Research ArticleNumerical Simulation of the Vibration Behavior ofCurved Carbon Nanotubes

Sadegh Imani Yengejeh1 Joatildeo M P Q Delgado2

Antonio G Barbosa de Lima3 and Andreas Oumlchsner45

1 Department of Solid Mechanics and Design Faculty of Mechanical Engineering Universiti Teknologi Malaysia (UTM)81310 Skudai Johor Malaysia

2 Laboratory of Building Physics (LFC) Civil Engineering Department Faculty of Engineering University of Porto 4200-465 PortoPortugal

3 Department of Mechanical Engineering Federal University of Campina Grande 58429-900 Campina Grande PB Brazil Brazil4 School of Engineering Griffith University Gold Coast Campus Southport QLS 4222 Australia5 School of Engineering The University of Newcastle Callaghan NSW 2308 Australia

Correspondence should be addressed to Sadegh Imani Yengejeh imanisdgmailcom

Received 20 February 2014 Accepted 22 April 2014 Published 29 May 2014

Academic Editor Bin Li

Copyright copy 2014 Sadegh Imani Yengejeh et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Several zigzag and armchair single-walled carbon nanotubes (CNTs) were modeled by a commercial finite element package andtheir vibrational behavior was studied Numerous computational tests with different boundary conditions and different bendingangles were performed Both computational and analytical results were compared It was shown that the computational resultsare in good agreement with the analytical calculations in the case of straight tubes In addition it was concluded that the naturalfrequency of straight armchair and zigzag CNTs increases by increasing the chiral number of both armchair and zigzag CNTs Itwas also revealed that the natural frequency of CNTs with higher chirality decreases by introducing bending angles Neverthelessthe influence of increasing bending angle on the natural frequency of armchair and zigzag CNTs with lower chiral number is almostnegligible

1 Introduction

Since the discovery of carbon nanotubes (CNTs) by Iijima in1991 [1] these nanostructures have been the focus of manyinvestigations CNTs are important to industrial applicationsbecause of their outstanding mechanical and physical prop-erties such as strength and lightness The investigations onCNTs can be divided into two groups of experimental andcomputational approaches Molecular dynamics (MD) andcontinuum mechanics approach such as the finite elementmethod (FEM) have been the most popular approaches tostudy the behavior for example the vibrational behavior ofthese nanoparticles [2] In the following the results of severalstudies on the evaluation of CNTs vibrational properties arepresented

In 2009 Georgantzinos et al [3] proposed a linear spring-based element formulation for the computation of vibrationalcharacteristics of single-walled carbon nanotubes (SWC-NTs)They developed three-dimensional nanoscale elementsand corresponding elemental equations for the numericaltreatment of the dynamic behavior that is appropriate stiff-ness and mass characteristics of SWCNTs They also assem-bled the elemental equations and constructed the dynamicequilibrium equation applying the atomic microstructure ofCNTs The developed elements in their investigations sim-ulated the relative translations and relation between atomsConsequently they could apply the molecular mechanicstheory directly due to the modeling of atomic bonds apply-ing physical variables such as bond stretching Based ontheir results new natural frequencies and mode shapes for

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2014 Article ID 815340 9 pageshttpdxdoiorg1011552014815340

2 Advances in Materials Science and Engineering

numerous CNTs under different boundary conditions werefoundThey also showed that their results indicated very goodagreement compared to numerical predictions from the liter-ature In 2010 Xia and Wang [4] investigated the vibrationcharacteristics of fluid-conveying carbon nanotubes with acurved longitudinal structure They could obtain the naturalfrequencies of curved carbon nanotubes conveying fluid andcompared them with those of straight CNTs Their findingsrevealed that CNTs with circular curved longitudinal shapeare unconditionally stable even for a system with sufficientlyhigh flow velocity In 2011 Arghavan and Singh [5] presenteda detailed numerical study on the free and forced vibration ofstraight SWCNTs They analyzed both armchair and zigzagCNTs with clamped-clamped and clamped-free boundaryconditions in order to find their natural frequencies andcorresponding mode shapes Their results indicated theappearance of thesemodes of vibration in the eigenvalues andeigenvectors without any distinction They finally concludedthat in the case of zigzag CNTs the axial modes appearedto be decoupled whereas the armchair nanotubes showcoupling between such modes Although they thoroughlyinvestigated the behavior of straight CNTs the study onthe behavior of curved nanotubes is only rarely addressedGhavamian and Ochsner [6] investigated the eigenvalues andeigenfrequencies of CNTs under the influence of defects in2013 After simulating armchair and zigzag configurationsof zigzag and armchair CNTs based on the FEM theyintroduced three most likely defects to the models in orderto represent defective forms of SWCNTs These defects arecarbon vacancy Si-doping and perturbation Finally theyexamined and also compared the vibrational behavior ofperfect and defective CNTs According to their findingsSWCNTs have a natural frequency in the range of 1869 to2401 GHz In addition it was pointed out that the existence ofany type of defects leads to a lower value of natural frequencyand vibrational stability The aim of this study is to continuethe previous investigations and to focus on the evaluationand comparison of vibrational behavior of curved CNTs withdifferent bending angles Since experimental observations(eg based on electronmicroscopes) show that CNTs are notusually straight but rather have certain degree of curvatureor waviness along the nanotubes length [7 8] we focus in thefollowing on the investigation of the effect of the curvatureon the dynamic behavior All our simulationswere performedwith the commercial finite element code MSCMARC (MSCSoftware Corporation Santa Ana CA USA)

2 Methodology

21 Geometric Definition CNTs are assumed to be some kindof hollow cylinder shape configurations as shown in Figure 1These special nanostructures can be imagined by rolling agraphene sheet into a cylinder They possess a length over10 120583m and diameters ranging from 1 to 50 nm The geometryof aCNT is defined by the chiral vector ℎ and the chiral angle120579 The chiral vector is presented by two unit vectors 1198861 and 1198862

and two integers 119898 and 119899 as it is presented by the followingequation [9]

ℎ = 119899 1198861 + 119898 1198862 (1)

The construction of CNTs is defined based on the chiralvector or angle by which the sheet is rolled into a cylinderin three different configurations including chiral zigzag andarmchair In the case of (120579 = 0∘) or (119898 = 0) the zigzag CNT isconstructed An armchair CNT is obtained in terms of chiralvector (119898 = 119899) or in terms of chiral angle (120579 = 30∘) andfinally a chiral CNT is shaped if 0∘ lt 120579 lt 30∘ or119898 = 119899 = 0 [9]

Based on the following equation the diameter of the CNTcan be calculated

119889CNT =1198860radic1198982 + 119898119899 + 1198992

120587 (2)

where 1198860 = radic3119887 and 119887 = 0142 nm is the length of the CndashCbond [9]

The modeling method presented in this study followsthe idea first suggested in [10] where the theory of classicalstructural mechanics was extended into the modeling ofCNTs In a CNT carbon atoms are bonded together bycovalent bonds which have their characteristic lengths andangles in a three-dimensional space Afterwards it wassuggested that CNTs when subjected to loading act as space-frame structuresTherefore the bonds between carbon atomsare considered as connecting load-carrying generalized beammembers while the carbon atoms behave as joints of themembers This idea is schematically shown in Figure 1

In this study the configurations of both armchair andzigzag SWCNTs were simulated by the CoNTub software[11] Defining the chirality and the length of the tubes thespatial coordinates of the C-atoms and the correspondingconnectivities (ie the primary bonds between two nearest-neighboring atoms) were calculated Then the gathered datawas transferred to a commercial finite element packagewhere CndashC bonds were modeled as circular beam elements[12] Afterwards the FE analyses were conducted and vibra-tional behavior of different armchair and zigzag CNTs wasevaluated

The natural frequency is the frequency of a vibratingsystem at which the system oscillates at greater amplitudedue to existence of the resonance phenomenon Geometrymass and applied boundary conditions are the factors whichinfluence these quantities The natural frequency is mainlyevaluated to examine the vibrational behavior of structuralmembers The first natural frequency of an Euler-Bernoullibeam element under different boundary condition is definedby the following equation [13]

119891 = (119860

2120587)radic119864119868

1198981198714 (3)

where 119864 119868 119898 and 119871 are the Youngrsquos modulus the secondmoment of area the mass per unit length and the length ofthe nanotube respectively The 119860 value for fixed-free fixed-fixed and free-free boundary condition is 35156 22373 and224 respectively

Advances in Materials Science and Engineering 3

Modeling

Approach

FE beam elementCNT

Carbon atom

CndashC bond

FE node

Figure 1 Side view of the (11 11) CNT as a space-frame structure

(a) (b)

y x

z

(c)

Figure 2 (16 0) zigzag CNT with (a) fixed-free (b) fixed-fixed and (c) free-free boundary conditions

22 Material Parameters and Boundary Conditions In thisstudy the vibrational behavior of numerous types of armchairand zigzag SWCNTs under fixed-free fixed-fixed and free-free boundary conditions is investigated in which one end iseither free or fully fixed while the other end is fixed in the firsttwo cases and for the third case both ends are completely freeas illustrated in Figure 2

Figure 3 illustrates an armchair CNT under fixed-freeboundary condition The models of CNTs were simulatedfrom straight CNT to curved CNT with 45∘ bending angleas shown in Figure 3

For defining the properties of the equivalent beam ele-ments for the CNT bonds no such classical examinationsor geometric derivations are available As a result the samevalues for the equivalent beam elements are assumed as inthe approach proposed in [10 15 16] These effective materialand geometrical properties were found in the mentionedreferences based on a molecular mechanics method Theirmotions are regulated by a force field which is generated byelectron-nucleus interactions and nucleus-nucleus interac-tions and frequently expressed in the form of steric potentialenergy This steric potential energy is in general the sumof contributions from bond stretch interaction bond anglebending dihedral angle torsion and improper (out of plane)torsion as illustrated in Figure 4

The introduced constants and the element properties ofCNTs (armchair and zigzag) are listed in Table 1

3 Results and Discussion

The vibrational behavior of different armchair and zigzagSWCNTs was investigated for different curvatures from 0∘to 45∘ bending angle (fixed-free fixed-fixed and free-freeboundary condition) After evaluating their natural frequen-cies from FEM the results were compared to analyticalcalculations given by (3) in the case of straight configurationsin order to have some kind of validation of the FE approachFigure 5 illustrates the first mode of vibration of a (16-0)armchair CNT under different boundary conditions

The relative difference between the analytical solution andFEM results of straight CNTs was defined by the followingequation

Relative difference in

=

10038161003816100381610038161003816100381610038161003816

FEM result minus analytical solutionanalytical solution

10038161003816100381610038161003816100381610038161003816times 100

(4)

This difference is listed in Table 2 for the case of thefixed-free boundary condition Based on the results itcan be concluded that the computational solutions are in

4 Advances in Materials Science and Engineering

y x

z

0∘ 5∘ 10∘ 15∘ 20∘ 25∘ 30∘ 35∘ 40∘ 45∘

Figure 3 (7-7) armchair CNTs with fixed-free boundary condition and different forms from 0∘ to 45∘ bending angle

ro Δr

Bond stretching(covalent)

Bond bending(covalent)

Bond torsion(covalent)

Pure tension of beamPure tension of beamPure tension of beam

120579o Δ120579

Δ0

TTMMPP

ΔllΔ120573

Figure 4 Equivalence of molecular mechanics and structural mechanics for covalent and noncovalent interactions between carbon atomsMolecular mechanics model (up) and structural mechanics model (down)

Table 1 Material and geometric properties of CndashC covalent bonds [14]

Corresponding force field constants

119896119903 = 65197 nNnm

119896120579 = 08758 nNnmrad2

119896120593 = 02780 nNnmrad2

119864 = Youngrsquos modulus =1198961199032119887

4120587119896120579

5484 times 10minus6Nnm2

119866 = shear modulus = 1198642(120592 + 1)

2159 times 10minus6Nnm2

119877119887 = bond radius = 2radic119896120579

119896119903

00733 nm

119868119909119909= 119868119910119910= second moments of area =

1205871198771198874

422661 times 10

minus5 nm4

Advances in Materials Science and Engineering 5

y

z

x

(a)

xy

z

(b)

y

z

x

(c)

Figure 5 First eigenmode of a (16-0) straight CNT under (a) fixed-free (b) fixed-fixed and (c) free-free boundary conditions

Table 2 Characteristics of simulated straight CNTs

CNT type Chirality (nm) Length (nm) Diameter (nm) Youngrsquos modulus (TPa)Natural frequency (GHz) Relative

difference in ()Analyticalsolution FEM result

Armchair (3 3) 1500 0407 1039 1001 847 1530Armchair (5 5) 1500 0678 1039 1430 1390 298Armchair (6 6) 1500 0814 1039 1659 1663 024Armchair (7 7) 1500 0949 1039 1901 1930 150Armchair (9 9) 1500 1220 1039 2391 2467 323Armchair (10 10) 1500 1356 1039 2637 2734 370Armchair (11 11) 1500 1492 1039 2885 2998 390Armchair (14 14) 1500 1898 1039 3637 3780 392Armchair (16 16) 1500 2171 1039 4142 4289 356Zigzag (8 0) 1505 0626 1021 1326 1256 527Zigzag (9 0) 1505 0705 1025 1459 1417 285Zigzag (10 0) 1505 0783 1028 1594 1576 111Zigzag (11 0) 1505 0861 1031 1731 1735 024Zigzag (12 0) 1505 0939 1032 1869 1894 135Zigzag (16 0) 1505 1253 1037 2433 2520 357Zigzag (20 0) 1505 1566 1039 3006 3136 431

good agreement with the analytical calculations for thecantilevered boundary condition

