Timoshenko-beam effects on transverse wave propagation
in carbon nanotubes
J. Yoon, C.Q. Ru
*
, A. Mioduchowski
Department of Mechanical Engineering, University of Alberta, Edmonton, Canada T6G 2G8
Received 15 July 2003; revised 18 August 2003; accepted 22 September 2003
Abstract
This paper studies effects of rotary inertia and shear deformation on transverse wave propagation in individual carbon nanotubes (CNTs)
within terahertz range. Detailed results are demonstrated for transverse wave speeds of doublewall CNTs, based on Timoshenko-beam model
and Euler-beam model, respectively. The present models predict some terahertz critical frequencies at which the number of wave speeds
changes. The effects of rotary inertia and shear deformation are negligible and transverse wave propagation can be described satisfactorily by
the existing single-Euler-beam model only when the frequency is far below the lowest critical frequency. When the frequency is below but
close to the lowest critical frequency, rotary inertia and shear deformation come to significantly affect the wave speed. Furthermore, when the
frequency is higher than the lowest critical frequency, more than one wave speed exists and transverse waves of given frequency could
propagate at various speeds that are considerably different than the speed predicted by the single-Euler-beam model. In particular, rotary
inertia and shear deformation have a significant effect on both the wave speeds and the critical frequencies especially for CNTs of larger radii.
Hence, terahertz transverse wave propagation in CNTs should be better modeled by Timoshenko-beam model, instead of Euler-beam model.
q 2003 Published by Elsevier Ltd.
Keywords: A. Nano-structures; B. Vibration; Timoshenko-beam
1. Introduction
Carbon nanotubes (CNTs) are among the most promising
new materials for nanotechnology [1–5], because of their
novel electronic and mechanical properties. In particular,
CNTs hold substantial promise as superfibers for nanocom-
posites [3– 5]. The study of vibration and wave propagation
in CNTs is a major topic of current interest. Among various
methods, continuum elastic-beam models have been
effectively used to study vibration [6 –9] and sound wave
propagation [10,11] in CNTs. In most previous
works, multiwall nanotubes (MWNTs) have been
modeled as single-Euler-elastic-beam [6,7,10],which
ignored non-coaxial intertube radial displacements and
assumed that the nested tubes of a MWNT deform coaxially
and thus can be described by a single deflection curve.
Recently, we have studied the role of non-
coaxial interlayer radial displacements in transverse
vibration [8,9] and wave propagation [11] in MWNTs
using the multiple-Euler-beam model [12]. Our results [8,9,
11] showed that non-coaxial intertube vibration and
transverse waves of MWNTs will be excited at ultrahigh
frequencies (above 1 THz), which would have substantial
effects on both the natural frequencies and the wave speed
of MWNTs. In view of growing interest in terahertz
vibrations and waves of nanoscale materials and devices
[13– 19], it is relevant to systematically study terahertz
wave propagation in individual MWNTs.
As stated in Ref. [5], in the terahertz range, the
characteristic wave-length of transverse waves in MWNTs
would be just few times the outermost diameter of MWNTs
[11]. In this case, a relevant issue to be clarified is the effects
of rotary inertia and shear deformation on transverse wave
propagation of MWNTs. It is well-known that rotary inertia
and shear deformation, which are ignored in the classic
Euler-beam model, would become essential for transverse
wave propagation of elastic beams at ultrahigh frequencies
when the characteristic wave-length downs to just few times
the diameter of the cross-section [20 –25]. For this reason,
the relevance of classic Euler-beam model to terahertz wave
propagation in CNTs is questionable. In the present paper,
1359-8368/$ - see front matter q 2003 Published by Elsevier Ltd.
doi:10.1016/j.compositesb.2003.09.002
Composites: Part B 35 (2004) 87–93
www.elsevier.com/locate/compositesb
*
Corresponding author.
E-mail address: c.ru@ualberta.ca (C.Q. Ru).