Skip to main content
SpringerLink
Log in

Vibration analysis of functionally graded microbeam under initial stress via a generalized thermoelastic model with dual-phase lags

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

The aim of this article is to investigate the thermal and mechanical vibration properties of functionally graded microbeams. The governing system of equations is formulated on the basis of classical Euler–Bernoulli beam model incorporating the generalized dual-phase lag model of thermoelasticity rather than the conventional steady-state Fourier heat conduction. In our analysis, it is also assumed that the mechanical and thermal properties such as modulus of elasticity, density and coefficient of thermal conductivity change through the thickness by an exponential law distribution, with the exception of Poisson’s ratio. The effects of system parameters on different field quantities have been depicted for initial stress, material gradient index and pulses duration . The numerical results are compared with benchmark findings, for the sake of verification. Presented results demonstrate the considerable effects of temperature and the material gradients on the dynamic behavior of microscopic structures. Finally, a comparison is conducted with the results discussed in the literature to justify the quality of the present technique.

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
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Zhang, J., Fu, Y.: Pull-in analysis of electrically actuated viscoelastic microbeams based on a modified couple stress theory. Meccanica 47, 1649–1658 (2012)

    MathSciNet  MATH  Google Scholar 

  2. Koizumi, M.: The concept of FGM. Ceramic transactions. . Funct. Grad. Mater. 34, 3–10 (1993)

    Google Scholar 

  3. Rahaeifard, M., Kahrobaiyan, M.H., Ahmadian, M.T., Firoozbakhsh, K.: Strain gradient formulation of functionally graded nonlinear beams. Int. J. Eng. Sci. 65, 49–63 (2013)

    MathSciNet  MATH  Google Scholar 

  4. Sedighi, H.M., Daneshmand, F., Abadyan, M.R.: Dynamic instability analysis of electrostatic functionally graded doubly-clamped nano-actuators. Compos. Struct. 124, 55–64 (2015)

    Google Scholar 

  5. Thai, H.T., Vo, T.P.: Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories. Int. J. Mech. Sci. 62, 57–66 (2012)

    Google Scholar 

  6. Shariati, A., Jung, D., Mohammad-Sedighi, H., Żur, K.K., Habibi, M., Safa, M.: On the vibrations and stability of moving viscoelastic axially functionally graded nanobeams. Materials 13, 1707 (2020)

    Google Scholar 

  7. Meradjah, M., Kaci, A., Houari, M.S.A., Tounsi, A., Mahmoud, S.R.: A new higher order shear and normal deformation theory for functionally graded beams. Steel Compos. Struct. 18(3), 793–809 (2015)

    Google Scholar 

  8. Ebrahimi, F., Dashti, S.: Free vibration analysis of a rotating non-uniform functionally graded beam. Steel Compos. Struct. 19(5), 1279–1298 (2015)

    Google Scholar 

  9. Bourada, M., Kaci, A., Houari, M.S.A., Tounsi, A.: A new simple shear and normal deformations theory for functionally graded beams. Steel Compos. Struct. 18(2), 409–423 (2015)

    Google Scholar 

  10. Ebrahimi and Barati: A nonlocal higher-order shear deformation beam theory for vibration analysis of size-dependent functionally graded nanobeams. Arab. J. Sci. Eng. 41(5), 1679–1690 (2016)

    MathSciNet  MATH  Google Scholar 

  11. Raminnea, M., Biglari, H., Vakili Tahami, F.: Nonlinear Dynamics Nonlinear higher order Reddy theory for temperature-dependent vibration and instability of embedded functionally graded pipes conveying fluid. Struct. Eng. Mech. 59(1), 153–186 (2016)

    Google Scholar 

  12. Abo-Bakr, H.M., Abo-Bakr, R.M., Mohamed. S.A., Eltaher, M.A., Weight optimization of axially functionally graded microbeams under buckling and vibration behaviors, Mech. Based Design Struct. Mach. https://doi.org/10.1080/15397734.2020.1838298.

  13. Shabani, S., Cunedioglu, Y.: Free vibration analysis of functionally graded beams with cracks. J. Appl. Comput. Mech. 6(4), 908–919 (2020)

    Google Scholar 

  14. Chen, M., Jin, G., Zhang, Y., NIu, F., Liu, Z, : Three dimensional vibration analysis of beams with axial functionally graded materials and variable thickness. Compos. Struct. 207, 314–322 (2019)

    Google Scholar 

  15. Abouelregal, A.E., Marin, M.: The size-dependent thermoelastic vibrations of nanobeams subjected to harmonic excitation and rectified sine wave heating. Mathematics 8(7), 1128 (2020)