Each mode shape shows a significant natural frequencyand eigenmode based on the boundary condition of theCNTs For instance the first four eigenmodes of a (7 7)armchair CNT are shown in Figure 6

Having a closer look on the results it was revealed thatthe natural frequency of straight armchair and zigzag CNTsincreases by increasing the chiral number of both armchairand zigzag CNTs as illustrated in Figure 7

It is also obvious that the natural frequency of botharmchair and zigzag CNTs varied in a high value rangeComparison with other computational approaches based ona linear spring-based element formulation for straight CNTs[3] indicates a similar range of the frequencies

It is clear that the existence of any geometrical imperfec-tion bending angle in particular in the structure of armchairand zigzag straight CNTs results in a reduction in the naturalfrequency by increasing the bending angle as illustrated inFigure 8 The natural frequency decreases for all cases of

6 Advances in Materials Science and Engineering

y x

z

(a)

xy

z

(b)

xy

z

(c)

xy

z

(d)

Figure 6 First four eigenmodes of a (7-7) straight CNT

0050

100150200250300350400450500

(3-3) (16-16)(14-14)(11-11)(10-10)(9-9)(7-7)(5-5)

Firs

t nat

ural

freq

uenc

y (G

Hz)

Straight armchair CNT

Computational valueAnalytical value

(a)

(8-0) (20-0)(16-0)(12-0)(11-0)(10-0)(9-0)Straight zigzag CNT

Computational valueAnalytical value

00

50

100

150

200

250

300

350

Firs

t nat

ural

freq

uenc

y (G

Hz)

(b)

Figure 7 (a) Change in natural frequency for straight armchair CNTs (b) Change in natural frequency for straight zigzag CNTsThemarkersrepresent the results of the computation the straight lines are linear regressions

boundary conditions However it should be noted that thistrend is more obvious for the CNTs with higher chiralityas the decrease of natural frequency for both armchair andzigzag CNTs with lower chirality is almost negligible It isalso indicated that the gradual decrease for the fixed-fixedand free-free case of boundary condition is more significantcompared to the fixed-free case at higher curvatures Inaddition this trend is more obvious for the armchair CNTs

Since higher natural frequencies may be important insome applications such as mass sensing at nanoscale [17] weexemplarily evaluated the second and third natural frequencyand the corresponding mode shapes for the case of threearmchair nanotubes see Figure 9

Comparison with Figure 8(a) shows that the secondnatural frequency is practically the same as the first one

However the third natural frequency is approximately fivetimes higher than the first natural frequency Furthermorethe trend line for the higher frequencies is similar to theone from the first natural frequency An example of thecorresponding mode shapes is presented in Figure 10

4 Conclusions

In this study numerous CNTs (zigzag and armchair) weresimulated by an FE approach and their vibrational behaviorwas examined through performing several computationaltestswith different boundary conditions and variable bendingangles Towards achieving the most accurate results thevibrational behaviors of these CNTs were evaluated ana-lytically and computationally and were compared together

Advances in Materials Science and Engineering 7

350

360

370

380

390

400

410

0 10 20 30 40

Firs

t nat

ural

freq

uenc

y (G

Hz)

(14-14)(15-15)

(16-16)

Fixed-free

Bending angle of armchair CNTs (∘)

(a)

Fixed-free

Bending angle of zigzag CNTs (∘)

(16-0)(20-0)

(26-0)

240

260

280

300

320

340

360

380

400

0 10 20 30 40

Firs

t nat

ural

freq

uenc

y (G

Hz)

(b)

Fixed-fixed

(14-14)(15-15)

(16-16)

Bending angle of armchair CNTs (∘)

0185

0190

0195

0200

0205

0210

0215

0220

0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz)

(c)

(16-0)(20-0)

(26-0)

Fixed-fixed

Bending angle of zigzag CNTs (∘)

014

015

016

017

018

019

020

021

022

023

0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz)

(d)

(14-14)(15-15)

(16-16)

Bending angle of armchair CNTs (∘)

0190

0195

0200

0205

0210

0215

0220

0225

0230

0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz) Free-free

(e)

Bending angle of zigzag CNTs (∘)0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz) Free-free

(16-0)(20-0)

(26-0)

014

015

016

017

018

019

020

021

022

023

(f)

Figure 8 Change in natural frequency with (a) armchair and (b) zigzag CNTs for fixed-free with (c) armchair and (d) zigzag CNTs forfixed-fixed and with (e) armchair and (f) zigzag for free-free boundary conditions The markers represent the results of the computation thestraight lines are linear regressions

8 Advances in Materials Science and Engineering

350

360

370

380

390

400

410

0 5 10 15 20 25 30 35 40 45

Seco

nd n

atur

al fr

eque

ncy

(GH

z)

(14-14)(15-15)(16-16)

Bending angle of armchair CNTs (∘)

(a)

1700

1800

1900

2000

2100

2200

2300

0 5 10 15 20 25 30 35 40 45

Third

nat

ural

freq

uenc

y (G

Hz)

(14-14)(15-15)(16-16)

Bending angle of armchair CNTs (∘)

(b)

Figure 9 Change in higher natural frequency with armchair CNTs for fixed-free boundary condition (a) second natural frequency and (b)third natural frequency The markers represent the results of the computation the straight lines are linear regressions

2nd mode1st mode 3rd mode

xy

z

xy

z

xy

z

Figure 10 First three eigenmodes of a (14-14) CNT with 10∘ bending angle

It was concluded that the natural frequency of straightCNTs increases by increasing the chiral number of botharmchair and zigzag CNTs It was also revealed that thenatural frequency of CNTs (zigzag and armchair) with higherchirality decreases by introducing bending angles for all casesof boundary conditions However the change of increasingthe bending angle on the natural frequency of armchairand zigzag CNTs with lower number of chirality is almostnegligible The finding of this study may have useful effectson further investigations of CNTs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Iijima ldquoHelicalmicrotubules of graphitic carbonrdquoNature vol354 no 6348 pp 56ndash58 1991

Advances in Materials Science and Engineering 9

[2] S Imani Yengejeh M Akbar Zadeh and A Ochsner ldquoOn thebuckling behavior of connected carbon nanotubes with parallellongitudinal axesrdquo Applied Physics A vol 115 no 4 pp 1335ndash1344 2014

[3] S K Georgantzinos G I Giannopoulos and N K AnifantisldquoAn efficient numerical model for vibration analysis of single-walled carbon nanotubesrdquo Computational Mechanics vol 43no 6 pp 731ndash741 2009

[4] W Xia and L Wang ldquoVibration characteristics of fluid-conveying carbon nanotubes with curved longitudinal shaperdquoComputationalMaterials Science vol 49 no 1 pp 99ndash103 2010

[5] S Arghavan andAV Singh ldquoOn the vibrations of single-walledcarbon nanotubesrdquo Journal of Sound and Vibration vol 330 no13 pp 3102ndash3122 2011

[6] A Ghavamian and A Ochsner ldquoNumerical modeling of eigen-modes and eigenfrequencies of single- andmulti-walled carbonnanotubes under the influence of atomic defectsrdquo Computa-tional Materials Science vol 72 pp 42ndash48 2013

[7] F N Mayoof and M A Hawwa ldquoChaotic behavior of a curvedcarbon nanotube under harmonic excitationrdquo Chaos Solitonsand Fractals vol 42 no 3 pp 1860ndash1867 2009

[8] M Farsadi A Ochsner and M Rahmandoust ldquoNumericalinvestigation of composite materials reinforced with wavedcarbon nanotubesrdquo Journal of Composite Materials vol 47 no11 pp 1425ndash1434 2013

[9] M S Dresselhaus G Dresselhaus and R Saito ldquoPhysics ofcarbon nanotubesrdquo Carbon vol 33 no 7 pp 883ndash891 1995

[10] C Li and T-W Chou ldquoA structural mechanics approach for theanalysis of carbon nanotubesrdquo International Journal of Solidsand Structures vol 40 no 10 pp 2487ndash2499 2003

[11] S Melchor F J Martin-Martinez and J A Dobado ldquoCoNTubv20mdashalgorithms for constructing C3-symmetric models ofthree-nanotube junctionsrdquo Journal of Chemical Information andModeling vol 51 no 6 pp 1492ndash1505 2011

[12] Z Kang M Li and Q Tang ldquoBuckling behavior of carbonnanotube-based intramolecular junctions under compressionmolecular dynamics simulation and finite element analysisrdquoComputational Materials Science vol 50 no 1 pp 253ndash2592010

[13] T Irvine Application of the Newton-Raphson Method to Vibra-tion Problems Vibration Data Publications Madison WisUSA 1999

[14] J P Lu ldquoElastic properties of carbon nanotubes and nanoropesrdquoPhysical Review Letters vol 79 no 7 pp 1297ndash1300 1997

[15] CW S To ldquoBending and shear moduli of single-walled carbonnanotubesrdquo Finite Elements in Analysis and Design vol 42 no5 pp 404ndash413 2006

[16] A L Kalamkarov A V Georgiades S K Rokkam V P Veeduand M N Ghasemi-Nejhad ldquoAnalytical and numerical tech-niques to predict carbon nanotubes propertiesrdquo InternationalJournal of Solids and Structures vol 43 no 22-23 pp 6832ndash6854 2006

[17] G I Giannopoulos ldquoFullerenes as mass sensors a numericalinvestigationrdquo Physica E vol 56 pp 36ndash42 2014

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2 Advances in Materials Science and Engineering

numerous CNTs under different boundary conditions werefoundThey also showed that their results indicated very goodagreement compared to numerical predictions from the liter-ature In 2010 Xia and Wang [4] investigated the vibrationcharacteristics of fluid-conveying carbon nanotubes with acurved longitudinal structure They could obtain the naturalfrequencies of curved carbon nanotubes conveying fluid andcompared them with those of straight CNTs Their findingsrevealed that CNTs with circular curved longitudinal shapeare unconditionally stable even for a system with sufficientlyhigh flow velocity In 2011 Arghavan and Singh [5] presenteda detailed numerical study on the free and forced vibration ofstraight SWCNTs They analyzed both armchair and zigzagCNTs with clamped-clamped and clamped-free boundaryconditions in order to find their natural frequencies andcorresponding mode shapes Their results indicated theappearance of thesemodes of vibration in the eigenvalues andeigenvectors without any distinction They finally concludedthat in the case of zigzag CNTs the axial modes appearedto be decoupled whereas the armchair nanotubes showcoupling between such modes Although they thoroughlyinvestigated the behavior of straight CNTs the study onthe behavior of curved nanotubes is only rarely addressedGhavamian and Ochsner [6] investigated the eigenvalues andeigenfrequencies of CNTs under the influence of defects in2013 After simulating armchair and zigzag configurationsof zigzag and armchair CNTs based on the FEM theyintroduced three most likely defects to the models in orderto represent defective forms of SWCNTs These defects arecarbon vacancy Si-doping and perturbation Finally theyexamined and also compared the vibrational behavior ofperfect and defective CNTs According to their findingsSWCNTs have a natural frequency in the range of 1869 to2401 GHz In addition it was pointed out that the existence ofany type of defects leads to a lower value of natural frequencyand vibrational stability The aim of this study is to continuethe previous investigations and to focus on the evaluationand comparison of vibrational behavior of curved CNTs withdifferent bending angles Since experimental observations(eg based on electronmicroscopes) show that CNTs are notusually straight but rather have certain degree of curvatureor waviness along the nanotubes length [7 8] we focus in thefollowing on the investigation of the effect of the curvatureon the dynamic behavior All our simulationswere performedwith the commercial finite element code MSCMARC (MSCSoftware Corporation Santa Ana CA USA)

2 Methodology

21 Geometric Definition CNTs are assumed to be some kindof hollow cylinder shape configurations as shown in Figure 1These special nanostructures can be imagined by rolling agraphene sheet into a cylinder They possess a length over10 120583m and diameters ranging from 1 to 50 nm The geometryof aCNT is defined by the chiral vector ℎ and the chiral angle120579 The chiral vector is presented by two unit vectors 1198861 and 1198862

and two integers 119898 and 119899 as it is presented by the followingequation [9]

ℎ = 119899 1198861 + 119898 1198862 (1)

