In this work, a recently proposed nonlocal theory of bending is used in the analysis of eigenfrequencies of single-walled carbon nanotubes (SWCNTs). The nanotube vibration is analyzed in the form of a homogenized continuum. Classical treatment where a nanotube is approximated by standard beam theory, is replaced by the more sophisticated nonlocal method of material interactions where a nonlocal parameter is used. The eigenfrequencies are computed by the combination of analytical as well as numerical methods for four different carbon nanotube (CNT) supports. Various types of supports are considered for the analysis: fixed–simply supported, fixed–free, simply–simply supported and fixed–fixed. Due to the huge amount of computed data, only outcomes of eigenfrequency computations for the nanobeams of armchair type with fixed and simply supported ends, and different nonlocal parameters are represented in the form of graphs at the end of the article. The study shows how the nanotube eigenfrequencies depend on nonlocal parameters as well as on the length and diameter of CNTs. The obtained results are in good agreement with the results published in papers which were gained by different procedures.

Functional materials and nanomaterials are attracting a lot of attention in the areas of modern engineering. The internal structure of such materials is relatively simple but the determination of their properties is often associated with difficulties. In many cases, the theoretical treatments with indirect measurements of conjugated behavior lead to the solution of a stated problem. In our case, we will deal with the determination of eigenfrequencies of carbon nanotubes. The eigenfrequencies are very important parameters of these structures because their measurement can serve as a base for the determination of elastic properties, e.g., Young's modulus of homogenized materials.

The massive investigation of carbon nanotubes started after publishing well-known article (Iijima, 1991). CNTs are large macromolecules composed exclusively of carbon atoms. Nanotubes can be formally obtained by rolling up a single-walled plane sheet of graphite, called graphene, into a cylindrical shape. Nanotubes have remarkable mechanical properties. The most important task in the design of a machine is to find an adequate material for a particular structure. In principle, the aim is to use strength and lightweight material. Nowadays, in mechanical engineering, the carbon nanotubes are candidates that fulfill such demands as parts of composite materials (Wu and Chou, 2012; Yayli, 2014). Besides of mechanical properties, the scientists focus their research on electrical, thermal and optical properties of CNTs. Many applications of CNTs are oriented to the area of biology and medicine.

Depending on the scale, the simulation of SWCNTs is realized by several methods: atomistic scale simulation, simulation by molecular dynamic and the macroscale continuum treatment. In this paper, the third method is applied to describe the bending eigenfrequencies of CNTs. As the elements of nanostructure have special properties, the homogenized continuum of CNTs has to reflect such specific behavior. The standard theory of elasticity is built on the principle of local action, i.e. the response at a point depends on actions in its neighborhood. However, this is not the case of nanostructures. Here, nonlocal influences can occur and therefore, the nonlocal theory has to be used (Eringen, 2002). According to the underlying theory, strain in every point of a body influences a stress level at the investigated point.

Different applications of the nonlocal theory can be found in many research papers. A review on the application of the nonlocal continuum theory for static and dynamic loadings of carbon nanotubes and graphene sheets is presented in the paper of Arash and Wang (2012). The nonlocal elastic beam model, the elastic shell model and the elastic plate model is established for modeling carbon nanotubes and graphene sheets. The different loading states as bending under transverse loading of CNTs, buckling analysis of axial loaded CNTs and free vibration of CNTs are described. Using the nonlocal continuum mechanics in nonlinear stability analysis of graphene sheets is presented in paper (Asemi et al., 2014). The Galerkin method with the nonlocal parameter for the analysis of simply supported orthotropic graphene sheets is used. The paper reports that the nonlocal parameter has significant effect on the postbuckling behavior of graphene sheets. Kirchhoff's plate theory with the nonlocal parameter is used for eigenfrequency investigation of nanoplates with elastic (Winkler–Pasternak) boundary conditions in the research of Chakraverty and Behera (2015). The effect of the aspect ratio, the elastic boundary conditions and the nonlocal parameter is presented. A forced vibration of a single- and double-walled carbon nanotube under excitation of a moving harmonic load has been analyzed using nonlocal elasticity theory by Rahmani et al. (2017, 2018).

