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Studying the Vibrational Behavior of Single-Walled Carbon Nanotubes under Different Boundary Conditions using the Rayleigh-Ritz Technique

The vibrational response of single-walled carbon nanotubes (SWCNTs) is studied in this paper. To take size effects into account, Eringen’s nonlocal elasticity equations are incorporated into the Donnell shell theory. The Rayleigh-Ritz method is employed in conjunction with the polynomial series as modal displacement functions to solve the problem. Four commonly used boundary conditions namely as simply supported-simply supported, clamped-clamped, clamped-simply supported, and clamped-free are considered. The fundamental frequencies of SWCNTs with various values of aspect ratios and nonlocal parameters are obtained. To propose the proper values of nonlocal parameter, the results of nonlocal shell model are matched with those of molecular dynamics (MD) simulations for armchair and zigzag SWCNTs through a nonlinear least square fitting procedure. The appropriate values of nonlocal parameter corresponding to each type of chirality and boundary condition are then derived. It is found that the results obtained via the presented nonlocal shell model with its proposed proper values of nonlocal parameter are in excellent agreement with those of MD simulations.


Introduction
After the discovery of carbon nanotubes (CNTs) by Iijima [1], the applications of these attractive nanostructures have been increasing in recent years due to their outstanding physical, mechanical, and electrical properties. They have provided new building block in various emerging fields of nanoscience and nanotechnology [2][3][4][5][6][7].
As the results predicted by the classical continuum models are independent of the influence of small scale of nanostructures, they become controversial to implement for the analyses of such nanoscale systems. Hence, the extension of the continuum mechanics to accommodate the size dependence of nanostructures becomes a topic of major concern. To incorporate the size effects that exist at nanoscale into the classical continuum theory, it is needed to refine the classical model. Modified continuum models are widely applied in nanomechanics due to their computational efficiency and the capability to produce accurate results which are comparable to those of atomistic models. The use of nonlocal elasticity for nanostructures was applied for the first time by Peddieson et al. [8]. After that, nonlocal continuum model has gained much popularity among the researchers because of its efficiency as well as simplicity to analyze the behavior of various nanostructures.
Ghorbanpour-Arani and Zarei [9] studied surface and small scale effects on free transverse vibration of a single-walled carbon nanotube with Y-junction at downstream end conveying viscose fluid based on the nonlocal Euler-Bernoulli beam theory. Hu et al. [10] investigated the transverse and torsional wave in SWCNT and DWCNT using nonlocal single and double elastic cylindrical shells. They found that the van der Waals interaction has little effect on the phase difference of transverse wave. Niknam and Aghdam [11] obtained a closed form solution for both natural frequency and buckling load of nonlocal functionally graded beams resting on nonlinear elastic foundation. Adhikari et al. [12] introduced the idea of nonlocal normal modes arising in the dynamic analysis of nanoscale structures. Ansari et al. [13] also developed a nonlocal finite element model which accounts for the small scale effects on free vibration of multi-layered graphene sheets. There are so many other research works in which the behaviors of nanostructures under various loading conditions have been predicted based on nonlocal elasticity continuum models [14][15][16][17][18][19][20][21][22][23][24][25].
The Rayleigh-Ritz technique is one of the well qualified numerical methods which has been employed in many studies. An investigation of the literature shows that there are many works in which the Rayleigh-Ritz technique has been applied to predict the vibrational behavior of various structures. Zhou [26] applied Rayleigh-Ritz method using a set of Timoshenko beam functions to analyze the free vibration of Mindlin rectangular plates with uniform edge constraints. Al-Obeid and Cooper [27] developed the Rayleigh-Ritz method using polynomial functions to study the vibration properties of composite laminates plates.
Continuing with the vibration problems, Roufaeil and Dawe [28]

Donnell Shell Equations based on Nonlocal Elasticity
The theory of nonlocal elasticity was first considered by Eringen [30]. In the nonlocal model, in contrast to the classical elasticity, the stress at a reference point x in an elastic body depends not only on the strains at x, but also on strains at all other points of the body [30]. The nonlocal constitutive equation is given by [31] (1 − ∇ ) = where μ is the nonlocal parameter used in nonlocal continuum; t is the macroscopic stress tensor at a point. In the limit when the characteristic length goes to zero, the nonlocal elasticity reduces to the local elasticity. The stress tensor is related to strain by generalized Hooke's law as = : here is the fourth-order elasticity tensor and ':' denotes the double dot product. Hooke's law for the stress and strain relation is thus expressed by Figure 1 shows the cylindrical shell of length L, radius R and thickness h with its coordinate system( , , ) as x-axis is along the length of the shell.
Based on Donnell shell theory [32], the displacements , , and the rotations , can be expressed in the following form where = 0.85 denotes the bending rigidity of the shell.
The governing equations on the basis of the Donnell shell theory can be given as [32]

Field equations for vibrations of SWCNTs
Substituting equations (6) into (7)  It is worth mentioning that the above field equations reduce to the classical Donnell shell theory by setting the nonlocal parameter equal to zero.

Implementation of the Rayleigh-Ritz Technique
One of the most important matters to apply the Rayleigh-Ritz method in an appropriate way in which Π ( , , , , ) = Using the Rayleigh-Ritz technique, we have Equation (14) leads to 5 × linear algebraic equations. By solving the resulting eigenvalue problem, the fundamental frequencies of SWCNTs can be obtained and the associated eigenvectors yield the corresponding mode shapes.

Numerical Results and Discussion
The successful application of the nonlocal continuum shell model to free vibration analysis of SWCNTs needs strongly the appropriate values of Young's modulus and effective thickness.
In the current study, the thickness of nonlocal shell modeled is assumed to be equal to the spacing of graphite (ℎ = 0.34 ). Also, the in-plane stiffness ( ℎ)of SWCNTs can be calculated from the variation of the potential energy with the value of applied compressive axial strain that provides the value of Young's modulus.  especially when its aspect ratio decreases. As the aspect ratio increases, resonant frequencies tend to decrease and the effect of small length scale diminishes so that the frequency envelopes tend to converge. This observation means that the classical continuum model would give a reasonable prediction in the study of nanotubes of high aspect ratios for which the whole structure can be homogenized into a continuum.

Conclusion
In this work, the vibrational behavior of SWCNTs was studied on the basis of nonlocal continuum elasticity. The Donnell shell theory was employed with Eringen's nonlocal elasticity equations for considering small scale effects. The Rayleigh-Ritz technique was also used to solve the problem corresponding to four commonly used boundary conditions. The fundamental frequencies of SWCNTs with various aspect ratios and nonlocal parameters were calculated.
To obtain the proper values of nonlocal parameter, MD simulations results were employed. It was found that in contrast to the chirality, the boundary conditions have a considerable influence on the appropriate values of nonlocal parameter. Also, it was observed that the present nonlocal continuum shell model with its proposed consistent values of nonlocal parameter has an excellent capability to predict the vibrational response of SWCNTs which is comparable to the results of MD simulation. :   Table 1: Values of , , , and for different boundary conditions