, Volume 45, Issue 1, pp 43–51 | Cite as

A structural mechanics approach for predicting the mechanical properties of carbon nanotubes

  • H. Wan
  • F. Delale


Based on molecular mechanics, a structural mechanics model of carbon nanotubes (CNTs) was developed with special consideration given to the bending stiffness of the graphite layer. The potentials associated with the atomic interactions within a CNT were evaluated by the strain energies of beam elements which serve as structural substitutions of covalent bonds in a CNT. In contrast to the original model developed by Li and Chou (Int. J. Solids Struct. 40(10):2487–2499, 2003), in the current model the out-of-plane deformation (inversion) of the bond was distinguished from the in-plane deformation by considering a rectangular cross-section for the beam element. Consequently, the model is able to study problems where the effect of local bending of the graphite layer in a carbon nanotube is significant. A closed-form solution of the sectional properties of the beam element was derived analytically. The model was verified through the analysis of rolling a graphite sheet into a carbon nanotube. Using the present model, the buckling behavior of nanotubes under bending is simulated. The predicted critical bending angle agrees well with molecular dynamics simulations.


Carbon nanotubes Mechanical properties Young’s modulus Bending stiffness Buckling 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Li C, Chou T-W (2003) A structural mechanics approach for the analysis of carbon nanotubes. Int J Solids Struct 40(10):2487–2499 MATHCrossRefGoogle Scholar
  2. 2.
    Harik VM (2001) Ranges of applicability for the continuum beam model in the mechanics of carbon nanotubes and nanorods. Solid State Commun 120:331–335 CrossRefADSGoogle Scholar
  3. 3.
    Yakobson BI, Brabec CJ, Bernholc J (1996) Nanomechanics of carbon tubes: Instabilities beyond linear response. Phys Rev Lett 76(14):2511 CrossRefADSGoogle Scholar
  4. 4.
    Ru CQ (2000) Effective bending stiffness of carbon nanotubes. Phys Rev B 62(15):9973 CrossRefADSGoogle Scholar
  5. 5.
    Shen L, Li J (2005) Equilibrium structure and strain energy of single-walled carbon nanotubes. Phys Rev B (Condens Matter Mater Phys) 71(16):165427-8 ADSGoogle Scholar
  6. 6.
    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 MATHCrossRefADSGoogle Scholar
  7. 7.
    Shen L, Li J (2004) Transversely isotropic elastic properties of single-walled carbon nanotubes. Phys Rev B (Condens Matter Mater Phys) 69(4):045414-10 ADSGoogle Scholar
  8. 8.
    Odegard GM, Gates TS, Nicholson LM, Wise KE (2002) Equivalent-continuum modeling of nano-structured materials. Compos Sci Technol 62:1869–1880 CrossRefGoogle Scholar
  9. 9.
    Huang M-Y, Chen H-B, Lu J-N, Lu P, Zhang P-Q (2006) A modified molecular structural mechanics method for analysis of carbon nanotubes. Chinese J Chem Phys 19(4):286–290 Google Scholar
  10. 10.
    Li C, Chou T-W (2004) Modeling of elastic buckling of carbon nanotubes by molecular structural mechanics approach. Mech Mater 36:1047–1055 CrossRefGoogle Scholar
  11. 11.
    Li C, Chou T-W (2005) Modeling of carbon nanotube clamping in tensile tests. Compos Sci Technol 65:2407–2415 CrossRefGoogle Scholar
  12. 12.
    Tu Z-C, Ou-Yang Z-C (2002) Single-walled and multiwalled carbon nanotubes viewed as elastic tubes with the effective Young’s moduli dependent on layer number. Phys Rev B 65:233407 CrossRefADSGoogle Scholar
  13. 13.
    Kundin KN, Scuseria GE, Yakobson BI (2001) C2f, bn, and c nanoshell elasticity from ab initio computations. Phys Rev B 64:235406 ADSGoogle Scholar
  14. 14.
    Pantano A, Parks DM, Boyce MC (2004) Mechanics of deformation of single- and multi-wall carbon nanotubes. J Mech Phys Solids 52(4):789–821 MATHCrossRefADSGoogle Scholar
  15. 15.
    Zhou X, Zhou JJ, Ou-Yang ZC (2000) Strain energy and Young’s modulus of single-wall carbon nanotubes calculated from electronic energy-band theory. Phys Rev B 62(20):13692–13696 CrossRefADSGoogle Scholar
  16. 16.
    Robertson DH, Brenner DW, Mintmire JW (1992) Energetics of nanoscale graphitic tubules. Phys Rev B 45(21):12592 CrossRefADSGoogle Scholar
  17. 17.
    Miyamoto Y, Rubio A, Louie SG, Cohen ML (1994) Electronic properties of tubule forms of hexagonal bc_{3}. Phys Rev B 50(24):18360 CrossRefADSGoogle Scholar
  18. 18.
    Kurti J, Kresse G, Kuzmany H (1998) First-principles calculations of the radial breathing mode of single-wall carbon nanotubes. Phys Rev B 58(14):R8869 CrossRefADSGoogle Scholar
  19. 19.
    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(19):12678 CrossRefADSGoogle Scholar
  20. 20.
    Hernandez E, Goze C, Bernier P, Rubio A (1998) Elastic properties of c and b x c y n z composite nanotubes. Phys Rev Lett 80(20):4502 CrossRefADSGoogle Scholar
  21. 21.
    Mylvaganam K, Vodenitcharova T, Zhang LC (2006) The bending-kinking analysis of a single-walled carbon nanotube—a combined molecular dynamics and continuum mechanics technique. J Mater Sci 41:3341–3347 CrossRefADSGoogle Scholar
  22. 22.
    Tu ZC, Ou-Yang ZC (2008) Elastic theory of low-dimensional continua and its applications in bio- and nano-structures. J Comput Theor Nanosci 5:422–448 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringThe City College of New YorkNew YorkUSA

Personalised recommendations