The vibrations of tensioned nanobeams using the nonlocal beam theory are investigated by Bagdatli (2015). The nonlinear frequencies for fixed–fixed and simply–simply supported Euler–Bernoulli nanobeams are presented. The vibration behavior of the nanobeams with small scale effects represented by the nonlocal parameter is studied in the research of Lim et al. (2010). The forced vibration of SWCNTs is investigated by the nonlocal theory in Şimşek (2010). The SWCNTs as nonlocal Euler–Bernoulli beams are modeled and the effects of aspect ratio and nonlocal parameter on vibration behavior of SWCNTs are discussed. The vibrational behavior of CNTs using wave propagation approach is studied by Hussain et al. (2017), Hussain and Nawaz (2017). Other applications of the nonlocal theory and the vibration behavior of carbon nanotubes with different boundary conditions can be found in papers (Narendar and Gopalakrishnan, 2012; Şimşek, 2011; Thongyothee et al., 2013; Wang et al., 2015; Yang et al., 2010). The properties of single-walled carbon nanotubes are well described in papers (Arash and Wang, 2012; Fu et al., 2012; Gupta and Batra, 2008; Harik, 2002; Karličić et al., 2015; Kumar and Srivastava, 2016; Lee and Chang, 2012).

In this paper, the bending frequencies of single-walled carbon nanotubes based on nonlocal stress theory are investigated. The carbon nanotubes are represented by the Euler–Bernoulli nanobeams with nonlocal parameter. The authors previously published bending eigenfrequencies for different types of boundary conditions and chirality (and corresponding diameters) in papers Bocko and Lengvarský (2014a, b, c). Novelty of paper lies in investigation of chirality effects to eigenfrequencies, comparision of semianalytical methods with the finite element method (FEM) for different nonlocal parameters for boundary condition of type fixed–free. Further, the eigenfrequencies are computed for four different supports (fixed–simply supported, fixed–free, simply–simply supported and fixed–fixed) of carbon nanotubes, the semianalytical results for fixed–simply supported armchair CNTs with different lengths and five nonlocal parameters are presented in graphical form at the end of the article. Finally, the comparison of the results with results from Imani Yengejeh et al. (2014), Lü et al. (2007), Zhang et al. (2009) is done.

The scheme of lattice structures

The elasticity theory of continuum media is
based on several fundamental assumptions: the principle of objectivity,
determinism, material symmetry, equipresence, causality, local action, etc.
The principle of local action represents the fact that stress at a material
point is related exclusively to the deformations in its immediate
surroundings. On the other hand, in the nonlocal continuum theory, this
assumption is abandoned, and the stress tensor at the investigated part of
continua is influenced by the movement of all points of continua (Eringen,
2002). The equation for nonlocal stress is written as:

Rolling up of a graphene sheet (Fig. 1a) can be accomplished in many ways and
in principle it can be described by a chiral vector

Chirality influences the response of carbon nanotubes to external
disturbances. Beside of chirality we will use another important
characteristic parameter of the carbon nanotube, its length

In this paper we have focused our attention on the CNTs with the following types of boundary conditions (Fig. 2): fixed–simply supported (F–S), fixed–free (F–Fr), simply–simply supported (S–S) and fixed–fixed (F–F).

The applied boundary conditions on nanobeams.

The application of boundary conditions to the differential equation results
in the following characteristic equations:

Because the nanotube is modeled as a continuum by homogenization method, the
characteristic equations are transcendental ones. The roots ^{®}. The first fourth values

The second moment of area

Solutions of frequency equations for different boundary conditions.

In this chapter, the nonlocal finite element formulation is presented. The mass and stiffness matrices of finite element are derived from Eq. (5).

The element stiffness matrix can be defined as:

Eigenfrequencies of carbon nanotubes with different chiral angles and boundary conditions.

The first eigenfrequency of carbon nanotubes for both methods with different nonlocal parameter.

The first eigenfrequencies of armchair nanotubes with the nonlocal
parameter

The first eigenfrequencies of armchair nanotubes with the nonlocal
parameter

The first eigenfrequencies of armchair nanotubes with length

The first four eigenfrequencies of fixed–simply supported armchair
nanotubes with nonlocal parameter

At first, the effect of chirality on the eigenfrequencies of CNTs is
investigated. The chirality angle varies from 0 to 30

The results for the tube chirality (5,5), length 10 nm and boundary condition of type F–Fr computed by the semianalytical as well as FEM are given in Table 3. The 100 elements are used for the FEM. It is clear from the results that both methods give almost the same results.