The construction of CNTs is defined based on the chiralvector or angle by which the sheet is rolled into a cylinderin three different configurations including chiral zigzag andarmchair In the case of (120579 = 0∘) or (119898 = 0) the zigzag CNT isconstructed An armchair CNT is obtained in terms of chiralvector (119898 = 119899) or in terms of chiral angle (120579 = 30∘) andfinally a chiral CNT is shaped if 0∘ lt 120579 lt 30∘ or119898 = 119899 = 0 [9]

Based on the following equation the diameter of the CNTcan be calculated

119889CNT =1198860radic1198982 + 119898119899 + 1198992

120587 (2)

where 1198860 = radic3119887 and 119887 = 0142 nm is the length of the CndashCbond [9]

The modeling method presented in this study followsthe idea first suggested in [10] where the theory of classicalstructural mechanics was extended into the modeling ofCNTs In a CNT carbon atoms are bonded together bycovalent bonds which have their characteristic lengths andangles in a three-dimensional space Afterwards it wassuggested that CNTs when subjected to loading act as space-frame structuresTherefore the bonds between carbon atomsare considered as connecting load-carrying generalized beammembers while the carbon atoms behave as joints of themembers This idea is schematically shown in Figure 1

In this study the configurations of both armchair andzigzag SWCNTs were simulated by the CoNTub software[11] Defining the chirality and the length of the tubes thespatial coordinates of the C-atoms and the correspondingconnectivities (ie the primary bonds between two nearest-neighboring atoms) were calculated Then the gathered datawas transferred to a commercial finite element packagewhere CndashC bonds were modeled as circular beam elements[12] Afterwards the FE analyses were conducted and vibra-tional behavior of different armchair and zigzag CNTs wasevaluated

The natural frequency is the frequency of a vibratingsystem at which the system oscillates at greater amplitudedue to existence of the resonance phenomenon Geometrymass and applied boundary conditions are the factors whichinfluence these quantities The natural frequency is mainlyevaluated to examine the vibrational behavior of structuralmembers The first natural frequency of an Euler-Bernoullibeam element under different boundary condition is definedby the following equation [13]

119891 = (119860

2120587)radic119864119868

1198981198714 (3)

where 119864 119868 119898 and 119871 are the Youngrsquos modulus the secondmoment of area the mass per unit length and the length ofthe nanotube respectively The 119860 value for fixed-free fixed-fixed and free-free boundary condition is 35156 22373 and224 respectively

Advances in Materials Science and Engineering 3

Modeling

Approach

FE beam elementCNT

Carbon atom

CndashC bond

FE node

Figure 1 Side view of the (11 11) CNT as a space-frame structure

(a) (b)

y x

z

(c)

Figure 2 (16 0) zigzag CNT with (a) fixed-free (b) fixed-fixed and (c) free-free boundary conditions

22 Material Parameters and Boundary Conditions In thisstudy the vibrational behavior of numerous types of armchairand zigzag SWCNTs under fixed-free fixed-fixed and free-free boundary conditions is investigated in which one end iseither free or fully fixed while the other end is fixed in the firsttwo cases and for the third case both ends are completely freeas illustrated in Figure 2

Figure 3 illustrates an armchair CNT under fixed-freeboundary condition The models of CNTs were simulatedfrom straight CNT to curved CNT with 45∘ bending angleas shown in Figure 3

For defining the properties of the equivalent beam ele-ments for the CNT bonds no such classical examinationsor geometric derivations are available As a result the samevalues for the equivalent beam elements are assumed as inthe approach proposed in [10 15 16] These effective materialand geometrical properties were found in the mentionedreferences based on a molecular mechanics method Theirmotions are regulated by a force field which is generated byelectron-nucleus interactions and nucleus-nucleus interac-tions and frequently expressed in the form of steric potentialenergy This steric potential energy is in general the sumof contributions from bond stretch interaction bond anglebending dihedral angle torsion and improper (out of plane)torsion as illustrated in Figure 4

The introduced constants and the element properties ofCNTs (armchair and zigzag) are listed in Table 1

3 Results and Discussion

The vibrational behavior of different armchair and zigzagSWCNTs was investigated for different curvatures from 0∘to 45∘ bending angle (fixed-free fixed-fixed and free-freeboundary condition) After evaluating their natural frequen-cies from FEM the results were compared to analyticalcalculations given by (3) in the case of straight configurationsin order to have some kind of validation of the FE approachFigure 5 illustrates the first mode of vibration of a (16-0)armchair CNT under different boundary conditions

The relative difference between the analytical solution andFEM results of straight CNTs was defined by the followingequation

Relative difference in

=

10038161003816100381610038161003816100381610038161003816

FEM result minus analytical solutionanalytical solution

10038161003816100381610038161003816100381610038161003816times 100

(4)

This difference is listed in Table 2 for the case of thefixed-free boundary condition Based on the results itcan be concluded that the computational solutions are in

4 Advances in Materials Science and Engineering

y x

z

0∘ 5∘ 10∘ 15∘ 20∘ 25∘ 30∘ 35∘ 40∘ 45∘

Figure 3 (7-7) armchair CNTs with fixed-free boundary condition and different forms from 0∘ to 45∘ bending angle

ro Δr

Bond stretching(covalent)

Bond bending(covalent)

Bond torsion(covalent)

Pure tension of beamPure tension of beamPure tension of beam

120579o Δ120579

Δ0

TTMMPP

ΔllΔ120573

Figure 4 Equivalence of molecular mechanics and structural mechanics for covalent and noncovalent interactions between carbon atomsMolecular mechanics model (up) and structural mechanics model (down)

Table 1 Material and geometric properties of CndashC covalent bonds [14]

Corresponding force field constants

119896119903 = 65197 nNnm

119896120579 = 08758 nNnmrad2

119896120593 = 02780 nNnmrad2

119864 = Youngrsquos modulus =1198961199032119887

4120587119896120579

5484 times 10minus6Nnm2

119866 = shear modulus = 1198642(120592 + 1)

2159 times 10minus6Nnm2

119877119887 = bond radius = 2radic119896120579

119896119903

00733 nm

119868119909119909= 119868119910119910= second moments of area =

1205871198771198874

422661 times 10

minus5 nm4

Advances in Materials Science and Engineering 5

y

z

x

(a)

xy

z

(b)

y

z

x

(c)

Figure 5 First eigenmode of a (16-0) straight CNT under (a) fixed-free (b) fixed-fixed and (c) free-free boundary conditions

Table 2 Characteristics of simulated straight CNTs

CNT type Chirality (nm) Length (nm) Diameter (nm) Youngrsquos modulus (TPa)Natural frequency (GHz) Relative

difference in ()Analyticalsolution FEM result

Armchair (3 3) 1500 0407 1039 1001 847 1530Armchair (5 5) 1500 0678 1039 1430 1390 298Armchair (6 6) 1500 0814 1039 1659 1663 024Armchair (7 7) 1500 0949 1039 1901 1930 150Armchair (9 9) 1500 1220 1039 2391 2467 323Armchair (10 10) 1500 1356 1039 2637 2734 370Armchair (11 11) 1500 1492 1039 2885 2998 390Armchair (14 14) 1500 1898 1039 3637 3780 392Armchair (16 16) 1500 2171 1039 4142 4289 356Zigzag (8 0) 1505 0626 1021 1326 1256 527Zigzag (9 0) 1505 0705 1025 1459 1417 285Zigzag (10 0) 1505 0783 1028 1594 1576 111Zigzag (11 0) 1505 0861 1031 1731 1735 024Zigzag (12 0) 1505 0939 1032 1869 1894 135Zigzag (16 0) 1505 1253 1037 2433 2520 357Zigzag (20 0) 1505 1566 1039 3006 3136 431

good agreement with the analytical calculations for thecantilevered boundary condition

Each mode shape shows a significant natural frequencyand eigenmode based on the boundary condition of theCNTs For instance the first four eigenmodes of a (7 7)armchair CNT are shown in Figure 6

Having a closer look on the results it was revealed thatthe natural frequency of straight armchair and zigzag CNTsincreases by increasing the chiral number of both armchairand zigzag CNTs as illustrated in Figure 7

It is also obvious that the natural frequency of botharmchair and zigzag CNTs varied in a high value rangeComparison with other computational approaches based ona linear spring-based element formulation for straight CNTs[3] indicates a similar range of the frequencies

It is clear that the existence of any geometrical imperfec-tion bending angle in particular in the structure of armchairand zigzag straight CNTs results in a reduction in the naturalfrequency by increasing the bending angle as illustrated inFigure 8 The natural frequency decreases for all cases of

6 Advances in Materials Science and Engineering

y x

z

(a)

xy

z

(b)

xy

z

(c)

xy

z

(d)

Figure 6 First four eigenmodes of a (7-7) straight CNT

0050

100150200250300350400450500

(3-3) (16-16)(14-14)(11-11)(10-10)(9-9)(7-7)(5-5)

Firs

t nat

ural

freq

uenc

y (G

Hz)

Straight armchair CNT

Computational valueAnalytical value

(a)

(8-0) (20-0)(16-0)(12-0)(11-0)(10-0)(9-0)Straight zigzag CNT

Computational valueAnalytical value

00

50

100

150

200

250

300

350

Firs

t nat

ural

freq

uenc

y (G

Hz)

(b)

Figure 7 (a) Change in natural frequency for straight armchair CNTs (b) Change in natural frequency for straight zigzag CNTsThemarkersrepresent the results of the computation the straight lines are linear regressions

boundary conditions However it should be noted that thistrend is more obvious for the CNTs with higher chiralityas the decrease of natural frequency for both armchair andzigzag CNTs with lower chirality is almost negligible It isalso indicated that the gradual decrease for the fixed-fixedand free-free case of boundary condition is more significantcompared to the fixed-free case at higher curvatures Inaddition this trend is more obvious for the armchair CNTs

Since higher natural frequencies may be important insome applications such as mass sensing at nanoscale [17] weexemplarily evaluated the second and third natural frequencyand the corresponding mode shapes for the case of threearmchair nanotubes see Figure 9

Comparison with Figure 8(a) shows that the secondnatural frequency is practically the same as the first one

However the third natural frequency is approximately fivetimes higher than the first natural frequency Furthermorethe trend line for the higher frequencies is similar to theone from the first natural frequency An example of thecorresponding mode shapes is presented in Figure 10

4 Conclusions

In this study numerous CNTs (zigzag and armchair) weresimulated by an FE approach and their vibrational behaviorwas examined through performing several computationaltestswith different boundary conditions and variable bendingangles Towards achieving the most accurate results thevibrational behaviors of these CNTs were evaluated ana-lytically and computationally and were compared together

Advances in Materials Science and Engineering 7

350

360

370

380

390

400

410

0 10 20 30 40

Firs

t nat

ural

freq

uenc

y (G

Hz)

(14-14)(15-15)

(16-16)

Fixed-free

Bending angle of armchair CNTs (∘)

(a)

Fixed-free

Bending angle of zigzag CNTs (∘)

(16-0)(20-0)

(26-0)

240

260

280

300

320

340

360

380

400

0 10 20 30 40

Firs

t nat

ural

freq

uenc

y (G

Hz)

(b)

Fixed-fixed

(14-14)(15-15)

(16-16)

Bending angle of armchair CNTs (∘)

0185

0190

0195

0200

0205

0210

0215

0220

0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz)

(c)

(16-0)(20-0)

(26-0)

Fixed-fixed

Bending angle of zigzag CNTs (∘)

014

015

016

017

018

019

020

021

022

023

0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz)

(d)

(14-14)(15-15)

(16-16)

Bending angle of armchair CNTs (∘)

0190

0195

0200

0205

0210

0215

0220

0225

0230

0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz) Free-free

(e)

Bending angle of zigzag CNTs (∘)0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz) Free-free

(16-0)(20-0)

(26-0)

014

015

016

017

018

019

020

021

022

023

(f)

Figure 8 Change in natural frequency with (a) armchair and (b) zigzag CNTs for fixed-free with (c) armchair and (d) zigzag CNTs forfixed-fixed and with (e) armchair and (f) zigzag for free-free boundary conditions The markers represent the results of the computation thestraight lines are linear regressions

8 Advances in Materials Science and Engineering

350

360

370

380

390

400

410

0 5 10 15 20 25 30 35 40 45

Seco

nd n

atur

al fr

eque

ncy

(GH

z)

(14-14)(15-15)(16-16)

Bending angle of armchair CNTs (∘)

(a)

1700

1800

1900

2000

2100

2200

2300

0 5 10 15 20 25 30 35 40 45

Third

nat

ural

freq

uenc

y (G

Hz)

(14-14)(15-15)(16-16)

Bending angle of armchair CNTs (∘)

(b)

Figure 9 Change in higher natural frequency with armchair CNTs for fixed-free boundary condition (a) second natural frequency and (b)third natural frequency The markers represent the results of the computation the straight lines are linear regressions

