Journal of Composites

Volume 2015, Article ID 750392, 11 pages

http://dx.doi.org/10.1155/2015/750392

## Rayleigh-Ritz Vibrational Analysis of Multiwalled Carbon Nanotubes Based on the Nonlocal Flügge Shell Theory

^{1}Department of Engineering Science, Faculty of Technology and Engineering (East of Guilan), University of Guilan, Vajargah, Rudsar 44891-63157, Iran^{2}Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran

Received 1 June 2015; Accepted 3 December 2015

Academic Editor: Federico Juan Sabina

Copyright © 2015 H. Rouhi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A nonlocal elastic shell model considering the small scale effects is developed to study the free vibrations of multiwalled carbon nanotubes subject to different types of boundary conditions. Based on the nonlocal elasticity and the Flügge shell theory, the governing equations are derived which include the interaction of van der Waals forces between adjacent and nonadjacent layers. To analytically solve the problem, the Rayleigh-Ritz method is employed. In the present analysis, different combinations of layerwise boundary conditions are taken into account. Some new intertube resonant frequencies and the associated noncoaxial vibrational modes are identified owing to incorporating circumferential modes into the shell model.

#### 1. Introduction

Nanotechnology is quickly becoming one of the fastest developing fields of research. This is largely due to numerous applications of carbon-based nanostructures in different fields. Among such nanostructures, carbon nanotubes (CNTs) discovered by Iijima [1] in 1991 have attracted the attention of researchers worldwide [2–4].

The theoretical models based on continuum mechanics can be efficiently employed for the analysis of nanostructures. Applications of the classical (local) continuum modeling to the study of CNTs have been suggested in several papers [5–11]. Because the local continuum models are incapable of capturing the influence of small scale of nanostructures, they become controversial to implement for the analysis of such nanoscale systems. So as to consider the small size effects, some other researchers have proposed the application of nonlocal continuum mechanics to study nanostructured materials.

The use of nonlocal elasticity to nanotechnology was first proposed by Peddieson et al. [12]. After that, nonlocal continuum models have been utilized by many researchers owing to their computational efficiency as well as their accuracy [13–22]. The vibrational analysis of CNTs based on the nonlocal continuum mechanics has been the focus of considerable research. Wang and Varadan [13] investigated the vibration of single-walled and double-walled CNTs (SWCNTs) based on a nonlocal beam model. It was shown that the nonlocal results are in good agreement with the experimental results. Li and Kardomateas [14] investigated the vibrations of multiwalled carbon nanotubes (MWCNTs) subject to simply supported boundary conditions via a nonlocal shell model. It was concluded that the beam models may be insufficient for analyzing the dynamics of nanotubes due to not considering the circumferential mode. Narendar and Gopalakrishnan [16] presented the influence of nonlocal scaling parameter on the wave propagation in MWCNTs using the nonlocal Timoshenko beam model. Arash and Ansari [20] utilized a nonlocal shell model to study the free vibration behavior of SWCNTs. The results indicated that although uncertainty exists in defining nanotube wall thickness, their developed nonlocal shell model is able to predict the results obtained from the molecular dynamics (MD) simulations.

Using shell models in the analysis of CNTs has attracted the attention of several researchers. There are different shell theories. The reader is referred to [23] for a comprehensive summary and discussion on shell theories. A pioneering work in this field is related to Love’s paper [24]. Using the first approximation theory of Love, Flügge shell theory [25] was established. The Flügge shell theory is according to the Kirchhoff-Love assumption for thin elastic shells. By means of this theory, the kinematical relations of the middle surface of a cylindrical shell can be derived. Donnell’s simplified theory can be also obtained by neglecting few terms in the equations of Flügge shell theory.

The Flügge shell theory is very accurate for thin shells. In addition, CNT is in fact a crystalline membrane with one atom thickness. From this viewpoint, the Flügge shell theory assumptions made in representing a CNT as a homogeneous thin shell can be satisfactory. Also, the Flügge shell theory is frequently used for the analysis of CNTs due to the relatively accurate results in spite of its theoretical simplicity. A marked increase in the volume of research papers dealing with the application of Flügge shell theory to different problems of CNTs stands as testimony to this claim [26–29]. In [27], the axisymmetric and beamlike vibrations of MWCNTs under simply supported boundary conditions were studied using a Flügge-type shell model.

The majority of the previous studies are limited to CNTs with all edges simply supported for which exact analytical solution exists. The boundary conditions can significantly affect the mechanical behavior of nanostructures. In addition, for MWCNTs, different combinations of layerwise boundary conditions are possible to be considered. In this regard, Xu et al. [30] commented that “the relevance of the existing model in which both tubes have the same boundary conditions for the vibration of double-walled CNTs is questionable.”

In this work, a nonlocal Flügge shell model is developed to study the vibrations of MWCNTs with different boundary conditions in a layerwise manner. The analytical Rayleigh-Ritz method is applied to the variational statement derived from the nonlocal Flügge-type vibration equations. Some new noncoaxial intertube resonances for MWCNTs are also predicted in this paper which is a matter of great technical importance in the study of vibrations of nested nanotubes.

#### 2. Elastic Shell Model in Nonlocal Elasticity

Based on Eringen’s nonlocal theory [31, 32], the nonlocal stress tensor at point is given bywhere denotes the macroscopic stress tensor and is the nonlocal modulus or attenuation function whose arguments are the Euclidean distance and a material constant with as an internal characteristic length, as an external characteristic length (e.g., crack length, wavelength), and as material constant. The differential form of (1) iswhere is the nonlocal parameter. Hooke’s generalized law states thatin which is the fourth-order elasticity tensor and “” denotes the double dot product. Using (3) leads towhere is Young’s modulus of the material and is the Poisson ratio. and are longitudinal and angular circumferential coordinates. Each tube of MWCNTs is described as an individual cylindrical shell of radius , length , and thickness , as shown in Figure 1. According to the classic shell theory, the three-dimensional displacement components , , and in , , and directions, respectively, are of the formwhere , , and are the reference surface displacements. The kinematical relations are given byThe stress and moment resultants can be obtained bywhere is the shell thickness.