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

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 derWaals 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.


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][3][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][6][7][8][9][10][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][14][15][16][17][18][19][20][21][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.

Elastic Shell Model in Nonlocal Elasticity
Based on Eringen's nonlocal theory [31,32], the nonlocal stress tensor  at point  is given by wavelength), and  0 as material constant.The differential form of ( 1) is where  0  is the nonlocal parameter.Hooke's generalized law states that t = S : in which S is the fourth-order elasticity tensor and ":" denotes the double dot product.Using (3) leads to where  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 ℎ, as shown in Figure 1.According to the classic shell theory, the three-dimensional displacement components   ,   , and   in , , and  directions, respectively, are of the form (, , , ) =  (, , , ) , where , V, and  are the reference surface displacements.The kinematical relations are given by The stress and moment resultants can be obtained by where ℎ is the shell thickness.Moreover, the stress and moment resultants are obtained as [25] in which  is the bending rigidity.The governing equations on the basis of the Flügge shell theory are given as [25] where  is the pressure exerted on tube  through the van der Waals (vdW) interaction forces, as indicated in Figure 2. The vdW model is formulated as [33] ( vdW in which vdW coefficients   representing the pressure increment contributing to layer  from layer  are given by where  = 1.42 Å is the C-C bond length,  is the depth of the potential,  is a parameter that is determined by the equilibrium distance and   is the radius of th layer, and    denotes the elliptic integral defined as here  is an integer and the coefficient   is given by

Implementation of the Rayleigh-Ritz Method with Beam
Functions.The field variables corresponding to the th tube, that is,  () , V () , and  () , can be expressed in the form () (, , ) =  ()  () cos () sin () , where  () ,  () , and  () are the constant parameters denoting the amplitudes of the vibrations,  is the circumferential wave number,  is the angular frequency of vibrating CNT, and () is the axial function which satisfies the geometric boundary conditions.The axial function () is chosen as the characteristics beam function as in which   ( = 1, . . ., 4) are constants with value 0, 1, or −1 depending on the tube ends,   is the roots of the transcendental equations obtained from the CNT boundary conditions, and   is the parameters corresponding to   .The geometric boundary conditions for clamped (C), free (F), and simply supported (S) boundary conditions can be expressed mathematically in terms of () as follows.

(23)
The above equations can be recast in the form of a generalized eigenvalue problem as where [K] is the stiffness matrix, [M] is the mass matrix of the CNT, and {X} is given by {X}  = { (1) ,  (1) ,  (1) ,  (2) ,  (2) ,  (2) , . . .,  () ,  () , By solving this eigenvalue problem, the natural frequencies of MWCNTs can be obtained and the associated eigenvectors yield the corresponding mode shapes.

Numerical Results and Discussion
The mechanical properties and thickness of each tube of MWCNTs are assumed to be  = 1TPa,  = 0.85 eV, ℎ = 0.34 nm, ] = 0.27, and  = 2300 Kg/m 3 .The configuration of layerwise boundary conditions, for example, will be denoted by (SS/CF/CC), where SS is related to the outermost tube and CC is related to the innermost tube.Also, for given intertube mode number , for convenience, the frequency associated with the th axial mode will be denoted by   ().

Validation of the Present Approach.
For validation, the present results are compared with the MD results given in [37].The first resonant frequencies of clamped and cantilever SWCNTs against nanotube aspect ratio (/) are depicted in Figures 3 and 4, respectively.The nonlocal parameter  0  should be calibrated.The calibrated values for  0  associated with clamped-clamped and clamped-free boundary conditions are 1.98 and 2 nm, respectively.It means that the calibrated value of  0  depends on end conditions.Moreover, Figure 3 provides a comparison between the nonlocal shell model and its local counterpart for clamped end conditions.It is observed that the local shell model ( 0  = 0) overestimates the frequencies, particularly for small aspect ratios.As the aspect ratio increases, resonant frequencies decrease and the small scale effect diminishes so that the frequency envelopes tend to converge.Nanotube aspect ratio L/R 1 significant for shorter nanotubes.Moreover, it is observed that the frequencies decrease by increasing  0 .As the ratio of length-to-innermost radius increases, resonant frequencies tend to decrease and the effects of small length scale and CNT end conditions diminish.

