In the current work, the vibration characteristics of single-walled carbon nanotubes (SWCNTs) under different boundary conditions are investigated. A nonlocal elastic shell model is utilized, which accounts for the small scale effects and encompasses its classical continuum counterpart as a particular case. The variational form of the Flugge type equations is constructed to which the analytical Rayleigh–Ritz method is applied. Comprehensive results are attained for the resonant frequencies of vibrating SWCNTs. The significance of the small size effects on the resonant frequencies of SWCNTs is shown to be dependent on the geometric parameters of nanotubes. The effectiveness of the present analytical solution is assessed by the molecular dynamics simulations as a benchmark of good accuracy. It is found that, in contrast to the chirality, the boundary conditions have a significant effect on the appropriate values of nonlocal parameter.

References

References
1.
Radushkevich
,
L. V.
, and
Lukyanovich
,
V. M.
, 1952, “
O Strukture Ugleroda, Obrazujucegosja Pri Termiceskom Razlozenii Okisi Ugleroda Na Zeleznom Kontakte
,”
Zurn. Fisic. Chim.
,
26
, pp.
88
95
.
2.
Iijima
,
S.
, 1991, “
Helical Microtubes of Graphitic Carbon
,”
Nature (London)
,
354
, pp.
56
58
.
3.
Iijima
,
S.
,
Brabec
,
C.
,
Maiti
,
A.
, and
Bernholc
,
J.
, 1996, “
Structural Flexibility of Carbon Nanotubes
,”
J. Chem. Phys.
,
104
, pp.
2089
2092
.
4.
Yakobson
,
B. I.
,
Campbell
,
M. P.
,
Brabecand
,
C. J.
, and
Bernholc
,
J.
, 1997, “
High Strain Rate Fracture and C-Chain Unraveling in Carbon Nanotubes
,”
Comput. Mater. Sci.
,
8
, pp.
341
348
.
5.
Hernandez
,
E.
,
Goze
,
C.
,
Bernier
,
P.
, and
Rubio
,
A.
, 1998, “
Elastic Properties of C and BxCyNz Composite Nanotubes
,”
Phys. Rev. Lett.
,
80
, pp.
4502
4505
.
6.
Sanchez-Portal
,
D.
,
Artacho
,
E.
,
Soler
,
J. M.
,
Rubio
,
A.
, and
Ordejón
,
P.
, 1999, “
Ab Initio Structural, Elastic, and Vibrational Properties of Carbon Nanotubes
,”
Phys. Rev. B
,
59
, pp.
12678
12688
.
7.
Qian
,
D.
,
Wagner
,
J. G.
,
Liu
,
W. K.
,
Yu
,
M. F.
, and
Ruoff
,
R. S.
, 2002, “
Mechanics of Carbon Nanotubes
,”
Appl. Mech. Rev.
,
55
, pp.
495
533
.
8.
Yakobson
,
B. I.
,
Brabec
,
C. J.
, and
Bernholc
,
J.
, 1996, “
Nanomechanics of Carbon Tubes: Instability Beyond Linear Response
,”
Phys. Rev. Lett.
,
76
, pp.
2511
2514
.
9.
Pantano
,
A.
,
Boyce
,
M. C.
, and
Parks
,
D. M.
, 2004, “
Mechanics of Axial Compression of Single and Multi-Wall Carbon Nanotubes
,”
ASME J. Eng. Mater. Technol.
,
126
, pp.
279
284
.
10.
Behfar
,
K.
, and
Naghdabadi
,
R.
, 2005, “
Nanoscale Vibrational Analysis of a Multi-Layered Graphene Sheet Embedded in an Elastic Medium
,”
Compos. Sci. Technol.
,
65
, pp.
1159
1164
.
11.
Yao
,
X.
, and
Han
,
Q.
, 2006, “
Buckling Analysis of Multiwalled Carbon Nanotubes Under Torsional Load Coupling With Temperature Change
,”
ASME J. Eng. Mater. Technol.
,
128
, pp.
419
427
.
12.
Wang
,
C. M.
,
Tan
,
V. B. C.
, and
Zhang
,
Y. Y.
, 2006, “
Timoshenko Beam Model for Vibration Analysis of Multi-Walled Carbon Nanotubes
,”
J. Sound Vib.
,
294
, pp.
1060
1072
.
13.
Wang
,
L.
,
Ni
,
Q.
,
Li
,
M.
, and
Qian
,
Q.
, 2008, “
The Thermal Effect on Vibration and Instability of Carbon Nanotubes Conveying Fluid
,”
Physica E
,
40
, pp.
3179
3182
.
14.
Ansari
,
R.
,
Hemmatnezhad
,
M.
, and
Ramezannezhad
,
H.
, 2009, “
Application of HPM to the Nonlinear Vibrations of Multiwalled Carbon Nanotubes
,”
Numer. Methods Partial Differ. Equ.
,
26
, pp.
490
500
.
15.
Lu
,
W. B.
,
Wu
,
J.
,
Feng
,
X.
,
Hwang
,
K. C.
, and
Huang
,
Y.
, 2010, “
Buckling Analyses of Double-Wall Carbon Nanotubes: A Shell Theory Based on the Interatomic Potential
,”
ASME J. Appl. Mech.
,
77
, p.
061016
.
16.
He
,
X. Q.
,
Kitipornchai
,
S.
,
Wang
,
C. M.
,
Xiang
,
Y.
, and
Zhou
,
Q.
, 2010, “
A Nonlinear Van Der Waals Force Model for Multiwalled Carbon Nanotubes Modeled by a Nested System of Cylindrical Shells
,”
ASME J. Appl. Mech.
,
77
, p.
061006
.
17.
Ansari
,
R.
, and
Rouhi
,
S.
, 2010, “
Atomistic Finite Element Model for Axial Buckling of Single-Walled Carbon Nanotubes
,”
Physica E
,
43
, pp.
58
69
.
18.
Ansari
,
R.
,
Hemmatnezhad
,
M.
, and
Rezapour
,
J.
, 2011, “
The Thermal Effect on Nonlinear Oscillations of Carbon Nanotubes with Arbitrary Boundary Conditions
,”
Curr. Appl. Phys.
,
11
, pp.
692
697
.
19.
Ansari
,
R.
, and
Hemmatnezhad
,
M.
, 2011, “
Nonlinear Vibrations of Embedded Multiwalled Carbon Nanotubes Using a Variational Approach
,”
Math. Comput. Model.
,
53
, pp.
927
38
.
20.
Eringen
,
A. C.
, 1983, “
On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves
,”
J. Appl. Phys.
,
54
, pp.
4703
4710
.
21.
Eringen
,
A. C.
, 2002,
Nonlocal Continuum Field Theories
,
Springer
,
New York.
22.
Aifantis
,
E. C.
, 1984, “
On the Microstructural Origin of Certain Inelastic Models
,”
ASME J. Eng. Mater. Technol.
,
106
, pp.
326
330
.
23.
Peddieson
,
J.
,
Buchanan
,
G. R.
, and
McNitt
,
R. P.
, 2003, “
Application of Nonlocal Continuum Models to Nanotechnology
,”
Int. J. Eng. Sci.
,
41
, pp.
305
312
.
24.
Sudak
,
L. J.
, 2003, “
Column Buckling of Multiwalled Carbon Nanotubes Using Nonlocal Continuum Mechanics
,”
J. Appl. Phys.
,
94
, pp.
7281
7287
.
25.
Li
,
R.
, and
Kardomateas
,
G. A.
, 2007, “
Vibration Characteristics of Multiwalled Carbon Nanotubes Embedded in Elastic Media by a Nonlocal Elastic Shell Model
,”
ASME J. Appl. Mech.
,
74
, pp.
1087
1094
.
26.
Li
,
R.
, and
Kardomateas
,
G. A.
, 2007, “
Thermal Buckling of Multi-Walled Carbon Nanotubes by Nonlocal Elasticity
,”
ASME J. Appl. Mech.
,
74
, pp.
399
405
.
27.
Hu
,
Y. G.
,
Liew
,
K. M.
,
Wang
,
Q.
,
He
,
X. Q.
, and
Yakobson
,
B. I.
, 2008, “
Nonlocal Shell Model for Elastic Wave Propagation in Single- and Double-Walled Carbon Nanotubes
,”
J. Mech. Phys. Solids
,
56
, pp.
3475
3485
.
28.
Shen
,
H. S.
, and
Zhang
,
C. L.
, 2010, “
Nonlocal Shear Deformable Shell Model for Post-Buckling of Axially Compressed Double-Walled Carbon Nanotubes Embedded in an Elastic Matrix
,”
ASME J. Appl. Mech.
,
77
, p.
041006
.
29.
Ansari
,
R.
,
Rajabiehfard
,
R.
, and
Arash
,
B.
, 2010, “
Nonlocal Finite Element Model for Vibrations of Embedded Multi-Layered Graphene Sheets
,”
Comput. Mater. Sci.
,
49
, pp.
831
838
.
30.
Arash
,
B.
, and
Ansari
,
R.
, 2010, “
Evaluation of Nonlocal Parameter in the Vibrations of Single-Walled Carbon Nanotubes with Initial Strain
,”
Physica E
,
42
, pp.
2058
2064
.
31.
Ansari
,
R.
,
Sahmani
,
S.
, and
Arash
,
B.
, 2010, “
Nonlocal Plate Model for Free Vibrations of Single-Layered Graphene Sheets
,”
Phys. Lett. A
,
375
, pp.
53
62
.
32.
Ansari
,
R.
,
Sahmani
,
S.
, and
Rouhi
,
H.
, 2011, “
Rayleigh–Ritz Axial Buckling Analysis of Single-Walled Carbon Nanotubes with Different Boundary Conditions
,”
Phys. Lett. A
,
375
, pp.
1255
1263
.
33.
Flugge
,
W.
, 1960,
Stresses in Shells
,
Springer
,
Berlin
.
34.
Adali
,
S.
, 2008, “
Variational Principles for Multi-Walled Carbon Nanotubes Undergoing Buckling Based on Nonlocal Elasticity Theory
,”
Phys. Lett. A
,
372
, pp.
5701
5705
.
35.
Loy
,
C. T.
, and
Lam
,
K. Y.
, 1997, “
Vibration of Cylindrical Shells with Ring Support
,”
Int. J. Mech. Sci.
,
39
, pp.
445
471
.
36.
Lennard-Jones
,
J. E.
, 1924, “
On the Determination of Molecular Fields II. From the Equation of State of a Gas
,” Proceedings of the Royal Society of London, the Royal Society,
106
, pp.
463
477
.
37.
Tersoff
,
J.
, 1989, “
Modeling Solid-State Chemistry: Interatomic Potentials for Multicomponent Systems
,”
Phys. Rev. B
,
39
, pp.
5566
5568
.
38.
Nanorex Inc., 2005, “NanoHive-1 v.1.2.0-b1,” www.nanoengineer-1.comwww.nanoengineer-1.com.
39.
Stuart
,
S. J.
,
Tutein
,
A. B.
, and
Harrison
,
J. A.
, 2000, “
A Reactive Potential for Hydrocarbons with Intermolecular Interactions
,”
J. Chem. Phys.
,
112
, pp.
6472
6486
.
40.
Batra
R. C.
, and
Gupta
,
S. S.
, 2008, “
Wall Thickness and Radial Breathing Modes of Single-Walled Carbon Nanotubes
,”
ASME J. Appl. Mech.
,
75
, p.
061010
.
41.
Wang
,
C. Y.
, and
Zhang
,
L. C.
, 2008, “
An Elastic Shell Model for Characterizing Single-Walled Carbon Nanotubes
,”
Nanotechnology
,
19
, p.
195704
.
42.
Yan
,
Y.
,
Wang
,
W. Q.
, and
Zhang
,
L. X.
, 2010, “
Nonlocal Effect on Axially Compressed Buckling of Triple-Walled Carbon Nanotubes Under Temperature Field
,”
Appl. Math. Model.
,
34
, pp.
3422
3429
.
You do not currently have access to this content.