Vibrational analysis of an irregular single - walled carbon nanotube incorporating initial stress

: In this work, an attempt is done to apply the eigenvalue approach as well as Donnell thin - shell theory to ﬁ nd out the vibrational analyses of an irregular single walled carbon ( ISWCNT ) incorporating initial stress e ﬀ ects. The e ﬀ ects of surface irregularity and initial stresses on natural frequency of vibration of nano materials, especially for single - walled carbon nanotubes ( SWCNTs ) , have not been investigated before, and most of the previous research have been carried for a regular and initial stress - free CNTs. Therefore, it must be emphasized that the vibrations of prestressed irregular SWCNT are novel and applicable for the design of nano oscillators and nanodevices, in which SWCNTs act as the most prevalent nanocomposite struc tural element. The surface irregularity is assumed in the parabolic form at the surface of SWCNT. A novel equation of motion and frequency equation is derived. The obtained numerical results provide a better representation of the vibration behavior of prestressed ISWCNTs. It has been observed that the presence of either surface irregularity or initial stress has notable e ﬀ ects on the natural frequency of vibration, particularly in the short - length SWCNTs. Numerical results show that the natural frequency of SWCNT decreases with increase in surface irregularity and initial stress parameters. To the authors ’ best knowl -edge, the e ﬀ ect of surface irregularity and initial stresses on the vibration behavior of SWCNTs has not yet been studied, and the present work is an attempt to ﬁ nd out this e ﬀ ec -tiveness. In addition, the results of the present analysis may serve as useful references for the application and the design of nano oscillators and nanodevices, in which SWCNTs act as the most prevalent nanocomposite struc tural element.


Introduction
The study of vibration analysis in the surfaced irregularity and prestressed single-walled carbon nanotubes (SWCNTs) may give a useful reference to predict frequencies to estimate the expected experimental values for potential application and design of nanoelectronics and nanodevices. Since Iijima [1] discovered the CNTs, they became widely used in scientific and practical applications; they were used in the manufacturing electrodes, in super capacitors and space elevator cables [2]. Moreover, CNTs are used to strengthen the components of nanocomposites as well as are used on a large scale in components of superconducting nanocomposites [3]. The vibrations of CNTs have a considerable importance in nanomechanical applications such as the design of nano oscillators and nanodevices. During the last decade, numerous studies on vibrations of SWCNTs using continuum shell models were performed, as it can be seen from the examples of ref. [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Based on Euler-Bernoulli beam, Tang and Yang investigated a novel model of fluid-conveying nanotubes made of bidirectional functionally graded materials (FGMs) and presented the dynamic behavior and stability of nanotubes [20].T o study the nonlinear free vibration of the bidirectional (2D) FGMs, Tang et al. [21] presented a novel model of Euler-Bernoulli beams. Nonlinear bending, buckling and vibration of bidirectional functionally graded nanobeams were studied by Yang et al. [22,23]. A novel model of fluid-conveying nanotubes made of bidirectional FGMs was presented by Tang and Yang [24] to investigate the dynamic behaviors and stability of the graded nanotubes. The post-buckling behavior and nonlinear vibration of a fluid-conveying pipe composed of an FGM were analytically studied by Tang and Yang [25]. Tang and Ding [26] presented the nonlinear hydrothermal dynamics of a bidirectional functionally graded beam with coupled transverse and longitudinal displacements. The fluidconveying pipes made of polymer-like materials are widely applied in engineering fields. The fractional dynamics of the pipes subjected to the excitation of the supporting foundation is studied by Tang et al. [27]. SWCNTs, also called graphene nanotubes have properties that make them the only all-purpose additive that can improve specific characteristics of a multitude of materials. Due to the importance of SWCNTs, which are known to be the strongest materials, there are many experimental and theoretical studies carried out about composites that contains SWCNT, as can be seen from the examples given in refs. [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47]. During the last decade, numerous studies were carried out on the vibration analysis of SWCNTs, in particular, to understand the dynamic behavior of torsional vibration of CNTs, and numerous researchers conducted the computational simulations to study the torsional vibrations of the nanotubes. Gheshlaghi and Hasheminejad [48] studied the free torsional responses of CNT using modified couple stress model to achieve the effect of the size dependency on the response of the nanotube. The torsional vibration of nonlinear homogenous and isotropic nano-cone was studied based on the strain gradient model theory and generalized differential quadrature method [49].T h e mechanic behavior of the CNTs embedded in an elastic medium were investigated by Arda and Aydogdu [50] based on the modern elasticity theory. Based on Kelvin-Voigt model, El-Borgi et al. [51] reported a novel model to investigate the vibrational behavior of the viscoelastic nanorods. Modern elasticity theory and Maxwell model were used by Zarezadeh et al. [52] to study the torsional vibration of functionally graded nanorods embedded in an elastic bulk medium under magnetic field. In addition, some studies focused on the mechanical behavior of CNTs, such as Ru [53], Shen [54] and Thai [55]. However, the models used in these studies could not be applied for predicting the vibrational behavior of prestressed irregular SWCNTs (ISWCNTs), due to technical limitations. CNTs acting as basic elements of nanostructures often occur in compressional initial stresses due to the mismatch between different materials or initially external axial loads. Moreover, the nature frequency of vibration of SWCNTs is influenced by the change in the surface structures of the nanotube. The irregularities in CNTs constructions may occur due to the consequence of manufacturing defect, reckless servicing, etc. Thus, it is of great importance to deal with surface irregularity to study the vibrational analysis of SWCNTs.
The effects of surface irregularity and initial stresses on the deformation of the static medium were studied by Selim [56]. The vibration analysis of ISWCNTs without initial stresses was investigated by Selim [57]. It was shown that the presence of surface irregularity has a notable effect on the vibration analysis of SWCNTs. The present study is an extension of the previous work to find the effectiveness of both surface irregularity and initial stresses on the natural frequency of vibrations of WCNTs, due to the similarity structures of CNTs with the cylindrical shells. Donnell thin-shell approach is capable of tackling vibrations studied of SWCNTs. The analytical technique used should be simple, easily applicable and yield results robustly. On the other hand, CNTs often suffer from initial compressional stresses due to residual stress, thermal effect, surface effect, mismatch between the material properties of nanoplates and the surrounding medium, initial external load and may be due to any other physical causes. It is obvious that the varying of the surface structures of the nanotube as well as the presence of the initial stresses influences the natural frequency of the nanotubes. Hence, it is of great importance to deal with various constructions to study the vibration behavior of irregular and prestressed CNTs. As can be seen from the above literature summary, to the best of our knowledge, the effect of both the surface irregularity and initial stresses on vibrational analysis of SWCNTs has not yet been investigated. The vibration of CNTs under the effects of surface irregularity and initial stresses is very important in terms of both predicting the results of the experiments and providing useful information for the next generation studies and accurate deigns of nanodevices. In this sense, the present study is much important within the framework of the Donnell thin-shell theory; and thus, the results can provide a useful information for the next generation of studies and accurate designs of nanomachines. It is pointed out that the present results can be used to analyze the surface irregularity and initial stresses effects on the natural frequency of vibration of SWCNTs. The numerical results are plotted in graphs for different values of initial stress and surface irregularity parameters and have been compared with the cases of uniform and initial stress-free SWCNTs. In this work, the obtained frequency equation and numerical results may give a useful reference to predict the natural frequency of vibrations of SWCNTs to estimate the expected experimental values for potential applications and designs of nano oscillators and nanodevices.