2nd mode1st mode 3rd mode

xy

z

xy

z

xy

z

Figure 10 First three eigenmodes of a (14-14) CNT with 10∘ bending angle

It was concluded that the natural frequency of straightCNTs increases by increasing the chiral number of botharmchair and zigzag CNTs It was also revealed that thenatural frequency of CNTs (zigzag and armchair) with higherchirality decreases by introducing bending angles for all casesof boundary conditions However the change of increasingthe bending angle on the natural frequency of armchairand zigzag CNTs with lower number of chirality is almostnegligible The finding of this study may have useful effectson further investigations of CNTs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Iijima ldquoHelicalmicrotubules of graphitic carbonrdquoNature vol354 no 6348 pp 56ndash58 1991

Advances in Materials Science and Engineering 9

[2] S Imani Yengejeh M Akbar Zadeh and A Ochsner ldquoOn thebuckling behavior of connected carbon nanotubes with parallellongitudinal axesrdquo Applied Physics A vol 115 no 4 pp 1335ndash1344 2014

[3] S K Georgantzinos G I Giannopoulos and N K AnifantisldquoAn efficient numerical model for vibration analysis of single-walled carbon nanotubesrdquo Computational Mechanics vol 43no 6 pp 731ndash741 2009

[4] W Xia and L Wang ldquoVibration characteristics of fluid-conveying carbon nanotubes with curved longitudinal shaperdquoComputationalMaterials Science vol 49 no 1 pp 99ndash103 2010

[5] S Arghavan andAV Singh ldquoOn the vibrations of single-walledcarbon nanotubesrdquo Journal of Sound and Vibration vol 330 no13 pp 3102ndash3122 2011

[6] A Ghavamian and A Ochsner ldquoNumerical modeling of eigen-modes and eigenfrequencies of single- andmulti-walled carbonnanotubes under the influence of atomic defectsrdquo Computa-tional Materials Science vol 72 pp 42ndash48 2013

[7] F N Mayoof and M A Hawwa ldquoChaotic behavior of a curvedcarbon nanotube under harmonic excitationrdquo Chaos Solitonsand Fractals vol 42 no 3 pp 1860ndash1867 2009

[8] M Farsadi A Ochsner and M Rahmandoust ldquoNumericalinvestigation of composite materials reinforced with wavedcarbon nanotubesrdquo Journal of Composite Materials vol 47 no11 pp 1425ndash1434 2013

[9] M S Dresselhaus G Dresselhaus and R Saito ldquoPhysics ofcarbon nanotubesrdquo Carbon vol 33 no 7 pp 883ndash891 1995

[10] C Li and T-W Chou ldquoA structural mechanics approach for theanalysis of carbon nanotubesrdquo International Journal of Solidsand Structures vol 40 no 10 pp 2487ndash2499 2003

[11] S Melchor F J Martin-Martinez and J A Dobado ldquoCoNTubv20mdashalgorithms for constructing C3-symmetric models ofthree-nanotube junctionsrdquo Journal of Chemical Information andModeling vol 51 no 6 pp 1492ndash1505 2011

[12] Z Kang M Li and Q Tang ldquoBuckling behavior of carbonnanotube-based intramolecular junctions under compressionmolecular dynamics simulation and finite element analysisrdquoComputational Materials Science vol 50 no 1 pp 253ndash2592010

[13] T Irvine Application of the Newton-Raphson Method to Vibra-tion Problems Vibration Data Publications Madison WisUSA 1999

[14] J P Lu ldquoElastic properties of carbon nanotubes and nanoropesrdquoPhysical Review Letters vol 79 no 7 pp 1297ndash1300 1997

[15] CW S To ldquoBending and shear moduli of single-walled carbonnanotubesrdquo Finite Elements in Analysis and Design vol 42 no5 pp 404ndash413 2006

[16] A L Kalamkarov A V Georgiades S K Rokkam V P Veeduand M N Ghasemi-Nejhad ldquoAnalytical and numerical tech-niques to predict carbon nanotubes propertiesrdquo InternationalJournal of Solids and Structures vol 43 no 22-23 pp 6832ndash6854 2006

[17] G I Giannopoulos ldquoFullerenes as mass sensors a numericalinvestigationrdquo Physica E vol 56 pp 36ndash42 2014

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

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International Journal of

Biomaterials

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

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MetallurgyJournal of

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BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 3

Modeling

Approach

FE beam elementCNT

Carbon atom

CndashC bond

FE node

Figure 1 Side view of the (11 11) CNT as a space-frame structure

(a) (b)

y x

z

(c)

Figure 2 (16 0) zigzag CNT with (a) fixed-free (b) fixed-fixed and (c) free-free boundary conditions

22 Material Parameters and Boundary Conditions In thisstudy the vibrational behavior of numerous types of armchairand zigzag SWCNTs under fixed-free fixed-fixed and free-free boundary conditions is investigated in which one end iseither free or fully fixed while the other end is fixed in the firsttwo cases and for the third case both ends are completely freeas illustrated in Figure 2

Figure 3 illustrates an armchair CNT under fixed-freeboundary condition The models of CNTs were simulatedfrom straight CNT to curved CNT with 45∘ bending angleas shown in Figure 3

For defining the properties of the equivalent beam ele-ments for the CNT bonds no such classical examinationsor geometric derivations are available As a result the samevalues for the equivalent beam elements are assumed as inthe approach proposed in [10 15 16] These effective materialand geometrical properties were found in the mentionedreferences based on a molecular mechanics method Theirmotions are regulated by a force field which is generated byelectron-nucleus interactions and nucleus-nucleus interac-tions and frequently expressed in the form of steric potentialenergy This steric potential energy is in general the sumof contributions from bond stretch interaction bond anglebending dihedral angle torsion and improper (out of plane)torsion as illustrated in Figure 4

The introduced constants and the element properties ofCNTs (armchair and zigzag) are listed in Table 1

3 Results and Discussion

The vibrational behavior of different armchair and zigzagSWCNTs was investigated for different curvatures from 0∘to 45∘ bending angle (fixed-free fixed-fixed and free-freeboundary condition) After evaluating their natural frequen-cies from FEM the results were compared to analyticalcalculations given by (3) in the case of straight configurationsin order to have some kind of validation of the FE approachFigure 5 illustrates the first mode of vibration of a (16-0)armchair CNT under different boundary conditions

The relative difference between the analytical solution andFEM results of straight CNTs was defined by the followingequation

Relative difference in

=

10038161003816100381610038161003816100381610038161003816

FEM result minus analytical solutionanalytical solution

10038161003816100381610038161003816100381610038161003816times 100

(4)

This difference is listed in Table 2 for the case of thefixed-free boundary condition Based on the results itcan be concluded that the computational solutions are in

4 Advances in Materials Science and Engineering

y x

z

0∘ 5∘ 10∘ 15∘ 20∘ 25∘ 30∘ 35∘ 40∘ 45∘

Figure 3 (7-7) armchair CNTs with fixed-free boundary condition and different forms from 0∘ to 45∘ bending angle

ro Δr

Bond stretching(covalent)

Bond bending(covalent)

Bond torsion(covalent)

Pure tension of beamPure tension of beamPure tension of beam

120579o Δ120579

Δ0

TTMMPP

ΔllΔ120573

Figure 4 Equivalence of molecular mechanics and structural mechanics for covalent and noncovalent interactions between carbon atomsMolecular mechanics model (up) and structural mechanics model (down)

Table 1 Material and geometric properties of CndashC covalent bonds [14]

Corresponding force field constants

119896119903 = 65197 nNnm

119896120579 = 08758 nNnmrad2

119896120593 = 02780 nNnmrad2

119864 = Youngrsquos modulus =1198961199032119887

4120587119896120579

5484 times 10minus6Nnm2

119866 = shear modulus = 1198642(120592 + 1)

2159 times 10minus6Nnm2

119877119887 = bond radius = 2radic119896120579

119896119903

00733 nm

119868119909119909= 119868119910119910= second moments of area =

1205871198771198874

422661 times 10

minus5 nm4

Advances in Materials Science and Engineering 5

y

z

x

(a)

xy

z

(b)

y

z

x

(c)

Figure 5 First eigenmode of a (16-0) straight CNT under (a) fixed-free (b) fixed-fixed and (c) free-free boundary conditions

Table 2 Characteristics of simulated straight CNTs

CNT type Chirality (nm) Length (nm) Diameter (nm) Youngrsquos modulus (TPa)Natural frequency (GHz) Relative

difference in ()Analyticalsolution FEM result

Armchair (3 3) 1500 0407 1039 1001 847 1530Armchair (5 5) 1500 0678 1039 1430 1390 298Armchair (6 6) 1500 0814 1039 1659 1663 024Armchair (7 7) 1500 0949 1039 1901 1930 150Armchair (9 9) 1500 1220 1039 2391 2467 323Armchair (10 10) 1500 1356 1039 2637 2734 370Armchair (11 11) 1500 1492 1039 2885 2998 390Armchair (14 14) 1500 1898 1039 3637 3780 392Armchair (16 16) 1500 2171 1039 4142 4289 356Zigzag (8 0) 1505 0626 1021 1326 1256 527Zigzag (9 0) 1505 0705 1025 1459 1417 285Zigzag (10 0) 1505 0783 1028 1594 1576 111Zigzag (11 0) 1505 0861 1031 1731 1735 024Zigzag (12 0) 1505 0939 1032 1869 1894 135Zigzag (16 0) 1505 1253 1037 2433 2520 357Zigzag (20 0) 1505 1566 1039 3006 3136 431

good agreement with the analytical calculations for thecantilevered boundary condition

Each mode shape shows a significant natural frequencyand eigenmode based on the boundary condition of theCNTs For instance the first four eigenmodes of a (7 7)armchair CNT are shown in Figure 6

Having a closer look on the results it was revealed thatthe natural frequency of straight armchair and zigzag CNTsincreases by increasing the chiral number of both armchairand zigzag CNTs as illustrated in Figure 7

It is also obvious that the natural frequency of botharmchair and zigzag CNTs varied in a high value rangeComparison with other computational approaches based ona linear spring-based element formulation for straight CNTs[3] indicates a similar range of the frequencies

It is clear that the existence of any geometrical imperfec-tion bending angle in particular in the structure of armchairand zigzag straight CNTs results in a reduction in the naturalfrequency by increasing the bending angle as illustrated inFigure 8 The natural frequency decreases for all cases of

6 Advances in Materials Science and Engineering

y x

z

(a)

xy

z

(b)

xy

z

(c)

xy

z

(d)

Figure 6 First four eigenmodes of a (7-7) straight CNT

0050

100150200250300350400450500

(3-3) (16-16)(14-14)(11-11)(10-10)(9-9)(7-7)(5-5)

Firs

t nat

ural

freq

uenc

y (G

Hz)

Straight armchair CNT

Computational valueAnalytical value

(a)

(8-0) (20-0)(16-0)(12-0)(11-0)(10-0)(9-0)Straight zigzag CNT

Computational valueAnalytical value

00

50

100

150

200

250

300

350

Firs

t nat

ural

freq

uenc

y (G

Hz)

(b)

Figure 7 (a) Change in natural frequency for straight armchair CNTs (b) Change in natural frequency for straight zigzag CNTsThemarkersrepresent the results of the computation the straight lines are linear regressions

boundary conditions However it should be noted that thistrend is more obvious for the CNTs with higher chiralityas the decrease of natural frequency for both armchair andzigzag CNTs with lower chirality is almost negligible It isalso indicated that the gradual decrease for the fixed-fixedand free-free case of boundary condition is more significantcompared to the fixed-free case at higher curvatures Inaddition this trend is more obvious for the armchair CNTs

Since higher natural frequencies may be important insome applications such as mass sensing at nanoscale [17] weexemplarily evaluated the second and third natural frequencyand the corresponding mode shapes for the case of threearmchair nanotubes see Figure 9

Comparison with Figure 8(a) shows that the secondnatural frequency is practically the same as the first one

However the third natural frequency is approximately fivetimes higher than the first natural frequency Furthermorethe trend line for the higher frequencies is similar to theone from the first natural frequency An example of thecorresponding mode shapes is presented in Figure 10

4 Conclusions

In this study numerous CNTs (zigzag and armchair) weresimulated by an FE approach and their vibrational behaviorwas examined through performing several computationaltestswith different boundary conditions and variable bendingangles Towards achieving the most accurate results thevibrational behaviors of these CNTs were evaluated ana-lytically and computationally and were compared together

Advances in Materials Science and Engineering 7

350

360

370

380

390

400

410

0 10 20 30 40

Firs

t nat

ural

freq

uenc

y (G

Hz)

(14-14)(15-15)

(16-16)

Fixed-free

Bending angle of armchair CNTs (∘)

(a)

Fixed-free

Bending angle of zigzag CNTs (∘)

(16-0)(20-0)

(26-0)

240

260

280

300

320

340

360

380

400

0 10 20 30 40

Firs

t nat

ural

freq

uenc

y (G

Hz)

(b)

Fixed-fixed

(14-14)(15-15)

(16-16)

Bending angle of armchair CNTs (∘)

0185

0190

0195

0200

0205

0210

0215

0220

0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz)

(c)

(16-0)(20-0)

(26-0)

Fixed-fixed

Bending angle of zigzag CNTs (∘)

014

015

016

017

018

019

020

021

022

023

0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz)

(d)

(14-14)(15-15)

