NONLOCAL ANISOTROPIC SHELL MODEL OF LINEAR VIBRATIONS OF MULTI-WALLED CARBON NANOTUBES

A simply-supported multi-walled carbon nanotube (MWCNT) is considered. Its vibrations will be studied in a cylindrical coordinate system. The elastic constants in Hooke's law depend on the CNT wall diameter, which is why each wall has its own elastic constants. CNT vibrations are described by the Sanders-Koiter shell theory. To derive partial differential equations (PDE) describing self-induced variations, a variational approach is used. The PDEs of vibrations are derived with respect to three projections of displacements. The model takes into account the Van der Waals forces between CNT walls. The three projections of displacements are expanded in basis functions. It was not possible to select the basis functions satisfying both geometric and natural boundary conditions. Therefore, selected are the basis functions that satisfy only geometric boundary conditions. To obtain a linear dynamic system with a finite number of degrees of freedom, the method of weighted residuals is used. To derive the basic relations of the method of weighted residuals, methods of variational calculus are used. The vibrational eigenfrequencies of single-walled (SW) CNTs are analyzed depending on the number of waves in the circumferential direction. With the number of waves in the circumferential direction from 2 to 4, the vibrational eigenfrequencies of CNTs are minimal. These numbers are smaller than those for the vibrational eigenfrequencies of engineering shells. Anisotropic models of trip-ple-walled (TW) CNTs were investigated. In their eigenforms, there is interaction between the basis functions and different numbers of waves in the longitudinal direction. This phenomenon was not observed in the isotropic CNT model. The appearance of such vibrations is a consequence of structural anisotropy.

A simply-supported multi-walled carbon nanotube (MWCNT) is considered. Its vibrations will be studied in a cylindrical coordinate system. The elastic constants in Hooke's law depend on the CNT wall diameter, which is why each wall has its own elastic constants. CNT vibrations are described by the Sanders-Koiter shell theory. To derive partial differential equations (PDE) describing self-induced variations, a variational approach is used. The PDEs of vibrations are derived with respect to three projections of displacements. The model takes into account the Van der Waals forces between CNT walls. The three projections of displacements are expanded in basis functions. It was not possible to select the basis functions satisfying both geometric and natural boundary conditions. Therefore, selected are the basis functions that satisfy only geometric boundary conditions. To obtain a linear dynamic system with a finite number of degrees of freedom, the method of weighted residuals is used. To derive the basic relations of the method of weighted residuals, methods of variational calculus are used. The vibrational eigenfrequencies of single-walled (SW) CNTs are analyzed depending on the number of waves in the circumferential direction. With the number of waves in the circumferential direction from 2 to 4, the vibrational eigenfrequencies of CNTs are minimal. These numbers are smaller than those for the vibrational eigenfrequencies of engineering shells. Anisotropic models of tripple-walled (TW) CNTs were investigated. In their eigenforms, there is interaction between the basis functions and different numbers of waves in the longitudinal direction. This phenomenon was not observed in the isotropic CNT model. The appearance of such vibrations is a consequence of structural anisotropy.

Introduction
CNT vibrations are extremely important for many nanomechanical devices, such as charge detectors, sensors, and devices for autoelectronic emission [1]. CNT vibrations are often observed during the processing and obtaining of nanocomposites. Wave processes in CNT-based nanodevices are studied in detail in articles [2,3,4]. CNT simulation approaches can be divided into two groups. The first group is a simulation based on molecular dynamics [5,6], which requires huge computer resources. The second group is the construction of continuum models based on the mechanics of deformable solids. There are not many works that are devoted to the construction of shell models of CNT vibrations. In [7], a linear shell model is used to describe the vibrations of simply-supported CNTs. In [8], a model of linear CNT vibrations was obtained on the basis of the theory of Flygge shells with account taken of nonlocal elasticity. In studying the geometrically nonlinear dynamic deformation of CNTs, rod models are mainly used. In [9], continuous nonlinear beam models are used to model the nonlinear vibrations of MWCNTs. Nonlinear vibrations of the MWCNTs embedded in an elastic medium are considered in [10]. Nonlinear vibrations with large amplitudes of double-walled (DW) CNTs were studied by the finite element method in [11]. Forced vibrations of DWCNTs are considered in [12].
In this article, an anisotropic shell model of vibrations of a MWCNT is constructed. This model is based on the Sanders-Koiter shell theory. The interaction between the walls is described by Van der Waals forces. A system of PDEs is derived, describing CNT vibrations. To obtain a dynamic system with a finite number of degrees of freedom, the generalized Galerkin method is used. Properties of the linear vibrations of MWCNTs are investigated.

Statement of the Problem and Equations of Linear Vibrations
A MWCNT is considered, (Fig. 1). The number of CNT walls is equal. This MWCNT's vibrations will be studied in the cylindrical coordinate system (x, θ, z), Fig. 2. The figure shows one of CNT walls, which we will denote by the number i.
The individual deformations of CNT walls are interconnected due to Van der Waals forces. This deformation model is explained by the fact that van der Waals forces are much smaller than the covalent bonds of neighboring carbon atoms. The vibrations of each CNT wall will be studied on the shell-model basis [13]. We denote the three projections of the displacements of points of the middle surface of the ith wall by (Fig. 2). Each CNT wall moves relative to the other walls. To describe wall elasticity, we use the nonlocal anisotropic Hooke's law [14,15]. The elastic constants in Hooke's law depend on the CNT wall diameter [16,17]. Therefore, each wall has its own elastic constants. Hooke's law for the ith wall has the following form:  12 11 thickness of each CNT wall; ( ) i jk Y are the anisotropic elastic constants of the ith wall. As follows from [16,17], the following relations hold: ( ) of the ith wall at a distance z from the middle surface are described as follows: The variation of the potential energy of the ith СNT wall is presented in the following form: where A is the region of the middle surface of the shell; ( ) ( ) ( ) are the specific power factors and moments that are defined as follows: 2 We introduce (2) into equation (3) and perform integration by parts. Then, as a result, we obtain the following expression for the variation of potential energy: where .
The variation of the kinetic energy of the ith CNT wall takes the following form: where ρ is the material density. The virtual work of external surface forces is defined as follows: where ( ) ( ) where t 1 , t 2 are some time values. We introduce (5, 6, 7) into equation (8) and obtain the following vibrational equations of the ith CNT wall: We introduce relations (4) into (9, 10, 11) and obtain the following system of vibrational equations of the ith CNT wall where are linear differential operators, which have the following form: The CNT under consideration consists of the N walls that are interconnected by Van der Waals forces. Following [18], the projections of the van der Waals forces, q i , acting on the ith CNT wall are determined as follows: , ε is the depth of the potential; σ is the parameter that determines the equilibrium; a is the C-C bond length; m is a positive integer. Now we write a PDE system describing the geometrically nonlinear deformation of MWNTs. Equations of motion (12)(13)(14) should be applied to each of the embedded CNT walls. Then we write the equations of motion for a MWCNT as We emphasize that the connection between the CNT walls during vibrations is realized through Van der Waals forces (15).
We write vibrational equations (16) with respect to dimensionless variables and parameters Relations (4) in dimensionless form will take the following form: Dynamical system (16) in dimensionless form looks like this: We represent these three PDEs in the following operator form: The dimensionless change in the curvature and torsion of the middle surface is defined as follows: The dimensionless Van der Waals forces acting on CNT walls take the following form: and natural boundary conditions

Equations of Motion with a Finite Number of Degrees of Freedom
To study vibrations, the Galerkin method is used [19]. Then it is necessary to satisfy both geometric (20) and natural boundary conditions (21). Taking into account relations (1), it is not possible to satisfy natural boundary conditions (21). Therefore, the generalized Galerkin method [19] is used, which is often called the method of weighted residuals [20]. We choose the expansions of displacements , satisfying only geometric boundary conditions (20) where is the vector of generalized coordinates of dimension N. In expansion (22), conjugate forms of vibrations must be present. If they are not taken into account in (22) and if such relations are introduced into PDEs (17-19), then conjugate forms will necessarily arise due to the presence of terms with ( ) i Y 13 and ( ) i Y 23 in Hooke's law (1). In the variation of potential energy (5), taken into account are natural boundary conditions, which are represented by the second integral. These relations are used in the generalized Galerkin method [19]. We introduce relations (5, 6, 7) into (8). In the resulting equation, we turn to dimensionless variables and parameters. Then we get the following relation: Then, from relations (24), we obtain a linear dynamic system with a finite number of degrees of freedom, which has the following matrix form: where M is the mass matrix; Kq is the product of the stiffness matrix and the vector of generalized coordinates, which describes the linear terms in equations (24). We emphasize that these terms are obtained by taking double integrals in (24). The linear terms K B q are obtained by taking the single integrals included in (24). From system (25), an eigenvalue problem is easily derived for calculating the vibrational eigenfrequencies and modes. System (25) describes the linear vibrations of CNTs with an arbitrary number of walls.

Numerical Vibrational Analysis of Single-walled СNTs
To verify the theory presented above, a numerical simulation of the eigenfrequencies of the linear vibrations of an isotropic CNT was carried out. A single-walled (SW) CNT with parameters [13] was considered: Eh=360 N/m, D=0.85 eV, ν=0.2, ρh=0.7718·10 -6 kg/m 2 , R 1 =0.65 nm, where D is the cylindrical stiffness; Eh is the longitudinal stiffness; ν is Poisson's ratio; ρh is the mass per unit length. To calculate vibrational eigenfrequencies, the problem of eigenvalues, which is derived from system (25), is solved.
The results of calculating vibrational eigenfrequencies are given in Table 1. The first column of the table shows the quantity L/R 1 . The calculations were carried out for different CNT lengths. The second column presents the first eigenfrequency published in [13]. In this article, the vibrations of CNT continuum shell models were studied using the Sanders-Koiter theory. The third column presents the first eigenfrequency in Hz, obtained on the basis of the approach considered in Section 3. Thus, the first eigenfrequencies obtained by the two methods are close. Eigenfrequencies of a simply-supported anisotropic SWCNT are studied. This SWCNT is described by the chiral indices, (n, m)= (15,15). The SWCNT parameters were taken as follows: K ρ =742 N/m, K θ =1.42 nN·nm, r 0 =0.142 nm, ρh=0.7718·10 -6 kg/m 2 , (26) where K ρ , K θ are constant forces associated with the stretching of the carbon-carbon bond and the angular curvature of bonds; r 0 is the carbon-carbon bond length. The parameters of Hooke's law (1) ( ) 1 ij Y were calculated using the method proposed in [16,17].
The results of calculating the eigenfrequencies of a simply-supported anisotropic SWCNT are presented in Fig. 3. Here, the eigenfrequencies for n=1 and n=2 expansions (22) are shown depending on the parameter πR 1 /L. The solid line represents the eigenfrequencies obtained using the approach presented in Section 3. The diamonds in Fig. 3 show the eigenfrequencies published in [14]. The proximity of the eigenfrequencies obtained by the two different methods is the evidence that the eigenfrequencies calculated by us are correct. As the parameter πR 1 /L increases, the eigenfrequencies increase.
We study the dependence of the vibrational eigenfrequencies of a simply-supported anisotropic SWCNT with parameters (26) on the number of waves in the circumferential direction of n expansion (22). The results of this analysis are shown in Fig. 4. It shows the dependence of the eigenfrequencies ω i on the number of waves in the circumferential direction. The calculations are presented for three parameter values πR 1 /L: 1, 2.5; 4. For the parameter values πR 1 /L=1; 2.5; 4, the minimum eigenfrequencies are observed at n=3; n=3; n=4, respectively.
As follows from [13], in isotropic CNT shell models, the minimum eigenfrequencies are observed at n=1 or n=2. In the shell considered here, the minimum eigenfrequencies are observed at somewhat larger n.
The results of calculating the eigenfrequencies of linear oscillations on the basis of an anisotropic model are shown in the second column of Table 2. The first two columns of the table show the number of waves in the circumferential direction n and the number m g of the only nonzero harmonic in expansion (22), respectively. The results of calculating the vibrational eigenfrequencies of this isotropic TWCNT (see the third column of the table) were published in [21,22]. So, the results of the calculations carried out by different methods are close.
The matrix Y of Hooke's law (1) is calculated on the basis of the molecular approach presented in [16,17]. For each CNT wall, the matrix Y has its own values. To calculate the eigenfrequencies, the approach proposed in Section 3 is used. The results of calculating the eigenfrequencies are presented in Table 3. The first column of the table shows the number of waves in the circumferential direction n in expansion (22). The second column shows the numbers of standing waves in the longitudinal direction m, which make a significant contribution to the decomposition of vibrational eigenmodes (22). The third column shows the first three eigenfrequencies obtained with the following parameters of expansion (22): To analyze the convergence of the obtained eigenfrequencies [12,13], they were calculated for a larger number of terms in expansion (22). The results of calculating the eigenfrequencies are shown in the fourth column of Table 3 at J 1 =J 2 =J 3 =4. So, the calculation results presented in the third and fourth columns are close, which indicates their convergence.