Mathematical formulation of the problem
The cylindrical coordinate system is used for describing the effectiveness of surface irregularity and initial stresses on the vibration analysis of SWCNT, which is defined in Figure 1. The x-coordinate is taken in the axial direction of the nanotube, the z-coordinate and θ-coordinate are taken in the radial and circumferential directions, respectively. The displacements are defined by u, v and w in the x-, θand z-directions, respectively. The SWCNT has a thickness of h, length of L and radius of R.
The compressional initial stress (−P) is effected on the x-direction and the surface irregularity is assumed in the parabola form at the lower boundary of the SWCNT (see Figure 1). The span of surface irregularity and maximum depth of irregularity are denoted by s and H′, respectively.
In case of parabolic irregularity, the boundary surface may be described by [58]: (1) where the maximum amplitude of irregular boundary is =≪ The governing equations of the vibration of prestressed ( =− S P xx ) ISWCNT can be written as [59]: (2)  (3) where 1 , h, R and ρ are the thickness, the radius of the nanotube and the mass density of the CNT, respectively; E, ν and Eh are the elastic modulus, the Poisson's ratio and the inplane rigidity, respectively; ρh, Pand t are the mass density per unit lateral area, the initial compressional stress and the time, respectively. In matrix form, the equations (2), (3) and (4) may be expressed as:  ΓΔ ΓΔ ΓΔ   0  0  0   ,   11  11  12  12  13  13   21  21  22  22  23  23   31  31  32  32 33 33 where Γ ij and (= ) ij Δ, 1 , 2 , 3 ij are the differential operators with respect to the coordinates axis x and θ, and are defined by,

