Skip to main content
SpringerLink
Log in

On size-dependent free vibration and thermal buckling of axially functionally graded nanobeams in thermal environment

  • Published:
Applied Physics A Aims and scope Submit manuscript

Abstract

This article aims to study the buckling and free vibrational behavior of axially functionally graded (AFG) nanobeam under thermal effect for the first time. The temperature is considered to be constant and variable along thickness and different boundary conditions. The governing equation is developed using the Hamilton’s principle considering the axial force. The Euler–Bernoulli beam theory is used to model the nanobeam, and Eringen’s nonlocal elasticity theory is utilized to consider the nano-size effect. The generalized differential quadrature method (GDQM) is used to solve the equations. The small-scale parameter, AFG power index, thermal distribution, different functions of temperature increase for different boundary conditions are given in detail.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

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

Similar content being viewed by others

References

  1. Q. Wang, K.M. Liew, Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures. Phys. Lett. A 363, 236–242 (2007)

    Article  ADS  Google Scholar 

  2. F. Najar, S. El-Borgi, J.N. Reddy, K. Mrabet, Nonlinear nonlocal analysis of electrostatic nanoactuators. Compos. Struct. 120, 117–128 (2015)

    Article  Google Scholar 

  3. N. Togun, Nonlinear vibration of nanobeam with attached mass at the free end via nonlocal elasticity theory. Microsyst. Technol. 22, 2349–2359 (2016)

    Article  Google Scholar 

  4. D. Karličić, P. Kozić, R. Pavlović, Nonlocal vibration and stability of a multiple-nanobeam system coupled by the Winkler elastic medium. Appl. Math. Modell. 40, 1599–1614 (2016)

    Article  MathSciNet  Google Scholar 

  5. F.L. Chaht, A. Kaci, M.S.A. Houari, A. Tounsi, O.A. Beg, S. Mahmoud, Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect. Steel Compos. Struct. 18, 425–442 (2015)

    Article  Google Scholar 

  6. Y. Li, P. Ma, W. Wang, Bending, buckling, and free vibration of magnetoelectroelastic nanobeam based on nonlocal theory. J. Intell. Mater. Syst. Struct. 27, 1139–1149 (2016)

    Article  Google Scholar 

  7. H. Niknam, M.M. Aghdam, A semi analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic foundation. Compos. Struct. 119, 452–462 (2015)

    Article  Google Scholar 

  8. H. Zeighampour, Y.T. Beni, Free vibration analysis of axially functionally graded nanobeam with radius varies along the length based on strain gradient theory. Appl. Math. Modell. 39, 5354–5369 (2015)

    Article  MathSciNet  Google Scholar 

  9. N. Shafiei, M. Kazemi, M. Safi, M. Ghadiri, Nonlinear vibration of axially functionally graded non-uniform nanobeams. Int. J. Eng. Sci. 106, 77–94 (2016)

    Article  Google Scholar 

  10. Y. Wang, D. Wu, Thermal effect on the dynamic response of axially functionally graded beam subjected to a moving harmonic load. Acta Astronaut. 127, 171–181 (2016)

    Article  ADS  Google Scholar 

  11. K. Sarkar, R. Ganguli, Closed-form solutions for axially functionally graded Timoshenko beams having uniform cross-section and fixed–fixed boundary condition. Compos. Part B 58, 361–370 (2014)

    Article  Google Scholar 

  12. M. Eltaher, A. Khairy, A. Sadoun, F.-A. Omar, Static and buckling analysis of functionally graded Timoshenko nanobeams. Appl. Math. Comput. 229, 283–295 (2014)

    Article  MathSciNet  Google Scholar 

  13. M. Eltaher, S.A. Emam, F. Mahmoud, Free vibration analysis of functionally graded size-dependent nanobeams. Appl. Math. Comput. 218, 7406–7420 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Shojaeian, Y.T. Beni, Size-dependent electromechanical buckling of functionally graded electrostatic nano-bridges. Sens. Actuators A 232, 49–62 (2015)

    Article  Google Scholar 

  15. A. Benzair, A. Tounsi, A. Besseghier, H. Heireche, N. Moulay, L. Boumia, The thermal effect on vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory. J. Phys. D 41, 225404 (2008)

    Article  ADS  Google Scholar 

  16. H.M. Youssef, K.A. Elsibai, Vibration of gold nanobeam induced by different types of thermal loading—a state-space approach. Nanoscale Microscale Thermophys. Eng. 15, 48–69 (2011)

