Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

54, 2, pp. 613-619, Warsaw 2016
DOI: 10.15632/jtam-pl.54.2.613

ON THE BUCKLING AND VIBRATIONAL RESPONSE OF CARBON

NANOTUBES WITH SPIRAL DEFORMATION

Sadegh Imani Yengejeh

Department of Solid Mechanics and Design, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia – UTM,

Skudai, Johor, Malaysia; e-mail: imani.sd@gmail.com

Seyedeh Alieh Kazemi

Department of Mechanical Engineering, The University of Birjand, Birjand, Iran; e-mail: sa.kazemi83@gmail.com

Andreas Öchsner

School of Engineering, Griffith University, Gold Coast Campus, Southport, Australia, and

School of Engineering, The University of Newcastle, Callaghan New South Wales, Australia

e-mail: andreas.oechsner@gmail.com

Perfect and spiral models of carbon nanotubes (CNTs) have been simulated based on the
finite elementmethod and their vibrational and buckling behavior has been investigated. In
order to evaluate their natural frequencyand critical buckling load, computational tests have
been conducted. It has been concluded that the existence of any geometricalmodification in
the configuration of perfect CNTs results in a remarkable reduction in the natural frequency
and critical buckling load of CNTs. It has been also revealed that the analytical solutions
are in good agreement with the finite element simulation results in the cases of perfect and
spiral CNTs.

Keywords: finite element method, carbon nanotubes, natural frequency, critical buckling
load, spiral shape

1. Introduction

The industrial application of carbon nanotubes (CNTs) is continuously increasing due to their
outstanding physical properties. Since their discovery (Iijima, 1991), these nanostructures have
attracted worldwide attention. CNTs are unique because of their outstanding mechanical and
physical properties such as strength, lightness and good conductivity (Dai et al., 1996; Saito et
al., 1997; Niu et al., 1997). The investigation of CNTs can be divided into two groups, i.e. expe-
rimental and computational approaches. Molecular dynamics (MD) and continuum mechanics
techniques such as the finite element method (FEM) have been themost popular approaches to
study the mechanical behavior of these nano-materials. In the following, the results of several
studies on vibrational characteristics and buckling behavior of CNTs are presented.

Previous investigations (Arghavan andSingh, 2011) conducted anumerical studyon free and
forced vibrations of single-walled carbonnanotubes (SWCNTs).They applied a simple approach
so that the proximity of themathematical model to the actual atomic structure of theCNTwas
considerably retained. Their results revealed that the appearance of these modes of vibration
in the eigenvectors and eigenvalues were indistinguishable. It was concluded that in the case of
zigzag nanotubes, the axial bending and torsional modes appeared to be decoupledwhereas the
armchair nanotubes showed couplingbetween suchmodes.Then, thevibrational behavior of two-
and three-junctioned CNTswas investigated with different geometries and boundary conditions
(SeyyedFakhrabadi et al., 2012).Theauthors appliedamolecularmechanics approach toanalyze



614 S. Imani Yengejeh et al.

the mentioned CNTs. They finally obtained natural frequencies and their corresponding mode
shapes of two–junctioned CNTs with different diameters and lengths. After that, the natural
frequency of CNTswas investigated (Ghavamian and Öchsner, 2013). They simulated numerous
forms of CNTs from single to 5-walled ones in their perfect form based on the FEM.Then, they
evaluated natural frequencies of the CNTs analytically and through the FEM and compared
their findings. Then, some specific defects were introduced to the perfect models and their
vibrational behavior and the influence of these defects on the vibrational stability of CNTswere
studied. They finally concluded that the existence and development of any type of defects in the
configuration of CNTs reduce the natural frequency and vibrational stability of perfect CNTs.
A study on the impact of vacancy defects on critical buckling loads and strains in CNTs was
conducted under axial compression (Parvaneh et al., 2009). Their results showed that vacancy
defects inCNTs canmost likely bemodeled as cutouts of the shells. They finally compared their
results of the structural model with those from MD simulations in which the outputs were in
good agreement with the presentmodel. The buckling characteristics of several curved forms of
SWCNTswas investigated performingMDsimulations (Wong andVijayaraghavan, 2012). They
concluded that the performance of the CNTunder compression can be changed by an inclusion
of a curvature along the tube axis. The effect of defects on the buckling behavior of CNTs
was investigated by Ghavamian and Öchsner (2012). Their study was based on the FEM. In
detail, theymodeled two basic CNTs in their perfect form.Then the buckling behavior of CNTs
was evaluated by comparing their critical loads obtained from the simulation and analytical
calculations. They concluded that the existence of any curvature in the structure of nanotubes
decreases their buckling strength. The aim of the actual research is to continue the previous
investigations and to study vibrational and buckling behavior of spiral CNTs.