(16-16)

Bending angle of armchair CNTs (∘)

0190

0195

0200

0205

0210

0215

0220

0225

0230

0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz) Free-free

(e)

Bending angle of zigzag CNTs (∘)0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz) Free-free

(16-0)(20-0)

(26-0)

014

015

016

017

018

019

020

021

022

023

(f)

Figure 8 Change in natural frequency with (a) armchair and (b) zigzag CNTs for fixed-free with (c) armchair and (d) zigzag CNTs forfixed-fixed and with (e) armchair and (f) zigzag for free-free boundary conditions The markers represent the results of the computation thestraight lines are linear regressions

8 Advances in Materials Science and Engineering

350

360

370

380

390

400

410

0 5 10 15 20 25 30 35 40 45

Seco

nd n

atur

al fr

eque

ncy

(GH

z)

(14-14)(15-15)(16-16)

Bending angle of armchair CNTs (∘)

(a)

1700

1800

1900

2000

2100

2200

2300

0 5 10 15 20 25 30 35 40 45

Third

nat

ural

freq

uenc

y (G

Hz)

(14-14)(15-15)(16-16)

Bending angle of armchair CNTs (∘)

(b)

Figure 9 Change in higher natural frequency with armchair CNTs for fixed-free boundary condition (a) second natural frequency and (b)third natural frequency The markers represent the results of the computation the straight lines are linear regressions

2nd mode1st mode 3rd mode

xy

z

xy

z

xy

z

Figure 10 First three eigenmodes of a (14-14) CNT with 10∘ bending angle

It was concluded that the natural frequency of straightCNTs increases by increasing the chiral number of botharmchair and zigzag CNTs It was also revealed that thenatural frequency of CNTs (zigzag and armchair) with higherchirality decreases by introducing bending angles for all casesof boundary conditions However the change of increasingthe bending angle on the natural frequency of armchairand zigzag CNTs with lower number of chirality is almostnegligible The finding of this study may have useful effectson further investigations of CNTs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Iijima ldquoHelicalmicrotubules of graphitic carbonrdquoNature vol354 no 6348 pp 56ndash58 1991

Advances in Materials Science and Engineering 9

[2] S Imani Yengejeh M Akbar Zadeh and A Ochsner ldquoOn thebuckling behavior of connected carbon nanotubes with parallellongitudinal axesrdquo Applied Physics A vol 115 no 4 pp 1335ndash1344 2014

[3] S K Georgantzinos G I Giannopoulos and N K AnifantisldquoAn efficient numerical model for vibration analysis of single-walled carbon nanotubesrdquo Computational Mechanics vol 43no 6 pp 731ndash741 2009

[4] W Xia and L Wang ldquoVibration characteristics of fluid-conveying carbon nanotubes with curved longitudinal shaperdquoComputationalMaterials Science vol 49 no 1 pp 99ndash103 2010

[5] S Arghavan andAV Singh ldquoOn the vibrations of single-walledcarbon nanotubesrdquo Journal of Sound and Vibration vol 330 no13 pp 3102ndash3122 2011

[6] A Ghavamian and A Ochsner ldquoNumerical modeling of eigen-modes and eigenfrequencies of single- andmulti-walled carbonnanotubes under the influence of atomic defectsrdquo Computa-tional Materials Science vol 72 pp 42ndash48 2013

[7] F N Mayoof and M A Hawwa ldquoChaotic behavior of a curvedcarbon nanotube under harmonic excitationrdquo Chaos Solitonsand Fractals vol 42 no 3 pp 1860ndash1867 2009

[8] M Farsadi A Ochsner and M Rahmandoust ldquoNumericalinvestigation of composite materials reinforced with wavedcarbon nanotubesrdquo Journal of Composite Materials vol 47 no11 pp 1425ndash1434 2013

[9] M S Dresselhaus G Dresselhaus and R Saito ldquoPhysics ofcarbon nanotubesrdquo Carbon vol 33 no 7 pp 883ndash891 1995

[10] C Li and T-W Chou ldquoA structural mechanics approach for theanalysis of carbon nanotubesrdquo International Journal of Solidsand Structures vol 40 no 10 pp 2487ndash2499 2003

[11] S Melchor F J Martin-Martinez and J A Dobado ldquoCoNTubv20mdashalgorithms for constructing C3-symmetric models ofthree-nanotube junctionsrdquo Journal of Chemical Information andModeling vol 51 no 6 pp 1492ndash1505 2011

[12] Z Kang M Li and Q Tang ldquoBuckling behavior of carbonnanotube-based intramolecular junctions under compressionmolecular dynamics simulation and finite element analysisrdquoComputational Materials Science vol 50 no 1 pp 253ndash2592010

[13] T Irvine Application of the Newton-Raphson Method to Vibra-tion Problems Vibration Data Publications Madison WisUSA 1999

[14] J P Lu ldquoElastic properties of carbon nanotubes and nanoropesrdquoPhysical Review Letters vol 79 no 7 pp 1297ndash1300 1997

[15] CW S To ldquoBending and shear moduli of single-walled carbonnanotubesrdquo Finite Elements in Analysis and Design vol 42 no5 pp 404ndash413 2006

[16] A L Kalamkarov A V Georgiades S K Rokkam V P Veeduand M N Ghasemi-Nejhad ldquoAnalytical and numerical tech-niques to predict carbon nanotubes propertiesrdquo InternationalJournal of Solids and Structures vol 43 no 22-23 pp 6832ndash6854 2006

[17] G I Giannopoulos ldquoFullerenes as mass sensors a numericalinvestigationrdquo Physica E vol 56 pp 36ndash42 2014

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

4 Advances in Materials Science and Engineering

y x

z

0∘ 5∘ 10∘ 15∘ 20∘ 25∘ 30∘ 35∘ 40∘ 45∘

Figure 3 (7-7) armchair CNTs with fixed-free boundary condition and different forms from 0∘ to 45∘ bending angle

ro Δr

Bond stretching(covalent)

Bond bending(covalent)

Bond torsion(covalent)

Pure tension of beamPure tension of beamPure tension of beam

120579o Δ120579

Δ0

TTMMPP

ΔllΔ120573

Figure 4 Equivalence of molecular mechanics and structural mechanics for covalent and noncovalent interactions between carbon atomsMolecular mechanics model (up) and structural mechanics model (down)

Table 1 Material and geometric properties of CndashC covalent bonds [14]

Corresponding force field constants

119896119903 = 65197 nNnm

119896120579 = 08758 nNnmrad2

119896120593 = 02780 nNnmrad2

119864 = Youngrsquos modulus =1198961199032119887

4120587119896120579

5484 times 10minus6Nnm2

119866 = shear modulus = 1198642(120592 + 1)

2159 times 10minus6Nnm2

119877119887 = bond radius = 2radic119896120579

119896119903

00733 nm

119868119909119909= 119868119910119910= second moments of area =

1205871198771198874

422661 times 10

minus5 nm4

Advances in Materials Science and Engineering 5

y

z

x

(a)

xy

z

(b)

y

z

x

(c)

Figure 5 First eigenmode of a (16-0) straight CNT under (a) fixed-free (b) fixed-fixed and (c) free-free boundary conditions

Table 2 Characteristics of simulated straight CNTs

CNT type Chirality (nm) Length (nm) Diameter (nm) Youngrsquos modulus (TPa)Natural frequency (GHz) Relative

difference in ()Analyticalsolution FEM result

Armchair (3 3) 1500 0407 1039 1001 847 1530Armchair (5 5) 1500 0678 1039 1430 1390 298Armchair (6 6) 1500 0814 1039 1659 1663 024Armchair (7 7) 1500 0949 1039 1901 1930 150Armchair (9 9) 1500 1220 1039 2391 2467 323Armchair (10 10) 1500 1356 1039 2637 2734 370Armchair (11 11) 1500 1492 1039 2885 2998 390Armchair (14 14) 1500 1898 1039 3637 3780 392Armchair (16 16) 1500 2171 1039 4142 4289 356Zigzag (8 0) 1505 0626 1021 1326 1256 527Zigzag (9 0) 1505 0705 1025 1459 1417 285Zigzag (10 0) 1505 0783 1028 1594 1576 111Zigzag (11 0) 1505 0861 1031 1731 1735 024Zigzag (12 0) 1505 0939 1032 1869 1894 135Zigzag (16 0) 1505 1253 1037 2433 2520 357Zigzag (20 0) 1505 1566 1039 3006 3136 431

good agreement with the analytical calculations for thecantilevered boundary condition

Each mode shape shows a significant natural frequencyand eigenmode based on the boundary condition of theCNTs For instance the first four eigenmodes of a (7 7)armchair CNT are shown in Figure 6

Having a closer look on the results it was revealed thatthe natural frequency of straight armchair and zigzag CNTsincreases by increasing the chiral number of both armchairand zigzag CNTs as illustrated in Figure 7

It is also obvious that the natural frequency of botharmchair and zigzag CNTs varied in a high value rangeComparison with other computational approaches based ona linear spring-based element formulation for straight CNTs[3] indicates a similar range of the frequencies

It is clear that the existence of any geometrical imperfec-tion bending angle in particular in the structure of armchairand zigzag straight CNTs results in a reduction in the naturalfrequency by increasing the bending angle as illustrated inFigure 8 The natural frequency decreases for all cases of

6 Advances in Materials Science and Engineering

y x

z

(a)

xy

z

(b)

xy

z

(c)

xy

z

(d)

Figure 6 First four eigenmodes of a (7-7) straight CNT

0050

100150200250300350400450500

(3-3) (16-16)(14-14)(11-11)(10-10)(9-9)(7-7)(5-5)

Firs

t nat

ural

freq

uenc

y (G

Hz)

Straight armchair CNT

Computational valueAnalytical value

(a)

(8-0) (20-0)(16-0)(12-0)(11-0)(10-0)(9-0)Straight zigzag CNT

Computational valueAnalytical value

00

50

100

150

200

250

300

350

Firs

t nat

ural

freq

uenc

y (G

Hz)

(b)

Figure 7 (a) Change in natural frequency for straight armchair CNTs (b) Change in natural frequency for straight zigzag CNTsThemarkersrepresent the results of the computation the straight lines are linear regressions

boundary conditions However it should be noted that thistrend is more obvious for the CNTs with higher chiralityas the decrease of natural frequency for both armchair andzigzag CNTs with lower chirality is almost negligible It isalso indicated that the gradual decrease for the fixed-fixedand free-free case of boundary condition is more significantcompared to the fixed-free case at higher curvatures Inaddition this trend is more obvious for the armchair CNTs

Since higher natural frequencies may be important insome applications such as mass sensing at nanoscale [17] weexemplarily evaluated the second and third natural frequencyand the corresponding mode shapes for the case of threearmchair nanotubes see Figure 9

Comparison with Figure 8(a) shows that the secondnatural frequency is practically the same as the first one

However the third natural frequency is approximately fivetimes higher than the first natural frequency Furthermorethe trend line for the higher frequencies is similar to theone from the first natural frequency An example of thecorresponding mode shapes is presented in Figure 10

4 Conclusions

In this study numerous CNTs (zigzag and armchair) weresimulated by an FE approach and their vibrational behaviorwas examined through performing several computationaltestswith different boundary conditions and variable bendingangles Towards achieving the most accurate results thevibrational behaviors of these CNTs were evaluated ana-lytically and computationally and were compared together

Advances in Materials Science and Engineering 7

350

360

370

380

390

400

410

0 10 20 30 40

Firs

t nat

ural

freq

uenc

y (G

Hz)

(14-14)(15-15)

(16-16)

Fixed-free

Bending angle of armchair CNTs (∘)

(a)

Fixed-free

Bending angle of zigzag CNTs (∘)

(16-0)(20-0)

(26-0)

240

260

280

300

320

340

360

380

400

0 10 20 30 40

Firs

t nat

ural

freq

uenc

y (G

Hz)

(b)

Fixed-fixed

(14-14)(15-15)

(16-16)

Bending angle of armchair CNTs (∘)

0185

0190

0195

0200

0205

0210

0215

0220

0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz)

(c)

(16-0)(20-0)

(26-0)

Fixed-fixed

Bending angle of zigzag CNTs (∘)

014

015

016

017

018

019

020

021

022

023

0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz)

(d)

(14-14)(15-15)

(16-16)

Bending angle of armchair CNTs (∘)

0190

0195

0200

0205

0210

0215

0220

0225

0230

0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz) Free-free

(e)

Bending angle of zigzag CNTs (∘)0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz) Free-free

(16-0)(20-0)

(26-0)

014

015

016

017

018

019

020

021

022

023

(f)

Figure 8 Change in natural frequency with (a) armchair and (b) zigzag CNTs for fixed-free with (c) armchair and (d) zigzag CNTs forfixed-fixed and with (e) armchair and (f) zigzag for free-free boundary conditions The markers represent the results of the computation thestraight lines are linear regressions

8 Advances in Materials Science and Engineering

350

360

370

380

390

400

410

0 5 10 15 20 25 30 35 40 45

Seco

nd n

atur

al fr

eque

ncy

(GH

z)

