1972) and was successfully used in solving the vibration (De Rosa and Lippiello 2017;Deng and Yang 2014;Ebrahimi and Nasirzadeh 2015;Hu et al 2012;Kiani 2013a, b;Xia and Wang 2010;Zhen et al 2011), wave propagation (Aydogdu 2014;Huang et al 2013;Narendar and Gopalakrishnan 2010;Narendar et al 2012;Wang et al 2006Wang et al , 2013Wang et al , 2015, and buckling (Adali 2008;Amara et al 2010;Khademolhosseini et al 2010;Robinson and Adali 2016;Setoodeh et al 2011) problems of CNTs. Several different nonlocal continuum theories, such as the nonlocal Euler-Bernoulli beam (Setoodeh et al 2011;Wang et al 2015;Zhen et al 2011), nonlocal Timoshenko beam (De Rosa and Lippiello 2017;Narendar and Gopalakrishnan 2010;Xia and Wang 2010), and nonlocal shell (Khademolhosseini et al 2010) theory, were adopted to study the mechanical behaviors of CNTs, in particular, because the CNT comprises a sheet of carbon atoms, which makes it perfect a candidate for nanocontainers to store gases and for nanotubes to convey fluids (Ansari et al 2016;Bahaadini and Hosseini 2016;SafarPour and Ghadiri 2017).…”