, Volume 48, Issue 6, pp 1355–1367 | Cite as

An accurate molecular mechanics model for computation of size-dependent elastic properties of armchair and zigzag single-walled carbon nanotubes



In this paper, an analytical solution based on a molecular mechanics model is developed to evaluate the mechanical properties of armchair and zigzag single-walled carbon nanotubes (SWCNTs). Adopting the Perdew–Burke–Ernzerhof (PBE) exchange correlation, the density functional theory (DFT) calculations are performed within the generalized gradient approximation (GGA) to evaluate force constants used in the molecular mechanics model. After that, based on the principle of molecular mechanics, explicit expressions are proposed to obtain surface Young’s modulus, Poisson’s ratio and surface shear modulus of SWCNTs corresponding to both types of armchair and zigzag chiralities. Based on the DFT calculations, it is found that the flexural rigidity of graphene is independent of the type of chirality which indicates the isotropic characteristic of this material. Moreover, it is observed that for the all values of nanotube diameter, surface Young’s modulus for the armchair nanotube is more than that of zigzag nanotube. It is shown that the trend predicted by the present model is in good agreement with other models which confirms the validity as well as the accuracy of the present molecular mechanics model.


Carbon nanotube Molecular mechanics model Density functional theory Graphene sheet Mechanical properties 


  1. 1.
    Iijima S (1991) Helical microtubes of graphite carbon. Nature 354:56–58 ADSCrossRefGoogle Scholar
  2. 2.
    Hernandez E, Goze C, Bernier P, Rubio A (1998) Elastic properties of C and BxCyNz composite nanotubes. Phys Rev Lett 80:4502–4505 ADSCrossRefGoogle Scholar
  3. 3.
    Sanchez-Portal D, Artacho E, Soler JM, Rubio A, Ordejon P (1999) Ab initio structural, elastic, and vibrational properties of carbon nanotubes. Phys Rev B 59:12678–12688 ADSCrossRefGoogle Scholar
  4. 4.
    Treacy MMJ, Ebbesen TW, Gibson JM (1996) Exceptionally high Young’s modulus observed for individual carbon nanotubes. Nature 381:678–680 ADSCrossRefGoogle Scholar
  5. 5.
    Krishnan A, Dujardin E, Ebbesen TW, Yianiolos PN, Treacy MMJ (1998) Young’s modulus of single-walled carbon nanotubes. Phys Rev B 58:14013–14019 ADSCrossRefGoogle Scholar
  6. 6.
    Demczyk BG, Wang YM, Cumings J, Hetman M, Han W, Zettl A, et al. (2002) Direct mechanical measurement of the tensile strength and elastic modulus of multiwalled carbon nanotubes. Mater Sci Eng A 334:173–178 CrossRefGoogle Scholar
  7. 7.
    Ke CH, Pugno N, Peng B, Espinosa HD (2005) Experiments and modeling of carbon nanotube-based NEMS devices. J Mech Phys Solids 53:1314–1333 ADSMATHCrossRefGoogle Scholar
  8. 8.
    Arroyo N, Belytschko T (2002) Large scale deformation atomistic-based continuum analysis of carbon nanotubes. In: AIAA, p 1317 Google Scholar
  9. 9.
    Arroyo M, Belytschko T (2003) Nonlinear mechanical response and rippling of thick multiwalled carbon nanotubes. Phys Rev Lett 91:215505 ADSCrossRefGoogle Scholar
  10. 10.
    Li YC, Chou TW (2003) A structural mechanics approach for the analysis of carbon nanotubes. Int J Solids Struct 40:2489–2499 Google Scholar
  11. 11.
    Kiani K (2010) A meshless approach for free transverse vibration of embedded single-walled nanotubes with arbitrary boundary conditions accounting for nonlocal effect. Int J Mech Sci 52:343–1356 MathSciNetCrossRefGoogle Scholar
  12. 12.
    Arash B, Ansari R (2010) Evaluation of nonlocal parameter in the vibrations of single-walled carbon nanotubes with initial strain. Physica E 42:2058–2064 ADSCrossRefGoogle Scholar
  13. 13.
    Ansari R, Sahmani S, Arash B (2010) Nonlocal plate model for free vibrations of single-layered graphene sheets. Phys Lett A 375:53–62 ADSCrossRefGoogle Scholar
  14. 14.
    Ansari R, Sahmani S, Rouhi H (2011) Rayleigh-Ritz axial buckling analysis of single-walled carbon nanotubes with different boundary conditions. Phys Lett A 375:1255–1263 ADSCrossRefGoogle Scholar
  15. 15.
    Ansari R, Sahmani S, Rouhi H (2011) Axial buckling analysis of single-walled carbon nanotubes in thermal environments via Rayleigh-Ritz technique. Comput Mater Sci 50:3050–3055 CrossRefGoogle Scholar
  16. 16.
    Ansari R, Sahmani S (2012) Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam models. Commun Nonlinear Sci Numer Simul 17:1965–1979 MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Hao MJ, Guo XM, Wang Q (2010) Small-scale effect on torsional buckling of multi-walled carbon nanotubes. Eur J Mech A, Solids 29:49–55 MathSciNetCrossRefGoogle Scholar
  18. 18.
    Yakobson BI, Brabec CJ, Bernholc J (1996) Nanomechanics of carbon nanotubes: instability beyond linear response. Phys Rev Lett 76:2511–2514 ADSCrossRefGoogle Scholar
  19. 19.
    Zhang LC, Vodenitcharova T (2003) Effective wall thickness of a single-walled carbon nanotubes. Phys Rev B 68:165401 ADSCrossRefGoogle Scholar
  20. 20.
    Zhou X, Zhou JJ, Ouyang ZC (2000) Strain energy and Young’s modulus of single-wall carbon nanotubes calculated from electronic energy-band theory. Phys Rev B 62:13692–13696 ADSCrossRefGoogle Scholar
  21. 21.
    Chang T, Gao H (2003) Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model. J Mech Phys Solids 51:1059–1074 ADSMATHCrossRefGoogle Scholar
  22. 22.
    Chang T, Li G, Guo X (2005) Elastic axial buckling of carbon nanotubes via a molecular mechanics model. Carbon 43:287–294 CrossRefGoogle Scholar
  23. 23.
    Fang SC, Chang WJ, Wang YH (2007) Computation of chirality- and size-dependent surface Young’s moduli for single-walled carbon nanotubes. Phys Lett A 371:499–503 ADSCrossRefGoogle Scholar
  24. 24.
    Rossi M, Meo M (2009) On the estimation of mechanical properties of single-walled carbon nanotubes by using a molecular-mechanics based FE approach. Compos Sci Technol 69:1394–1398 CrossRefGoogle Scholar
  25. 25.
    Wan H, Delale F (2010) A structural mechanics approach for predicting the mechanical properties of carbon nanotubes. Meccanica 45:43–51 MATHCrossRefGoogle Scholar
  26. 26.
    Szabo A, Ostlund NS (1989) Modern quantum chemistry. McGraw-Hill, New York Google Scholar
  27. 27.
    Hedin L (1965) New method for calculating the one-particle Green’s function with application to the electron-gas problem. Phys Rev A 139:796 ADSGoogle Scholar
  28. 28.
    Perdew JP, Burke K, Ernzerhof M (1996) Generalized gradient approximation made simple. Phys Rev Lett 77:3865–3868 ADSCrossRefGoogle Scholar
  29. 29.
    Perdew JP, Burke K, Wang Y (1996) Generalized gradient approximation for the exchange-correlation hole of a many-electron system. Phys Rev B 54:16533–16539 ADSCrossRefGoogle Scholar
  30. 30.
    Baroni S, Corso DA, Gironcoli S, Giannozzi P, Cavazzoni C, Ballabio G, Scandolo S, Chiarotti G, Focher P, Pasquarello A, Laasonen K, Trave A, Car R, Marzari N, Kokalj A.
  31. 31.
    Topsakal M, Cahangirov S, Ciracil S (2010) The response of mechanical and electronic properties of graphene to the elastic strain. Appl Phys Lett 96:091912 ADSCrossRefGoogle Scholar
  32. 32.
    Pack JD (1976) Special points for Brillouin-zone integrations. Phys Rev B 13:5188–5192 MathSciNetADSCrossRefGoogle Scholar
  33. 33.
    Zhao K, Zhao M, Wang Z, Fan Y (2010) Tight-binding model for the electronic structures of SiC and BN nanoribbons. Phys Rev E 43:440–445 Google Scholar
  34. 34.
    Grosso G, Parravicini GP (2000) Solid state physics. Academic Press, San Diego Google Scholar
  35. 35.
    Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev B 136:864–871 MathSciNetADSCrossRefGoogle Scholar
  36. 36.
    Kohn W, Sham LJ (1965) Self-consistent equations including exchange and correlation effects. Phys Rev A 140:1133–1138 MathSciNetADSGoogle Scholar
  37. 37.
    Schlüter M, Hamann DR, Chiang C (1979) Norm-conserving pseudopotentials Phys Rev Lett 43:1494–1497 ADSCrossRefGoogle Scholar
  38. 38.
    Troullier N, Martins JL (1991) Efficient pseudopotentials for planewave calculations. Phys Rev B 43:15221 Google Scholar
  39. 39.
    Lee C, Wei X, Kysar JW, Hone J (2008) Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science 321:385–392 ADSCrossRefGoogle Scholar
  40. 40.
    Liu F, Ming P, Li J (2007) Ab initio calculation of ideal strength and phonon instability of graphene under tension. Phys Rev B 76:064120 ADSCrossRefGoogle Scholar
  41. 41.
    Goze C, Vaccarini L, Henrard L, Bernier P, Hernandez E, Rubio A (1999) Elastic and mechanical properties of carbon nanotubes. Synth Met 103:2500–2501 CrossRefGoogle Scholar
  42. 42.
    Popov VN, Van Doren VE, Balkanski M (2000) Elastic properties of single walled carbon nanotubes. Phys Rev B 61:3078–3084 ADSCrossRefGoogle Scholar
  43. 43.
    Lu JP (1977) Elastic properties of carbon nanotubes and nanopores. Phys Rev Lett 79:1297–1300 ADSCrossRefGoogle Scholar
  44. 44.
    Van Lier G, Van Alsenoy C, Van Doren V, Geerlings P (2000) Ab initio study of the elastic properties of single-walled carbon nanotubes and graphene. Chem Phys Lett 326:181 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of GuilanRashtIran

Personalised recommendations