(14-14)(15-15)(16-16)

Bending angle of armchair CNTs (∘)

(a)

1700

1800

1900

2000

2100

2200

2300

0 5 10 15 20 25 30 35 40 45

Third

nat

ural

freq

uenc

y (G

Hz)

(14-14)(15-15)(16-16)

Bending angle of armchair CNTs (∘)

(b)

Figure 9 Change in higher natural frequency with armchair CNTs for fixed-free boundary condition (a) second natural frequency and (b)third natural frequency The markers represent the results of the computation the straight lines are linear regressions

2nd mode1st mode 3rd mode

xy

z

xy

z

xy

z

Figure 10 First three eigenmodes of a (14-14) CNT with 10∘ bending angle

It was concluded that the natural frequency of straightCNTs increases by increasing the chiral number of botharmchair and zigzag CNTs It was also revealed that thenatural frequency of CNTs (zigzag and armchair) with higherchirality decreases by introducing bending angles for all casesof boundary conditions However the change of increasingthe bending angle on the natural frequency of armchairand zigzag CNTs with lower number of chirality is almostnegligible The finding of this study may have useful effectson further investigations of CNTs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Iijima ldquoHelicalmicrotubules of graphitic carbonrdquoNature vol354 no 6348 pp 56ndash58 1991

Advances in Materials Science and Engineering 9

[2] S Imani Yengejeh M Akbar Zadeh and A Ochsner ldquoOn thebuckling behavior of connected carbon nanotubes with parallellongitudinal axesrdquo Applied Physics A vol 115 no 4 pp 1335ndash1344 2014

[3] S K Georgantzinos G I Giannopoulos and N K AnifantisldquoAn efficient numerical model for vibration analysis of single-walled carbon nanotubesrdquo Computational Mechanics vol 43no 6 pp 731ndash741 2009

[4] W Xia and L Wang ldquoVibration characteristics of fluid-conveying carbon nanotubes with curved longitudinal shaperdquoComputationalMaterials Science vol 49 no 1 pp 99ndash103 2010

[5] S Arghavan andAV Singh ldquoOn the vibrations of single-walledcarbon nanotubesrdquo Journal of Sound and Vibration vol 330 no13 pp 3102ndash3122 2011

[6] A Ghavamian and A Ochsner ldquoNumerical modeling of eigen-modes and eigenfrequencies of single- andmulti-walled carbonnanotubes under the influence of atomic defectsrdquo Computa-tional Materials Science vol 72 pp 42ndash48 2013

[7] F N Mayoof and M A Hawwa ldquoChaotic behavior of a curvedcarbon nanotube under harmonic excitationrdquo Chaos Solitonsand Fractals vol 42 no 3 pp 1860ndash1867 2009

[8] M Farsadi A Ochsner and M Rahmandoust ldquoNumericalinvestigation of composite materials reinforced with wavedcarbon nanotubesrdquo Journal of Composite Materials vol 47 no11 pp 1425ndash1434 2013

[9] M S Dresselhaus G Dresselhaus and R Saito ldquoPhysics ofcarbon nanotubesrdquo Carbon vol 33 no 7 pp 883ndash891 1995

[10] C Li and T-W Chou ldquoA structural mechanics approach for theanalysis of carbon nanotubesrdquo International Journal of Solidsand Structures vol 40 no 10 pp 2487ndash2499 2003

[11] S Melchor F J Martin-Martinez and J A Dobado ldquoCoNTubv20mdashalgorithms for constructing C3-symmetric models ofthree-nanotube junctionsrdquo Journal of Chemical Information andModeling vol 51 no 6 pp 1492ndash1505 2011

[12] Z Kang M Li and Q Tang ldquoBuckling behavior of carbonnanotube-based intramolecular junctions under compressionmolecular dynamics simulation and finite element analysisrdquoComputational Materials Science vol 50 no 1 pp 253ndash2592010

[13] T Irvine Application of the Newton-Raphson Method to Vibra-tion Problems Vibration Data Publications Madison WisUSA 1999

[14] J P Lu ldquoElastic properties of carbon nanotubes and nanoropesrdquoPhysical Review Letters vol 79 no 7 pp 1297ndash1300 1997

[15] CW S To ldquoBending and shear moduli of single-walled carbonnanotubesrdquo Finite Elements in Analysis and Design vol 42 no5 pp 404ndash413 2006

[16] A L Kalamkarov A V Georgiades S K Rokkam V P Veeduand M N Ghasemi-Nejhad ldquoAnalytical and numerical tech-niques to predict carbon nanotubes propertiesrdquo InternationalJournal of Solids and Structures vol 43 no 22-23 pp 6832ndash6854 2006

[17] G I Giannopoulos ldquoFullerenes as mass sensors a numericalinvestigationrdquo Physica E vol 56 pp 36ndash42 2014

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 5

y

z

x

(a)

xy

z

(b)

y

z

x

(c)

Figure 5 First eigenmode of a (16-0) straight CNT under (a) fixed-free (b) fixed-fixed and (c) free-free boundary conditions

Table 2 Characteristics of simulated straight CNTs

CNT type Chirality (nm) Length (nm) Diameter (nm) Youngrsquos modulus (TPa)Natural frequency (GHz) Relative

difference in ()Analyticalsolution FEM result

Armchair (3 3) 1500 0407 1039 1001 847 1530Armchair (5 5) 1500 0678 1039 1430 1390 298Armchair (6 6) 1500 0814 1039 1659 1663 024Armchair (7 7) 1500 0949 1039 1901 1930 150Armchair (9 9) 1500 1220 1039 2391 2467 323Armchair (10 10) 1500 1356 1039 2637 2734 370Armchair (11 11) 1500 1492 1039 2885 2998 390Armchair (14 14) 1500 1898 1039 3637 3780 392Armchair (16 16) 1500 2171 1039 4142 4289 356Zigzag (8 0) 1505 0626 1021 1326 1256 527Zigzag (9 0) 1505 0705 1025 1459 1417 285Zigzag (10 0) 1505 0783 1028 1594 1576 111Zigzag (11 0) 1505 0861 1031 1731 1735 024Zigzag (12 0) 1505 0939 1032 1869 1894 135Zigzag (16 0) 1505 1253 1037 2433 2520 357Zigzag (20 0) 1505 1566 1039 3006 3136 431

good agreement with the analytical calculations for thecantilevered boundary condition

Each mode shape shows a significant natural frequencyand eigenmode based on the boundary condition of theCNTs For instance the first four eigenmodes of a (7 7)armchair CNT are shown in Figure 6

Having a closer look on the results it was revealed thatthe natural frequency of straight armchair and zigzag CNTsincreases by increasing the chiral number of both armchairand zigzag CNTs as illustrated in Figure 7

It is also obvious that the natural frequency of botharmchair and zigzag CNTs varied in a high value rangeComparison with other computational approaches based ona linear spring-based element formulation for straight CNTs[3] indicates a similar range of the frequencies

It is clear that the existence of any geometrical imperfec-tion bending angle in particular in the structure of armchairand zigzag straight CNTs results in a reduction in the naturalfrequency by increasing the bending angle as illustrated inFigure 8 The natural frequency decreases for all cases of

6 Advances in Materials Science and Engineering

y x

z

(a)

xy

z

(b)

xy

z

(c)

xy

z

(d)

Figure 6 First four eigenmodes of a (7-7) straight CNT

0050

100150200250300350400450500

(3-3) (16-16)(14-14)(11-11)(10-10)(9-9)(7-7)(5-5)

Firs

t nat

ural

freq

uenc

y (G

Hz)

Straight armchair CNT

Computational valueAnalytical value

(a)

(8-0) (20-0)(16-0)(12-0)(11-0)(10-0)(9-0)Straight zigzag CNT

Computational valueAnalytical value

00

50

100

150

200

250

300

350

Firs

t nat

ural

freq

uenc

y (G

Hz)

(b)

Figure 7 (a) Change in natural frequency for straight armchair CNTs (b) Change in natural frequency for straight zigzag CNTsThemarkersrepresent the results of the computation the straight lines are linear regressions

boundary conditions However it should be noted that thistrend is more obvious for the CNTs with higher chiralityas the decrease of natural frequency for both armchair andzigzag CNTs with lower chirality is almost negligible It isalso indicated that the gradual decrease for the fixed-fixedand free-free case of boundary condition is more significantcompared to the fixed-free case at higher curvatures Inaddition this trend is more obvious for the armchair CNTs

Since higher natural frequencies may be important insome applications such as mass sensing at nanoscale [17] weexemplarily evaluated the second and third natural frequencyand the corresponding mode shapes for the case of threearmchair nanotubes see Figure 9

Comparison with Figure 8(a) shows that the secondnatural frequency is practically the same as the first one

However the third natural frequency is approximately fivetimes higher than the first natural frequency Furthermorethe trend line for the higher frequencies is similar to theone from the first natural frequency An example of thecorresponding mode shapes is presented in Figure 10

4 Conclusions

In this study numerous CNTs (zigzag and armchair) weresimulated by an FE approach and their vibrational behaviorwas examined through performing several computationaltestswith different boundary conditions and variable bendingangles Towards achieving the most accurate results thevibrational behaviors of these CNTs were evaluated ana-lytically and computationally and were compared together

Advances in Materials Science and Engineering 7

350

360

370

380

390

400

410

0 10 20 30 40

Firs

t nat

ural

freq

uenc

y (G

Hz)

(14-14)(15-15)

(16-16)

Fixed-free

Bending angle of armchair CNTs (∘)

(a)

Fixed-free

Bending angle of zigzag CNTs (∘)

(16-0)(20-0)

(26-0)

240

260

280

300

320

340

360

380

400

0 10 20 30 40

Firs

t nat

ural

freq

uenc

y (G

Hz)

(b)

Fixed-fixed

(14-14)(15-15)

(16-16)

Bending angle of armchair CNTs (∘)

0185

0190

0195

0200

0205

0210

0215

0220

0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz)

(c)

(16-0)(20-0)

(26-0)

Fixed-fixed

Bending angle of zigzag CNTs (∘)

014

015

016

017

018

019

020

021

022

023

0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz)

(d)

(14-14)(15-15)

(16-16)

Bending angle of armchair CNTs (∘)

0190

0195

0200

0205

0210

0215

0220

0225

0230

0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz) Free-free

(e)

Bending angle of zigzag CNTs (∘)0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz) Free-free

(16-0)(20-0)

(26-0)

014

015

016

017

018

019

020

021

022

023

(f)

Figure 8 Change in natural frequency with (a) armchair and (b) zigzag CNTs for fixed-free with (c) armchair and (d) zigzag CNTs forfixed-fixed and with (e) armchair and (f) zigzag for free-free boundary conditions The markers represent the results of the computation thestraight lines are linear regressions

8 Advances in Materials Science and Engineering

350

360

370

380

390

400

410

0 5 10 15 20 25 30 35 40 45

Seco

nd n

atur

al fr

eque

ncy

(GH

z)

(14-14)(15-15)(16-16)

Bending angle of armchair CNTs (∘)

(a)

1700

1800

1900

2000

2100

2200

2300

0 5 10 15 20 25 30 35 40 45

Third

nat

ural

freq

uenc

y (G

Hz)

(14-14)(15-15)(16-16)

Bending angle of armchair CNTs (∘)

(b)

Figure 9 Change in higher natural frequency with armchair CNTs for fixed-free boundary condition (a) second natural frequency and (b)third natural frequency The markers represent the results of the computation the straight lines are linear regressions

2nd mode1st mode 3rd mode

xy

z

xy

z

xy

z

Figure 10 First three eigenmodes of a (14-14) CNT with 10∘ bending angle

It was concluded that the natural frequency of straightCNTs increases by increasing the chiral number of botharmchair and zigzag CNTs It was also revealed that thenatural frequency of CNTs (zigzag and armchair) with higherchirality decreases by introducing bending angles for all casesof boundary conditions However the change of increasingthe bending angle on the natural frequency of armchairand zigzag CNTs with lower number of chirality is almostnegligible The finding of this study may have useful effectson further investigations of CNTs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Iijima ldquoHelicalmicrotubules of graphitic carbonrdquoNature vol354 no 6348 pp 56ndash58 1991

Advances in Materials Science and Engineering 9

[2] S Imani Yengejeh M Akbar Zadeh and A Ochsner ldquoOn thebuckling behavior of connected carbon nanotubes with parallellongitudinal axesrdquo Applied Physics A vol 115 no 4 pp 1335ndash1344 2014

[3] S K Georgantzinos G I Giannopoulos and N K AnifantisldquoAn efficient numerical model for vibration analysis of single-walled carbon nanotubesrdquo Computational Mechanics vol 43no 6 pp 731ndash741 2009

[4] W Xia and L Wang ldquoVibration characteristics of fluid-conveying carbon nanotubes with curved longitudinal shaperdquoComputationalMaterials Science vol 49 no 1 pp 99ndash103 2010

