Displacement time history analysis and radial wave propagation velocity in pressurized multiwall carbon nanotubes S.T. Talebian a , M. Tahani a, * , S.M. Hosseini b , M.H. Abolbashari c a Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, P.O. Box 91775-1111, Mashhad, Iran b Department of Industrial Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, P.O. Box 91775-1111, Mashhad, Iran c Department of Mechanical Engineering, Lean Production Engineering Research Center, Ferdowsi University of Mashhad, P.O. Box 91775-1111, Mashhad, Iran article info Article history: Received 10 January 2010 Received in revised form 21 March 2010 Accepted 4 May 2010 Available online 8 June 2010 Keywords: Multiwall carbon nanotube Radial wave propagation velocity Dynamic load factor Van der Walls force Equivalent continuum structure abstract In this paper, radial wave propagation velocity in multiwall carbon nanotubes (MWCNTs) is investigated using continuum model as the multiple elastic isotropic shells. The effect of van der Waals forces between tubes is taken into account as a nonlinear function of inter-layer spacing of MWCNTs. To solve the gov- erning equations, the finite element method is used. By using time history of displacement, the mean velocity of radial wave propagation from outermost tube to innermost tube is calculated. Also, the effects of geometric parameters such as inner radius and thickness of nanotubes on dynamic behaviors are investigated. Furthermore, dynamic load factors for each tube of MWCNTs are calculated using presented solution. The effects of hydrostatic pressures applied on bounding surfaces of MWCNTs on radial wave propagation are discussed. The presented solution and results are verified by comparing against those reported in published literatures. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction Since the discovery of Iijima [1] in 1991, there has been a signif- icant attention in characterizing mechanical properties of both sin- glewall carbon nanotubes (SWCNTs) and MWCNTs. The symmetric structure of carbon nanotubes (CNTs) results in extraordinary mechanical properties, such as high specific strength and resil- ience, together with enormous electrical and thermal conductivi- ties. A recent trend in research demonstrates a growing interest in dynamic analysis of CNTs due to their various applications in nanoscale devices, superconductivity, transport, and optical phe- nomena. Mir et al. [2] employed a structural mechanics approach along with the finite element analysis for determining the natural frequencies and their corresponding modes for two types of nano- tubes, zigzag and armchair. In numerical and analytical researches of CNTs, continuum models, such as elastic shell and beam models, were employed because molecular simulations are expensive and almost inapplicable for large-scale problems. Fu et al. [3] investi- gated the nonlinear free vibration of embedded CNTs considering intertube radial displacement and related internal degrees of free- dom based on the continuum mechanics and a multiple-elastic beam model. Natsuki et al. [4] proposed a theoretical approach to vibration characteristic analysis of doublewall carbon nanotubes (DWCNTs) with simply supported boundary conditions. Wang and Cai [5] reported an investigation of the influence of initial stress on the flexural vibration of an individual MWCNTs with simply sup- ported ends, based on a laminated elastic beam model considering the van der Waals force interaction between two adjacent nano- tubes. Taking advantage of the multiple-elastic shell model, Wang et al. [6] studied the free vibration of MWCNTs and examined the role of the inter-layer van der Waals interaction in free vibration of MWCNTs. Batra and Gupta [7] analyzed vibrations of free–free SWCNTs of various chiralities and diameters using molecular mechanics 3 (MM3) potential and compared their frequencies with those of their equivalent continuum structures (ECSs). They ascer- tained values of the thickness and the elastic moduli of the ECSs and hence of the SWCNTs. Raman spectroscopy, in particular resonant Raman, has played a key role in understanding the vibration and electronic properties of SWCNTs. Teredesai et al. [8] reported high-pressure Raman studies on SWCNT bundles carried out up to 25.9 GPa pressure. They found that the intensity of the radial modes decreases more drastically as compared to that of the tangential modes. Araujo et al. [9] used resonance Raman scattering to determine the radial breathing mode (RBM) frequency dependence on tube diameter for SWCNTs. Lucas and Young [10] investigated the RBM intensities of SWCNTs by Raman spectroscopy as a function of applied strain. Xia et al. [11] assigned the chiralities of DWCNTs using two radial breathing modes of the DWCNTs unambiguously, which are calculated theo- retically by using the modified continuum model. Wang et al. [12] 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.05.008 * Corresponding author. Tel./fax: +98 511 876 3304. E-mail address: mtahani@ferdowsi.um.ac.ir (M. Tahani). Computational Materials Science 49 (2010) 283–292 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci