This paper reviews recent research studies on the buckling of carbon nanotubes. The structure and properties of carbon nanotubes are introduced to the readers. The various buckling behaviors exhibited by carbon nanotubes are also presented herein. The main factors, such as dimensions, boundary conditions, temperature, strain rate, and chirality, influencing the buckling behaviors are also discussed, as well as a brief introduction of the two most used methods for analyzing carbon nanotubes, i.e., continuum models and atomistic simulations. Summary and recommendations for future research are also given. Finally, a large body of papers is given in the reference section. It is hoped that this paper provides current knowledge on the buckling of carbon nanotubes, reviews the computational methods for determining the buckling loads, and inspires researchers to further investigate the buckling properties of carbon nanotubes for practical applications.
Issue Section:
Review Articles
1.
Iijima
, S.
, 1991, “Helical Microtubules of Graphitic Carbon
,” Nature (London)
0028-0836, 354
, pp. 56
–58
.2.
Iijima
, S.
, and Ichihashi
, T.
, 1993, “Single-Shell Carbon Nanotubes of 1-nm Diameter
,” Nature (London)
0028-0836, 363
, pp. 603
–605
.3.
Bethune
, D. S.
, Kiang
, C. H.
, de Vries
, M. S.
, Gorman
, G.
, Savoy
, R.
, Vazquez
, J.
, and Beyers
, R.
, 1993, “Cobalt-Catalyzed Growth of Carbon Nanotubes With Single-Atomic-Layer Walls
,” Nature (London)
0028-0836, 363
, pp. 605
–607
.4.
Dresselhaus
, M. S.
, Dressehaus
, G.
, and Saito
, R.
, 1995, “Physics of Carbon Nanotubes
,” Carbon
0008-6223, 33
(7
), pp. 883
–891
.5.
Ebbesen
, T. W.
, 1996, “Carbon Nanotubes
,” Phys. Today
0031-9228, 49
(6
), pp. 26
–32
.6.
Yakobson
, B. I.
, and Richard
, E. S.
, 1997, “Fullerene Nanotubes: C1,000,000 and Beyond
,” Am. Sci.
0003-0996, 85
, pp. 324
–337
.7.
Ball
, P.
, 2001, “Roll Up for the Revolution
,” Nature (London)
0028-0836, 414
, pp. 142
–144
.8.
Thostenson
, E. T.
, Ren
, Z. F.
, and Chou
, T. W.
, 2001, “Advances in the Science and Technology of Carbon Nanotubes and Their Composites: A Review
,” Compos. Sci. Technol.
0266-3538, 61
, pp. 1899
–1912
.9.
Baughman
, R. H.
, Zakhidov
, A. A.
, and de Heer
, W. A.
, 2002, “Carbon Nanotube-Route Toward Applications
,” Science
0036-8075, 297
, pp. 787
–792
.10.
Lau
, K. T.
, Gu
, C.
, and Hui
, D.
, 2006, “A Critical Review on Nanotube and Nanotube/Nanoclay Related Polymer Composite Materials
,” Composites, Part B
1359-8368, 37
, pp. 425
–436
.11.
Qian
, D.
, Wagner
, G. J.
, Liu
, W. K.
, Yu
, M. F.
, and Ruoff
, R. S.
, 2002, “Mechanics of Carbon Nanotubes
,” Appl. Mech. Rev.
0003-6900, 55
, pp. 495
–533
.12.
Saito
, Y.
, Yoshikawa
, T.
, Bandow
, S.
, Tomita
, M.
, and Hayashi
, T.
, 1993, “Interlayer Spacing in Carbon Nanotubes
,” Phys. Rev. B
0163-1829, 48
, pp. 1907
–1909
.13.
Charlier
, C.
, and Michenaud
, J. P.
, 1993, “Energetics of Multilayered Carbon Tubules
,” Phys. Rev. Lett.
0031-9007, 70
, pp. 1858
–1861
.14.
Kiang
, C. H.
, Endo
, M.
, Ajayan
, P. M.
, Dresselhaus
, G.
, and Dresselhaus
, M. S.
, 1998, “Size Effects in Carbon Nanotubes
,” Phys. Rev. Lett.
0031-9007, 81
, pp. 1869
–1872
.15.
Yu
, M. F.
, Yakobson
, B. I.
, and Ruoff
, R. S.
, 2000, “Controlled Sliding and Pullout of Nested Shells in Individual Multiwalled Carbon Nanotubes
,” J. Phys. Chem. B
1089-5647, 104
, pp. 8764
–8767
.16.
Kolmogorov
, A. N.
, and Crespi
, V. H.
, 2000, “Smoothest Bearings: Interlayer Sliding in Multiwalled Carbon Nanotubes
,” Phys. Rev. Lett.
0031-9007, 85
, pp. 4727
–4730
.17.
Cumings
, J.
, and Zettl
, A.
, 2000, “Low-Friction Nanoscale Linear Bearing Realized From Multiwall Carbon Nanotubes
,” Science
0036-8075, 289
, pp. 602
–604
.18.
Lide
, D. R.
, 1994, CRC Handbook of Chemistry and Physics
, 75th ed., CRC
, Boca Raton, FL
.19.
Dresselhaus
, M. S.
, Dressulhaus
, G.
, and Avouris
, P.
, 2001, Carbon Nanotubes
, Springer
, Heidelberg
.20.
Collins
, P. G.
, and Avouris
, P.
, 2000, “Nanotubes for Electronics
,” Sci. Am.
0036-8733, 283
, pp. 62
–69
.21.
Che
, J. W.
, Cagin
, T.
, and Goddard
, W. A.
, 2000, “Thermal Conductivity of Carbon Nanotubes
,” Nanotechnology
0957-4484, 11
, pp. 65
–69
.22.
Berber
, S.
, Kwon
, Y. K.
, and Tománek
, D.
, 2000, “Unusually High Thermal Conductivity of Carbon Nanotubes
,” Phys. Rev. Lett.
0031-9007, 84
(20
), pp. 4613
–4616
.23.
Osman
, M. A.
, and Srivastava
, D.
, 2001, “Temperature Dependence of the Thermal Conductivity of Carbon Nanotubes
,” Nanotechnology
0957-4484, 12
(1
), pp. 21
–22
.24.
Kim
, P.
, and Lieber
, C. M.
, 1999, “Nanotube Nanotweezers
,” Science
0036-8075, 286
, pp. 2148
–2150
.25.
Dai
, H. J.
, Hafner
, J. H.
, Rinzler
, A. G.
, Colbert
, D. T.
, and Smalley
, R. E.
, 1996, “Nanotubes as Nanoprobes in Scanning Probe Microscopy
,” Nature (London)
0028-0836, 384
, pp. 147
–150
.26.
Sharke
, P.
, 2001, “Hybrid NEMS
,” ASME Mechanical Engineering-CIME
, Vol. 123
, pp. 42
–45
.27.
Majumdar
, A.
, 2001, “Not Without Engineering—Promises and Challenges of Nanotechnology
,” ASME Mechanical Engineering
, Vol. 123
, pp. 46
–49
.28.
Maney
, K.
, 2000, “Nanotech Is Near: Celebrate or Take Cover? Scientists Explore New Frontier of Promising Microscopic Technology
,” USA Today, Oct. 24.29.
Zhao
, Q.
, Gan
, Z. H.
, and Zhuang
, O. K.
, 2002, “Electrochemical Sensors Based on Carbon Nanotubes
,” Electroanalysis
1040-0397, 14
, pp. 1609
–1613
.30.
Mahar
, B.
, Laslau
, C.
, Yip
, R.
, and Sun
, Y.
, 2007, “Development of Carbon Nanotube-Based Sensors—A Review
,” IEEE Sens. J.
1530-437X, 7
(2
), pp. 266
–284
.31.
Baughman
, R. H.
, Cui
, C. X.
, Zakhidov
, A. A.
, Iqbal
, Z.
, Barisci
, J. N.
, Spinks
, G. M.
, Wallace
, G. G.
, Mazzoldi
, A.
, De Rossi
, D.
, Rinzler
, A. G.
, Jaschinski
, O.
, Roth
, S.
, and Kertesz
, M.
, 1999, “Carbon Nanotube Acuators
,” Science
0036-8075, 284
(5418
), pp. 1340
–1344
.32.
Minett
, A.
, Fraysse
, J.
, Gang
, G.
, Kim
, G. T.
, and Roth
, S.
, 2002, “Nanotube Actuators for Nanomechanics
,” Curr. Appl. Phys.
1567-1739, 2
(1
), pp. 61
–64
.33.
Yang
, X. B.
, and Ni
, J.
, 2006, “Calculations of Hydrogen Coverage on Single-Walled Carbon Nanotubes: Dependence on Nanotube Size, Temperature, and Pressure
,” Phys. Rev. B
0163-1829, 74
, p. 195437
.34.
Ye
, X.
, Gu
, X.
, Gong
, X. G.
, Shing
, T. K. M.
, and Liu
, Z. F.
, 2007, “A Nanocontainer for the Storage of Hydrogen
,” Carbon
0008-6223, 45
(2
), pp. 315
–320
.35.
Treacy
, M. M. J.
, Ebbesen
, T. W.
, and Gibson
, J. M.
, 1996, “Exceptionally High Young’s Modulus Observed for Individual Carbon Nanotubes
,” Nature (London)
0028-0836, 381
, pp. 678
–680
.36.
Wong
, E. W.
, Sheehan
, P. E.
, and Lieber
, C. M.
, 1997, “Nanobeam Mechanics: Elasticity, Strength, and Toughness of Nanorods and Nanotubes
,” Science
0036-8075, 277
, pp. 1971
–1975
.37.
Lu
, J. P.
, 1997, “Elastic Properties of Carbon Nanotubes and Nanoropes
,” Phys. Rev. Lett.
0031-9007, 79
, pp. 1297
–1300
.38.
Krishnan
, A.
, Dujardin
, E.
, Ebbesen
, T. W.
, Yianilos
, P. N.
, and Treacy
, M. M. J.
, 1998, “Young’s modulus of single-walled nanotubes
,” Phys. Rev. B
0163-1829, 58
, pp. 14013
–14019
.39.
Yao
, N.
, and Lordi
, V.
, 1998, “Young’s Modulus of Single-Walled Carbon Nanotubes
,” J. Appl. Phys.
0021-8979, 84
, pp. 1939
–1943
.40.
Hernández
, E.
, Goze
, C.
, Bernier
, P.
, and Rubio
, A.
, 1998, “Elastic Properties of C and BxCyNz Composite Nanotubes
,” Phys. Rev. Lett.
0031-9007, 80
, pp. 4502
–4505
.41.
Salvetat
, J. P.
, Bonard
, J. M.
, Thomson
, N. H.
, Kulik
, A. J.
, Forro
, L.
, Benoit
, W.
, and Zuppiroli
, L.
, 1999, “Mechanical Properties of Carbon Nanotubes
,” Appl. Phys. A
0721-7250, 69
, pp. 255
–260
.42.
Sánchez-Portal
, D.
, Artacho
, D.
, Soler
, J. M.
, Rubio
, A.
, and Ordejon
, P.
, 1999, “Ab Initio Structural, Elastic, and Vibrational Properties of Carbon Nanotubes
,” Phys. Rev. B
0163-1829, 59
, p. 12678
.43.
Popov
, V. N.
, Van Doren
, V. E.
, and Balkanski
, M.
, 2000, “Elastic Properties of Single-Walled Carbon Nanotubes
,” Phys. Rev. B
0163-1829, 61
, pp. 3078
–3084
.44.
Kudin
, K. N.
, Scuseria
, G. E.
, and Yakobson
, B. I.
, 2001, “C2F, BN, and C Nanoshell Elasticity From Ab Initio Computations
,” Phys. Rev. B
0163-1829, 64
, p. 235406
.45.
Li
, C. Y.
, and Chou
, T. W.
, 2003, “A Structural Mechanics Approach for the Analysis of Carbon Nanotubes
,” Int. J. Solids Struct.
0020-7683, 40
, pp. 2487
–2499
.46.
Sears
, A.
, and Batra
, R. C.
, 2004, “Macroscopic Properties of Carbon Nanotubes From Molecular-Mechanics Simulations
,” Phys. Rev. B
0163-1829, 69
, p. 235406
.47.
Li
, C. Y.
, and Chou
, T. W.
, 2004, “Elastic Properties of single-Walled Carbon Nanotubes in Transverse Directions
,” Phys. Rev. B
0163-1829, 69
, p. 073401
.48.
Shen
, L. X.
, and Li
, J.
, 2005, “Transversely Isotropic Elastic Properties of Multiwalled Carbon Nanotubes
,” Phys. Rev. B
0163-1829, 71
, p. 035412
.49.
Yakobson
, B. I.
, Brabec
, C. J.
, and Bernholc
, J.
, 1996, “Nanomechanics of Carbon Tubes: Instabilities Beyond Linear Response
,” Phys. Rev. Lett.
0031-9007, 76
, pp. 2511
–2514
.50.
Xiao
, T.
, Xu
, X. J.
, and Liao
, K.
, 2004, “Characterization of Nonlinear Elasticity and Elastic Instability in Single-Walled Carbon Nanotubes
,” J. Appl. Phys.
0021-8979, 95
, pp. 8145
–8148
.51.
Shen
, H. S.
, and Zhang
, C. L.
, 2007, “Postbuckling of Double-Walled Carbon Nanotubes With Temperature Dependent Properties and Initial Defects Under Combined Axial and Radial Mechanical Loads
,” Int. J. Solids Struct.
0020-7683, 44
, pp. 1461
–1487
.52.
Wang
, Q.
, Duan
, W. H.
, Liew
, K. M.
, and He
, X. Q.
, 2007, “Inelastic Buckling of Carbon Nanotubes
,” Appl. Phys. Lett.
0003-6951, 90
, p. 033110
.53.
Yu
, M. F.
, Lourie
, O.
, Dyer
, M. J.
, Moloni
, K.
, Kelly
, T. F.
, and Ruoff
, R. S.
, 2000, “Strength and Breaking Mechanism of Multi-Walled Carbon Nanotubes Under Tensile Load
,” Science
0036-8075, 287
, pp. 637
–640
.54.
Yakobson
, B. I.
, Campbell
, M. P.
, Brabec
, C. J.
, and Bernholc
, J.
, 1997, “High Strain Rate Fracture and C-Chain Unraveling in Carbon Nanotubes
,” Comput. Mater. Sci.
0927-0256, 8
, pp. 341
–348
.55.
Yakobson
, B. I.
, 1998, “Mechanical Relaxation and ‘Intramolecular Plasticity’ in Carbon Nanotubes
,” Appl. Phys. Lett.
0003-6951, 72
, pp. 918
–920
.56.
Dumitrică
, T.
, Belytschko
, T.
, and Yakobson
, B. I.
, 2003, “Bond-Breaking Bifurcation States in Carbon Nanotube Fracture
,” J. Chem. Phys.
0021-9606, 118
, pp. 9485
–9488
.57.
Belytschko
, T.
, Xiao
, S. P.
, Schatz
, G. C.
, and Ruoff
, R. S.
, 2002, “Atomistic Simulation of Nanotubes Frature
,” Phys. Rev. B
0163-1829, 65
, p. 235430
.58.
Zhang
, P.
, Jiang
, H.
, Huang
, Y.
, Geubelle
, P. H.
, and Huang
, K. C.
, 2004, “An Atomistic-Based Continuum Theory for Carbon Nanotubes: Analysis of Fracture Nucleation
,” J. Mech. Phys. Solids
0022-5096, 52
, pp. 977
–998
.59.
Dresselhaus
, M. S.
, and Eklund
, P. C.
, 2000, “Phonons in Carbon Nanotubes
,” Adv. Phys.
0001-8732, 49
, pp. 705
–814
.60.
Gibson
, R. F.
, Ayorinde
, E. O.
, and Wen
, Y. F.
, 2007, “Vibrations of Carbon Nanotubes and Their Composites: A Review
,” Compos. Sci. Technol.
0266-3538, 67
, pp. 1
–28
.61.
Iijima
, S.
, Brabec
, C.
, Maiti
, A.
, and Bernholc
, J.
, 1996, “Structural Flexibility of Carbon Nanotubes
,” J. Chem. Phys.
0021-9606, 104
, pp. 2089
–2092
.62.
Falvo
, M. R.
, Clary
, G. J.
, Taylor
, R. M.
, Chi
, V.
, Brooks
, F. P.
, Jr., Washburn
, S.
, and Superfine
, R.
, 1997, “Bending and Buckling of Carbon Nanotubes Under Large Strain
,” Nature (London)
0028-0836, 389
, pp. 582
–584
.63.
Knechtel
, W. H.
, Dusberg
, G. S.
, Blau
, W. J.
, Hernandez
, E.
, and Rubio
, A.
, 1998, “Reversible Bending of Carbon Nanotubes Using a Transmission Electron Microscope
,” Appl. Phys. Lett.
0003-6951, 73
, pp. 1961
–1963
.64.
Lourie
, O.
, Cox
, D. M.
, and Wagner
, H. D.
, 1998, “Buckling and Collapse of Embedded Carbon Nanotubes
,” Phys. Rev. Lett.
0031-9007, 81
, pp. 1638
–1641
.65.
Bower
, C.
, Rosen
, R.
, Jin
, L.
, Han
, J.
, and Zhou
, O.
, 1999, “Deformation of Carbon Nanotubes in Nanotube-Polymer Composites
,” Appl. Phys. Lett.
0003-6951, 74
, pp. 3317
–3319
.66.
Poncharal
, P.
, Wang
, Z. L.
, Ugarte
, D.
, and de Heer
, W. A.
, 1999, “Electrostatic Deflections and Electromechanical Resonances of Carbon Nanotubes
,” Science
0036-8075, 283
, pp. 1513
–1516
.67.
Peters
, M. J.
, McNeil
, L. E.
, Lu
, J. P.
, and Kahan
, D.
, 2000, “Structural Phase Transition in Carbon Nanotube Bundles Under Pressure
,” Phys. Rev. B
0163-1829, 61
, pp. 5939
–5944
.68.
Venkateswaran
, U. D.
, Brandsen
, E. A.
, Schlecht
, U.
, Eklund
, P. C.
, Rao
, A. M.
, Richter
, E.
, Loa
, L.
, Syassen
, K.
, and Eklund
, P. C.
, 2001, “High Pressure Studies of the Raman-Active Phonons in Carbon Nanotubes
,” Phys. Status Solidi B
0370-1972, 223
, pp. 225
–236
.69.
Gaál
, R.
, Salvetat
, J. P.
, and Forro
, L.
, 2000, “Pressure Dependence of the Resistively of Single-Wall Carbon Nanotube Ropes
,” Phys. Rev. B
0163-1829, 61
, pp. 7320
–7323
.70.
Tang
, D. S.
, Bao
, Z. X.
, Wang
, L. J.
, Chen
, L. C.
, Sun
, L. F.
, Liu
, Z. Q.
, Zhou
, W. Y.
, and Xie
, S. S.
, 2000, “The Electrical Behavior of Carbon Nanotubes Under High Pressure
,” J. Phys. Chem. Solids
0022-3697, 61
, pp. 1175
–1178
.71.
Ni
, B.
, Sinnott
, S. B.
, Mikulski
, P. T.
, and Harrison
, J. A.
, 2002, “Compression of Carbon Nanotubes Filled With C60, CH4, or Ne: Predictions From Molecular Dynamics Simulations
,” Phys. Rev. Lett.
0031-9007, 88
, p. 205505
.72.
Buehler
, M. J.
, Kong
, J.
, and Gao
, H. J.
, 2004, “Deformation Mechanism of Very Long Single-Wall Carbon Nanotubes Subject to Compressive Loading
,” ASME J. Eng. Mater. Technol.
0094-4289, 126
, pp. 245
–249
.73.
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.
0094-4289, 126
, pp. 279
–284
.74.
Ru
, C. Q.
, 2000, “Effect of Van Der Waals Forces on Axial Buckling of a Doublewalled Carbon Nanotube
,” J. Appl. Phys.
0021-8979, 87
, pp. 7227
–7231
.75.
Ru
, C. Q.
, 2000, “Effective Bending Stiffness of Carbon Nanotubes
,” Phys. Rev. B
0163-1829, 62
, pp. 9973
–9976
.76.
Ru
, C. Q.
, 2001, “Axially Compressed Buckling of a Doublewalled Carbon Nanotube Embedded in an Elastic Medium
,” J. Mech. Phys. Solids
0022-5096, 49
, pp. 1265
–1279
.77.
Ru
, C. Q.
, 2001, “Degraded Axial Buckling Strain of Multiwalled Carbon Nanotubes Due to Interlayer Slips
,” J. Appl. Phys.
0021-8979, 89
, pp. 3426
–3433
.78.
Ru
, C. Q.
, 2000, “Column Buckling of Multiwalled Carbon Nanotubes With Interlayer Radial Displacements
,” Phys. Rev. B
0163-1829, 62
, pp. 16962
–16967
.79.
Waters
, J. F.
, Riester
, L.
, Jouzi
, M.
, Guduru
, P. R.
, and Xu
, J. M.
, 2004, “Buckling Instabilities in Multiwalled Carbon Nanotubes Under Uniaxial Compression
,” Appl. Phys. Lett.
0003-6951, 85
, pp. 1787
–1789
.80.
Waters
, J. F.
, Guduru
, P. R.
, Jouzi
, M.
, Xu
, J. M.
, Hanlon
, T.
, and Suresh
, S.
, 2005, “Shell Buckling of Individual Multi-Walled Carbon Nanotubes Using Nanoindentation
,” Appl. Phys. Lett.
0003-6951, 87
, p. 103109
.81.
Duan
, X. J.
, Tang
, C.
, Zhang
, J.
, Guo
, W. L.
, and Liu
, Z. F.
, 2007, “Two Distinct Buckling Modes in Carbon Nanotube Bending
,” Nano Lett.
1530-6984, 7
, pp. 143
–148
.82.
Shibutani
, Y.
, and Ogata
, S.
, 2004, “Mechanical Integrity of Carbon Nanotubes for Bending and Torsion
,” Modell. Simul. Mater. Sci. Eng.
0965-0393, 12
, pp. 599
–610
.83.
Wang
, Q.
, Liew
, K. M.
, He
, X. Q.
, and Xiang
, Y.
, 2007, “Local Buckling of Carbon Nanotubes Under Bending
,” Appl. Phys. Lett.
0003-6951, 91
, p. 093128
.84.
Kutana
, A.
, and Giapis
, K. P.
, 2006, “Transient Deformation Regime in Bending of Single-Walled Carbon Nanotubes
,” Phys. Rev. Lett.
0031-9007, 97
, p. 245501
.85.
Yang
, H. K.
, and Wang
, X.
, 2006, “Bending Stability of Multi-Wall Carbon Nanotubes Embedded in an Elastic Medium
,” Modell. Simul. Mater. Sci. Eng.
0965-0393, 14
, pp. 99
–116
.86.
Wang
, X.
, and Yang
, H. K.
, 2006, “Bending Stability of Multiwalled Carbon Nanotubes
,” Phys. Rev. B
0163-1829, 73
, p. 085409
.87.
Wang
, Q.
, 2008, “Torsional Buckling of Double-Walled Carbon Nanotubes
,” Carbon
0008-6223, 46
, pp. 1172
–1174
.88.
