On the static stability of nonlocal nanobeams using higher-order beam theories



Eltaher, M.A.;Khater, M.E.;Park, S.;Abdel-Rahman, E.;Yavuz, M.

  • 투고 : 2016.01.16
  • 심사 : 2016.03.18
  • 발행 : 2016.03.25


This paper investigates the effects of thermal load and shear force on the buckling of nanobeams. Higher-order shear deformation beam theories are implemented and their predictions of the critical buckling load and post-buckled configurations are compared to those of Euler-Bernoulli and Timoshenko beam theories. The nonlocal Eringen elasticity model is adopted to account a size-dependence at the nano-scale. Analytical closed form solutions for critical buckling loads and post-buckling configurations are derived for proposed beam theories. This would be helpful for those who work in the mechanical analysis of nanobeams especially experimentalists working in the field. Results show that thermal load has a more significant impact on the buckling behavior of simply-supported beams (S-S) than it has on clamped-clamped (C-C) beams. However, the nonlocal effect has more impact on C-C beams that it does on S-S beams. Moreover, it was found that the predictions obtained from Timoshenko beam theory are identical to those obtained using all higher-order shear deformation theories, suggesting that Timoshenko beam theory is sufficient to analyze buckling in nanobeams.


buckling;nanobeams;higher-order beam theories;nonlocal elasticity;thermal loads


  1. Adali, S. (2008), "Variational principles for multi-walled carbon nanotubes undergoing buckling based on nonlocal elasticity theory", Phys. Lett. A, 372(35), 5701-5705. https://doi.org/10.1016/j.physleta.2008.07.003
  2. Adali, S. (2012), "Variational formulation for buckling of multi-walled carbon nanotubes modelled as nonlocal Timoshenko beams", J. Theo. Appl. Mech., 50 (1), 321-333.
  3. Ansari, R., Gholami, R. and Sahmani, S. (2013), "Prediction of compressive post-buckling behavior of single-walled carbon nanotubes in thermal environments", Appl. Phys. A, 113(1), 145-153. https://doi.org/10.1007/s00339-012-7502-5
  4. Amirian, B., Hosseini-Ara, R. and Moosavi, H. (2013), "Thermo-mechanical vibration of short carbon nanotubes embedded in pasternak foundation based on nonlocal elasticity theory", Shock Vib., 20(4), 821-832. https://doi.org/10.1155/2013/281676
  5. Belabed, Z., Houari, M.S.A., Tounsi, A., Mahmoud, S.R. and Beg, O.A. (2014), "An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates", Compos. Part B: Eng., 60, 274-283. https://doi.org/10.1016/j.compositesb.2013.12.057
  6. Benguediab, S., Tounsi, A., Zidour, M. and Semmah, A. (2014), "Chirality and scale effects on mechanical buckling properties of zigzag double-walled carbon nanotubes", Compos. Part B: Eng., 57, 21-24. https://doi.org/10.1016/j.compositesb.2013.08.020
  7. Benzair, A., Besseghier, A., Heireche, H., Bousahla, A.A. and Tounsi, A. (2015), "Nonlinear vibration properties of a zigzag single-walled carbon nanotube embedded in a polymer matrix", Adv. Nano Res., 3(1), 029. https://doi.org/10.12989/anr.2015.3.1.029
  8. Bessaim, A., Houari, M.S., Tounsi, A. and Mahmoud, S.R., (2013), "A new higher-order shear and normal deformation theory for the static and free vibration analysis of sandwich plates with functionally graded isotropic face sheets", J. Sandw. Struct. Mater., 15(6), 671-703. https://doi.org/10.1177/1099636213498888
  9. Chang, T. (2011), "Thermal-nonlocal vibration and instability of single-walled carbon nanotubes conveying fluid", J. Mech., 27 (04), 567-573. https://doi.org/10.1017/jmech.2011.59
  10. Chang, T. (2012), "Thermal-mechanical vibration and instability of a fluid-conveying single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory", Appl. Math. Model., 36(5), 1964-1973. https://doi.org/10.1016/j.apm.2011.08.020
  11. Eltaher, M., Khater, M., Abdel-Rahman, E. and Yavuz, M. (2014), "A Model for Nano Bonding Wires under Thermal Loading", IEEE-Nano 2014, Toronto, Canada.
  12. Eltaher, M., Emam, S. and Mahmoud, F. (2013), "Static and stability analysis of nonlocal functionally graded nanobeams", Compos. Struct., 96, 82-88. https://doi.org/10.1016/j.compstruct.2012.09.030
  13. Eltaher, M., Khairy, A., Sadoun, A. and Omar, F. (2014), "Static and buckling analysis of functionally graded Timoshenko nanobeams", Appl. Math. Comput., 229, 283-295.
  14. Eltaher, M.A., Khater, M.E. and Emam, S.A. (2016), "A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams", Appl. Math. Model., 40(5-6), 4109-4128. https://doi.org/10.1016/j.apm.2015.11.026
  15. Emam, S. (2013), "A general nonlocal nonlinear model for buckling of nanobeams", Appl. Math. Model., 37(10), 6929-6939. https://doi.org/10.1016/j.apm.2013.01.043
  16. Eringen, A. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54(9), 4703-4710. https://doi.org/10.1063/1.332803
  17. Ghasemi, A., Dardel, M., Ghasemi, M. and Barzegari, M. (2013), "Analytical analysis of buckling and postbuckling of fluid conveying multi-walled carbon nanotubes", Appl. Math. Model., 37(7), 4972-4992. https://doi.org/10.1016/j.apm.2012.09.061
  18. Hebali, H., Tounsi, A., Houari, M.S.A., Bessaim, A. and Bedia, E.A.A. (2014), "New quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates", J. Eng. Mech., 140(2), 374-383. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000665
  19. Heireche, H., Tounsi, A., Benzair, A., Maachou, M. and Bedia, E.A. (2008), "Sound wave propagation in single-walled carbon nanotubes using nonlocal elasticity", Physica E, 40(8), 2791-2799. https://doi.org/10.1016/j.physe.2007.12.021
  20. Houari, M.S.A., Tounsi, A. and Beg, O.A. (2013), "Thermoelastic bending analysis of functionally graded sandwich plates using a new higher order shear and normal deformation theory", Int. J. Mech. Sci., 76, 102-111. https://doi.org/10.1016/j.ijmecsci.2013.09.004
  21. Janghorban, M. (2012), "Two different types of differential quadrature methods for static analysis of microbeams based on nonlocal thermal elasticity theory in thermal environment", Arch. Appl. Mech., 82(5), 669-675. https://doi.org/10.1007/s00419-011-0582-4
  22. Lim, C., Yang, Q. and Zhang, J. (2012), "Thermal buckling of nanorod based on non-local elasticity theory", Int. J. Nonlin. Mech., 47(5), 496-505. https://doi.org/10.1016/j.ijnonlinmec.2011.09.023
  23. Narendar, S. and Gopalakrishnan, S. (2011), "Critical buckling temperature of single-walled carbon nanotubes embedded in a one-parameter elastic medium based on nonlocal continuum mechanics", Physica E, 43(6), 1185-1191. https://doi.org/10.1016/j.physe.2011.01.026
  24. Nayfeh, A. and Emam, S. (2008), "Exact solution and stability of postbuckling configurations of beams", Nonlin. Dyn., 54(4), 395-408. https://doi.org/10.1007/s11071-008-9338-2
  25. Meziane, M.A.A., Abdelaziz, H.H. and Tounsi, A. (2014), "An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions", J. Sandw. Struct. Mater., 16(3), 293-318. https://doi.org/10.1177/1099636214526852