[5] S Arghavan andAV Singh ldquoOn the vibrations of single-walledcarbon nanotubesrdquo Journal of Sound and Vibration vol 330 no13 pp 3102ndash3122 2011

[6] A Ghavamian and A Ochsner ldquoNumerical modeling of eigen-modes and eigenfrequencies of single- andmulti-walled carbonnanotubes under the influence of atomic defectsrdquo Computa-tional Materials Science vol 72 pp 42ndash48 2013

[7] F N Mayoof and M A Hawwa ldquoChaotic behavior of a curvedcarbon nanotube under harmonic excitationrdquo Chaos Solitonsand Fractals vol 42 no 3 pp 1860ndash1867 2009

[8] M Farsadi A Ochsner and M Rahmandoust ldquoNumericalinvestigation of composite materials reinforced with wavedcarbon nanotubesrdquo Journal of Composite Materials vol 47 no11 pp 1425ndash1434 2013

[9] M S Dresselhaus G Dresselhaus and R Saito ldquoPhysics ofcarbon nanotubesrdquo Carbon vol 33 no 7 pp 883ndash891 1995

[10] C Li and T-W Chou ldquoA structural mechanics approach for theanalysis of carbon nanotubesrdquo International Journal of Solidsand Structures vol 40 no 10 pp 2487ndash2499 2003

[11] S Melchor F J Martin-Martinez and J A Dobado ldquoCoNTubv20mdashalgorithms for constructing C3-symmetric models ofthree-nanotube junctionsrdquo Journal of Chemical Information andModeling vol 51 no 6 pp 1492ndash1505 2011

[12] Z Kang M Li and Q Tang ldquoBuckling behavior of carbonnanotube-based intramolecular junctions under compressionmolecular dynamics simulation and finite element analysisrdquoComputational Materials Science vol 50 no 1 pp 253ndash2592010

[13] T Irvine Application of the Newton-Raphson Method to Vibra-tion Problems Vibration Data Publications Madison WisUSA 1999

[14] J P Lu ldquoElastic properties of carbon nanotubes and nanoropesrdquoPhysical Review Letters vol 79 no 7 pp 1297ndash1300 1997

[15] CW S To ldquoBending and shear moduli of single-walled carbonnanotubesrdquo Finite Elements in Analysis and Design vol 42 no5 pp 404ndash413 2006

[16] A L Kalamkarov A V Georgiades S K Rokkam V P Veeduand M N Ghasemi-Nejhad ldquoAnalytical and numerical tech-niques to predict carbon nanotubes propertiesrdquo InternationalJournal of Solids and Structures vol 43 no 22-23 pp 6832ndash6854 2006

[17] G I Giannopoulos ldquoFullerenes as mass sensors a numericalinvestigationrdquo Physica E vol 56 pp 36ndash42 2014

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

6 Advances in Materials Science and Engineering

y x

z

(a)

xy

z

(b)

xy

z

(c)

xy

z

(d)

Figure 6 First four eigenmodes of a (7-7) straight CNT

0050

100150200250300350400450500

(3-3) (16-16)(14-14)(11-11)(10-10)(9-9)(7-7)(5-5)

Firs

t nat

ural

freq

uenc

y (G

Hz)

Straight armchair CNT

Computational valueAnalytical value

(a)

(8-0) (20-0)(16-0)(12-0)(11-0)(10-0)(9-0)Straight zigzag CNT

Computational valueAnalytical value

00

50

100

150

200

250

300

350

Firs

t nat

ural

freq

uenc

y (G

Hz)

(b)

Figure 7 (a) Change in natural frequency for straight armchair CNTs (b) Change in natural frequency for straight zigzag CNTsThemarkersrepresent the results of the computation the straight lines are linear regressions

boundary conditions However it should be noted that thistrend is more obvious for the CNTs with higher chiralityas the decrease of natural frequency for both armchair andzigzag CNTs with lower chirality is almost negligible It isalso indicated that the gradual decrease for the fixed-fixedand free-free case of boundary condition is more significantcompared to the fixed-free case at higher curvatures Inaddition this trend is more obvious for the armchair CNTs

Since higher natural frequencies may be important insome applications such as mass sensing at nanoscale [17] weexemplarily evaluated the second and third natural frequencyand the corresponding mode shapes for the case of threearmchair nanotubes see Figure 9

Comparison with Figure 8(a) shows that the secondnatural frequency is practically the same as the first one

However the third natural frequency is approximately fivetimes higher than the first natural frequency Furthermorethe trend line for the higher frequencies is similar to theone from the first natural frequency An example of thecorresponding mode shapes is presented in Figure 10

4 Conclusions

In this study numerous CNTs (zigzag and armchair) weresimulated by an FE approach and their vibrational behaviorwas examined through performing several computationaltestswith different boundary conditions and variable bendingangles Towards achieving the most accurate results thevibrational behaviors of these CNTs were evaluated ana-lytically and computationally and were compared together

Advances in Materials Science and Engineering 7

350

360

370

380

390

400

410

0 10 20 30 40

Firs

t nat

ural

freq

uenc

y (G

Hz)

(14-14)(15-15)

(16-16)

Fixed-free

Bending angle of armchair CNTs (∘)

(a)

Fixed-free

Bending angle of zigzag CNTs (∘)

(16-0)(20-0)

(26-0)

240

260

280

300

320

340

360

380

400

0 10 20 30 40

Firs

t nat

ural

freq

uenc

y (G

Hz)

(b)

Fixed-fixed

(14-14)(15-15)

(16-16)

Bending angle of armchair CNTs (∘)

0185

0190

0195

0200

0205

0210

0215

0220

0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz)

(c)

(16-0)(20-0)

(26-0)

Fixed-fixed

Bending angle of zigzag CNTs (∘)

014

015

016

017

018

019

020

021

022

023

0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz)

(d)

(14-14)(15-15)

(16-16)

Bending angle of armchair CNTs (∘)

0190

0195

0200

0205

0210

0215

0220

0225

0230

0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz) Free-free

(e)

Bending angle of zigzag CNTs (∘)0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz) Free-free

(16-0)(20-0)

(26-0)

014

015

016

017

018

019

020

021

022

023

(f)

Figure 8 Change in natural frequency with (a) armchair and (b) zigzag CNTs for fixed-free with (c) armchair and (d) zigzag CNTs forfixed-fixed and with (e) armchair and (f) zigzag for free-free boundary conditions The markers represent the results of the computation thestraight lines are linear regressions

8 Advances in Materials Science and Engineering

350

360

370

380

390

400

410

0 5 10 15 20 25 30 35 40 45

Seco

nd n

atur

al fr

eque

ncy

(GH

z)

(14-14)(15-15)(16-16)

Bending angle of armchair CNTs (∘)

(a)

1700

1800

1900

2000

2100

2200

2300

0 5 10 15 20 25 30 35 40 45

Third

nat

ural

freq

uenc

y (G

Hz)

(14-14)(15-15)(16-16)

Bending angle of armchair CNTs (∘)

(b)

Figure 9 Change in higher natural frequency with armchair CNTs for fixed-free boundary condition (a) second natural frequency and (b)third natural frequency The markers represent the results of the computation the straight lines are linear regressions

2nd mode1st mode 3rd mode

xy

z

xy

z

xy

z

Figure 10 First three eigenmodes of a (14-14) CNT with 10∘ bending angle

It was concluded that the natural frequency of straightCNTs increases by increasing the chiral number of botharmchair and zigzag CNTs It was also revealed that thenatural frequency of CNTs (zigzag and armchair) with higherchirality decreases by introducing bending angles for all casesof boundary conditions However the change of increasingthe bending angle on the natural frequency of armchairand zigzag CNTs with lower number of chirality is almostnegligible The finding of this study may have useful effectson further investigations of CNTs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Iijima ldquoHelicalmicrotubules of graphitic carbonrdquoNature vol354 no 6348 pp 56ndash58 1991

Advances in Materials Science and Engineering 9

[2] S Imani Yengejeh M Akbar Zadeh and A Ochsner ldquoOn thebuckling behavior of connected carbon nanotubes with parallellongitudinal axesrdquo Applied Physics A vol 115 no 4 pp 1335ndash1344 2014

[3] S K Georgantzinos G I Giannopoulos and N K AnifantisldquoAn efficient numerical model for vibration analysis of single-walled carbon nanotubesrdquo Computational Mechanics vol 43no 6 pp 731ndash741 2009

[4] W Xia and L Wang ldquoVibration characteristics of fluid-conveying carbon nanotubes with curved longitudinal shaperdquoComputationalMaterials Science vol 49 no 1 pp 99ndash103 2010

[5] S Arghavan andAV Singh ldquoOn the vibrations of single-walledcarbon nanotubesrdquo Journal of Sound and Vibration vol 330 no13 pp 3102ndash3122 2011

[6] A Ghavamian and A Ochsner ldquoNumerical modeling of eigen-modes and eigenfrequencies of single- andmulti-walled carbonnanotubes under the influence of atomic defectsrdquo Computa-tional Materials Science vol 72 pp 42ndash48 2013

[7] F N Mayoof and M A Hawwa ldquoChaotic behavior of a curvedcarbon nanotube under harmonic excitationrdquo Chaos Solitonsand Fractals vol 42 no 3 pp 1860ndash1867 2009

[8] M Farsadi A Ochsner and M Rahmandoust ldquoNumericalinvestigation of composite materials reinforced with wavedcarbon nanotubesrdquo Journal of Composite Materials vol 47 no11 pp 1425ndash1434 2013

[9] M S Dresselhaus G Dresselhaus and R Saito ldquoPhysics ofcarbon nanotubesrdquo Carbon vol 33 no 7 pp 883ndash891 1995

[10] C Li and T-W Chou ldquoA structural mechanics approach for theanalysis of carbon nanotubesrdquo International Journal of Solidsand Structures vol 40 no 10 pp 2487ndash2499 2003

[11] S Melchor F J Martin-Martinez and J A Dobado ldquoCoNTubv20mdashalgorithms for constructing C3-symmetric models ofthree-nanotube junctionsrdquo Journal of Chemical Information andModeling vol 51 no 6 pp 1492ndash1505 2011

[12] Z Kang M Li and Q Tang ldquoBuckling behavior of carbonnanotube-based intramolecular junctions under compressionmolecular dynamics simulation and finite element analysisrdquoComputational Materials Science vol 50 no 1 pp 253ndash2592010

[13] T Irvine Application of the Newton-Raphson Method to Vibra-tion Problems Vibration Data Publications Madison WisUSA 1999

[14] J P Lu ldquoElastic properties of carbon nanotubes and nanoropesrdquoPhysical Review Letters vol 79 no 7 pp 1297ndash1300 1997

[15] CW S To ldquoBending and shear moduli of single-walled carbonnanotubesrdquo Finite Elements in Analysis and Design vol 42 no5 pp 404ndash413 2006

[16] A L Kalamkarov A V Georgiades S K Rokkam V P Veeduand M N Ghasemi-Nejhad ldquoAnalytical and numerical tech-niques to predict carbon nanotubes propertiesrdquo InternationalJournal of Solids and Structures vol 43 no 22-23 pp 6832ndash6854 2006

[17] G I Giannopoulos ldquoFullerenes as mass sensors a numericalinvestigationrdquo Physica E vol 56 pp 36ndash42 2014

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 7

350

360

370

380

390

400

410

0 10 20 30 40

Firs

t nat

ural

freq

uenc

y (G

Hz)

(14-14)(15-15)

(16-16)

Fixed-free

Bending angle of armchair CNTs (∘)

(a)

Fixed-free

Bending angle of zigzag CNTs (∘)

(16-0)(20-0)

(26-0)

240

260

280

300

320

340

360

380

400

0 10 20 30 40

Firs

t nat

ural

freq

uenc

y (G

Hz)

(b)

Fixed-fixed

(14-14)(15-15)

(16-16)

Bending angle of armchair CNTs (∘)

0185

0190

0195

0200

0205

0210

0215

0220

0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz)

(c)

(16-0)(20-0)

(26-0)

Fixed-fixed

Bending angle of zigzag CNTs (∘)

014

015

016

017

018

019

020

021

022

023

0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz)

(d)

(14-14)(15-15)

(16-16)

Bending angle of armchair CNTs (∘)

0190

0195

0200

0205

0210

0215

0220

0225

0230

0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz) Free-free

(e)

Bending angle of zigzag CNTs (∘)0 5 10 15 20 25 30 35 40 45

Firs

t nat

ural

freq

uenc

y (G

Hz) Free-free

(16-0)(20-0)

(26-0)

014

015

016

017

018

019

020

021

022

023

(f)

Figure 8 Change in natural frequency with (a) armchair and (b) zigzag CNTs for fixed-free with (c) armchair and (d) zigzag CNTs forfixed-fixed and with (e) armchair and (f) zigzag for free-free boundary conditions The markers represent the results of the computation thestraight lines are linear regressions

8 Advances in Materials Science and Engineering

350

360

370

380

390

400

410

0 5 10 15 20 25 30 35 40 45

Seco

nd n

atur

al fr

eque

ncy

(GH

z)

(14-14)(15-15)(16-16)

