Journal of Nanomaterials

Volume 2015, Article ID 958138, 11 pages

http://dx.doi.org/10.1155/2015/958138

## Curved Nanotube Structures under Mechanical Loading

Concordia Centre for Composites (CONCOM), Department of Mechanical and Industrial Engineering, Concordia University, Montreal, QC, Canada H3G 1M8

Received 25 April 2015; Accepted 2 July 2015

Academic Editor: Hassan Karimi-Maleh

Copyright © 2015 Hamidreza Yazdani Sarvestani and Ali Naghashpour. 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

Configuration of carbon nanotube (CNT) has been the subject of research to perform theoretical development for analyzing nanocomposites. A new theoretical solution is developed to study curved nanotube structures subjected to mechanical loadings. A curved nanotube structure is considered. A nonlocal displacement-based solution is proposed by using a displacement approach of Toroidal Elasticity based on Eringen’s theory of nonlocal continuum mechanics. The governing equations of curved nanotube structures are developed in toroidal coordinate system. The method of successive approximation is used to discretize the displacement-based governing equations and find the general solution subjected to bending moment. The numerical results show that all displacement components increase with increasing the nonlocal parameter. The present theoretical study highlights the significance of the geometry and nonlocal parameter effects on mechanical behavior of nanotube structures.

#### 1. Introduction

Due to superior properties of CNT which offer unprecedented potential to produce multifunctional nanocomposites, many investigations have been performed on different applications of these nanostructures such as nanomechanical [1, 2] and nanobiological [3, 4] applications and optic [5] and molecular engineering [6–8]. CNTs appear in different shapes. Each shape has its own special properties and potential applications. CNT with curved configuration is one of their shapes. Actually, a single nanotube naturally curves during growth if no external forces exist [9]. Continuum mechanics approach has been employed by many researchers to study mechanical behavior of curved nanostructures.

The continuum mechanics approach has been used to investigate several aspects of mechanical analysis of CNT such as the static [10–13], the buckling [14–16], free vibration [17, 18], wave propagation [19–23], and thermomechanical analysis of CNT [24].

The effect of initial stress on the vibration of an individual multiwall carbon nanotube (MWCNT) with simply supported ends was investigated based on a laminated elastic beam model [12]. Wang et al. [14] proposed nonlocal elastic beam and shell models. They applied the proposed method to study the small-scale effect on buckling analysis of CNT under compression. Based on the continuum mechanics and the molecular dynamics simulation, the investigation on the wave propagation in a single-walled carbon nanotube (SWCNT) was performed [18]. Hu et al. [19] used nonlocal elastic beam theories to study the scale effect of the CNT on the wave dispersion. They compared the results obtained by nonlocal elastic Timoshenko beam theory with the results obtained by molecular dynamics simulations. Based on thermal elasticity mechanics and nonlocal elasticity theory, a single-elastic beam model was presented to investigate the thermal vibration of SWCNT [20]. Murmu and Pradhan [20] studied the buckling of SWCNT. They considered the effect of temperature change and surrounding elastic medium.

Compared with the classical continuum theory, the nonlocal elasticity theory uses the constitutive assumptions that the stress at a point is a function of strains at all points in the continuum. Nonlocal elasticity theory was used by many researchers to analyze nanostructures advanced by Eringen [25, 26]. Peddieson et al. [27] used a version of nonlocal elasticity theory to propose a nonlocal Bernoulli-Euler beam model. Zhang et al. [28] presented a nonlocal double-elastic beam model for the free vibrations of double-walled carbon nanotubes (DWCNTs) based on theory of nonlocal elasticity and by considering the effect of small length scale. The elastic buckling analysis of nanotubes was analyzed based on Eringen’s nonlocal elasticity Timoshenko beam theories [29, 30]. The small-scale effect was considered through Eringen’s nonlocal elasticity theory while the effect of transverse shear deformation was considered through Timoshenko beam theory. Frequency equations and modal shape functions of beam structures were derived based on a nonlocal Bernoulli-Euler beam [31].

The above review shows that little work has been done to study behavior of curved nanotube structures under mechanical loadings. A new displacement-based solution is proposed to study theoretically mechanical behavior of curved nanotube structures based on Toroidal Elasticity that includes the full three-dimensional constitutive relations. Governing equations of curved nanotube are developed by using the constitutive relations of Eringen. Then, the method of successive approximation is employed to find the general solution for mechanical behavior of curved nanotube structures subjected to pure bending moment. The effects of nonlocal parameter, nanotube length, and thickness on the mechanical behavior of nanotubes are studied.

#### 2. Nonlocal Theory (Constitutive Relations)

According to Eringen [25, 26], the stress field at a point in an elastic continuum depends not only on the strain field at the point but also on strains at all other points of the body. Eringen attributed this fact to the atomic theory of lattice dynamics and experimental observations on phonon dispersion. Therefore, the following equivalent differential form is represented based on Eringen [25, 26]:where and are the Laplacian operator and nonlocal parameter, respectively. In addition, and are a material constant and internal characteristic length, respectively.

#### 3. Governing Equations for Curved Nanotube Structures

The nonclassical toroidal coordinate system is shown in Figure 1. The curved nanotube has a bend radius and an annular cross section bounded by radii and (see Figure 1). A general point in a constant thickness curved nanotube is represented easily by the nonclassical toroidal coordinate system , , and , where and are polar coordinates in the plane of the nanotube cross section and defines the position of the nanotube cross section. is a nondimensional radial coordinate in the toroidal coordinate, and is a reference length to be specified later. For convenience, nondimensional displacements , , and are defined asNondimensional stress components are defined aswhere and are modulus of elasticity and Poisson ratio for nanotubes. The kinematics relations for the case of small displacements are [34]where , , and represent the displacement components in the , , and directions, respectively. From both considerations of (1) and Hooke’s law, it is written thatBy considering (5), (3), and strain displacement relations (4), the nondimensional stress components may be expressed in terms of the three nondimensional displacement components and nonlocal parameter aswhereThree equilibrium equations in stress form in the toroidal coordinate are represented as [34]where is the radial cylindrical coordinate of the general point. Substituting the stress-displacement relations (6) into the equilibrium equations (8), the government Navier equations in toroidal coordinate for the curved nanotube are obtained aswhereThe three Navier equations serve as the fundamental equations for the curved nanotube. These equations are composed of three parts. The first part is independent of . The second and the third parts are the linear and nonlinear parts of . As it is impossible to find an exact solution for the Navier equations, the method of successive approximation is used to obtain an approximate solution.