Physics Letters A 374 (2010) 2670–2674 Contents lists available at ScienceDirect Physics Letters A www.elsevier.com/locate/pla Free vibration characteristics of double-walled carbon nanotubes embedded in an elastic medium Toshiaki Natsuki a,∗ , Xiao-Wen Lei a , Qing-Qing Ni a , Morinobu Endo b a Faculty of Textile Science & Technology, Shinshu University, 3-15-1 Tokida, Ueda-shi, 386-8567, Japan b Faculty of Engineering, Shinshu University, 4-17-1 Wakasato, Nagano-shi, 380-8553, Japan article info abstract Article history: Received 4 March 2010 Received in revised form 12 April 2010 Accepted 15 April 2010 Available online 18 April 2010 Communicated by R. Wu In this Letter, a theoretical analysis of the resonant vibration of double-walled carbon nanotubes (DWCNTs) and the DWCNTs embedded in an elastic medium is presented based on Euler–Bernoulli beam model and Winkler spring model. The vibration modes of DWCNTs are quite different from those of single-walled carbon nanotubes (SWCNTs). The resonant vibrations of DWCNTs are found to have in- phase and anti-phase modes, in which the deflections of the inner and outer nanotubes occur in the same and opposite directions, respectively. For the vibration of DWCNTs with the same harmonic numbers, the resonant frequencies of anti-phase mode are larger than the ones of in-phase mode. Moreover, influence of the surrounding medium on the resonant vibrations is investigated using the Winkler spring model. The results show that surrounding medium makes a strong impact on the vibration frequencies of in- phase mode, but little on those of anti-phase mode.  2010 Elsevier B.V. All rights reserved. 1. Introduction The discovery of carbon nanotubes (CNTs) has stimulated ex- tensive research activities in science and engineering devoted to carbon nanostructures and nanocomposites [1–4]. Researches on the mechanical properties of CNTs have been proposed, and the results show that CNTs have exhibited superior mechanical stiff- ness and strength [5,6]. The study of vibration in CNTs is a major topic of current interest, which can be used to further understand their dynamic mechanical behavior. Since controlled experiments at nanometer scale are quite difficult and molecular dynamics (MD) simulations [7–9] are limited to system size of small atom numbers, the continuum mechanics approach has been widely and successfully used to study the mechanical behaviors of CNTs, such as the stress simulation [10,11], the buckling instability [12–15], the resonant vibration, [16,17] and the wave propagation [18–20]. Recently, growing interests in terahertz physics of nanoscale ma- terials and devices [21–23] opened a new topic on wave charac- teristics of CNTs, especially on terahertz frequency range. Some researches have been reported for this topic, but mainly for the study of single-walled carbon nanotubes (SWCNTs) [24–26]. The study of vibrational properties in CNTs is of great current interest [27–30], which can be used to measure the elastic modu- lus of individual CNTs [16,31]. The CNT dimensions and vibration amplitudes were measured by electron micrographs, and it was * Corresponding author. Tel.: +81 0268 21 5421; fax: +81 0268 21 5482. E-mail address: natsuki@shinshu-u.ac.jp (T. Natsuki). assumed that the vibration modes were driven stochastically and chosen of clamped cantilevers [31]. For vibration analysis of one- dimensional beam-like structures, Euler–Bernoulli beam model and Timoshenko beam model were usually employed, which assumed that the cross-section of beams remained plane under bending de- formation [27,32,33]. In the Euler beam model, the cross-section of beam was perpendicular to the neutral axis, whereas this as- sumption was removed in the Timoshenko beam model to account for the shear and rotary effects. The Timoshenko theory was nor- mally able to provide more accurate wave solution at higher wave, although it was more complicated than the Euler–Bernoulli the- ory [27]. Up to now, most of studies have been done on the mechanical behaviors of CNTs, and attention was recently turned to those of CNTs embedded in polymer or metal matrix [34–39]. CNT compos- ites under compressive loads, the influence of surrounding elastic medium on buckling behavior of CNTs is of practical importance. For CNTs with surrounding elastic medium, Winkler-type model was usually used to investigate the buckling stability and the vi- bration property of CNTs [40–42]. Lee and Chang [43] analyzed the influences of viscosity effect, aspect ratio, and elastic medium on the vibration frequency of fluid-conveying SWCNTs embedded in an elastic medium. Based on nonlocal elasticity theory and Winkler-type model, vibration analysis of layered grapheme sheets embedded in an elastic medium has been reported by Murmu et al. [44] Moreover, they also investigated the buckling property [45] and the thermo-mechanical vibration [46] of SWCNTs embedded in an elastic medium using the nonlocal elasticity and the Timo- shenko beam theories. 0375-9601/$ – see front matter  2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.04.040