Skip to main content
SpringerLink
Log in

Mechanics of Carbon Nanotubes

A Review of Basic Models and New Nanoscale Effects

  • Chapter
  • First Online:
Trends in Nanoscale Mechanics

  • 1402 Accesses

Abstract

This chapter reviews basic models and new effects in the still emerging field of Nanoscale Mechanics and one of its essential parts: Mechanics of Carbon Nanotubes . Experiments with carbon nanotubes, theoretical models and modeling (i.e., molecular dynamics simulations), classification of carbon nanotubes into four classes (i.e., thin and thick lattice shells , long high-aspect-ratio nanotubes and beam-like carbon nanotube crystals of small radii) have been reviewed. Classification of carbon nanotubes is important for the safety of nanotechnology and evaluation of health effects. Interfacial sliding of the adjacent lattice shells in the multi wall carbon nanotubes (MWNT) has been discussed along with a nanoscale analog of the Newton’s friction law and the effect of spatial exclusion of electrons (ESEE) at the interface , which effectively can be viewed as a nanoscale analog of the Pauli’s exclusion principle. Examples of lattice waves, i.e., phonons, in carbon nanotubes have been presented. Ranges of applicability of estimates for the effective thickness of carbon nanotubes varying between 0.66 and 3.4 Å have been examined along with their dependence on the balance between the elastic interactions and van der Waals forces.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Classification of new types of materials is important in any field of science, especially, for the highly promising carbon nanotubes, which can be separated into four distinct classes associated with quite distinct geometric parameters and some similarities with asbestos though.

  2. 2.

    This research results have been first published at NASA and its ICASE Institute; see Harik, V.M., 2001. Ranges of applicability for the continuum-beam model in the constitutive analys is of carbon nanotubes: nanotubes or nano-beams? (NASA/CR-2001-211013, NASA Langley Research Center), Hampton, Virginia, USA. Harik, V.M., 2001. Mechanics of carbon nanotubes © 2001. ASME Education Institute (Notes for a Short Course, a 2002 CD and a 2001 video), American Society of Mechanical Engineers, New York, NY.

  3. 3.

    For more historical perspectives and some epistemological notes about the concepts of emerging Nanoscale Mechanics see author’s footnotes for the references cited in this chapter.

  4. 4.

    Nanoscale homogenization itself and nanoscale homogenization criteria [11, 12], in particular, are very important for the application of continuum concepts (e.g., continuous surface or a properly-defined number of representative volume elements for the volume-averaging for the uniquely-defined material properties of any material having a discrete atomic lattice structure) to the CNT lattice structures (for more details, see the next part of this chapter).

  5. 5.

    V.M. Harik et al. 2002. Applicability of the Continuum-shell Theories to the Mechanics of Carbon Nanotubes. (NASA/CR-2002-211460/ICASE Report No. 2002–2007, ICASE Institute) NASA Langley Research Center, Hampton, Virginia. In this NASA report model applicability map [12] for the continuum shell models has been presented.

  6. 6.

    V.M. Harik, New Trends in Mechanics of Carbon Nanotubes and Applications, Technical Report TR-2012-2, Nanodesigns Consulting, Newark, Delaware, 2012.

  7. 7.

    Initiation of the so called SEE effect is associated with the need of at least one π-electron to overcome the registry potential of an opposing C–C bond and the associated Coulomb repulsion within the so called spatial exclusion zone (SEZ) for electrons. The size of the spatial exclusion zone depends on the local atomic lattice configuration (e.g., orientation of C–C bonds), the registry potential barriers, the nanoscale Coulomb repulsion proportional to 1/r2, and the nanoscale repulsion proportional to −1/r12. The combined effect results in the so called SEE effect . The nanoscale analog of Pauli exclusion-repulsion for electrons stems from the quantum Pauli principle for the identical electrons, i.e. particles with the spin ½ (fermions); the two identical particles cannot occupy the same energy state, as their combined wave function, ψ, is anti-symmetric. The nanoscale analog of Pauli repulsion and the quantum Pauli principle for the electrons both affect the precise dimensions of the spatial exclusion zone for interfacial electrons.

  8. 8.

    S. Iijima was awarded the 1996 Nobel Prize in Chemistry for “discovering fullerenes”, which also include carbon nanotubes.

  9. 9.

    V.M. Harik, Mechanics of Carbon Nanotubes © (2001), a short course, the Annual ASME Congress (2001, 2004) and the 2002 Nanosystems Conference (Berkeley, Califormia).

  10. 10.

    In the late 15th century Leonardo da Vinci had identified the three important parts of friction as follows. “Friction is divided into three parts: these are simple, compound and disordered.” Simple friction is due to the motion and dragging; the compound friction is “between two immovable things” and the irregular friction is associated with the “corners of different sides.” For more details see the notebooks of Leonardo da Vinci [76].

References

  1. S. Iijima, Helical microtubules of graphitic carbon. Nature, 354, 56 (1991) (Note that in 1991 S. Iijima has first referred to carbon nanotubes as ‘microtubules’ of graphitic carbon, which he and his colleague later changed to nanotubes in 1993 [2]. From the epistemological point of view this change from ‘microtubules’ to nanotubes indicates one of the first origins of the nanoscale concepts for new nanostructures, i.e., one of the first nanoparticles. Note that Buckminster ball, i.e., molecule C60, or the buckyball, was discovered in 1985 by R. Smalley.)

    Google Scholar 

  2. S. Iijima, T. Ichihashi, Single-shell (Note that in 1993 S. Iijima and T. Ichihachi have first proposed to consider carbon nanotubes as an atomic lattice ‘shell’. The concept of a ‘shell’ was later used by B. Yakobson and his colleagues in their analysis of the carbon nanotube buckling [7, 8]. This was important from the epistemological point of view and for the development of new models for carbon nanotubes.) carbon nanotubes of low diameters. Nature, 363, 603 (1993)

    Google Scholar 

  3. R.S. Ruoff, J. Tersoff, D.C. Lorents, S. Subramoney, B. Chan, Radial deformation of carbon nanotubes by van der Waals’ forces. Nature, 364, 514–516 (1993)

    Google Scholar 

  4. J. Tersoff, R.S. Ruoff, Structural properties of a carbon nanotube crystal, (Note that in 1993 J. Tersoff and R. S. Ruoff have first recognized the importance of the carbon nanotube crystals (or bundles), which are important for the manufacturing of centimeter long carbon nanotubes based fibers more than fifteen years later.) Phys. Rev. Lett. 73, 676 (1994)

    Google Scholar 

  5. M.M.J. Treacy, T.W. Ebbesen, J.M. Gibson, Exceptionally high Young’s modulus observed for individual carbon nanotubes. Nature 381, 680 (1996)

    Article  Google Scholar 

  6. E.W. Wong, P.E. Sheehan, C.M. Lieber, Nanobeam mechanics: (Note that in 1997 C. Lieber and his colleagues have introduced the concept of nanobeams (as opposed to nanowires, for example) and an area in Nanoscale Mechanics: Nanobeam Mechanics or Mechanics of Nanobeams) elasticity, strength, and toughness of nanorods and nanotubes. Science, 277, 1971 (1997)

    Google Scholar 

  7. B.I. Yakobson, C.J. Brabec, J. Bernholc, Nanomechanics (Note that in 1996 B. Yakobson and his colleagues were among the first scientists, who have proposed to use a new term Nanomechanics, which has heralded the emerging field of Nanoscale Mechanics. They have also used the term “carbon tubes” for carbon nanotubes while they have utilized the concepts of macroscopic shell theory in the analysis of carbon nanotube buckling.) of carbon tubes: instabilities beyond linear response. Phys. Rev. Lett. 76, 2511 (1996)

    Google Scholar 

  8. B.I. Yakobson, T. Dimitrica, Chapter 1, in Trends in Nanoscale Mechanics, ed. by V.M. Harik, M. Salas (Kluwer Academic Publishers, Dordrecht, 2003), pp. 3–33

    Google Scholar 

  9. R. Saito, M. Fujita, G. Dresselhaus, M.S. Dresselhaus, Electronic-structure of chiral graphene tubules. (Note that in 1992 (and 1991) the use of terms of “tubules” and “microtubules” for carbon nanotubes was present as an alternative term during the early stages of the emerging Nanoscale Materials Science and Nanoscale Science of Nanostructured Materials. The cited paper has also introduced the nomenclature for the physical description of carbon nanotubes and their chirality.) Appl. Phys. Lett. 60, 18, 2204–2206 (1992)

    Google Scholar 

  10. V.M. Harik, Ranges of applicability (Note that this paper was among the first addressing the ranges of applicability (and limitations) of continuum models to carbon nanotubes. Also note the criticism of the use of multiple macroscopic shells in the analysis of continuum cross-sectional area by S.J. Frankland, V.M. Harik, Analysis of carbon nanotube pull-out from a polymer matrix. Mat. Res. Soc. Symp. Proc. 733 E, T6.2.1 (2002). In 1997 in their paper [13], B. I. Yakobson and R. Smaley expressed their hope that continuum mechanics can be applied at the level of “a few atoms”, which was rather optimistic as was later demonstrated in the aforementioned papers.) for the continuum beam model in the mechanics of carbon nanotubes and nanorods. Solid State Commun. 120 (7–8), 331–335 (2001)

    Google Scholar 

  11. V.M. Harik, Mechanics of carbon nanotubes: (Note that in 2001 [10], and in 2002, a particular area of Nanoscale Mechanics: Mechanics of Carbon Nanotubes was defined and its key basic concepts were outlined in a first nationally-distributed short course by V.M. Harik: Mechanics of Carbon Nanotubes © 2001 (a Short Curse of ASME Education Institute, American Society of Mechanical Engineers (ASME), New York).) applicability of the continuum-beam models. Comput. Mater. Sci. 24(3), 328–342 (2002)

    Google Scholar 

  12. V.M. Harik, Mechanics of Carbon Nanotubes (Nanodesigns Press, Newark, 2011)

    Google Scholar 

  13. X. Zhou, J.J. Zhou, Z.C. Ou-Yang, Phys. Rev. B, 62, 13692 (2000)

    Google Scholar 

  14. Z.C. Tu, Z.C. Ou-Yang, Phys. Rev. B, 65, 233407–233414 (2002)

    Google Scholar 

  15. A. Pantano, M.C. Boyce, D.M. Parks, Phys. Rev. Lett. 91, 145504 (2004)

    Google Scholar 

  16. A. Pantano, D.M. Parks, M.C. Boyce, J. Mech. Phys. Solids 52, 789–821 (2004)

    Google Scholar 

  17. S.V. Goupalov, Phys. Rev. B, 71, 085420 (2005)

    Google Scholar 

  18. P. Avouris, T. Hertel, R. Martel, T. Schmidt, H.R. Shea, R.E. Walkup, Appl. Surf. Sci. 141, 20 (1999)

    Google Scholar 

  19. T. Halicioglu, Stress calculations for carbon nanotubes. Thin Solid Films, 312, 11 (1998)

    Google Scholar 

  20. L. Wang, Q. Zheng, J.Z. Liu, Q. Jiang, Phys. Rev. Lett. 95, 105501 (2005)

    Google Scholar 

  21. Y. Huang, J. Wu, K.C. Hwang, Phys. Rev. B, 74(24), 24541 (2006)

    Google Scholar 

  22. K.N. Kudin, C.E. Scuseria, B.I. Yakobson, Phys. Rev. B, 64, 235406 (2001)

    Google Scholar 

  23. A. Sears, R.C. Batra, Phys. Rev. B. 69, 235406 (2004)

    Google Scholar 

  24. C.Q. Ru, Effective bending stiffness of carbon nanotubes. Phys. Rev. B, 62(15), 9973 (2000)

    Google Scholar 

  25. S. Govindjee, J.L. Sackman, Solid State Commun. 110, 227, 1999

    Google Scholar 

  26. S.P. Timoshenko, J.M. Gere, Theory of Elastic Stability (McGraw-Hill, New York, 1961)

    Google Scholar 

  27. M. Arroyo, T. Belytschko, J. Mech. Phys. Solids, 50, 1941 (2002)

    Google Scholar 

  28. M. Arroyo, T. Belytschko, Mech. Mater. 35(3–6), 193–215 (2003)

    Article  Google Scholar 

  29. M. Arroyo, T. Belytschko, Phys. Rev. B, No. 69, Paper N 115415 (2004)

    Google Scholar 

  30. D. Stojkovic, P. Zhang, V.H. Crespi, Phys. Rev. Lett. 87(12), 125502 (2001) (In 2001 V. H. Crespi [33] and his group at the Penn State University and V. M. Harik [34] at NASA Langley Research Center have independently predicted degeneration of CNT lattice shells into nano-beams around the critical value of the normalized CNT radius, R NT /a ≈ 1, Crespi predicted “breaking the symmetry of sp3 bonds in tubular geometries” in the smallest nanotubes.)

    Google Scholar 

  31. V.M. Harik, Ranges of Applicability for the Continuum-Beam Model in the Constitutive Analysis of Carbon Nanotubes: Nanotubes or Nano-beams? NASA/CR-2001-211013, NASA Langley Research Center, Hampton, Virginia (2001)

    Google Scholar 

  32. B.I. Yakobson, R.E. Smalley, (In 1997 B.I. Yakobson and R. Smaley after a very successful paper on the origins of Nanomechanics [7, 8] have expressed rather optimistic hopes that the concepts of macroscopic continuum mechanics can be applied to nanostructures at the scale of “a few atoms”, which was later corrected by the nanoscale homogenization criterion as often happens in an emerging new science.) Am. Sci. 85, 324 (1997)

    Google Scholar 

  33. C.Y. Li, T.-W. Chou, Int. J. Solids Struct. 40, 2487–2499 (2003)

    Article  MATH  Google Scholar 

  34. R.C. Batra, S.S. Gupta, ASME J. Appl. Mech. 75, 061010 (2008)

    Google Scholar 

  35. A.N. Kolmogorov, V.H. Crespi, Phys. Rev. Lett. 85(22), 4727 (2000)

    Google Scholar 

  36. K. Liao, S. Li, Appl. Phys. Lett. 79(25), 4225 (2001)

    Google Scholar 

  37. A.N. Kolmogorov, V.H. Crespi, Phys. Rev. B 71, 235415 (2005)

    Google Scholar 

  38. N. Marom, J. Bernstein, J. Garel, A. Tkatchenko, E. Joselevich, L. Kronik, O. Hod, Phys. Rev. Lett. 105, 046801 (2010)

    Google Scholar 

  39. S.J.V. Frankland, V.M. Harik, Surf. Sci. Lett. 525, L103 (2003)

    Google Scholar 

  40. S.J.V. Frankland, V.M. Harik, Mat. Res. Soc. Symp. Proc. 733 E, T6.2.1 (2002). (This research on the nanoscale friction modeling was supported in part by NASA and the Princeton University based NASA URETI Institute for the Bio-inspired Nanostructured Multifunctional Materials (see the Princeton University based web site: http://bimat.org, award No. NCC-1-02037 by the NASA Langley Research Center, Hampton, Virginia).)

  41. S.J.V. Frankland, A. Caglar, D.W. Brenner, M. Griebel, J. Phys. Chem. B 106(12), 3046–3048 (2002)

    Article  Google Scholar 

  42. Q.Y. Li, K-S. Kim, Proc. R. Soc. A 464, 1319 (2008)

    Google Scholar 

  43. M.S. Dresselhaus, P.C. Eklund, Adv. Phys. 49(6), 705 (2000)

    Google Scholar 

  44. A.N. Cleland, Foundations of Nanomechanics (Springer, Berlin, 2003)

    Google Scholar 

  45. P. Tangney, M.L. Cohen, S.G. Louie, Phys. Rev. Lett. 97(19), 195901 (2006)

    Google Scholar 

  46. X.H. Zhang, G.E. Santoro, U. Tartaglino, E. Tosatti, Phys. Rev. Lett. 102(12), 125502 (2009)

    Google Scholar 

  47. V.M. Harik, R.A. Cairncross, Mech. Mater. 32, 807 (2000)

    Google Scholar 

  48. M.F. Yu, B.I. Yakobson, R.S. Ruoff, J. Phys. Chem. 104, 8764 (2000)

    Google Scholar 

  49. J. Buchoux, J.-P. Aime, R. Boisgard, C.V. Nguyen, L. Buchaillot, S. Marsaudon, Nanotechnology 20, 475701 (2009)

    Article  Google Scholar 

  50. H. Dai, Nature 384, 147 (1996)

    Article  Google Scholar 

  51. K. Jensen, C. Girit, W. Mickelson, A. Zettl, Phys. Rev. Lett. 96 (21), 215503 (2006)

    Google Scholar 

  52. R. Stevens et al., Appl. Phys. Lett. 77, 3453 (2000)

    Article  Google Scholar 

  53. H. Nishijima, Appl. Phys. Lett. 74, 4061 (1999)

    Google Scholar 

  54. J.H. Hafner, C.L. Cheung, T. Oosterkamp, C.M. Lieber, J. Phys. Chem. 105, 743 (2001)

    Google Scholar 

  55. A. Garg, S.B. Sinnott, Chem. Phys. Lett. 295, 273–278 (1998)

    Google Scholar 

  56. I.-L. Chang, B.-C. Chiang, J. Appl. Phys. 106, 114313 (2009)

    Article  Google Scholar 

  57. C.Y. Li, T.-W. Chou, Mech. Mater. 36(11), 1047–1055 (2004)

    Google Scholar 

  58. L.S. Schadler, S.C. Giannaris, P.M. Ajayan, Appl. Phys. Lett. 73, 3842 (1998)

    Google Scholar 

  59. S.B. Sinnott, O.A. Shenderova, C.T. White, D.W. Brenner, Carbon, 36(1–2), 1–9 (1998)

    Google Scholar 

  60. H.D. Wagner, O. Lourie, Y. Feldman, R. Tenne, Appl. Phys. Lett. 72, 188 (1998)

    Article  Google Scholar 

  61. P.M. Ajayan, L.S. Schadler, S.C. Giannaris, A. Rubio, Adv. Mater. 12, 750 (2000)

    Article  Google Scholar 

  62. C.Y. Li, Chou, J. Nanosci. Nanotechnol. 3(5), 423–430 (2000)

    Google Scholar 

  63. S.B. Sinnott, J. Nanosci. Nanotechnol. 2(2), 113–123 (2002)

    Article  Google Scholar 

  64. S.J.V. Frankland, V.M. Harik, G.M. Odegard, D.W. Brenner, T.S. Gates, Compos. Sci. Technol. 63(11), 1655–1661 (2003)

    Google Scholar 

  65. Y. Hu, I. Jang, S.B. Sinnott, Compos. Sci. Technol. 63(11), 1663–1669 (2003)

    Google Scholar 

  66. Z. Qunaies, C. Park, K.E. Wise, E.J. Siochi, J.S. Harrison, Compos. Sci. Technol. 63(11), 1637–1646 (2003)

    Google Scholar 

  67. D. Srivastava, C. Wei, K. Cho, Appl. Mech. Rev. 56(2), 215–230 (2003)

    Google Scholar 

  68. A.N. Cleland, M.L. Roukes, Nature, 320, 160–161 (1998)

    Google Scholar 

  69. V.M. Harik, L.-S. Luo (eds.), Micromechanics and Nanoscale Effects (Kluwer Academic Publishers, The Netherlands, 2004)

    Google Scholar 

  70. T.W. Chou, Microstructural Design of Fiber Composites (Cambridge University Press, Cambridge, 1992)

    Google Scholar 

  71. E.T. Thostenson, C. Li, T.-W. Chou, Compos. Sci. Tech. 65, 491–516 (2005)

    Google Scholar 

  72. R.C. Picu, A. Sarvestani, M.S. Ozmusul, Chapter 3, in V.M. Harik, M.D. Salas (eds.) Trends in Nanoscale Mechanics (Kluwer Academic Publishers, The Netherlands, 2003), pp. 61–87

    Google Scholar 

  73. Y. Hu, O.A. Shenderova, Z. Hu, C.W. Padgett, D.W. Brenner, Rep. Prog. Phys. 69, 1847 (2006)

    Google Scholar 

  74. B.N.J. Persson, Sliding Friction: Physical Principles and Applications (Springer, Berlin, 1998)

    Google Scholar 

  75. B.N.J. Persson, Surf. Sci. Reports 33, 83 (1999)

    Google Scholar 

  76. E. MacCurdy (ed.), The Notebooks of Leonardo da Vinci (Konecky and Konecky printing, Duckworth, London, 1906)

    Google Scholar 

  77. S. Gorantla, S. Avdoshenko, F. Börrnert, A. Bachmatiuk, M. Dimitrakopoulou, F. Schäffel, R. Schönfelder, J. Thomas, T. Gemming, J. H. Warner, G. Cuniberti, J. Eckert, B. Büchner, M. H. Rümmeli, Enhanced π–π Interactions Between a C60 Fullerene and a Buckle Bend on a Double-Walled Carbon Nanotube. Nano. Res. 3, 92–97 (2010)

    Google Scholar 

  78. S.J. Frankland, V.M. Harik, Analysis of carbon nanotube pull-out from a polymer matrix. Mat. Res. Soc. Symp. Proc. 733 E, T6.2.1 (2002)

    Google Scholar 

  79. C. Semaan, G. Pecastaings, M. Schappacher, A. Soum, Polym. Bull. 68, 465–481 (2012)

    Google Scholar 

  80. T. Vodenitcharova, L.C. Zhang, Phys. Rev. B, 68, 165401 (2003)

    Google Scholar 

  81. M.-F. Yu, B.S. Files, S. Arepalli, R.S. Ruoff, Phys. Rev. Lett. 84, 5552 (2000)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Nanodesigns Consulting, P.O. Box 5303, Wilmington, DE, 19808-5303, USA

    Vasyl Harik

Authors
  1. Vasyl Harik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vasyl Harik .

Editor information

Editors and Affiliations

  1. Nanodesigns Consulting, Wilmington, Delaware, USA

    Vasyl Harik

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Harik, V. (2014). Mechanics of Carbon Nanotubes. In: Harik, V. (eds) Trends in Nanoscale Mechanics. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9263-9_2

Download citation

  • DOI: https://doi.org/10.1007/978-94-017-9263-9_2

  • Published:

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-017-9262-2

  • Online ISBN: 978-94-017-9263-9

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation