Low-frequency linear vibrations of single-walled carbon nanotubes: Analytical and numerical models

Abstract

Low-frequency vibrations of single-walled carbon nanotubes with various boundary conditions are considered in the framework of the Sanders–Koiter thin shell theory. Two methods of analysis are proposed. The first approach is based on the Rayleigh–Ritz method, a double series expansion in terms of Chebyshev polynomials and harmonic functions is considered for the displacement fields; free and clamped edges are analysed. This approach is partially numerical. The second approach is based on the same thin shell theory, but the goal is to obtain an analytical solution useful for future developments in nonlinear fields; the Sanders–Koiter equations are strongly simplified neglecting in-plane circumferential normal strains and tangential shear strains. The model is fully validated by means of comparisons with experiments, molecular dynamics data and finite element analyses obtained from the literature. Several types of nanotubes are considered in detail by varying aspect ratio, chirality and boundary conditions. The analyses are carried out for a wide range of frequency spectrum. The strength and weakness of the proposed approaches are shown; in particular, the model shows great accuracy even though it requires minimal computational effort

Vibrations of an elastic cylindrical shell near the lowest cut-off frequency

“…They refine the assumption of zero shear and circumferential mid-surface deformations underlying the semimembrane theory. It is also demonstrated in this section that the earlier proposed adhoc equation [22], based on the hypotheses in the semi-membrane theory but also taking into account some extra terms in the related energy functional, is virtually identical to its asymptotic counterpart to within the values of constant coefficients. …”

“…Consistency of the proposed theory is illustrated by comparison with the dispersion relation in the Sanders-Koiter theory. The asymptotic behaviour of shell displacements established in the paper enables refining the traditional assumption on zero shear and circumferential mid-surface deformations underlying both the semi-membrane theory and the more general adhoc theory in [22].…”

A Shell Model for Free Vibration Analysis of Carbon Nanoscroll

“…For instance, in some studies, the distance between layers of graphite is assumed as wall thickness of equivalent shell and isotropic mechanical parameters are used. However, these equivalent models fail to accurately predict the bending rigidity of CNTs [42]. Previous studies show that the equivalent parameters $E=5.5TPa$, $\nu =0.19$, $\rho =11700kg/m^3$, and $h=0.066$ nm are suitable for modeling a SWCNT with equivalent continuum shell [42].…”

“…However, these equivalent models fail to accurately predict the bending rigidity of CNTs [42]. Previous studies show that the equivalent parameters $E=5.5TPa$, $\nu =0.19$, $\rho =11700kg/m^3$, and $h=0.066$ nm are suitable for modeling a SWCNT with equivalent continuum shell [42]. This mass density for CNT is calculated using equation, $\rho =\sigma /h$, which $\sigma =7.718\times {10}^{-7}kg/m^2$ denotes the surface density of graphite.…”

Chaos prediction in nano-resonators based on nonlocal elasticity theory

“…Many researches on the local and nonlocal theories have been carried out not only considering the beam model but also considering the thin shell model, in the linear and nonlinear fields, with particular application to carbon nanotubes [7,8].…”