As most of the investigations have been performed on perfect CNTs, it is necessary to
pay more attention to geometrical imperfections, e.g. twists and spirals of CNTs, in order to
realistically examine their mechanical properties (see Fig. 1). Spiral deformations were reported
in (Faria et al., 2013; Liuyue et al., 2013; Faria et al., 2013) where the CNTs revealed a shape
similar to a springwhich is deformedaround its longitudinal axis.Thepurpose of this research is
toderive computationalmodels ofCNTsand investigate their vibrational andbucklingbehavior.

Fig. 1. General view of a spiral CNT

2. Methodology

2.1. Geometric definition

The atomic structure of CNTs can be imagined as a graphene sheet that has been rolled
into a tube. The thickness of the tube wall is generally considered to be 0.34nm, which is very



On the buckling and vibrational response of carbon nanotubes... 615

close to that of a graphene sheet (Li and Chou, 2003; To, 2006). These nano-particles possess a
length of over 10µmand diameters ranging from 0.4 to 3.0nm.
We followed themodelingmethodwhichwas first proposed byLi andChou (2003) where the

theory of classical structural mechanics was extended to themodeling of CNTs. It was assumed
that CNTs, when subjected to loading, behave like space-frame structures. Based on Fig. 2,
the bonds between carbon atoms are considered as connecting load-carrying generalized beam
members, while the carbon atoms act as joints of the members.

Fig. 2. Front view and single ring of a SWCNT as a space-frame structure

The configuration of CNTs in this study has been generated by the CoNTub software (Mel-
chor and Dobado, 2004; Melchor et al., 2011), a computer program for determining the coordi-
nates of CNTs. Then, the finite element analyses have been conducted in order to investigate
the vibrational and buckling behavior of different types of perfect and spiral CNTs.

2.2. Boundary conditions

Vibrational and bucking behavior of perfect and spiral CNTs under cantilevered boundary
conditions are investigated, where for the vibrational behavior one end is fully fixed and the
other end is completely free; and for the buckling behavior one end is fully fixed and the other
end is exposed to a compressive axial load. Different angles for spiral CNTs have been taken
into consideration, as shown in Fig. 3.

3. Results and discussion

In order to evaluate the critical buckling load of CNTs, the numerical finite element approach
allows the introduction of arbitrary compressive point loads to one of theCNT’s ends.The simu-
lation of the vibrational behavior does not require the introduction of any loads. The obtained
numerical finite element results are compared to simple analytical predictions to check if these
simple design equations provide reasonable values.