Zhang
, Y. Y.
, and Wang
, C. M.
, 2008, “Torsional Responses of Double-Walled Carbon Nanotubes via Molecular Dynamics Simulations
,” J. Phys.: Condens. Matter
0953-8984, 20
, p. 455214
.89.
Jeong
, B. W.
, Lim
, J. K.
, and Sinnott
, S. B.
, 2007, “Elastic Torsional Response of Carbon Nanotube Systems
,” J. Appl. Phys.
0021-8979, 101
, p. 084309
.90.
Jeong
, B. W.
, Lim
, J. K.
, and Sinnott
, S. B.
, 2007, “Turning Stiffness of Carbon Nanotube Systems
,” Appl. Phys. Lett.
0003-6951, 91
, p. 093102
.91.
Wang
, Q.
, 2009, “Atomic Transportation via Carbon Nanotubes
,” Nano Lett.
1530-6984, 9
, pp. 245
–249
.92.
Wang
, Q.
, 2009, “Transportation of Hydrogen Molecules Using Carbon Nanotube in Torsion
,” Carbon
0008-6223, 47
, pp. 1870
–1873
.93.
Han
, Q.
, and Lu
, G. X.
, 2003, “Torsional Buckling of a Double-Walled Carbon Nanotube Embedded in an Elastic Medium
,” Eur. J. Mech. A/Solids
0997-7538, 22
, pp. 875
–883
.94.
Wang
, X.
, Yang
, H. K.
, and Dong
, K.
, 2005, “Torsional Buckling of Multi-Walled Carbon Nanotubes
,” Mater. Sci. Eng., A
0921-5093, 404
, pp. 314
–322
.95.
Han
, Q.
, Yao
, X. H.
, and Li
, L. F.
, 2006, “Theoretical and Numerical Study of Torsional Buckling of Multiwall Carbon Nanotubes
,” Mech. Adv. Mater. Structures
, 13
, pp. 329
–337
.96.
Yao
, X. H.
, and Han
, Q.
, 2006, “Buckling Analysis of Multiwalled Carbon Nanotubes Under Torsional Load Coupling With Temperature Changes
,” ASME J. Eng. Mater. Technol.
0094-4289, 128
, pp. 419
–427
.97.
Yang
, H. K.
, and Wang
, X.
, 2007, “Torsional Buckling of Multi-Wall Carbon Nanotubes Embedded in an Elastic Medium
,” Compos. Struct.
0263-8223, 77
, pp. 182
–192
.98.
Wang
, Q.
, Quek
, S. T.
, and Varadan
, V. K.
, 2007, “Torsional Buckling of carBon Nanotubes
,” Phys. Lett. A
0375-9601, 367
, pp. 135
–139
.99.
Zhang
, C. L.
, and Shen
, H. S.
, 2007, “Buckling and Postbuckling of Single-Walled Carbon Nanotubes Under Combined Axial Compression and Torsion in Thermal Environments
,” Phys. Rev. B
0163-1829, 75
, p. 045408
.100.
Jeong
, B. W.
, Lim
, J. K.
, and Sinnott
, S. B.
, 2008, “Tuning the Torsional Properties of Carbon Nanotube Systems With Axial Prestress
,” Appl. Phys. Lett.
0003-6951, 92
, p. 253114
.101.
Wang
, X.
, Sun
, B.
, and Yang
, H. K.
, 2006, “Stability of Multi-Walled Carbon Nanotubes Under Combined Bending and Axial Compression Loading
,” Nanotechnology
0957-4484, 17
, pp. 815
–823
.102.
Lu
, Y. J.
, and Wang
, X.
, 2006, “Combined Torsional Buckling of Multi-Walled Carbon Nanotubes
,” J. Phys. D
0022-3727, 39
, pp. 3380
–3387
.103.
Wang
, X.
, Lu
, G. X.
, and Lu
, Y. J.
, 2007, “Buckling of Embedded Multi-Walled Carbon Nanotubes Under Combined Torsion and Axial Loading
,” Int. J. Solids Struct.
0020-7683, 44
, pp. 336
–351
.104.
Srivastava
, D.
, Menon
, M.
, and Cho
, K.
, 1999, “Nanoplasticity of Single-Wall Carbon Nanotubes Under Uniaxial Compression
,” Phys. Rev. Lett.
0031-9007, 83
, pp. 2973
–2976
.105.
Huang
, J. Y.
, Chen
, S.
, Wang
, Z. Q.
, Kempa
, K.
, Wang
, Y. M.
, Jo
, S. H.
, Chen
, G.
, Dresselhaus
, M. S.
, and Ren
, Z. F.
, 2006, “Superplastic Carbon Nanotubes: Conditions Discovered That Allow Extensive Deformation of Rigid Single-Walled Nanotubes
,” Nature (London)
0028-0836, 439
, p. 281
.106.
Huang
, J. Y.
, Chen
, S.
, Ren
, Z. F.
, Wang
, Z.
, Kempa
, K.
, Naughton
, M. J.
, Chen
, G.
, and Dresselhaus
, M. S.
, 2007, “Enhanced ductile behavior of Tensile-Elongated Individual Double-Walled and Triple-Walled Carbon Nanotubes at High Temperatures
,” Phys. Rev. Lett.
0031-9007, 98
, p. 185501
.107.
Donnell
, L. H.
, 1976, Beams, Plates and Shells
, McGraw-Hill
, New York
.108.
Flugge
, W.
, 1960, Stresses in Shells
, Springer-Verlag
, Berlin
.109.
Li
, C. Y.
, and Chou
, T. W.
, 2004, “Modeling of Elastic Buckling of Carbon Nanotubes by Molecular Structural Mechanics Approach
,” Mech. Mater.
0167-6636, 36
, pp. 1047
–1055
.110.
Chang
, T. C.
, Li
, G. Q.
, and Guo
, X. M.
, 2005, “Elastic Axial Buckling of Carbon Nanotubes via a Molecular Mechanics Model
,” Carbon
0008-6223, 43
, pp. 287
–294
.111.
Chang
, T. C.
, Guo
, W. L.
, and Guo
, X. M.
, 2005, “Buckling of Multiwalled Carbon Nanotubes Under Axial Compression and Bending via a Molecular Mechanics Model
,” Phys. Rev. B
0163-1829, 72
, p. 064101
.112.
Wang
, C. Y.
, Ru
, C. Q.
, and Mioduchowski
, A.
, 2003, “Axially Compressed Buckling of Pressured Multiwall Carbon Nanotubes
,” Int. J. Solids Struct.
0020-7683, 40
, pp. 3893
–3911
.113.
Liew
, K. M.
, Wong
, C. H.
, He
, X. Q.
, Tan
, M. J.
, and Meguid
, S. A.
, 2004, “Nanomechanics of Single and Multiwalled Carbon Nanotubes
,” Phys. Rev. B
0163-1829, 69
, p. 115429
.114.
Sears
, A.
, and Batra
, R. C.
, 2006, “Buckling of Multiwalled Carbon Nanotubes Under Axial Compression
,” Phys. Rev. B
0163-1829, 73
, p. 085410
.115.
Zhang
, Y. Y.
, Wang
, C. M.
, and Tan
, V. B. C.
, 2008, “Examining the Effects of Wall Numbers on Buckling Behavior and Mechanical Properties of Multiwalled Carbon Nanotubes via Molecular Dynamics Simulations
,” J. Appl. Phys.
0021-8979, 103
, p. 053505
.116.
Thostenson
, E. T.
, and Chou
, T. W.
, 2004, “Nanotube Buckling in Aligned Multi-Wall Carbon Nanotube Composites
,” Carbon
0008-6223, 42
, pp. 3015
–3018
.117.
Wang
, C. Y.
, and Mioduchowski
, A.
, 2007, “The Effect of Dimensional Factors on Buckling of Multiwall Carbon Nanotubes
,” J. Appl. Phys.
0021-8979, 101
, p. 014306
.118.