Solution procedure
The functions u, v and w are used to designate their respective displacement functions and u 0 is the midline stretching. Therefore, for modal deformation, the displacements are assumed ( ||≤ /) xs for 2 as: where U V , mm and W m represent three vibration amplitude coefficients in axial, circumferential and radial directions, respectively. m and n are the axial half and the circumferential wave numbers, respectively, where k m is the axial wave number related with an end condition. The angular frequency is denoted by ω and the natural frequency is =/ f ω π 2 . In view of the differential operators in equations (7)-(17) and using the u, v and w given in equations (18)- (21), the matrix form of the vibration frequency equation of the SWCNTs with surface irregularity under initial compression stresses is as follows: where the lowest root of equation (36) indicates that the eigenvalues are associated with vibration frequencies of prestressed ISWCNT and the vector ( ) UVW ,, where which agrees with the result of Hussain [60].
In this section, numerical results for vibration of the simply supported SWCNTs under the effects of surface irregularity are carried out using the simulation parameters presented in Table 1 [5].
An examination of the studies concerned with fabrication procedures of CNTs reveals that perfectly straight CNTs without any deviation in geometry and orientation are very difficult to achieve [61]. Therefore, understanding the influences of surface irregularity as well as initial stresses on the vibration analysis of SWCNT may serve as useful references for the applications and designs of nano oscillators and nanodevices, in which SWCNTs act as the most prevalent nanocomposite structural element.
In order to show the effects of compressive initial stresses as well as surface irregularity on the torsional natural frequency of SWCNT, the relation between the torsional frequency and axial wave number is calculated.  The effect of surface irregularity becomes obvious for the higher values of the irregularity parameter ( = ) ε 0.56   as shown in Figure 4. From this figure, one can observe that the natural frequency of nanotube influenced by the increase in the surface irregularity parameter and the curve takes opposite direction compared with the result obtained in Figure 3 at = ε 0.28. This means that the increase in irregularity parameter decreases the natural frequency of the vibration and this decrease is most pronounced when the nanotube is long. However, the natural frequency of the nanotube is more affected by increasing the value of irregularity parameter compared with the cases of lower values of this parameter ( = ε 0.28 and = ε 0.0). From Figures 2 and 3, one can get the information that the presence of surface irregularities in the SWCNTs has notable effects on the natural frequency of the nanotubes. Figure 5 shows the effects of initial stress on the natural frequency of uniform nanotube with various aspect ratio at ( = ) ζ 0.01, 0.025 x . From Figure 5, it can be seen that small increase in the initial stress parameter (0.015) makes a notable change in the natural frequency of the vibrating nanotube. This means that the natural frequency is independent of the initial stresses present in the nanotube.
The effects of irregularity and initial stress parameters on the natural frequency of SWCNT are shown in Figures 6 and 7.      Figure 5 for uniform nanotube at the same values of the initial stress parameter ( = ) ζ 0.01, 0.025 x . The effect of surface irregularity and initial stresses becomes obvious for the higher values of the irregularity parameter ( = ε 0.56) as shown in Figure 7. From this figure, we can observe that the natural frequency of nanotube is influenced by the increase in the surface irregularity and initial stress parameters, and the curve takes the opposite direction. This means that the increase in surface irregularity with an increase in initial stress parameter decreases the natural frequency of the vibration, and this decrease is most pronounced when the nanotube is long. However, the natural frequency of the nanotube is more affected by the surface irregularity and initial stresses at shorter nanotubes.
From Figures 6 and 7, one can get the information that the presence of surface irregularity and initial stresses in SWCNTs, has notable effects on the natural frequency of vibration.
From the numerical results, it is observed that the presence of surface irregularity as well as compressive initial stresses in the SWCNTs has notable effects on torsional natural frequency. In addition, we conclude that the increase in compressive initial stress parameter increases the torsional natural frequency of SWCNTs no matter whether the surface irregularity is increasing or not.

Conclusions
In this article, a novel frequency equation of vibration of pre-stressed ISWCNT based on Donnell thin-shell theory is reported. Taking into account the effects of surface irregularity and initial stresses, vibration of SWCNTs is analyzed and the numerical results are plotted in several figures to examine the dependence of natural frequency of vibration on the surface irregularity and initial stress parameters. Some conclusions are drawn as follows: • Based on a theoretical model for vibration of SWCNT dependence of the surface irregularity and initial stresses, novel equation of motion and natural frequency equation are derived. • The natural frequencies are calculated for various surface irregularity and initial stress parameters, with detailed comparison to the results, which neglect the effects of surface irregularity or the initial stresses on the vibration. • The presence of surface irregularity and initial stresses has notable effects on the natural frequency of SWCNTs. • To the authors' best knowledge, the effect of surface irregularity and initial stresses on the vibration behavior of SWCNTs has not yet been studied, and the present work is an attempt to find out this effectiveness, and the results of the present analysis may serve as useful references for the application and design of nano oscillators and nanodevices, in which SWCNTs act as the most prevalent nanocomposite structural element.