Bending angle of armchair CNTs (∘)

(a)

1700

1800

1900

2000

2100

2200

2300

0 5 10 15 20 25 30 35 40 45

Third

nat

ural

freq

uenc

y (G

Hz)

(14-14)(15-15)(16-16)

Bending angle of armchair CNTs (∘)

(b)

Figure 9 Change in higher natural frequency with armchair CNTs for fixed-free boundary condition (a) second natural frequency and (b)third natural frequency The markers represent the results of the computation the straight lines are linear regressions

2nd mode1st mode 3rd mode

xy

z

xy

z

xy

z

Figure 10 First three eigenmodes of a (14-14) CNT with 10∘ bending angle

It was concluded that the natural frequency of straightCNTs increases by increasing the chiral number of botharmchair and zigzag CNTs It was also revealed that thenatural frequency of CNTs (zigzag and armchair) with higherchirality decreases by introducing bending angles for all casesof boundary conditions However the change of increasingthe bending angle on the natural frequency of armchairand zigzag CNTs with lower number of chirality is almostnegligible The finding of this study may have useful effectson further investigations of CNTs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Iijima ldquoHelicalmicrotubules of graphitic carbonrdquoNature vol354 no 6348 pp 56ndash58 1991

Advances in Materials Science and Engineering 9

[2] S Imani Yengejeh M Akbar Zadeh and A Ochsner ldquoOn thebuckling behavior of connected carbon nanotubes with parallellongitudinal axesrdquo Applied Physics A vol 115 no 4 pp 1335ndash1344 2014

[3] S K Georgantzinos G I Giannopoulos and N K AnifantisldquoAn efficient numerical model for vibration analysis of single-walled carbon nanotubesrdquo Computational Mechanics vol 43no 6 pp 731ndash741 2009

[4] W Xia and L Wang ldquoVibration characteristics of fluid-conveying carbon nanotubes with curved longitudinal shaperdquoComputationalMaterials Science vol 49 no 1 pp 99ndash103 2010

[5] S Arghavan andAV Singh ldquoOn the vibrations of single-walledcarbon nanotubesrdquo Journal of Sound and Vibration vol 330 no13 pp 3102ndash3122 2011

[6] A Ghavamian and A Ochsner ldquoNumerical modeling of eigen-modes and eigenfrequencies of single- andmulti-walled carbonnanotubes under the influence of atomic defectsrdquo Computa-tional Materials Science vol 72 pp 42ndash48 2013

[7] F N Mayoof and M A Hawwa ldquoChaotic behavior of a curvedcarbon nanotube under harmonic excitationrdquo Chaos Solitonsand Fractals vol 42 no 3 pp 1860ndash1867 2009

[8] M Farsadi A Ochsner and M Rahmandoust ldquoNumericalinvestigation of composite materials reinforced with wavedcarbon nanotubesrdquo Journal of Composite Materials vol 47 no11 pp 1425ndash1434 2013

[9] M S Dresselhaus G Dresselhaus and R Saito ldquoPhysics ofcarbon nanotubesrdquo Carbon vol 33 no 7 pp 883ndash891 1995

[10] C Li and T-W Chou ldquoA structural mechanics approach for theanalysis of carbon nanotubesrdquo International Journal of Solidsand Structures vol 40 no 10 pp 2487ndash2499 2003

[11] S Melchor F J Martin-Martinez and J A Dobado ldquoCoNTubv20mdashalgorithms for constructing C3-symmetric models ofthree-nanotube junctionsrdquo Journal of Chemical Information andModeling vol 51 no 6 pp 1492ndash1505 2011

[12] Z Kang M Li and Q Tang ldquoBuckling behavior of carbonnanotube-based intramolecular junctions under compressionmolecular dynamics simulation and finite element analysisrdquoComputational Materials Science vol 50 no 1 pp 253ndash2592010

[13] T Irvine Application of the Newton-Raphson Method to Vibra-tion Problems Vibration Data Publications Madison WisUSA 1999

[14] J P Lu ldquoElastic properties of carbon nanotubes and nanoropesrdquoPhysical Review Letters vol 79 no 7 pp 1297ndash1300 1997

[15] CW S To ldquoBending and shear moduli of single-walled carbonnanotubesrdquo Finite Elements in Analysis and Design vol 42 no5 pp 404ndash413 2006

[16] A L Kalamkarov A V Georgiades S K Rokkam V P Veeduand M N Ghasemi-Nejhad ldquoAnalytical and numerical tech-niques to predict carbon nanotubes propertiesrdquo InternationalJournal of Solids and Structures vol 43 no 22-23 pp 6832ndash6854 2006

[17] G I Giannopoulos ldquoFullerenes as mass sensors a numericalinvestigationrdquo Physica E vol 56 pp 36ndash42 2014

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

8 Advances in Materials Science and Engineering

350

360

370

380

390

400

410

0 5 10 15 20 25 30 35 40 45

Seco

nd n

atur

al fr

eque

ncy

(GH

z)

(14-14)(15-15)(16-16)

Bending angle of armchair CNTs (∘)

(a)

1700

1800

1900

2000

2100

2200

2300

0 5 10 15 20 25 30 35 40 45

Third

nat

ural

freq

uenc

y (G

Hz)

(14-14)(15-15)(16-16)

Bending angle of armchair CNTs (∘)

(b)

Figure 9 Change in higher natural frequency with armchair CNTs for fixed-free boundary condition (a) second natural frequency and (b)third natural frequency The markers represent the results of the computation the straight lines are linear regressions

2nd mode1st mode 3rd mode

xy

z

xy

z

xy

z

Figure 10 First three eigenmodes of a (14-14) CNT with 10∘ bending angle

It was concluded that the natural frequency of straightCNTs increases by increasing the chiral number of botharmchair and zigzag CNTs It was also revealed that thenatural frequency of CNTs (zigzag and armchair) with higherchirality decreases by introducing bending angles for all casesof boundary conditions However the change of increasingthe bending angle on the natural frequency of armchairand zigzag CNTs with lower number of chirality is almostnegligible The finding of this study may have useful effectson further investigations of CNTs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Iijima ldquoHelicalmicrotubules of graphitic carbonrdquoNature vol354 no 6348 pp 56ndash58 1991

Advances in Materials Science and Engineering 9

[2] S Imani Yengejeh M Akbar Zadeh and A Ochsner ldquoOn thebuckling behavior of connected carbon nanotubes with parallellongitudinal axesrdquo Applied Physics A vol 115 no 4 pp 1335ndash1344 2014

[3] S K Georgantzinos G I Giannopoulos and N K AnifantisldquoAn efficient numerical model for vibration analysis of single-walled carbon nanotubesrdquo Computational Mechanics vol 43no 6 pp 731ndash741 2009

[4] W Xia and L Wang ldquoVibration characteristics of fluid-conveying carbon nanotubes with curved longitudinal shaperdquoComputationalMaterials Science vol 49 no 1 pp 99ndash103 2010

[5] S Arghavan andAV Singh ldquoOn the vibrations of single-walledcarbon nanotubesrdquo Journal of Sound and Vibration vol 330 no13 pp 3102ndash3122 2011

[6] A Ghavamian and A Ochsner ldquoNumerical modeling of eigen-modes and eigenfrequencies of single- andmulti-walled carbonnanotubes under the influence of atomic defectsrdquo Computa-tional Materials Science vol 72 pp 42ndash48 2013

[7] F N Mayoof and M A Hawwa ldquoChaotic behavior of a curvedcarbon nanotube under harmonic excitationrdquo Chaos Solitonsand Fractals vol 42 no 3 pp 1860ndash1867 2009

[8] M Farsadi A Ochsner and M Rahmandoust ldquoNumericalinvestigation of composite materials reinforced with wavedcarbon nanotubesrdquo Journal of Composite Materials vol 47 no11 pp 1425ndash1434 2013

[9] M S Dresselhaus G Dresselhaus and R Saito ldquoPhysics ofcarbon nanotubesrdquo Carbon vol 33 no 7 pp 883ndash891 1995

[10] C Li and T-W Chou ldquoA structural mechanics approach for theanalysis of carbon nanotubesrdquo International Journal of Solidsand Structures vol 40 no 10 pp 2487ndash2499 2003

[11] S Melchor F J Martin-Martinez and J A Dobado ldquoCoNTubv20mdashalgorithms for constructing C3-symmetric models ofthree-nanotube junctionsrdquo Journal of Chemical Information andModeling vol 51 no 6 pp 1492ndash1505 2011

[12] Z Kang M Li and Q Tang ldquoBuckling behavior of carbonnanotube-based intramolecular junctions under compressionmolecular dynamics simulation and finite element analysisrdquoComputational Materials Science vol 50 no 1 pp 253ndash2592010

[13] T Irvine Application of the Newton-Raphson Method to Vibra-tion Problems Vibration Data Publications Madison WisUSA 1999

[14] J P Lu ldquoElastic properties of carbon nanotubes and nanoropesrdquoPhysical Review Letters vol 79 no 7 pp 1297ndash1300 1997

[15] CW S To ldquoBending and shear moduli of single-walled carbonnanotubesrdquo Finite Elements in Analysis and Design vol 42 no5 pp 404ndash413 2006

[16] A L Kalamkarov A V Georgiades S K Rokkam V P Veeduand M N Ghasemi-Nejhad ldquoAnalytical and numerical tech-niques to predict carbon nanotubes propertiesrdquo InternationalJournal of Solids and Structures vol 43 no 22-23 pp 6832ndash6854 2006

[17] G I Giannopoulos ldquoFullerenes as mass sensors a numericalinvestigationrdquo Physica E vol 56 pp 36ndash42 2014

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 9

[2] S Imani Yengejeh M Akbar Zadeh and A Ochsner ldquoOn thebuckling behavior of connected carbon nanotubes with parallellongitudinal axesrdquo Applied Physics A vol 115 no 4 pp 1335ndash1344 2014

[3] S K Georgantzinos G I Giannopoulos and N K AnifantisldquoAn efficient numerical model for vibration analysis of single-walled carbon nanotubesrdquo Computational Mechanics vol 43no 6 pp 731ndash741 2009

[4] W Xia and L Wang ldquoVibration characteristics of fluid-conveying carbon nanotubes with curved longitudinal shaperdquoComputationalMaterials Science vol 49 no 1 pp 99ndash103 2010

[5] S Arghavan andAV Singh ldquoOn the vibrations of single-walledcarbon nanotubesrdquo Journal of Sound and Vibration vol 330 no13 pp 3102ndash3122 2011

[6] A Ghavamian and A Ochsner ldquoNumerical modeling of eigen-modes and eigenfrequencies of single- andmulti-walled carbonnanotubes under the influence of atomic defectsrdquo Computa-tional Materials Science vol 72 pp 42ndash48 2013

[7] F N Mayoof and M A Hawwa ldquoChaotic behavior of a curvedcarbon nanotube under harmonic excitationrdquo Chaos Solitonsand Fractals vol 42 no 3 pp 1860ndash1867 2009

[8] M Farsadi A Ochsner and M Rahmandoust ldquoNumericalinvestigation of composite materials reinforced with wavedcarbon nanotubesrdquo Journal of Composite Materials vol 47 no11 pp 1425ndash1434 2013

[9] M S Dresselhaus G Dresselhaus and R Saito ldquoPhysics ofcarbon nanotubesrdquo Carbon vol 33 no 7 pp 883ndash891 1995

[10] C Li and T-W Chou ldquoA structural mechanics approach for theanalysis of carbon nanotubesrdquo International Journal of Solidsand Structures vol 40 no 10 pp 2487ndash2499 2003

[11] S Melchor F J Martin-Martinez and J A Dobado ldquoCoNTubv20mdashalgorithms for constructing C3-symmetric models ofthree-nanotube junctionsrdquo Journal of Chemical Information andModeling vol 51 no 6 pp 1492ndash1505 2011

[12] Z Kang M Li and Q Tang ldquoBuckling behavior of carbonnanotube-based intramolecular junctions under compressionmolecular dynamics simulation and finite element analysisrdquoComputational Materials Science vol 50 no 1 pp 253ndash2592010

[13] T Irvine Application of the Newton-Raphson Method to Vibra-tion Problems Vibration Data Publications Madison WisUSA 1999

[14] J P Lu ldquoElastic properties of carbon nanotubes and nanoropesrdquoPhysical Review Letters vol 79 no 7 pp 1297ndash1300 1997

[15] CW S To ldquoBending and shear moduli of single-walled carbonnanotubesrdquo Finite Elements in Analysis and Design vol 42 no5 pp 404ndash413 2006

[16] A L Kalamkarov A V Georgiades S K Rokkam V P Veeduand M N Ghasemi-Nejhad ldquoAnalytical and numerical tech-niques to predict carbon nanotubes propertiesrdquo InternationalJournal of Solids and Structures vol 43 no 22-23 pp 6832ndash6854 2006

[17] G I Giannopoulos ldquoFullerenes as mass sensors a numericalinvestigationrdquo Physica E vol 56 pp 36ndash42 2014

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials