A finite-deformation shell theory for carbon nanotubes (CNTs) is established directly from the interatomic potential for carbon to account for the effect of bending and curvature. Its constitutive relation accounts for the nonlinear multibody atomistic interactions and therefore can model the important effect of CNT chirality and radius. The equilibrium equations and boundary conditions are obtained for the symmetric stresses and bending moments, which are different from many existing shell theories that involve asymmetric stress and bending moments. The theory is used in Part II of this paper to study the instability of carbon nanotubes subjected to different loadings.
Issue Section:
Research Papers
1.
Srivastava
, D.
, Menon
, M.
, and Cho
, K. J.
, 2001, “Computational Nanotechnology With Carbon Nanotubes and Fullerenes
,” Comput. Sci. Eng.
1521-9615, 3
, pp. 42
–55
.2.
Yakobson
, B. I.
, and Avouris
, P.
, 2001, “Mechanical Properties of Carbon Nanotubes
,” Top. Appl. Phys.
0303-4216, 80
, pp. 287
–329
.3.
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
.4.
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
.5.
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
.6.
Muster
, J.
, Burghard
, M.
, Roth
, S.
, Duesberg
, G. S.
, HernMandez
, E.
, and Rubio
, A.
, 1998, “Scanning Force Microscopy Characterization of Individual Carbon Nanotubes on Electrode Arrays
,” J. Vac. Sci. Technol. B
1071-1023, 16
, pp. 2796
–2801
.7.
Salvetat
, J.-P.
, Briggs
, G. A. D.
, Bonard
, J.-M.
, Bacsa
, R. R.
, and Kulik
, A. J.
, 1999, “Elastic and Shear Moduli of Single-Walled Carbon Nanotube Ropes
,” Phys. Rev. Lett.
0031-9007, 82
, pp. 944
–947
.8.
Tombler
, T. W.
, Zhou
, C.
, Alexseyev
, L.
, Kong
, J.
, Dai
, H.
, Liu
, L.
, Jayanthi
, C. S.
, Tang
, M.
, and Wu
, S.-Y.
, 2000, “Reversible Electromechanical Characteristics of Carbon Nanotubes Under Local-Probe Manipulation
,” Nature (London)
0028-0836, 405
, pp. 769
–772
.9.
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
.10.
Overney
, G.
, Zhong
, W.
, and TomManek
, D.
, 1993, “Structural Rigidity and Low-Frequency Vibrational Modes of Long Carbon Tubules
,” Z. Phys. D: At., Mol. Clusters
0178-7683, 27
, pp. 93
–96
.11.
Molina
, J. M.
, Savinsky
, S. S.
, and Khokhriakov
, N. V.
, 1996, “A Tight-Binding Model for Calculations of Structures and Properties of Graphitic Nanotubes
,” J. Chem. Phys.
0021-9606, 104
, pp. 4652
–4656
.12.
Halicioglu
, T.
, 1998, “Stress Calculations for Carbon Nanotubes
,” Thin Solid Films
0040-6090, 312
, pp. 11
–14
.13.
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
.14.
Goze
, C.
, Vaccarini
, L.
, Henrard
, L.
, Bernier
, P.
, Hernandez
, E.
, and Rubio
, A.
, 1999, “Elastic and Mechanical Properties of Carbon Nanotubes
,” Synth. Met.
0379-6779, 103
, pp. 2500
–2501
.15.
Sánchez-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
0163-1829, 59
, pp. 12678
–12688
.16.
Van Lier
, G.
, Van Alsenoy
, C.
, Van Doran
, V.
, and Geerlings
, P.
, 2000, “Ab Initio Study of the Elastic Properties of Single-Walled Carbon Nanotubes and Graphene
,” Chem. Phys. Lett.
0009-2614, 326
, pp. 181
–185
.17.
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
.18.
Prylutskyy
, Y. I.
, Durov
, S. S.
, Ogloblya
, O. V.
, Buzaneva
, E. V.
, and Scharff
, P.
, 2000, “Molecular Dynamics Simulations of Mechanical, Vibrational and Electronic Properties of Carbon Nanotubes
,” Comput. Mater. Sci.
0927-0256, 17
, pp. 352
–355
.19.
Vaccarini
, L.
, Goze
, C.
, Henrard
, L.
, HernMandez
, E.
, Bernier
, P.
, and Rubio
, A.
, 2000, “Mechanical and Electronic Properties of Carbon and Boron-Nitride Nanotubes
,” Carbon
0008-6223, 38
, pp. 1681
–1690
.20.
Zhou
, G.
, Duan
, W.
, and Gu
, B.
, 2001, “First-Principles Study on Morphology and Mechanical Properties of Single-Walled Carbon Nanotube
,” Chem. Phys. Lett.
0009-2614, 333
, pp. 344
–349
.21.
Liu
, B.
, Huang
, Y.
, Jiang
, H.
, Qu
, S.
, and Hwang
, K. C.
, 2004, “The Atomic-Scale Finite Element Method
,” Comput. Methods Appl. Mech. Eng.