The semianalytical computations for armchair single-walled carbon nanotubes
with F–S, F–Fr, S–S and F–F boundary conditions were accomplished in
commercial program system MATLAB^{®}. The
chirality of armchair carbon nanotubes was changed from (4, 4) to (20, 20)
which corresponds to the range of diameters from

The first four eigenfrequencies of fixed–simply supported armchair
nanotubes with nonlocal parameter

The first four eigenfrequencies of fixed–simply supported armchair
nanotubes with nonlocal parameter

The first four eigenfrequencies of fixed–simply supported armchair
nanotubes with nonlocal parameter

The first four eigenfrequencies of fixed–simply supported armchair
nanotubes with nonlocal parameter

The graphs conclude that increasing diameter of the carbon nanotube leads to higher first eigenfrequency and the eigenfrequencies of ten times longer carbon nanotubes are hundred times lower. Comparison of Figs. 3 and 4 shows that ten times higher value of nonlocal parameter leads to approximately three times lower eigenfrequencies. The value of nonlocal parameter affects more the eigenfrequencies of the carbon nanotubes with a smaller diameter.

As there is a huge number of possible combinations of parameters, we have
focused our attention on the first eigenfrequencies for the diameters

Due to complexity, the effect of the nonlocal parameter on only first four
eigenfrequencies of armchair CNTs was studied. The graphs in Figs. 5–9 show
relations between eigenfrequencies and length

The graphs show that the eigenfrequencies decrease with increasing length and nonlocal parameter. The eigenfrequencies for the nonlocal parameter equal 0.01 are approximately three times lower than for the nonlocal parameter 0.1.

Because different authors use different parameters for nanotube description in the literature (density, Young's modulus, nominal thickness of nanotube, boundary conditions), we compared our results only with those where the authors used similar inputs. As a number of such inputs was published for the boundary conditions of type fixed–free, Table 5 gives the comparison of the first eigenfrequencies computed by different authors and methods with the result of method in the article. The biggest difference in this comparison does not exceed 15 %.

The comparison of the first eigenfrequencies with published results.

We discussed the analytical computation of eigenfrequencies of SWCNTs based
on nonlocal beam theory and the numerical results are given for armchair
SWCNTs. In the nonlocal theory, the response of a structure at the point of a
question is influenced by all particles of the structure. The small scale
length effect of the CNTs was represented by the nonlocal parameter in the
nonlocal beam theory. The transcendental frequency equations resulting from
four different boundary conditions have been solved by computer system
MATLAB^{®} and parametric studies of results for
the fixed–simply supported beam are given in the graphic form. Five
different values of nonlocal parameter were used and its influence on the
computed eigenfrequencies is shown.

It can be stated that:

The effect of the chirality of SWCNTs can be neglected.

The nanotube eigenfrequencies depend on nonlocal parameters, as well as on the length and diameter of CNTs.

Increase in the diameter of a carbon nanotube leads to higher eigenfrequencies and the eigenfrequencies of ten times longer carbon nanotubes are hundred times lower.

Higher nonlocal parameter leads to the smaller eigenfrequencies of CNTs. Ten times higher value of nonlocal parameter leads to approximately three times smaller eigenfrequencies.

The eigenfrequencies of shorter and, in general, smaller carbon nanotubes are more affected by the value of nonlocal parameter.

The presented results are in good agreement with the results published in other papers.

The experimentally measured eigenfrequencies can be used for the determination of Young's modulus of homogenized single-walled carbon nanotubes.

In the future work, the finite element method for modeling carbon nanotubes and the effects of boundary conditions, nonlocal parameters and vacancies on mechanical parameters will be investigated.

Data can be made available upon reasonable request. Please contact Pavol Lengvarský (pavol.lengvarsky@tuke.sk).

BJ formulated theoretical description of work and prepared analytical formulations; LP performed semianalytical computations; HR accomplished FEM computations; ŠJ wrote the paper with contributions from all co-authors.

The authors declare that they have no conflict of interest.

The authors gratefully acknowledge the support given by the Slovak Grant Agency VEGA under the grant no. 1/0731/16 Development of Modern Numerical and Experimental Methods of Mechanical System Analysis. no. 1/0355/18 The use of experimental methods of mechanics for refinement and verification of numerical models of mechanical systems with a focus on composite materials and ITMS: 26220120060 supported by the Research and Development Operational Programme funded by the ERDF. Edited by: Anders Eriksson Reviewed by: two anonymous referees