Vibrational Behavior of Single-Walled Carbon Nanotubes Based on Donnell Shell Theory Using Wave Propagation Approach

2.1. Cylindrical shell model for the vibration of SWCNT

Carbon nanotubes have two kinds, which are single-walled carbon nanotubes and multi-walled carbon nanotubes. Actually multi-walled carbon nanotubes are singled walled carbon nanotubes that are coaxially interposed with different radii. When a graphene sheet rolled up into one time, then it becomes a SWCNTs to produce a hollow cylinder but with end caps. A schema of graphene sheet and single-walled carbon nanotube are shown in Figure 1.

Armchair and zigzag nanotubes are made when chiral angle is equal to 0 and 30 respectively and both are the limiting cases with (m, m) and (m, 0). The structure of single-walled carbon nanotubes is similar to the circular cylinders with regard to geometrical shapes as shown in Figure 2. So, the motion equations for cylindrical shells are utilized for studying the free vibrations of SWCNTs. According to the Donnell thin shell theory (He et al. [24]), the governing equation of motion for free vibration of a CNTs is used. Where v1, v2, and v3are the longitudinal, circumferential, and radial displacements of the shell, R is the radius of the shell, Eh is the in-plane rigidity, ρh is the mass density per unit lateral area, t is the time and ν is the Poisson ratio.

It is assumed that for the representation of the modal deformation displacement functions in the axial, circumferential and radial directions are v1xθt, v2xθtand v3xθtcorrespondingly. The three unknown displacement functions for SWCNTs executing vibration, a system of PDE is given as:

where D=Eh3121−v2denotes the effective bending stiffness.

2.2. Applications of the wave propagation approach

An efficient and a simple technique which corporate as wave propagation approach is employed for the solution of CNT problem in the form of differential equation. Before this, present method has been successively used for the study of shell vibrations [25, 26, 27]. The axial coordinate and time variable are denoted by x, tcorrespondingly and the circumferential coordinate signifies by θ. The functions v1xθt, v2xθtand v3xθtare used to designate their respective displacement deformation function. So for modal deformation displacements are written in the assumed expression as:

where pm, qmand rmstand for three vibration amplitude coefficients in the axial, circumferential and radial directions. The axial half and the circumferential wave numbers are denoted by m and n respectively and angular frequency is designated by ω. The formula for fundamental frequency fwhich is written as: f=ω/2π.Where kmis the axial wave number related with an end conditions. Using the expressions for v1xθt, v2xθt, v3xθtand their partial derivatives in applying the product method by substituting the modal displacement functions for partial differential equations, the space and time variable are split.

After putting Eqs. (4)–(6), into Eqs. (1)–(3), the above equations is transmuted in matrix representation after the arrangement of terms, and to designate the vibration frequency equation for SWCNTs, an eigenvalue problem is formed:

The form of non-zero solution of pmqmrmyields the vibration frequency and associated modes for SWCNTs. The expressions for the terms Lij‘s are given in the Appendix-I. Where the roots of the equation furnish the frequencies. The lowest root corresponds to the frequency of vibration. It is clear that the frequency should be minimized with respect to the wave numbers m, n in order to obtain the frequency of vibration.

3.1. Vibration of clamped-clamped and clamped-free SWCNTs

In the application of micro-oscillators and micro or nano-strain sensors, the carbon nanotube sensor is generally clamped with both ends [30]. The clamped-clamped and clamped-free single-walled carbon nanotubes have been performed by atomistic simulations [20, 21, 31]. In this section, the distinctive first and third mode frequencies for the set of clamped-clamped single-walled carbon nanotubes is given by present models for their vibration frequencies and compared with molecular dynamic simulations and Timoshenko beam model with thicknesses h = 0.34 nm. The results are in good agreement with the MD and Timoshenko beam model results showing same trend in the open literature.

The first and third mode natural frequencies accessed by present model and compared with MD simulations are depicted graphically in Figure 3 respectively. It can be observed from Tables 1 and 2 that the length-to-diameter ratio of the set of C-C SWCNTs is somewhat dissimilar from that of C-F SWCNTs. For the prediction of mechanical characters [7, 19, 20, 21] from atomistic studies and the experimental studies [5, 6, 33, 34, 35, 36, 37, 38] are often used for the clamped-free carbon nanotubes. The frequencies for the first and third modes obtained from present model which is compared with molecular dynamic simulation and Timoshenko beam model are shown in Figure 3. It can be readily seen that higher frequencies are produced by higher modes and when length-to-diameter ratio rises at each mode then frequency falls down smoothly as shown in Figure 3. The relationship between the length-to-diameter ratios and natural frequencies is inversely proportional indicates that the vibrations are very sensitive due to long tube and since the SWCNTs are of almost the same diameter. The results for SWCNTs given by MD are little bit higher than the frequencies investigated by the present model. In MD simulation, the frequencies of length-to-radius ratio are 8.28 is 0.0793 and at 20.89 is 0.0138. But in the present model, the frequencies at 8.28 are 0.05566 and at 20.89 is 0.00883, when compared to the MD results.