Zhang
, Y. Y.
, Wang
, C. M.
, Duan
, W. H.
, Xiang
, Y.
, and Zong
, Z.
, 2009, “Assessment of Continuum Mechanics Models in Predicting Buckling Strains of Single-Walled Carbon Nanotubes
,” Nanotechnology
0957-4484, 20
, p. 395707
.119.
Chang
, T. C.
, and Hou
, J.
, 2006, “Molecular Dynamics Simulations on Buckling of Multiwalled Carbon Nanotubes Under Bending
,” J. Appl. Phys.
0021-8979, 100
, p. 114327
.120.
Li
, X. Y.
, Yang
, W.
, and Liu
, B.
, 2007, “Bending Induced Rippling and Twisting of Multiwalled Carbon Nanotubes
,” Phys. Rev. Lett.
0031-9007, 98
, p. 205502
.121.
Chang
, T. C.
, 2007, “Torsional Behavior of Chiral Single-Walled Carbon Nanotubes in Loading Direction Dependent
,” Appl. Phys. Lett.
0003-6951, 90
, p. 201910
.122.
Wang
, C. M.
, Ma
, Y. Q.
, Zhang
, Y. Y.
, and Ang
, K. K.
, 2006, “Buckling of Double-Walled Carbon Nanotubes Modeled by Solid Shell Elements
,” J. Appl. Phys.
0021-8979, 99
, p. 114317
.123.
Wang
, Y.
, Wang
, X. X.
, Ni
, X. G.
, and Wu
, H. A.
, 2005, “Simulation of the Elastic Response and the Buckling Modes of Single-Walled Carbon Nanotubes
,” Comput. Mater. Sci.
0927-0256, 32
, pp. 141
–146
.124.
Leung
, A. Y. T.
, Guo
, X.
, He
, X. Q.
, Jiang
, H.
, and Huang
, Y.
, 2006, “Postbuckling of Carbon Nanotubes by Atomic-Scale Finite Element
,” J. Appl. Phys.
0021-8979, 99
, p. 124308
.125.
Cornwell
, C. F.
, and Wille
, L. T.
, 1997, “Elastic Properties of Single-Walled Carbon Nanotubes in Compression
,” Solid State Commun.
0038-1098, 101
, pp. 555
–558
.126.
Cao
, G. X.
, and Chen
, X.
, 2006, “Buckling of Single-Walled Carbon Nanotubes Upon Bending: Molecular Dynamics Simulations and Finite Element Method
,” Phys. Rev. B
0163-1829, 73
, p. 155435
.127.
Pantano
, A.
, Boyce
, M. C.
, and Parks
, D. M.
, 2003, “Nonlinear Structural Mechanics Based Modeling of Carbon Nanotube Deformation
,” Phys. Rev. Lett.
0031-9007, 91
, p. 145504
.128.
Akita
, S.
, Nishio
, M.
, and Nakayama
, Y.
, 2006, “Buckling of Multiwall Carbon Nanotubes Under Axial Compression
,” Jpn. J. Appl. Phys.
0021-4922, 45
, pp. 5586
–5589
.129.
Chang
, T. C.
, Geng
, J. Y.
, and Guo
, X. M.
, 2005, “Chirality- and Size-Dependent Elastic Properties of Single-Walled Carbon Nanotubes
,” Appl. Phys. Lett.
0003-6951, 87
, p. 251929
.130.
Shen
, L. X.
, and Li
, J.
, 2004, “Transversely Isotropic Elastic Properties of Single-Walled Carbon Nanotubes
,” Phys. Rev. B
0163-1829, 69
, p. 045414
.131.
Huang
, Y.
, Wu
, J.
, and Hwang
, K. C.
, 2006, “Thickness of Graphene and Single-Wall Carbon Nanotubes
,” Phys. Rev. B
0163-1829, 74
, p. 245413
.132.
Ozaki
, T.
, Iwasa
, Y.
, and Mitani
, T.
, 2000, “Stiffness of Single-Walled Carbon Nanotube Under Large Strain
,” Phys. Rev. Lett.
0031-9007, 84
, pp. 1712
–1715
.133.
Xiao
, J. R.
, Lopatnikov
, S. L.
, Gama
, B. A.
, and Gillespie
, J. W.
, Jr., 2006, “Nanomechanics on the Deformation of Single- and Multi-Walled Carbon Nanotubes Under Radial Pressure
,” Mater. Sci. Eng., A
0921-5093, 416
, pp. 192
–204
.134.
Zhang
, Y. Y.
, Tan
, V. B. C.
, and Wang
, C. M.
, 2006, “Effect of Chirality on Buckling Behavior of Single-Walled Carbon Nanotubes
,” J. Appl. Phys.
0021-8979, 100
, p. 074304
.135.
He
, X. Q.
, Kitipornchai
, S.
, and Liew
, K. M.
, 2005, “Buckling Analysis of Multi-Walled Carbon Nanotubes: A Continuum Model Accounting for Van Der Waals Interaction
,” J. Mech. Phys. Solids
0022-5096, 53
, pp. 303
–326
.136.
Qian
, H.
, and Xu
, K. Y.
, 2006, “Curvature Effects on Pressure-Induced Buckling of Empty or Filled Double-Walled Carbon Nanotubes
,” Acta Mech.
0001-5970, 187
, pp. 55
–73
.137.
Gao
, R. P.
, Wang
, Z. L.
, Bai
, Z. G.
, de heer
, W. A.
, Dai
, L. M.
, and Gao
, M.
, 2000, “Nanomechanics of Aligned Carbon Nanotube Arrays
,” Phys. Rev. Lett.
0031-9007, 85
, pp. 622
–625
.138.
Sammalkorpi
, M.
, Krasheninnikov
, A.
, Kuronen
, A.
, Nordlund
, K.
, and Kaski
, K.
, 2004, “Mechanical Properties of Carbon Nanotubes With Vacancies and Related Defects
,” Phys. Rev. B
0163-1829, 70
, p. 245416
.139.
Guduru
, P. R.
, and Xia
, Z.
, 2007, “Experiments and Analysis of Buckling in Imperfect Multiwalled Carbon Nanotubes
,” Exp. Mech.
0014-4851, 47
, pp. 153
–161
.140.
Xia
, Y. Y.
, Zhao
, M. W.
, Ma
, Y.
, Ying
, M.
, Liu
, X.
, Liu
, P.
, and Mei
, L.
, 2002, “Tensile Strength of Single-Walled Carbon Nanotubes With Defects Under Hydrostatic Pressure
,” Phys. Rev. B
0163-1829, 65
, p. 155415
.141.
Mielke
, S. L.
, Troya
, D.
, Zhang
, S. L.
, Li
, J. J.
, Xiao
, S. P.
, Car
, R.
, Ruoff
, R. S.
, Schatz
, G. C.
, and Belytschko
, T.
, 2004, “The Role of Vacancy Defects and Holes in the Fracture of Carbon Nanotubes
,” Chem. Phys. Lett.
0009-2614, 390
, pp. 413
–420
.142.
Jeng
, Y. R.
, Tsai
, P. C.
, and Fang
, T. H.
, 2004, “Effects of Temperature and Vacancy Defects on Tensile Deformation of Single-Walled Carbon Nanotubes
,” J. Phys. Chem. Solids
0022-3697, 65
, pp. 1849
–1856
.143.
Zhang
, S. L.
, Mielke
, S. L.
, Khare
, R.
, Troya
, D.
, Ruoff
, R. S.
, Schatz
, G. C.
, and Belytschko
, T.
, 2005, “Mechanics of Defects in Carbon Nanotubes: Atomistic and Multiscale Simulations
,” Phys. Rev. B
0163-1829, 71
, p. 115403
.144.
Hirai
, Y.
, Nishimaki
, S.
, Mori
, H.
, Kimoto
, Y.
, Akita
, S.
, Nakayama
, Y.
, and Tanaka
, Y.
, 2003, “Molecular Dynamics Studies on Mechanical Properties of Carbon Nano Tubes With Pinhole Defects
,” Jpn. J. Appl. Phys.
0021-4922, 42
, pp. 4120
–4123
.145.
Shen
, G. A.
, Namilae
, S.
, and Chandra
, N.
, 2006, “Load Transfer Issues in the Tensile and Compressive Behavior of Multiwall Carbon Nanotubes
,” Mater. Sci. Eng., A
0921-5093, 429
, pp. 66
–73
.146.
Chandra
, N.
, and Namilae
, S.
, 2006, “Tensile and Compressive Behavior of Carbon Nanotubes: Effects of functionalization and Topological Defects
,” Mech. Adv. Mater. Structures
, 13
, pp. 115
–127
.147.
Xin
, H.
, Han
, Q.
, and Yao
, X. H.
, 2007, “Buckling and Axially Compressive Properties of Perfect and Defective Single-Walled Carbon Nanotubes
,” Carbon
0008-6223, 45
, pp. 2486
–2495
.148.
Xia
, Z. H.
, Guduru
, P. R.
, and Curtin
, W. A.
, 2007, “Enhancing Mechanical Properties of Multiwall Carbon Nanotubes via sp3 Interwall Bridging
,” Phys. Rev. Lett.
0031-9007, 98
, pp. 245501
.149.
Huq
, A. M. A.
, Goh
, K. L.
, Zhou
, Z. R.
, and Liao
, K.
, 2008, “On Defect Interactions in Axially Loaded Single-Walled Carbon Nanotubes
,” J. Appl. Phys.
0021-8979, 103
, p. 054306
.150.
Wang
, Q.
, Varadan
, V. K.
, Xiang
, Y.
, Han
, Q. K.
, and Wen
, B. C.
, 2008, “On Stability of Single-Walled Carbon Nanotubes With a Vacancy Defect
,” Int. J. Struct. Stab. Dyn.
0219-4554, 8
, pp. 357
–366
.151.
Zhang
, Y. Y.
, Xiang
, Y.
, and Wang
, C. M.
, 2009, “Buckling of Defective Single-Walled Carbon Nanotubes
,” J. Appl. Phys.
0021-8979, 106
, p. 113503
.152.
Shen
, H. S.
, and Zhang
, C. L.
, 2006, “Postbuckling Prediction of Axially Loaded Double-Walled Carbon Nanotubes With Temperature Dependent Properties and Initial Defects
,” Phys. Rev. B
0163-1829, 74
, p. 035410
.153.
Thomsen
, C.
, and Reich
, S.
, 2001, “The Pressure Dependence of the High-Energy Raman Modes in Empty and Filled Multiwalled Carbon Nanotubes
,” Phys. Status Solidi B
0370-1972, 225
, pp. R18
–R19
.154.
Zhu
, J.
, Pan
, Z. Y.
, Wang
, Y. X.
, Zhou
, L.
, and Jiang
, O.
, 2007, “The Effects of Encapsulating C-60 Fullerenes on the Bending Flexibility of Carbon Nanotubes
,” Nanotechnology
0957-4484, 18
, p. 275702
.155.
Zhou
, L.
, Zhu
, B. E.
, Pan
, Z. Y.
, Wang
, Y. X.
, and Zhu
, J.
, 2007, “Reduction of the Buckling Strength of Carbon Nanotubes Resulting From Encapsulation of C60 Fullerenes
,” Nanotechnology
0957-4484, 18
, p. 275709
.156.
Hu
, Z. L.
, Guo
, X. M.
, and Ru
, C. Q.
, 2008, “Enhanced Critical Pressure Fore Buckling of Carbon Nanotubes Due to an Inserted Linear Carbon Chain
,” Nanotechnology
0957-4484, 19
, p. 305703
.157.
Král
, P.
, and Tomanek
, D.
, 1999, “Laser-Driven Atomic Pump
,” Phys. Rev. Lett.
0031-9007, 82
, pp. 5373
–5376
.158.
Bulson
, P. S.
, 1985, Buried Structure
, Chapman and Hall
, London
.159.
Vlasov
, V. Z.
, and Leont’ev
, U. N.
, 1966, Beams, Plates and Shells on Elastic Foundations
, Israel Program for Scientific Translations
, Jerusalem
.160.
Kitipornchai
, S.
, He
, X. Q.
, and Liew
, K. M.
, 2005, “Buckling Analysis of Triple-Walled Carbon Nanotubes Embedded in an Elastic Matrix
,” J. Appl. Phys.
0021-8979, 97
, p. 114318
.161.
Pasternak
, P. L.
, 1954, On a New Method of Analysis of an Elastic Foundation by Means of Two Foundation Constants
, Gos. Izd, Lit. Po Strait I Arkh
, Moscow
, (in Russian).162.
Zhang
, Y. Q.
, Liu
, G. R.
, Qian
, H. F.
, and Li
, G. Y.
, 2006, “Investigation of Buckling of Double-Walled Carbon Nanotubes Embedded in an Elastic Medium Using the Energy Method
,” Int. J. Mech. Sci.
0020-7403, 48
, pp. 53
–61
.163.
Yao
, X. H.
, and Han
, Q.
, 2007, “Postbuckling Prediction of Double-Walled Carbon Nanotubes Under Axial Compression
,” Eur. J. Mech. A/Solids
0997-7538, 26
, pp. 20
–32
.164.
Garg
, A.
, and Sinnott
, S. B.
, 1998, “Effect of Chemical Functionalization on Mechanical Properties of Carbon Nanotubes
,” Chem. Phys. Lett.
0009-2614, 295
, pp. 273
–278
.165.
Namilae
, S.
, and Chandra
, N.
, 2006, “Role of Atomic Scale Interfaces in the Compressive Behavior of Carbon Nanotubes in Composites
,” Compos. Sci. Technol.
0266-3538, 66
, pp. 2030
–2038
.166.
Hadjiev
, V. G.
, Lagoudas
, D. C.
, Oh
, E. -S.
, Thakre
, P.
, Davis
, D.
, Files
, B. S.
, Yowell
, L.
, Arepalli
, S.
, Bahr
, J. L.
, and Tour
, J. M.
, 2006, “Buckling Instabilities of Octadecylamine Functionalized Carbon Nanotubes Embedded in Epoxy
,” Compos. Sci. Technol.
0266-3538, 66
, pp. 128
–136
.167.
Zhang
, W.
, Suhr
, J.
, and Koratkar
, N. A.
, 2006, “Observation of High Buckling Stability in Carbon Nanotube Polymer Composites
,” Adv. Mater.
0935-9648, 18
, pp. 452
–456
.168.
Zhang
, C. L.
, and Shen
, H. S.
, 2006, “Temperature-Dependent Elastic Properties of Single-Walled Carbon Nanotubes: Prediction From Molecular Dynamics Simulation
,” Appl. Phys. Lett.
0003-6951, 89
, p. 081904
.169.
Wang
, Y.
, Ni
, X. G.
, Wang
, X. X.
, and Wu
, H. A.
, 2003, “Effect of Temperature on Deformation of Carbon Nanotube Under Compression
,” Chin. Phys.
1009-1963, 12
, pp. 1007
–1010
.170.
Zhang
, C. L.
, and Shen
, H. S.
, 2006, “Buckling and Postbuckling Analysis of Single-Walled Carbon Nanotubes in Thermal Environments via Molecular Dynamics Simulation
,” Carbon
0008-6223, 44
, pp. 2608
–2616
.171.
Tang
, C.
, Guo
, W. L.
, and Chen
, C. F.
, 2008, “Mechanism for Superelongation of Carbon Nanotubes at High Temperature
,” Phys. Rev. Lett.
0031-9007, 100
, p. 175501
.172.
Zhang
, Y. Y.
, Wang
, C. M.
, and Tan
, V. B. C.
, 2009, “Buckling of Carbon Nanotubes at High Temperatures
,” Nanotechnology
0957-4484, 20
, p. 215702
.173.
Das
, P. S.
, and Wille
, L. T.
, 2002, “Atomistic and Continuum Studies of Carbon Nanotubes Under Pressure
,” Comput. Mater. Sci.
0927-0256, 24
, pp. 159
–162
.174.
Garg
, A.
, Han
, J.
, and Sinnott
, S. B.
, 1998, “Interactions of Carbon-Nanotubule Proximal Probe Tips With Diamond and Graphene
,” Phys. Rev. Lett.
0031-9007, 81
, pp. 2260
–2263
.175.
Wei
, C. Y.
, Cho
, K.
, and Srivastava
, D.
, 2003, “Tensile Strength of Carbon Nanotubes Under Realistic Temperature and Strain Rate
,” Phys. Rev. B
0163-1829, 67
, p. 115407
.176.
Cao
, G. X.
, and Chen
, X.
, 2006, “The Effect of the Displacement Increment on the Axial Compressive Buckling Behaviours of Single-Walled Carbon Nanotubes
,” Nanotechnology
0957-4484, 17
, pp. 3844
–3855
.177.
Zhang
, Y. Y.
, Tan
, V. B. C.
, and Wang
, C. M.
, 2007, “Effect of Strain Rate on the Buckling Behavior of Single- and Double-Walled Carbon Nanotubes
,” Carbon
0008-6223, 45
, pp. 514
–523
.178.
Wang
, C. Y.
, Zhang
, Y. Y.
, Wang
, C. M.
, and Tan
, V. B. C.
, 2007, “Buckling of Carbon Nanotubes: A Literature Survey
,” J. Nanosci. Nanotechnol.
1533-4880, 7
, pp. 4221
–4247
.179.
Davydov
, A. S.
, 1965, Quantum Mechanics
, Pergamon
, Oxford, New York
.180.
Ohno
, K.
, Esfarjani
, K.
, and Kawazoe
, Y.
, 1999, Computational Materials Sciences: From Ab Initio to Monte Carlo Methods
, Springer
, Berlin
.181.
Clementi
, E.
, 2000, “Ab Initio Computations in Atoms and Molecules
,” IBM J. Res. Dev.
0018-8646, 44
, pp. 228
–245
.182.
Hohenberg
, P.
, and Kohn
, W.
, 1964, “Inhomogeneous Electron Gas
,” Phys. Rev.
0096-8250, 136
, pp. B864
–B871
.183.
Slater
, J. C.
, and Koster
, G. F.
, 1954, “Simplified LCAO Method for the Periodic Potential Problem
,” Phys. Rev.
0096-8250, 94
, pp. 1498
–1524
.184.
Harrison
, W. A.
, 1989, Electronic Structure and the Properties of Solids: The Physics of Chemical Bond
, Dover
, New York
.185.
Van Lier
, G.
, Van Alsenoy
, C.
, Van Doren
, V.
, and Geerlings
, P.
, 2000, “Ab Initio Study of the Elastic Properties of Singlewalled Carbon Nanotubes and Grapheme
,” Chem. Phys. Lett.
0009-2614, 326
, pp. 181
–185
.186.
Wang
, L. F.
, Zheng
, Q. S.
, Liu
, J. Z.
, and Jiang
, Q.
, 2005, “Size Dependence of the Thin-Shell Model for Carbon Nanotubes
,” Phys. Rev. Lett.
0031-9007, 95
, p. 105501
.187.
Pullen
, A.
, Zhao
, G. L.
, Bagayoko
, D.
, and Yang
, L.
, 2005, “Structural, Elastic, and Electronic Properties of Deformed Carbon Nanotubes Under Uniaxial Strain
,” Phys. Rev. B
0163-1829, 71
, p. 205410
.188.
Hasegawa
, M.
, and Nishidate
, K.
, 2006, “Radial Deformation and Stability of Single-Wall Carbon Nanotubes Under Hydrostatic Pressure
,” Phys. Rev. B
0163-1829, 74
, p. 115401
.189.
Dereli
, G.
, and Ozdogan
, C.
, 2003, “Structural Stability and Energetics of Single-Walled Carbon Nanotubes Under Uniaxial Strain
,” Phys. Rev. B
0163-1829, 67
, p. 035416
.190.
Yeak
, S. H.
, Ng
, T. Y.
, and Liew
, K. M.
, 2005, “Multiscale Modeling of Carbon Nanotubes Under Axial Tension and Compression
,” Phys. Rev. B
0163-1829, 72
, p. 165401
.191.
Allen
, M. P.
, and Tildesley
, D. J.
, 1987, Computer Simulation of Liquids
, Clarendon
, Oxford
.192.
Frenkel
, D.
, and Smit
, B.
, 1996, Understanding Molecular Simulation: From Algorithms to Applications
, Academic
, New York
.193.
Woodcock
, L. V.
, 1971, “Isothermal Molecular Dynamics Calculations for Liquid Salts
,” Chem. Phys. Lett.
0009-2614, 10
, pp. 257
–261
.194.
Berendsen
, H. J. C.
, Postma
, J. P. M.
, van Gunsteren
, W. F.
, Dinola
, A.
, and Haak
, J. R.
, 1984, “Molecular Dynamics With Coupling to an External Bath
,” J. Chem. Phys.
0021-9606, 81
, pp. 3684
–3690
.195.
Schneider
, T.
, and Stoll
, E.
, 1978, “Molecular-Dynamics Study of a 3-Dimensional One-Component Model for Distortive Phase-Transitions
,” Phys. Rev. B
0163-1829, 17
, pp. 1302
–1322
.196.
Nosé
, S.
, 1984, “A Unified Formulation of the Constant Temperature Molecular-Dynamics Methods
,” J. Chem. Phys.
0021-9606, 81
, pp. 511
–519
.197.
Hoover
, W. G.
, 1985, “Canonical Dynamics: Equilibrium Phase-Space Distributions
,” Phys. Rev. A
1050-2947, 31
, pp. 1695
–1697
.198.
Hunenberger
, P. H.
, 2005, “Thermostat Algorithms for Molecular Dynamics Simulations
,” Adv. Polym. Sci.
0065-3195, 173
, pp. 105
–149
.199.
Heo
, S. J.
, and Sinnott
, S. B.
, 2007, “Investigation of the Influence of Thermostat Configurations on the Mechanical Properties of Carbon Nanotubes in Molecular Dynamics Simulations
,” J. Nanosci. Nanotechnol.
1533-4880, 7
, pp. 1518
–1524
.200.
Mylvaganam
, K.
, and Zhang
, L. C.
, 2004, “Important Issues in a Molecular Dynamics Simulation for Characterizing the Mechanical Properties of Carbon Nanotubes
,” Carbon
0008-6223, 42
, pp. 2025
–2032
.201.
Brenner
, D. W.
, Shenderova
, O. A.
, Harrison
, J. A.
, Stuart
, S. J.
, Ni
, B.
, and Sinnott
, S. B.
, 2002, “A Second-Generation Reactive Empirical Bond Order (REBO) Potential Energy Expression for Hydrocarbons
,” J. Phys.: Condens. Matter
0953-8984, 14
, pp. 783
–802
.202.
Allinger
, N. L.
, 1977, “Conformational-Analysis.130. MM2—Hydrocarbon Force-Field Utilizing V1 and V2 Torsional Terms
,” J. Am. Chem. Soc.
0002-7863, 99
, pp. 8127
–8134
.203.
Allinger
, N. L.
, Yuh
, Y. H.
, and Lii
, J. H.
, 1989, “The MM3 Force Field for Hydrocarbons
,” J. Am. Chem. Soc.
0002-7863, 111
, pp. 8551
–8566
.204.
Sun
, H.
, Ren
, P.
, and Fried
, J. R.
, 1998, “The COMPASS Force Field: Parameterization and Validation for Phosphazenes
,” Comput. Theor. Polym. Sci.