  26. Murmu, T. and Pradhan, S. (2010), "Thermal effects on the stability of embedded carbon nanotubes", Comput. Mater. Sci., 47(3), 721-726. https://doi.org/10.1016/j.commatsci.2009.10.015
  27. Peddieson, J., Buchanan, G. and McNitt, R.P. (2003), "Application of nonlocal continuum models to nanotechnology", Int. J. Eng. Sci., 41, 305-312. https://doi.org/10.1016/S0020-7225(02)00210-0
  28. Pradhan, S. and Mandal, U. (2013), "Finite element analysis of CNTs based on nonlocal elasticity and Timoshenko beam theory including thermal effect", Physica E, 53, 223-232. https://doi.org/10.1016/j.physe.2013.04.029
  29. Reddy, J. (2007), "Nonlocal theories for bending, buckling and vibration of beams", Int. J. Eng. Sci., 45(2), 288-307. https://doi.org/10.1016/j.ijengsci.2007.04.004
  30. Reddy, J.N. and El-Borgi, S. (2014), "Eringen's nonlocal theories of beams accounting for moderate rotations", Int. J. Eng. Sci., 82, 159-177. https://doi.org/10.1016/j.ijengsci.2014.05.006
  31. Setoodeh, A., Khosrownejad, M. and Malekzadeh, P. (2011), "Exact nonlocal solution for postbuckling of single-walled carbon nanotubes", Physica E, 43(9), 1730-1737. https://doi.org/10.1016/j.physe.2011.05.032
  32. Simsek, M. (2010), "Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories", Nucl. Eng. Des., 240(4), 697-705. https://doi.org/10.1016/j.nucengdes.2009.12.013
  33. Sudak, L. (2003), "Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics", J. Appl. Phys., 94(11), 7281-7287. https://doi.org/10.1063/1.1625437
  34. Tounsi, A., Semmah, A. and Bousahla, A. (2013), "Thermal buckling behavior of nanobeams using an efficient higher-order nonlocal beam theory", J. Nanomech. Micromech., 3(3), 37-42. https://doi.org/10.1061/(ASCE)NM.2153-5477.0000057
  35. Tounsi, A., Houari, M.S.A. and Benyoucef, S. (2013), "A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates", Aerosp. Sci. Tech., 24(1), 209-220. https://doi.org/10.1016/j.ast.2011.11.009
  36. Tounsi, A., Benguediab, S., Adda, B., Semmah, A. and Zidour, M. (2013), "Nonlocal effects on thermal buckling properties of double-walled carbon nanotubes", Adv. Nano Res., 1(1), 1-11. https://doi.org/10.12989/anr.2013.1.1.001
  37. Wang, C., Zhang, Y., Xiang, Y. and Reddy, J. (2010), "Recent studies on buckling of carbon nanotubes", Appl. Mech. Rev., 63(3), 030804. https://doi.org/10.1115/1.4001936
  38. Wang, L., Ni, Q., Li, M. and Qian, Q. (2008), "The thermal effect on vibration and instability of carbon nanotubes conveying fluid", Physica E, 40(10), 3179-3182. https://doi.org/10.1016/j.physe.2008.05.009
  39. Zhen, Y. and Fang, B. (2010), "Thermal-mechanical and nonlocal elastic vibration of single-walled carbon nanotubes conveying fluid", Comput. Mater. Sci., 49(2), 276-282. https://doi.org/10.1016/j.commatsci.2010.05.007
  40. Zidi, M., Tounsi, A., Houari, M.S.A. and Beg, O.A. (2014), "Bending analysis of FGM plates under hygrothermo-mechanical loading using a four variable refined plate theory", Aerosp. Sci. Tech., 34, 24-34. https://doi.org/10.1016/j.ast.2014.02.001

피인용 문헌

  1. 1. A new refined nonlocal beam theory accounting for effect of thickness stretching in nanoscale beams vol.4, pp.4, 2016, doi:10.12989/anr.2016.4.1.051
  2. 2. Effect of higher order terms of Maclaurin expansion in nonlinear analysis of the Bernoulli beam by single finite element vol.58, pp.6, 2016, doi:10.12989/anr.2016.4.1.051