0045-7825, 193
, pp. 1849
–1864
.22.
Liu
, B.
, Yu
, M. F.
, and Huang
, Y.
, 2004, “Role of Lattice Registry in the Full Collapse and Twist Formation of Carbon Nanotubes
,” Phys. Rev. B
0163-1829, 70
, p. 161402
.23.
Arroyo
, M.
, and Belytschko
, T.
, 2002, “An Atomistic-Based Finite Deformation Membrane for Single Layer Crystalline Films
,” J. Mech. Phys. Solids
0022-5096, 50
, pp. 1941
–1977
.24.
Zhang
, P.
, Huang
, Y.
, Geubelle
, P. H.
, Klein
, P. A.
, and Hwang
, K. C.
, 2002, “The Elastic Modulus of Single-Wall Carbon Nanotubes: A Continuum Analysis Incorporating Interatomic Potentials
,” Int. J. Solids Struct.
0020-7683, 39
, pp. 3893
–3906
.25.
Zhang
, P.
, Jiang
, H.
, Huang
, Y.
, Geubelle
, P. H.
, and Hwang
, K. C.
, 2004, “An Atomistic-Based Continuum Theory for Carbonnanotubes: Analysis of Fracture Nucleation
,” J. Mech. Phys. Solids
0022-5096, 52
, pp. 977
–998
.26.
Yang
, J. Z.
, and E
, W.
, 2006, “Generalized Cauchy–Born Rules for Elastic Deformation of Sheets, Plates, and Rods: Derivation of Continuum Models From Atomistic Models
,” Phys. Rev. B
0163-1829, 74
, p. 184110
.27.
Weiner
, J. H.
, 1983, Statistical Mechanics of Elasticity
, Wiley
, New York
.28.
Tadmor
, E. B.
, Smith
, G. S.
, Bernstein
, N.
, and Kaxiras
, E.
, 1999, “Mixed Finite Element and Atomistic Formulation for Complex Crystals
,” Phys. Rev. B
0163-1829, 59
, pp. 235
–245
.29.
Brenner
, D. W.
, 1990, “Empirical Potential for Hydrocarbons for Use in Simulating the Chemical Vapor Deposition of Diamond Films
,” Phys. Rev. B
0163-1829, 42
, pp. 9458
–9471
.30.
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
.31.
Wu
, J.
, Hwang
, K. C.
, and Huang
, Y.
, 2008, “An Atomistic-Based Finite-Deformation Shell Theory for Single-Wall Carbon Nanotubes
,” J. Mech. Phys. Solids
0022-5096, 56
, 279
–292
.32.
Born
, M.
, and Huang
, K.
, 1959, Dynamical Theory of Crystal Lattices
, Oxford University Press
, New York
.33.
Sanders
, J. L.
, 1963, “Nonlinear Theories for Thin Shells
,” Q. Appl. Math.
, 21
, pp. 21
–36
. 0033-569X34.
Koiter
, W. T.
, 1966, “On the Nonlinear Theory of Thin Elastic Shells, I, II, III
,” Proc. K. Ned. Akad. Wet., Ser. B: Phys. Sci.
, 69
, pp. 1
–54
. 0033-569X35.
Niordson
, F.
, 1985, Shell Theory
(North-Holland Series in Applied Mathematics and Mechanics)
Vol. 29
, North-Holland
, Amsterdam
.36.
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
.37.
Lu
, J. P.
, 1997, “Elastic Properties of Carbon Nanotubes and Nanoropes
,” Phys. Rev. Lett.
0031-9007, 79
, pp. 1297
–1300
.38.
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
.39.
Jin
, Y.
, and Yuan
, F. G.
, 2003, “Simulation of Elastic Properties of Single-Walled Carbon Nanotubes
,” Compos. Sci. Technol.
0266-3538, 63
, pp. 1507
–1515
.40.
Zhou
, X.
, Zhou
, J. J.
, and Ou-yang
, Z. C.
, 2000, “Strain Energy and Young’s Modulus of Single-Wall Carbon Nanotubes Calculated From Electronic Energy-Band Theory
,” Phys. Rev. B
0163-1829, 62
, pp. 13692
–13696
.41.
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
.42.
Tu
, Z. C.
, and Ou-yang
, Z.
, 2002, “Single-Walled and Multi-Walled Carbon Nanotubes Viewed as Elastic Tubes With the Effective Young’s Moduli Dependent on Layer Number
,” Phys. Rev. B
0163-1829, 65
, p. 233407
.43.
Pantano
, A.
, Parks
, D. M.
, and Boyce
, C. M.
, 2004, “Mechanics of Deformation of Single- and Multi-Wall Carbon Nanotubes
,” J. Mech. Phys. Solids
0022-5096, 52
, pp. 789
–821
.44.
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
.45.
Tersoff
, J.
, 1988, “New Empirical Approach for the Structure and Energy of Covalent Systems
,” Phys. Rev. B
0163-1829, 37
, pp. 6991
–7000
.Copyright © 2008
by American Society of Mechanical Engineers
You do not currently have access to this content.