1089-3156, 8
, pp. 229
–246
.205.
Timoshenko
, S. P.
, 1974, Vibration Problems in Engineering
, Wiley
, New York
.206.
Doyle
, J. F.
, 1989, Wave Propagation in Structure
, Sprinter-Velag
, Berlin
.207.
Gere
, J. M.
, 2001, Mechanics of Materials
, 5th ed., Brooke/Cole
, Belmont, CA
.208.
Girifalco
, L. A.
, and Lad
, R. A.
, 1956, “Energy of Cohesion, Compressibility, and the Potential Energy Functions of the Graphite System
,” J. Chem. Phys.
0021-9606, 25
, pp. 693
–697
.209.
Timoshenko
, S. P.
, and Gere
, J. M.
, 1961, Theory of Elastic Stability
, McGraw-Hill
, New York
.210.
Zhang
, Y. Y.
, Wang
, C. M.
, and Tan
, V. B. C.
, 2006, “Buckling of Multiwalled Carbon Nanotubes Using Timoshenko Beam Theory
,” J. Eng. Mech.
0733-9399, 132
, pp. 952
–958
.211.
Robertson
, D. H.
, Brenner
, D. W.
, and Mintmire
, J. W.
, 1992, “Energetics of Nanoscale Graphitic Tubules
,” Phys. Rev. B
0163-1829, 45
, pp. 12592
–12595
.212.
Timoshenko
, S. P.
, and Woinowsky-Krieger
, S.
, 1976, Theory of Plates and Shells
, McGraw-Hill
, New York
.213.
Ru
, C. Q.
, 2004, “Elastic Models for Carbon Nanotubes
,” Encyclopedia of Nanosci. Nanotechnol.
, 2
, pp. 731
–744
.214.
Wang
, C. Y.
, Ru
, C. Q.
, and Mioduchowski
, A.
, 2004, “Applicability and Limitations of Simplified Elastic Shell Equations for Carbon Nanotubes
,” ASME J. Appl. Mech.
0021-8936, 71
, pp. 622
–631
.215.
Wang
, C. Y.
, Ru
, C. Q.
, and Miduchowski
, A.
, 2005, “Axisymmetric and Beamlike Vibrations of Multiwall Carbon Nanotubes
,” Phys. Rev. B
0163-1829, 72
, p. 075414
.216.
Wang
, C. Y.
, Ru
, C. Q.
, and Miduchowski
, A.
, 2003, “Elastic Buckling of Multiwall Carbon Nanotubes Under High Pressure
,” J. Nanosci. Nanotechnol.
1533-4880, 3
, pp. 199
–208
.217.
He
, X. Q.
, Kitipornchai
, S.
, Wang
, C. M.
, and Liew
, K. M.
, 2005, “Modelling Van Der Waals Forces for Infinitesimal Deformation of Multi-Walled Carbon Nanotubes
,” Int. J. Solids Struct.
0020-7683, 42
, pp. 6032
–6047
.218.
Liew
, K. M.
, He
, X. Q.
, and Kitipornchai
, S.
, 2006, “Buckling Characteristics of Embedded Multi-Walled Carbon Nanotubes
,” Proc. R. Soc. London, Ser. A
0950-1207, 461
, pp. 3785
–3805
.219.
Sun
, C. T.
, and Zhang
, H.
, 2003, “Size-Dependent Elastic Module of Platelike Nanotumaterials
,” J. Appl. Phys.
0021-8979, 93
, pp. 1212
–1218
.220.
Eringen
, A. C.
, 1972, “Nonlocal Polar Elastic Continua
,” Int. J. Eng. Sci.
0020-7225, 10
, pp. 1
–16
.221.
Eringen
, A. C.
, and Edelen
, D. G. B.
, 1972, “On Nonlocal Elasticity
,” Int. J. Eng. Sci.
0020-7225, 10
, pp. 233
–248
.222.
Eringen
, A. C.
, 1983, “On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves
,” J. Appl. Phys.
0021-8979, 54
, pp. 4703
–4710
.223.
Sudak
, L. J.
, 2003, “Column Buckling of Multiwalled Carbon Nanotubes Using Nonlocal Continuum Mechanics
,” J. Appl. Phys.
0021-8979, 94
, pp. 7281
–7287
.224.
Zhang
, Y. Q.
, Liu
, G. R.
, and Wang
, J. S.
, 2004, “Small-Scale Effects on Buckling of Multiwalled Carbon Nanotubes Under Axial Compression
,” Phys. Rev. B
0163-1829, 70
, p. 205430
.225.
Zhang
, Y. Q.
, Liu
, G. R.
, and Han
, X.
, 2006, “Effect of Small Length Scale on Elastic Buckling of Multi-Walled Carbon Nanotubes Under Radial Pressure
,” Phys. Lett. A
0375-9601, 349
, pp. 370
–376
.226.
Xie
, G. Q.
, Han
, X.
, Liu
, G. R.
, and Long
, S. Y.
, 2006, “Effect of Small Size-Scale on the Radial Buckling Pressure of a Simply Supported Multi-Walled Carbon Nanotube
,” Smart Mater. Struct.
0964-1726, 15
, pp. 1143
–1149
.227.
Wang
, C. M.
, Zhang
, Y. Y.
, Ramesh
, S. S.
, and Kitipornchai
, S.
, 2006, “Buckling Analysis of Micro- and Nano-Rods/Tubes Based on Nonlocal Timoshenko Beam Theory
,” J. Phys. D
0022-3727, 39
, pp. 3904
–3909
.228.
Wang
, C. M.
, Zhang
, Y. Y.
, and He
, X. Q.
, 2007, “Vibration of Nonlocal Timoshenko Beams
,” Nanotechnology
0957-4484, 18
, p. 105401
.229.
Wang
, C. M.
, Zhang
, Y. Y.
, and Kitipornchai
, S.
, 2007, “Vibration of Initially Stressed Micro and Nano-Beams
,” Int. J. Struct. Stab. Dyn.
0219-4554, 7
, pp. 555
–570
.230.
Wang
, L. F.
, and Hu
, H. Y.
, 2005, “Flexural Wave Propagation in Single-Walled Carbon Nanotubes
,” Phys. Rev. B
0163-1829, 71
, p. 195412
.231.
Wang
, L. F.
, Hu
, H. Y.
, and Guo
, W. L.
, 2006, “Validation of the Non-Local Elastic Shell Model for Studying Longitudinal Waves in Single-Walled Carbon Nanotubes
,” Nanotechnology
0957-4484, 17
, pp. 1408
–1415
.232.
Silvestre
, N.
, Wang
, C. M.
, Zhang
, Y. Y.
, and Xiang
, Y.
, “Sanders Shell Model for Buckling of Single-Walled Carbon Nanotubes With Small Length-to-Diameter Ratios
,” Comput. Mater. Sci.
0927-0256, submitted233.
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.
0021-8936, 77
, p. 041006
.234.
Shen
, H. S.
, and Zhang
, C. L.
, 2010, “Torsional Buckling and Postbuckling of Double-Walled Carbon Nanotubes by Nonlocal Shear Deformable Shell Model
,” Compos. Struct.
0263-8223, 92
, pp. 1073
–1084
.235.
Challamel
, N.
, and Wang
, C. M.
, 2008, “Small Length Scale Effect in Non-Local Cantilever Beams: Paradox Solved
,” Nanotechnology
0957-4484, 19
, p. 345703
.236.
Zhang
, Y. Y.
, Wang
, C. M.
, and Challamel
, N.
, 2010, “Bending, Buckling, and Vibration of Micro/Nanobeams by Hybrid Nonlocal Beam Model
,” J. Eng. Mech.
0733-9399, 136
, pp. 562
–574
.237.
Chang
, T. C.
, 2008, “Dominoes in Carbon Nanotubes
,” Phys. Rev. Lett.
0031-9007, 101
, p. 175501
.Copyright © 2010
by American Society of Mechanical Engineers
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