Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Buckling and post-buckling behaviors of higher order carbon nanotubes using energy-equivalent model

  • 60 Accesses


This paper aims to investigate the size scale effect on the buckling and post-buckling of single-walled carbon nanotube (SWCNT) rested on nonlinear elastic foundations using energy-equivalent model (EEM). CNTs are modelled as a beam with higher order shear deformation to consider a shear effect and eliminate the shear correction factor, which appeared in Timoshenko and missed in Euler–Bernoulli beam theories. Energy-equivalent model is proposed to bridge the chemical energy between atoms with mechanical strain energy of beam structure. Therefore, Young’s and shear moduli and Poisson’s ratio for zigzag (n, 0), and armchair (n, n) carbon nanotubes (CNTs) are presented as functions of orientation and force constants. Conservation energy principle is exploited to derive governing equations of motion in terms of primary displacement variable. The differential–integral quadrature method (DIQM) is exploited to discretize the problem in spatial domain and transformed the integro-differential equilibrium equations to algebraic equations. The static problem is solved for critical buckling loads and the post-buckling deformation as a function of applied axial load, CNT length, orientations and elastic foundation parameters. Numerical results show that effects of chirality angle, boundary conditions, tube length and elastic foundation constants on buckling and post-buckling behaviors of armchair and zigzag CNTs are significant. This model is helpful especially in mechanical design of NEMS manufactured from CNTs.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6


  1. 1.

    Aydogdu M (2008) Vibration of multi-walled carbon nanotubes by generalized shear deformation theory. Int J Mech Sci 50(4):837–844

  2. 2.

    Baghdadi H, Tounsi A, Zidour M, Benzair A (2014) Thermal effect on vibration characteristics of armchair and zigzag single-walled carbon nanotubes using nonlocal parabolic beam theory. Fuller Nanotub Carbon Nanostruct 23(3):266–272

  3. 3.

    Bedia WA, Benzair A, Semmah A, Tounsi A, Mahmoud SR (2015) On the thermal buckling characteristics of armchair single-walled carbon nanotube embedded in an elastic medium based on nonlocal continuum elasticity. Braz J Phys 45(2):225–233

  4. 4.

    Benguediab S, Tounsi A, Zidour M, Semmah A (2014) Chirality and scale effects on mechanical buckling properties of zigzag double-walled carbon nanotubes. Compos B Eng 57:21–24

  5. 5.

    Besseghier A, Heireche H, Bousahla AA, Tounsi A, Benzair A (2015) Nonlinear vibration properties of a zigzag single-walled carbon nanotube embedded in a polymer matrix. Adv Nano Res 3(1):029

  6. 6.

    Dehghan M, Ebrahimi F, Vinyas M (2019) Wave dispersion characteristics of fluid-conveying magneto-electro-elastic nanotubes. Eng Comput. https://doi.org/10.1007/s00366-019-00790-5

  7. 7.

    Ebrahimi F, Hosseini SHS (2019) Nonlinear vibration and dynamic instability analysis nanobeams under thermo-magneto-mechanical loads: a parametric excitation study. Eng Comput. https://doi.org/10.1007/s00366-019-00830-0

  8. 8.

    Ebrahimi F, Karimiasl M, Mahesh V (2019) Chaotic dynamics and forced harmonic vibration analysis of magneto-electro-viscoelastic multiscale composite nanobeam. Eng Comput. https://doi.org/10.1007/s00366-019-00865-3

  9. 9.

    Eltaher MA, Agwa MA (2016) Analysis of size-dependent mechanical properties of CNTs mass sensor using energy equivalent model. Sens Actuators A 246:9–17

  10. 10.

    Eltaher MA, El-Borgi S, Reddy JN (2016) Nonlinear analysis of size-dependent and material-dependent nonlocal CNTs. Compos Struct 153:902–913

  11. 11.

    Eltaher MA, Agwa M, Kabeel A (2018) Vibration analysis of material size-dependent CNTs using energy equivalent model. J Appl Comput Mech 4(2):75–86

  12. 12.

    Eltaher MA, Mohamed N, Mohamed S, Seddek LF (2019a) Postbuckling of curved carbon nanotubes using energy equivalent model. J Nano Res 57:136–157 (Trans Tech Publications)

  13. 13.

    Eltaher MA, Mohamed N, Mohamed SA, Seddek LF (2019) Periodic and nonperiodic modes of postbuckling and nonlinear vibration of beams attached to nonlinear foundations. Appl Math Model 75:414–445

  14. 14.

    Eltaher MA, Almalki TA, Ahmed KI, Almitani KH (2019) Characterization and behaviors of single walled carbon nanotube by equivalent-continuum mechanics approach. Adv Nano Res 7(1):39–49

  15. 15.

    Eltaher MA, Almalki TA, Almitani KH, Ahmed KIE (2019) Participation factor and vibration of carbon nanotube with vacancies. J Nano Res 57:158–174 (Trans Tech Publications)

  16. 16.

    Eltaher MA, Almalki TA, Almitani KH, Ahmed KIE, Abdraboh AM (2019) Modal participation of fixed–fixed single-walled carbon nanotube with vacancies. Int J Adv Struct Eng 11(2):151–163

  17. 17.

    Emam SAJCS (2011) Analysis of shear-deformable composite beams in postbuckling. Compos Struct 94(1):24–30

  18. 18.

    Emam SA, Eltaher MA, Khater ME, Abdalla WS (2018) Postbuckling and free vibration of multilayer imperfect nanobeams under a pre-stress load. Appl Sci 8(11):2238

  19. 19.

    Ghadyani G, Öchsner A (2015) On a thickness free expression for the stiffness of carbon nanotubes. Solid State Commun 209:38–44

  20. 20.

    Gholami R, Ansari R, Gholami Y (2017) Nonlinear resonant dynamics of geometrically imperfect higher-order shear deformable functionally graded carbon-nanotube reinforced composite beams. Compos Struct 174:45–58

  21. 21.

    Heshmati M, Yas MH, Daneshmand F (2015) A comprehensive study on the vibrational behavior of CNT-reinforced composite beams. Compos Struct 125:434–448

  22. 22.

    Iijima S (1991) Helical microtubules of graphitic carbon. Nature 354(6348):56–58

  23. 23.

    Joshi UA, Sharma SC, Harsha SP (2012) A multiscale approach for estimating the chirality effects in carbon nanotube reinforced composites. Phys E 45:28–35

  24. 24.

    Karimiasl M, Ebrahimi F, Mahesh V (2019) Postbuckling analysis of piezoelectric multiscale sandwich composite doubly curved porous shallow shells via Homotopy Perturbation Method. Eng Comput. https://doi.org/10.1007/s00366-019-00841-x

  25. 25.

    Khater ME, Eltaher MA, Abdel-Rahman E, Yavuz M (2014) Surface and thermal load effects on the buckling of curved nanowires. Eng Sci Technol Int J 17(4):279–283

  26. 26.

    Kordkheili SAH, Mousavi T, Bahai H (2018) Nonlinear dynamic analysis of SWNTs conveying fluid using nonlocal continuum theory. Struct Eng Mech 66(5):621–629

  27. 27.

    Leung AYT, Guo X, He XQ, Kitipornchai S (2005) A continuum model for zigzag single-walled carbon nanotubes. Appl Phys Lett 86(8):083110

  28. 28.

    Maneshi MA, Ghavanloo E, Fazelzadeh SA (2018) Closed-form expression for geometrically nonlinear large deformation of nano-beams subjected to end force. Eur Phys J Plus 133(7):256

  29. 29.

    Mayoof FN, Hawwa MA (2009) Chaotic behavior of a curved carbon nanotube under harmonic excitation. Chaos Solitons Fractals 42(3):1860–1867

  30. 30.

    Mikata Y (2007) Complete solution of elastica for a clamped-hinged beam, and its applications to a carbon nanotube. Acta Mech 190(1–4):133–150

  31. 31.

    Mohamed N, Eltaher MA, Mohamed SA, Seddek LF (2018) Numerical analysis of nonlinear free and forced vibrations of buckled curved beams resting on nonlinear elastic foundations. Int J Non-Linear Mech 101:157–173

  32. 32.

    Mohamed N, Eltaher MA, Mohamed SA, Seddek LF (2019) Energy equivalent model in analysis of postbuckling of imperfect carbon nanotubes resting on nonlinear elastic foundation. Struct Eng Mech 70(6):737–750

  33. 33.

    Mohammadi H, Mahzoon M, Mohammadi M, Mohammadi M (2014) Postbuckling instability of nonlinear nanobeam with geometric imperfection embedded in elastic foundation. Nonlinear Dyn 76(4):2005–2016

  34. 34.

    Nasdala L, Kempe A, Rolfes R (2012) Are finite elements appropriate for use in molecular dynamic simulations? Compos Sci Technol 72(9):989–1000

  35. 35.

    Rappé AK, Casewit CJ, Colwell KS, Goddard Iii WA, Skiff WM (1992) UFF, a full periodic table force field for molecular mechanics and molecular dynamics simulations. J Am Chem Soc 114(25):10024–10035

  36. 36.

    Reddy JN (1984) A simple higher-order theory for laminated composite plates. J Appl Mech 51(4):745–752

  37. 37.

    She GL, Yuan FG, Ren YR (2017) Thermal buckling and post-buckling analysis of functionally graded beams based on a general higher-order shear deformation theory. Appl Math Model 47:340–357

  38. 38.

    Shodja HM, Delfani MR (2011) A novel nonlinear constitutive relation for graphene and its consequence for developing closed-form expressions for Young’s modulus and critical buckling strain of single-walled carbon nanotubes. Acta Mech 222(1–2):91–101

  39. 39.

    Shokrieh MM, Rafiee R (2010) Prediction of Young’s modulus of graphene sheets and carbon nanotubes using nanoscale continuum mechanics approach. Mater Des 31(2):790–795

  40. 40.

    Wang B, Deng Z, Zhang K, Zhou J (2012) Dynamic analysis of embedded curved double-walled carbon nanotubes based on nonlocal Euler-Bernoulli Beam theory. Multidiscip Model Mater Struct 8(4):432–453

  41. 41.

    Wang B, Deng ZC, Zhang K (2013) Nonlinear vibration of embedded single-walled carbon nanotube with geometrical imperfection under harmonic load based on nonlocal Timoshenko beam theory. Appl Math Mech 34:269–280

  42. 42.

    Wu Y, Zhang X, Leung AYT, Zhong W (2006) An energy-equivalent model on studying the mechanical properties of single-walled carbon nanotubes. Thin-Walled Struct 44(6):667–676

Download references


This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. (D-423-135-1441). The authors, therefore, acknowledge with thanks DSR technical and financial support.

Author information

Correspondence to M. A. Eltaher.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mohamed, N., Mohamed, S.A. & Eltaher, M.A. Buckling and post-buckling behaviors of higher order carbon nanotubes using energy-equivalent model . Engineering with Computers (2020). https://doi.org/10.1007/s00366-020-00976-2

Download citation


  • Differential–integral quadrature method
  • Carbon nanotube
  • Energy-equivalent model
  • Static post-buckling instability
  • Nonlinear integro-differential equation