3.2. Vibration of SWCNTs with dimensionless frequency

Furthermore, the parametric study for the vibrational behavior of SWCNTs with dimensionless is carried out and presented in Figure 4. Alibeigloo et al. [39, 40] and Soldatos et al. [41] used the dimensionless frequency for multi-walled carbon nanotubes and for thin cylindrical shell with respect to length-to-radius ratio respectively. This frequency is associated with frequency Ωthrough the following formula: Ω=ωRρE.A variation of non-dimensional frequency versus length-to-diameter ratio is presented in Figure 4. This figure shows that, increasing the value of length-to-diameter ratio as well as there is a decrease in dimensionless frequency. From the physical point of view, it is noted that when the length of SWCNTs becomes small, the effect of the atomic interactions among a reference point and all other atoms becomes significant.

Figure 4 shows an armchair and zigzag CNT, vary the length-to-diameter ratio from 8.3 to 20.9 will change the dimensionless frequency from 0.0385 to 0.0063 THz in case of clamped-clamped boundary condition. Likewise, in clamped-free condition it changes from 1.2827 to 0.5094 THz in armchair case. It may be seen from the above Figure 4 that the resulting value of dimensionless frequency decreases with the increase in length-to-diameter ratio. Now in zigzag CNT, changing the length-to-diameter ratio from 4.86 to 35.53, the dimensionless frequency changes from 0.0303 to 0.0049 THz in case of clamped-clamped boundary condition. Likewise, in clamped-free condition it varies from 1.16605 to 0.4609 THz.

3.3. Vibration of SWCNTs with in-plane rigidity

For the results generated so far, the nanotube in-plane rigidity has been taken to be Eh=278.25 Gpa·nm. However, there exist some inconsistencies concerning this quantity in the literature. The reported CNT in-plane stiffness is largely scattered, ranging from Eh=300 Gpa·nm to Eh= 400 Gpa·nm [42]. Both set of Figure 5 is presented to investigate the influence of the in-plane rigidity Ehvariation on the dimensionless frequency of a (7, 7) armchair and (9, 0) Zigzag SWCNT with different boundary conditions likewise as clamped-clamped and clamped-free boundary conditions. These figures shows that for all the selected boundary conditions, dimensionless frequency calculated via shell model are sensitive to the nanotube in-plane rigidity Ehand also the larger the in-plane rigidity in-plane rigidity Eh, the higher the dimensionless frequency. The difference is more considerable for shorter length CNTs.

Previous study reveals that the bending rigidity of SWCNTs should be considered as an independent material parameter not linked to the representative thickness by the classic bending rigidity formula, i.e., D=Eh3/121−ν2and the actual bending rigidity of SWCNTs is lesser than its classical counterpart [43, 44]. For shorter length-to-diameter ratio, the value of dimensionless frequencies for clamped-clamped at Eh= 300 Gpa·nm, Eh= 400 Gpa·nm is 0.04863, 0.05788, respectively which shows that a slight increase in frequency due to increase of in-plane rigidity Eh. Same trend is observed for dimensionless frequency. For the present shell model with in-plane rigidity Eh, the values of the C-F single-walled carbon nanotubes respectively, which are a little lower than those of corresponding CC SWCNTs with bending rigidity are plotted in Figure 5.

3.4. Vibration with mass density per unit lateral area

Figure 6 is presented to investigate the influence of the mass density per unit lateral area variation on the dimensionless frequency of a (12, 12) armchair and (14, 0) zigzag SWCNT with boundary conditions: clamped-clamped and clamped-free. These figures shows that for all the selected boundary conditions, the frequency calculated via shell model are sensitive to the nanotube mass density and also the larger the mass density per unit lateral area ρh, lower the frequency. It is observed that applying the mass density per unit lateral area ρhto the present shell model, yields the slight decrease of the frequency. For shorter length-to-diameter ratio, the value of dimensionless frequencies at ρh=740.52 nm, 800.64 nm and 820.80 nm is 0.3667, 0.03342, 0.03128 respectively for clamped-clamped and 1.74285, 1.71298, 1.6001 respectively for clamped-free, which shows that decreases in frequency. For the present shell model, the values of length-to-diameter ratio for C-C SWCNTs, which are a little higher than those corresponding C-F SWCNTs values with mass density per unit lateral area are plotted in Figure 6.