Dispersion of Transverse Waves in Single-Wall Carbon Nanotubes

Abstract

In this work we investigate the influence of initial stress on the transverse waves propagation in single-wall carbon nanotubes with small scale effects under ultrahigh frequency (above k = 1nm −1 ), using the nonlocal continuum theory. The phase and group velocities are derived in a simple way. The effects of wave numbers, scale coefficients and initial stress parameter on phase and group velocity are analyzed. The results show that the dispersion properties of the transverse wave propagation in a single-wall carbon nanotube are induced by small scale effects, which cannot appear in classical models. Also, it is clear that the increase of the scale coefficient and wave number could be strengthened by the dispersion degree of the phase and group velocity of transverse waves. In particular, the analysis shows that the compression initial stress in a single-wall carbon nanotube does not only affect the number of transverse wave speeds and the magnitude of transverse wave speeds, but also the critical value of the wave number at the wave speed changes. The effects of the initial stress and small scale on the transverse wave propagation in single-wall carbon nanotubes may result in some complex dynamic phenomena of terahertz sound waves propagation in the carbon nanotubes, which can be used as a useful reference for the designs of nanodrive devices, nanooscillators and nanosensors.

Dispersion relation for transverse waves in pre-stressed irregular single-walled carbon nanotubes

“…The real parts of equations ( 25) and (26) give the phase velocity of optical (C Om ) and acoustic (C Am ) transverse waves for different response modes (m), respectively. The respective imaginary parts of equations (25) and (26) give the damping velocities. In the present study, we suppose the nanotube is simply supported in its ends, so the axial wave number p = k m L m [31].…”

“…In this section, numerical results for the dispersion relation of transverse waves in a pre-stressed irregular single-walled carbon nanotube are carried out using the simulation parameters presented in table 1 [32]. For time consuming, our calculations are confined to the dispersion curves of the phase velocities of optical (higher) transverse waves for the case of simply supported-simply supported SWCNT using equation (25).…”

“…Selim [24,25] has showed how to construct and analyze the propagation of transverse waves in a pre-stressed plate and single-wall carbon nanotube using local and nonlocal scale effects.…”