    Article  Google Scholar 

  17. M. Maachou, M. Zidour, H. Baghdadi, N. Ziane, A. Tounsi, A nonlocal Levinson beam model for free vibration analysis of zigzag single-walled carbon nanotubes including thermal effects. Solid State Commun. 151, 1467–1471 (2011)

    Article  ADS  Google Scholar 

  18. K. Liao-Liang, W. Yue-Sheng, Thermoelectric-mechanical vibration of piezoelectric nanobeams based on the nonlocal theory. Smart Mater. Struct. 21, 025018 (2012)

    Article  ADS  Google Scholar 

  19. M. Zidour, K.H. Benrahou, A. Semmah, M. Naceri, H.A. Belhadj, K. Bakhti et al., The thermal effect on vibration of zigzag single walled carbon nanotubes using nonlocal Timoshenko beam theory. Comput. Mater. Sci. 51, 252–260 (2012)

    Article  Google Scholar 

  20. C.W. Lim, Q. Yang, J.B. Zhang, Thermal buckling of nanorod based on non-local elasticity theory. Int. J. Non Linear Mech. 47, 496–505 (2012)

    Article  ADS  Google Scholar 

  21. A. Tounsi, A. Semmah, A.A. Bousahla, Thermal buckling behavior of nanobeams using an efficient higher-order nonlocal beam theory. J. Nanomech. Micromech. 3, 37–42 (2013)

    Article  Google Scholar 

  22. H.M. Youssef, Vibration of gold nanobeam with variable thermal conductivity: state-space approach. Appl. Nanosci. 3, 397–407 (2013)

    Article  ADS  Google Scholar 

  23. A. Ghorbanpour Arani, S. Amir, Electro-thermal vibration of visco-elastically coupled BNNT systems conveying fluid embedded on elastic foundation via strain gradient theory. Phys. B 419, 1–6 (2013)

    Article  ADS  Google Scholar 

  24. B. Amirian, R. Hosseini-Ara, H. Moosavi, Surface and thermal effects on vibration of embedded alumina nanobeams based on novel Timoshenko beam model. Appl. Math. Mech. 35, 875–886 (2014)

    Article  MathSciNet  Google Scholar 

  25. M. Mohammadi, A. Farajpour, A. Moradi, M. Ghayour, Shear buckling of orthotropic rectangular graphene sheet embedded in an elastic medium in thermal environment. Compos. Part B 56, 629–637 (2014)

    Article  Google Scholar 

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

    Article  Google Scholar 

  27. D.S. Mashat, A.M. Zenkour, M. Sobhy, Investigation of vibration and thermal buckling of nanobeams embedded in an elastic medium under various boundary conditions. J. Mech. 32, 277–287 (2016)

    Article  Google Scholar 

  28. Y.-G. Wang, H.-F. Song, W.-H. Lin, J.-K. Wang. Large amplitude free vibration of micro/nano beams based on nonlocal thermal elasticity theory. Lat. Am. J.Solids Struct. 12, 1918–1933, (2015).

  29. Y.J. Yu, Z.-N. Xue, C.-L. Li, X.-G. Tian, Buckling of nanobeams under nonuniform temperature based on nonlocal thermoelasticity. Compos. Struct. 146, 108–113 (2016)

    Article  Google Scholar 

  30. R. Ansari, T. Pourashraf, R. Gholami, S. Sahmani, Postbuckling behavior of functionally graded nanobeams subjected to thermal loading based on the surface elasticity theory. Meccanica 52, 283–297 (2017)

    Article  MathSciNet  Google Scholar 

  31. H. Baghdadi, A. Tounsi, M. Zidour, A. Benzair, Thermal effect on vibration characteristics of armchair and zigzag single-walled carbon nanotubes using nonlocal parabolic beam theory. Fuller. Nanotub. Carbon Nanostruct. 23, 266–272 (2015)

    Article  ADS  Google Scholar 

  32. W.A. Bedia, A. Benzair, A. Semmah, A. Tounsi, S.R. Mahmoud, 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, 225–233 (2015)

    Article  ADS  Google Scholar 

  33. M. Zarepour, S.A. Hosseini, A semi analytical method for electro-thermo-mechanical nonlinear vibration analysis of nanobeam resting on the Winkler-Pasternak foundations with general elastic boundary conditions. Smart Mater. Struct. 25, 085005 (2016)

    Article  ADS  Google Scholar 

  34. N. Shafiei, M. Kazemi, M. Ghadiri, Nonlinear vibration behavior of a rotating nanobeam under thermal stress using Eringen’s nonlocal elasticity and DQM. Appl. Phys. A 122, 1–18 (2016)

    Article  Google Scholar 

  35. M. Mohammadi, M. Safarabadi, A. Rastgoo, A. Farajpour, Hygro-mechanical vibration analysis of a rotating viscoelastic nanobeam embedded in a visco-Pasternak elastic medium and in a nonlinear thermal environment. Acta Mech. 227, 2207–2232 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. A.M. Zenkour, M. Sobhy, A simplified shear and normal deformations nonlocal theory for bending of nanobeams in thermal environment. Phys. E 70, 121–128 (2015)

    Article  Google Scholar 

  37. R. Ansari, M. Faraji Oskouie, R. Gholami, F. Sadeghi, Thermo-electro-mechanical vibration of postbuckled piezoelectric Timoshenko nanobeams based on the nonlocal elasticity theory. Compos. Part B 89, 316–327 (2016)

    Article  Google Scholar 

  38. Y.S. Touloukian, C. Ho. Thermal Expansion. Nonmetallic solids. Thermophysical properties of matter-The TPRC Data Series. Y.S. Touloukian, C.Y. Ho, (eds.) (New York, IFI/Plenum, 1970)

    Google Scholar 

  39. J. Yang, H.-S. Shen, Vibration characteristics and transient response of shear-deformable functionally graded plates in thermal environments. J. Sound Vib. 255, 579–602 (2002)

    Article  ADS  Google Scholar 

  40. N. Shafiei, M. Kazemi, M. Ghadiri, Comparison of modeling of the rotating tapered axially functionally graded Timoshenko and Euler–Bernoulli microbeams. Phys. E 83, 74–87 (2016)

    Article  Google Scholar 

  41. M.A. Eltaher, M.E. Khater, S.A. Emam, A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams. Appl. Math. Modell. 40, 4109–4128 (2016)

    Article  MathSciNet  Google Scholar 

  42. A.C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)

    Article  ADS  Google Scholar 

  43. F. Fazzolari, E. Carrera, Thermal stability of FGM sandwich plates under various through-the-thickness temperature distributions. J. Therm. Stress. 37, 1449–1481 (2014)

    Article  Google Scholar 

  44. R. Bellman, J. Casti, Differential quadrature and long-term integration. J. Math. Anal. Appl. 34, 235–238 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  45. R. Bellman, B. Kashef, J. Casti, Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. J. Comput. Phys. 10, 40–52 (1972)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. C. Shu. Differential Quadrature and its Application in Engineering. (Springer-Verlag, London 2000)

    Book  MATH  Google Scholar 

  47. C.M. Wang, Y.Y. Zhang, X.Q. He, Vibration of nonlocal Timoshenko beams. Nanotechnology 18, 105401 (2007)

    Article  ADS  Google Scholar 

  48. F. Ebrahimi, E. Salari, Thermo-mechanical vibration analysis of nonlocal temperature-dependent FG nanobeams with various boundary conditions. Compos. Part B 78, 272–290 (2015)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran

    Seyed Sajad Mirjavadi

  2. Department of Information Technology, Engineering, Payame Noor University (PNU), PO Box 19395-3697, Tehran, Iran

    Samira Rabby

  3. Department of Mechanical Engineering, Payame Noor University (PNU), PO Box 19395-3697, Tehran, Iran

    Navvab Shafiei

  4. School of Mechanical Engineering, College of Engineering, Sharif University of Technology, Tehran, Iran

    Behzad Mohasel Afshari

  5. Hoonam Sanat Farnak Engineering and Technology Company, PO Box 6931876647, Ilam, Iran

    Mohammad Kazemi

Authors
  1. Seyed Sajad Mirjavadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Samira Rabby
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Navvab Shafiei
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Behzad Mohasel Afshari
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Mohammad Kazemi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Navvab Shafiei.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mirjavadi, S.S., Rabby, S., Shafiei, N. et al. On size-dependent free vibration and thermal buckling of axially functionally graded nanobeams in thermal environment. Appl. Phys. A 123, 315 (2017). https://doi.org/10.1007/s00339-017-0918-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00339-017-0918-1

Keywords

Search

Navigation