3.1. Analytical approach to the vibrational behavior

Thenatural frequency is the frequency of a vibrating systematwhich the systemoscillates at
a greater amplitude.This phenomenonoccurs because of the existence of resonance.Thenatural



616 S. Imani Yengejeh et al.

Fig. 3. Spiral CNTwith (a) 2◦, (b) 42◦ and (c) 62◦ twisting angle

frequency ismostly investigated to examine the vibrational response of structuralmembers.The
first natural frequency of a fixed-free helical spring is definedby the following analytical equation
(Renno andMace, 2012)

fn =
1

4

√

k

m
(3.1)

wherem and k are the value of mass per unit of the spring and the spring rate, respectively. In
the case of spiral CNTs, we assumed an average value for the diameter as the whole structure
does not possess a constant diameter along its length. The value of the mass per unit length m
can be obtained from the following equation

m=
mtotal

ltotal
(3.2)

where mtotal is the total mass of the CNT carbon atoms and ltotal is the total length of the
nanotube. Figure 4 illustrates the first five natural frequencies of a spiral CNT obtained from
the numerical finite element simulation.

Fig. 4. First five eigenmodes of a spiral CNT



On the buckling and vibrational response of carbon nanotubes... 617

3.2. Analytical approach to the buckling behavior

The phenomenon of buckling is in its simplest form a particular kind of elastic instability
in a slender configuration that occurs under certain compressive loads. In the basic theory
of elasticity, the critical buckling deflection of a compressive spring is presented by Eq. (3.3)
(Pearson, 1982) as

δcr

Lf
=0.812

[

1±

√

1−6.87
(2Dm
Lf

)2
]

(3.3)

where Lf is the free length or unloaded length of the spring, Dm is the mean diameter of the
spring, and δcr is the critical deflection of the spring.

Figure 5 shows a spiral CNT under the cantilevered boundary condition in its original and
buckled shape obtained from the numerical finite element calculation.

Fig. 5. A spiral CNT’s first mode under buckling load in the original and buckled formwith the
cantilevered boundary condition

It should be indicated that the proposed approach in this paper deals with the evaluation
of mechanical properties, i.e. vibrational and buckling behavior of spiral CNTs, whereby the
CNTs are deformed as a spring around their axis. By manipulating the twisting angle of these
spiral CNTs from 02◦ to 102◦, the behavior of different models with different conditions could
be studied. This study tried to continue and broaden the investigations on only slightly twisted
CNTs around their straight vertical axis.

It could be shown that the computational and analytical values of the natural frequency as
well as critical buckling load of perfect CNTs are reduced by introducing spiral imperfections
to the configuration of these perfect nano-structures, as illustrated in Fig. 6. Comparing all
cases, it is clear that a decrease in the natural frequency and the critical buckling load is
more visible in the case of armchair spiral CNTs. Figure 6a-c shows a decrease in the natural
frequency of armchair, zigzag and chiral CNTs as a result of increasing the twisting angle of
spiral configurations. It is clear that the natural frequency of all CNTs reduces to less than
12GHz at the angle of 102◦. Figure 6d-f illustrates a significant change in the critical buckling
load of armchair, zigzag and chiral models by increasing the twisting angle of spiral CNTs.
Based on the obtained calculations, it is concluded that the finite element values are in good
agreement with analytical results, where themaximum difference for the natural frequency and
critical buckling load is 17 and 23%, respectively.



618 S. Imani Yengejeh et al.

Fig. 6. Change in the natural frequency with different twisting angles for (a) armchair, (b) zigzag and
(c) chiral spiral CNTs. Change in the critical buckling load with different twisting angles for

(d) armchair, (e) zigzag and (f) chiral spiral CNTs

4. Conclusions

In this study, perfect and spiral CNTs (armchair, zigzag and chiral) have been simulated and
their vibrational andbucklingbehaviorhasbeen studied throughperformingcomputational tests
with cantilevered boundarycondition.Both computational and analytical calculations have been
compared in the cases of perfect and spiral CNTs. It has been shown that the finite element
simulation are in good agreement with the analytical solutions in the case of perfect and spiral
CNTs. It has been shown that in all cases, the analytical values are slightly smaller than the
computational results. It has been also concluded that the existence of any spiral imperfection
in the configuration of perfect CNTs results in a significant reduction in the natural frequency
and critical buckling load of these nano-configurations.

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Manuscript received September 29, 2014; accepted for print October 16, 2015