As follows from the results given in Table 3, only one longitudinal half-wave with the number m g prevails in its vibrational eigenmode (22). So, there is no interaction between the vibrational modes presented in expansion (22). This is due to the fact that the parameters included in Hooke's law (1) As follows from the further numerical analysis, at ( ) ( ) i i Y , Y 23 13 other than zero, there is an interaction between the vibrational modes in expansion (22).
Note that in all the eigenforms presented in Table 3, all three CNT walls vibrate in phase, i.e., they do not move in opposite radial directions.
The results of calculating the first three eigenfrequencies of a TWCNT are presented in Fig. 5. As follows from Fig. 5, a, with an increase in n starting from 1, the first eigenfrequency is constantly growing. The minimum value of the second and third eugenfrequencies is observed at n=2.
Consider the polychiral DWCNT whose electronic structure is discussed in [24]. Its chiral indices are: (9, 6) (15,10). This CNT has the following wall radii: R 1 =0.5119 nm; R 2 =0.8532 nm; R 2 -R 1 =0.34 nm. The matrix Y of Hooke's law (1) was calculated on the basis of the molecular approach that was proposed in [16,17]. The results of calculating the matrix Y for the two walls are presented in Table 4. The remaining CNT parameters were taken as follows: L=9R 2 ; a=0.142 nm; σ=0.3407 nm; ε=2.968 meV.
To calculate the eigenfrequencies, the approach presented in Section 3 was used. An analysis was made of the convergence of eigenfrequencies. Then, in expansions (22), the quantity J 1 =J 2 =J 3 changes. The influence of the quantity J 1 on the values of eigenfrequencies was investigated. The results of this analysis are given in Table 5. The first column of this table shows the number of terms in expansion (22). The second column shows the numbers of waves in the circumferential direction. The third, fourth, and fifth columns show the first three vibrational eigenfrequencies in THz. As follows from this table, at J 1 =J 2 =J 3 =3 and 4 the eigen frequencies coincide, which indicates the convergence of results. Now consider the properties of the eigenfrequencies obtained. The first eigenfrequency ω 1 increases with the number n of waves in the circumferential direction. The minimum value of the second and third eigenfrequencies is observed at n=2. With increasing n, the spectrum of the eigenfrequencies (ω 1 , ω 2 , ω 3 ) becomes tight. At n=1 and n=2 the spectrum of the eigenfrequencies is not tight, whereas at n=3 and n=4 it becomes tighter.
The results of the analysis of the eigenforms that correspond to the eigenfrequencies (Table 5) are systematized in Table 6. The first column of the table shows the number of waves in the circumferential direction n. In the second column, the numbers of vibrational eigenforms, N , are presented. In the third one, the numbers of nonzero vibrational eigenmodes, m g , that are present in expansion (22). With the number of waves in the circumferential direction n=1 and n=2, the vibrational eigenmodes (22) are represented as single-mode expansions. We emphasize that the number of a nonzero mode equals the number of the vibrational eigenmode. At n=3 and n=4 in the expansion of vibrational eigenmodes (22), several vibrational eigenmodes are present, which is shown in Table 6. Such an interaction between the vibrational modes is explained by the nonzero values of the parameters ( ) ( ) i i Y Y 23 13 , in Hooke's law (1). Thus, if all the elements of the matrix Y of Hooke's law (1) are nonzero, then, in expansion (22), there is an interaction between the modes at linear vibrations.

Conclusion
This article provides a model of free CNT vibrations, which is expressed by a system of ordinary differential equations. To derive this dynamical system, we use a PDF system describing the deformation of MWCNTs and the method of weighted residuals. Thanks to the use of this method, the basis  functions over which the vibrations are expanded do not have to satisfy all the boundary conditions of the problem. It is these basis functions that are used in this paper. The mathematical model takes into account the CNT chirality, which is described by the anisotropic elastic constants of the model. Moreover, the nonlocal elasticity and Van der Waals forces between the CNT walls are taken into account in the model obtained.
The vibrational eigenfrequencies of SWCNTs are analyzed depending on the number of waves in the circumferential direction n. With the number of waves in the circumferential direction from 2 to 4, the minimum vibrational eigenfrequencies of CNTs are observed. These numbers are smaller than those for the vibrational eigenfrequencies of engineering shells.