Explanatory Examples
Example 2. Since the effective thicknesses of nanotubes are scattered in the range of 0.066−0.34nm, Figure 6 is presented to demonstrate the influence of thickness variation on the fundamental resonant frequencies of a triple-walled CNT with clamped end conditions (CC/CC/CC).From this figure, the frequency difference can be observed due to this effect in both local and nonlocal models.Furthermore, resonant frequencies of CNTs with ℎ = 0.066 nm are higher than those of CNTs with ℎ = 0.34 nm when aspect ratio decreases.
Example 3. Figure 7 depicts the natural frequency of a simply supported triple-walled CNT versus circumferential mode number.Different values of nonlocal parameter have been considered ranging from 0 to 1.5 nm.Once again, it is found that the natural frequency diminishes with increasing nonlocal parameter.Physical interpretation is that the small scale effects in the nonlocal model make nanotubes more flexible.It is further observed that the magnitude of decrease in natural frequencies corresponding to higher circumferential modes is considerably higher than those corresponding to lower ones.This reveals that the influence of the small scale becomes more important for shorter wavelength at higher modes.
Example 4. Figure 8 shows the frequency ratio of a triplewalled (SS/SS/SS) CNT for several length-to-innermost radius ratios.It can be found that the effects of the small length scale are more significant for shorter length CNTs.For example, in the case of frequencies of nanotube with / 1 = 5 a relative error equal to 26.5% for nonlocal parameter  0  = 2 nm is obtained.This relative error reduces to about 9% as / 1 increases by 20.
Example 5. To show the ability of the present shell model in predicting new intertube frequencies and corresponding noncoaxial vibrational modes, mode shapes of a simply supported triple-walled CNT are shown in Figure 9.As revealed  in this figure, noncoaxial vibrational modes are predictable.Furthermore, noncoaxial vibrational modes may shift to the ones corresponding to higher circumferential mode numbers as the radius of MWCNT increases.The three-dimensional vibrational mode shape of a simply supported triple-walled CNT associated with the first intertube and the fifth axial and circumferential modes is also plotted in Figure 10.This figure is accompanied by a cross-sectional view in the middle of the nanotube.

Innermost tube Middle tube
Outermost tube  Example 6.This example provides a comparison between the present shell model and the beam model given by Xu et al. [30].The first three natural frequencies of a double-walled CNT with different layerwise boundary conditions obtained by the present shell model and by the beam model [30] versus nanotube length are tabulated in Table 2.It is observed that, for long double-walled CNTs for which the beam-like vibrations are dominant, the two models agree reasonably well.However, the results obtained by the beam model for double-walled CNTs of finite length are overestimated due to not taking circumferential modes into consideration.Moreover, this reduction in the size of nanotubes brings the effects of small scale and boundary conditions into focus and accordingly makes the present nonlocal shell model more preferable than the local beam counterpart.
Example 7. Presented graphically in Figure 11 are the frequency ratios related to a simply supported CNT with different number of walls.It is seen that the effects of the small length scale are dependent on the number of tubes so that the relative error in resonant frequencies decreases by increasing the number of walls.For  0  = 2 nm as an example, the relative errors corresponding to the double-walled and fivewalled CNTs are approximately 19% and 12.5%, respectively.

Concluding Remarks
Considering various layerwise boundary conditions, this paper probed the free vibrations of MWCNTs based on a nonlocal elastic shell model.Using the Flügge shell theory, the displacement field equations coupled by vdW forces were derived.The variational form of the Flügge type equations was constructed to which the analytical Rayleigh-Ritz method was applied.Among the more significant conclusions  to be obtained, the following findings may be summarized from the present study:

Figure 2 :
Figure 2: Cross-sectional view of an MWCNT under the vdW interactions.

Figure 5 :
Figure 5: Fundamental resonant frequency curves for a triplewalled CNT corresponding to various end conditions and nonlocal parameters ( 1 = 8.5 nm).

Fundamental 1 Nanotube aspect ratio L/R 1 Figure 6 :
Figure 6: Effect of the nanotube thickness variation on the fundamental resonant frequencies of a fully clamped triple-walled CNT ( 1 = 0.35 nm).

Figure 10 :
Figure 10: Three-dimensional mode shape associated with the first intertube and the fifth axial and circumferential modes for a triplewalled CNT with simply supported boundary conditions ( 1 = 8.5 nm, / 1 = 5).

Figure 11 :
Figure 11: Effect of the small length scale on the frequency ratio for a simply supported CNT with different number of walls (  = 8.5 nm, /  = 10).
(i) The efficiency of the present shell model was checked by the MD simulation and nonlocal parameters were calibrated for clamped and clamped-free SWCNTs.(ii) The small scale effects in the nonlocal continuum model reduce the frequencies of CNT as competed to the predictions of classical model.(iii) It was shown that the significance of the small size effects on the natural frequencies of MWCNTs is dependent on the geometric parameters of CNTs, vibrational modes, boundary conditions, and number of walls.(iv) The elastic beam model tends to overestimate the resonant frequencies of CNTs as compared to its shell counterpart, due to not incorporating circumferential mode number into the model, especially for shorter CNTs.(v) The results generated provide valuable information concerning new noncoaxial modes affecting the properties of MWCNTs.

Table 1 :
Values of   ,   , and   for SS, CC, FF, CS, CF, and FS boundary conditions.

Table 2 :
[30]arison between the present shell model and the beam model given in[30]for the first three natural frequencies of a doublewalled CNT with different boundary conditions ( 1 = 0.35 nm,  0  = 0).
[30]sed on the present shell model.bBasedon the beam model given by Xu et al.[30].