
International Journal of ChemTech Research CODEN (USA): IJCRGG, ISSN: 09744290, ISSN(Online):24559555 Vol.10 No.3, pp 339346, 2017

Vibration response of doublewalled carbon nanotubes embedded in an elastic medium with intertube Vander waals forces
Ravi kumar B*
School of Mechanical Engineering, SASTRA University, Tirumalaisamudram, Thanjavur, Tamilnadu, INDIA – 613401.
Abstract : The study of vibration in carbon nanotubes(CNTs) is currently a major topic of interest that increases understanding of their dynamic mechanical behavior. In this work differential transform method(DTM) is used to study the vibrational behavior of the double walled carbon nanotubes(DWCNT) for various boundary conditions. Elastic continuum models are used to study the vibrational behavior of CNTs to avoid the difficulties encountered during experimental characterization of nanotubes as well as the timeconsuming nature of computational atomistic simulations. To calculate the resonant vibration of doublewalled carbon nanotubes embedded in an elastic medium, a theoretical analysis is presented based on EulerBernoulli beam model and Winkler spring model.
Keywords : Aerospace , BuvnovGalerkin, DTM, DWCNT, MATLAB, PetrovGalerkin.
Introduction
Iijima’s discovery paper on multiwalled carbon nanotubes in 1991 [1] led to a major revolution in the area of nanoscience and nanotechnology. Carbon nanotubes (CNTs) have subjected to much attention as a result of their extending applications in the different emerging fields of nanotechnology.
In aerospace industries, there is a great need for new materials which exhibit improved mechanical properties i.e, high strength at reduced weight. Carbon nanotubes (CNTs) are allotropes of carbon with a cylindrical nanostructure. Nanotubes have been constructed with lengthtodiameter ratio of up to 132,000,000:1[2], significantly larger than any other material. The structure of an SWCNT can be conceptualized by wrapping a oneatomthick layer of graphite called graphene into a seamless cylinder [2], [3] and [6]. Singlewalled nanotubes are the most likely candidate for miniaturizing electronics beyond the micro electromechanical scale currently used in electronics [9]. Singlewalled nanotubes are an important variety of carbon nanotube because they exhibit electric properties that are not shared by the multiwalled carbon nanotube (MWNT) variants[2].Carbon nanotubes are the strongest and stiffest materials yet discovered in terms of tensile strength and Elastic Modulus respectively[7]. Since carbon nanotubes have a low density for a solid of 1.3 to 1.4 g·cm−3[13]. This strength results from the covalent sp2 bonds formed between the individual carbon atoms.
sizedependent continuumbased methods [5–7] are becoming popular in modeling small sized structures as it offers much faster and accurate solutions. Sudak [8] carried out buckling analysis of multiwalled carbon nanotubes. Wang and Varadhan [9] analyzed the small scale effect of CNT and shell model. Yakobson et al. [10] introduced an atomistic model for axially compressed SWCNT and compared it to a simple continuum shell
model. Sears and Batra[11] proposed a comprehensive buckling analysis of single walled and multiwalled CNTs by molecular mechanics simulations and continuum mechanics models.
In the present work, DTM has been used to study the vibration of CNTs embedded in an elastic medium. Zhou [16] proposed differential transformation method to solve both linear and nonlinear initial value problems in electric circuit analysis. Later Chen and Ho [17] applied this method to eigen value problems. Arikoglu and ozkol [18] applied differential transformation method to solve the intergro – differential equation.
The Differential transform method is a semianalytical method based on the Taylor series expansion. In this method, certain transformation rules are applied and the governing differential equations and the boundary conditions of the system are transformed into a set of algebraic equations in terms of the differential transforms of the original functions. The solution of these algebraic equations gives the desired solution of the problem.
The differential transformation of the kth derivative of the function u(x) is defined as follows:
(1)
And the differential inverse transformation of U(k) is expressed as
(2)
In real application function, u(x) is expressed as finite series and equation (2) can be written as:
(3)
Now using certain transformation rules we can convert the governing differential equation and associated Boundary Conditions into some algebraic equations and after solving them we can get the desired results. We can use the following transformation table for this purpose.
Table 1: Differential Transformations for Mathematical Equations
Original Function Transformed Function 



The continuum mechanics method has been successfully applied to analyze the dynamic responses of individual carbon nanotubes. Based on the Euler–Bernoulli beam model, the governing equation of motion of a beam is given by [18]
(4)
Where x and t are the axial coordinate and time, respectively. w(x,t) is the deflection of carbon nanotubes and p is the distributed transverse force acted on the nanotubes. E and I are the elastic modulus and the moment of inertia of a crosssection, respectively. A is the crosssectional area and ρ is the mass density of nanotubes.
For the DWCNTs, the interaction between inner and outer nanotubes is considered to be coupled together through the Vander Waals (vdW) forces. Equation (4) can be used to each layer of the inner and outer nanotubes of the DWCNTs. Assuming that the inner and outer tubes have the same thickness and effective material constants. Based on the EulerBernoulli beam model, we have:
(5)
(6)
Where, the subscripts 1 and 2 denote the quantities associated with the inner and outer nanotubes respectively. (j=1,2) are the pressures exerted on inner and outer nanotubes.
The pressure P1 acting on the inner nanotube caused by vdW interaction is given by
(7)
Where, c is the vdW interaction coefficient between inner and outer nanotubes.
Fig.1 shows the analysis model CNTs embedded in an elastic medium. The pressure acting on the outermost layer due to the surrounding elastic medium can be given by
(8)
Where negative sign indicates that is opposite to the deflection of nanotubes. k is the spring constant.
Thus, for the embedded DWCNTs, the pressure of the outermost nanotube contacting with the elastic medium is given by
(9)
In the simulation vdW interaction coefficient (c) can be obtained from the interlayer energy potential, given as [13]
(10)
where,
R1= Radius of the inner nanotube.
Thus,
(11)
(12)
In this analysis, we consider the deflection of DWCNTs has different vibrational modes , , j = 1,2 for the inner and outer nanotubes. The displacements of the vibrational solution in DWCNTs can be given by
(13)
Which can be further simplified as:
(14)
( (15)
Where,
,,,,
For the simply supported CNT beam boundary conditions at both ends are defined mathematically as
(16)
For clampedclamped CNT case, the boundary conditions at both ends are defined as:
(17)
For clampedhinged CNT case, the boundary conditions are defined as
At x=0
(18)
At x=L
(19)
In this study, we consider double walled carbon nanotubes embedded in an elastic (Winkler) medium having the inner and outer diameters of 0.7nm and 1.4 nm, respectively. The effective thickness of each nanotube
is taken to be that of graphite sheet with 0.34 nm. The CNT has an elastic modulus of 1 TPa and the density of 2.3[13,18].
By using the DTM as the numerical method the natural frequency for DWCNTs has been computed. Results are compared with Elishakoff & penataras et al [18] study in which he used BuvnovGalerkin and PetrovGalerkin methods for analyzing vibration response of DWCNTs. Also, Results are compared with Xu et al [26] and exact results. Very good agreement is observed with the exact solution. We have taken n= 50 so that the result converges up to four decimal places. Where n is the number of iterations required to converge the result.
Table 2:  Simply supported (SS) DWCNTs Fundamental frequency in THz
L/d 
10 
12 
14 
16 
18 
20 
Present[DTM] 
0.4683 
0.3252 
0.2389 
0.1829 
0.1446 
0.1171 
Exact[18] 
0.4683 
0.3252 
0.2389 
0.1829 
0.1446 
0.1171 
Bubnov[18] 
0.4721 
0.3279 
0.24093 
0.1844 
0.1457 
0.1180 
Petrov [18] 
0.4688 
0.3256 
0.2392 
0.1831 
0.1447 
0.1172 
Xu et al[26] 
0.46 
……. 
……. 
…….. 
…….. 
0.11 
Table 3:  ClampedClamped (CC) DWCNTs Fundamental Frequency in THz
L/d 
10 
12 
14 
16 
18 
20 
Present[DTM] 
1.0640 
0.7368 
0.5425 
0.4137 
0.3265 
0.2654 
Bubnov[18] 
1.0798 
0.7506 
0.5517 
0.4224 
0.3338 
0.2704 
Petrov[18] 
1.0647 
0.7308 
0.5434 
0.4113 
0.3250 
0.2633 
Xu et al[26] 
1.0636 
…… 
…… 
……. 
……. 
0.2660 
Table 4: ClampedHinged (CH) DWCNTs Fundamental Frequency in THz
L/d 
10 
12 
14 
16 
18 
20 
Present[DTM] 
0.7314 
0.5085 
0.3728 
0.2858 
0.2258 
0.1829 
Buvnov[18] 
0.7327 
0.5090 
0.3740 
0.2864 
0.2263 
0.1833 
Petrov[18] 
0.7284 
0.5060 
0.3718 
0.2847 
0.2249 
0.1822 
Xu et al[26] 
0.728 
……. 
……. 
……. 
…….. 
0.1834 
Clearly it is observed that fundamental frequency of DWCNTs decreasing with increasing aspect ratio (L/d, where d=diameter of the outer nanotube) of nanotubes.
Now if we change the value of Winkler Elasticity constant (k) from 0 300 GPa and L=20 nm, we can obtain different values of vibration frequencies which are given below.
Fig 1: Influence of Winkler foundation on Vibration frequencies for simply supported DWCNTs
Fig 2: Influence of Winkler foundation on Vibration frequencies for ClampedClamped DWCNTs
Influence of the surrounding medium on the vibration frequency is investigated based on the Winkler spring model. It is found that vibration frequencies of the embedded double walled carbon nanotubes are larger than those of the nested nanotubes. Especially, the influences of surrounding medium on the vibration frequency are significant for the first inphase modes. On the other hand stiffness of surrounding medium impacts very little on the frequencies of the antiphase modes.
In this study, the vibration analysis of DWCNTs embedded in an elastic medium for various boundary conditions like clampedclamped, simply supported, and clamped hinged are studied by a semianalytical numerical technique called the differential transform method in a simple and accurate way. The solution of the present vibration analysis problem using the DTM includes transforming the governing equations of motion into algebraic equations and solving the transformed equations. Results indicate that phase modes have a strong
influence on vibration frequencies of CNTs. The stiffness of surrounding medium affects the resonant frequencies of DWCNTs, especially for the first inphase modes. The investigation presented may be helpful in the application of CNTs such as highfrequency oscillators, dynamic mechanical analysis and mechanical sensors.
Acknowledgements
I am very much thankful to my guide Dr. S.C. Pradhan for his continuous guidance, support and constant encouragement right from the inception of the problem to the successful completion of this study.
References
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