    Google Scholar 

  16. Abo Dahab, S.M., Abouelregal, A.E., Marin, M.: Generalized thermoelastic functionally graded on a thin slim strip non-Gaussian laser beam. Symmetry 12(7), 1094 (2020)

    Google Scholar 

  17. Abo-Bakr, R.M., Eltaher, M.A., Attia, M.A.: Pull-in and freestanding instability of actuated functionally graded nanobeams including surface and stiffening effects. Eng. Comput. (2020). https://doi.org/10.1007/s00366-020-01146-0

    Article  Google Scholar 

  18. Chen, W., Chen, C., Chang, H.: Thermal buckling analysis of functionally graded Euler-Bernoulli beams with temperature-dependent properties. J. Appl. Comput. Mech. 6(3), 457–470 (2020)

    MathSciNet  Google Scholar 

  19. Abouelregal, A.E., Mohammed, W.W.: Effects of nonlocal thermoelasticity on nanoscale beams based on couple stress theory. Math. Methods Appl. Sci. (2020). https://doi.org/10.1002/mma.6764

    Article  Google Scholar 

  20. Abouelregal, A.E., Marin, M.: The response of nanobeams with temperature-dependent properties using state-space method via modified couple stress theory. Symmetry 12(8), 1276 (2020)

    Google Scholar 

  21. Abouelregal, A.-E., Mohamed, B.O.: Fractional order thermoelasticity for a functionally graded thermoelastic nanobeam induced by a sinusoidal pulse heating. J. Comput. Theor. Nanosci. 15, 1233–1242 (2018)

    Google Scholar 

  22. Korznikova, G., Korneva, A., Korznikova, E.: Application of combined load for obtaining ultra-fine grained structure in magnetic alloys of the Fe-Cr-Co system. Rep. Mech. Eng. 1(1), 1–9 (2020)

    Google Scholar 

  23. Li, S.-R., Ma, H.-K.: Analysis of free vibration of functionally graded material micro-plates with thermoelastic damping. Arch. Appl. Mech. 90, 1285–1304 (2020)

    Google Scholar 

  24. Ebrahimi, F., Barati, M.R., Civalek, O.: Application of Chebyshev-Ritz method for static stability and vibration analysis of nonlocal microstructure-dependent nanostructures. Eng. Comput. 36, 953–964 (2020)

    Google Scholar 

  25. Gurses, M., Akgoz, B., Civalek, O.: Mathematical modeling of vibration problem of nano-sized annular sector plates using the nonlocal continuum theory via eight-node discrete singular convolution transformation. Appl. Math. Comput. 219, 3226–3240 (2012)

    MathSciNet  MATH  Google Scholar 

  26. Uzun, B., Civalek, O.: Nonlocal FEM formulation for vibration analysis of nanowires on elastic matrix with different materials. Math. Comput. Appl. 24(2), 38 (2019)

    MathSciNet  Google Scholar 

  27. Civalek, O., Uzun, B., Yaylı, M.O., Akgöz, B.: Size-dependent transverse and longitudinal vibrations of embedded carbon and silica carbide nanotubes by nonlocal finite element method. Eur. Phys. J. Plus 135, 381 (2020). https://doi.org/10.1140/epjp/s13360-020-00385-w

    Article  Google Scholar 

  28. Jalaei, M., Civalek, Ӧ: On dynamic instability of magnetically embedded viscoelastic porous FG nanobeam. Int. J. Eng. Sci. 143, 14–32 (2019)

    MathSciNet  MATH  Google Scholar 

  29. Biot, M.A.: Mechanics of Incremental Deformations. Wiley, New York (1965)

    Google Scholar 

  30. Abouelregal, A.E.: Zenkour AM (2019) Dynamic characteristics of initially stressed viscoelastic microbeams induced by ultra-intense lasers. Indian J. Phys. 94, 779–788 (2020)

    Google Scholar 

  31. Abouelregal, A.E., Zenkour, A.M.: Fractional viscoelastic Voigt’s model for initially stressed microbeams induced by ultrashort laser heat source. Waves Random Complex Media (2019). https://doi.org/10.1080/17455030.2018.1554927

    Article  Google Scholar 

  32. Wang, S.C.M., Zhang, Y.Y., Kitipornchai, S.: Vibration of initially stressed micro- and nano-beams Int. J. Str. Stab. Dyn. 7, 555 (2007)

    MATH  Google Scholar 

  33. Güven, U.: Transverse vibrations of single-walled carbon nanotubes with initial stress under magnetic field. Comp. Struct. 114, 92–98 (2014)

    Google Scholar 

  34. Xu, X.J., Deng, Z.C.: Adsorption-induced frequency analysis using nonlocal Euler-Bernoulli beam theory with initial axial stress. Multid. Mod. Mater. Struct. 9, 116–127 (2013)

    Google Scholar 

  35. Lord, H.W., Shulman, Y.H.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15(5), 299–309 (1967)

    MATH  Google Scholar 

  36. Tzou, D.Y.: Thermal shock phenomena under high rate response in solids. Annual Rev. Heat Transf. 4, 4 (1992)

    Google Scholar 

  37. Tzou, D.Y.: A unified field approach for heat conduction from macro-to micro-scales. J. Heat Transf. 117(1), 8–16 (1995)

    Google Scholar 

  38. Tzou, D.Y.: The generalized lagging response in small-scale and high-rate heating. Int. J. Heat Mass Transf. 38(17), 3231–3240 (1995)

    Google Scholar 

  39. Abouelregal, A.E.: Two-temperature thermoelastic model without energy dissipation including higher order time-derivatives and two phase-lags. Mater. Res. Express 6(11), 116535 (2019)

    Google Scholar 

  40. Abouelregal, A.E.: On Green and Naghdi thermoelasticity model without energy dissipation with higher order time differential and phase-lags. J. Appl. Comput. Mech. 6(3), 445–456 (2020)

    Google Scholar 

  41. Abouelregal, A.E.: A novel generalized thermoelasticity with higher-order time-derivatives and three-phase lags. Multidisc. Model. Mater. Struct. (2019). https://doi.org/10.1108/MMMS-07-2019-0138

    Article  Google Scholar 

  42. Abouelregal, A.E.: Three-phase-lag thermoelastic heat conduction model with higher-order time-fractional derivatives. Indian J. Phys. (2019). https://doi.org/10.1007/s12648-019-01635-z

    Article  Google Scholar 

  43. Allam, M.N.M., Abouelregal, A.E.: The thermoelastic waves induced by pulsed laser and varying heat of inhomogeneous microscale beam resonators. J. Therm. Stresses 37(4), 455–470 (2014)

    Google Scholar 

  44. Narayanan, G.V., Beskos, D.E.: Numerical operational methods for time-dependent linear problems. Int. J. Numer. Meth. Eng. 18, 1829–1854 (1982)

    MathSciNet  MATH  Google Scholar 

  45. Durbin, F.: Numerical inversion of Laplace transforms: an efficient improvement to Duber and Abates method. Computer J. 17, 371–376 (1974)

    MathSciNet  MATH  Google Scholar 

  46. Tzou, D.: Macro-to-Micro Heat Transfer. Taylor & Francis, Washington, D.C. (1996)

    Google Scholar 

  47. Ruhi, M., Angoshtari, A., Naghdabadi, R.: Thermoelastic analysis of thick-walled finite-length cylinders of functionally graded materials. J. Therm. Stresses 28(4), 391–408 (2005)

    Google Scholar 

  48. Ziaee, S.: Small scale effect on linear vibration of buckled size-dependent FG nanobeams. Ain Shams Engineering Journal 6(2), 587–598 (2015)

    Google Scholar 

  49. Mohammadi, H., Mahzoon, M.: Thermal effects on postbuckling of nonlinear microbeams based on the modified strain gradient theory. Compos Struct 106, 764–776 (2013)

    Google Scholar 

  50. Li, C., Lim, C.W., Yu, J.L., Zeng, Q.C.: Analytical solutions for vibration of simply supported nonlocal nanobeams with an axial force. Int. J. Struct. Stab. Dyn. 11(02), 257–271 (2011)

    MathSciNet  MATH  Google Scholar 

  51. Lu, P.: Dynamic analysis of axially prestressed micro/nanobeam structures based on nonlocal beam theory. J. Appl. Phys. 101(7), 073504 (2007)

    Google Scholar 

  52. Achenbach, J.D.: The influence of heat conduction on propagating stress jumps. J. Mech. Phys. Solids 16(4), 273–282 (1968)

    Google Scholar 

  53. Dhaliwal, R.S., Singh, A.: Dynamic Coupled Thermoelasticity. Hindustan Publishing Corporation, New Delhi, India (1980)

    Google Scholar 

Download references

Acknowledgements

Hamid Mohammad-Sedighi is grateful to the Research Council of Shahid Chamran University of Ahvaz for its financial support (Grant No. SCU.EM99.98).

Author information

Authors and Affiliations

  1. Department of Mathematics, College of Science and Arts, Jouf University, Al-Qurayyat, Saudi Arabia

    A. E. Abouelregal

  2. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

    A. E. Abouelregal & W. W. Mohammed

  3. Department of Mathematics, Faculty of Science, University of Ha’il, Ha’il, 2440, Saudi Arabia

    W. W. Mohammed

  4. Mechanical Engineering Department, Faculty of Engineering, Shahid Chamran University of Ahvaz, 61357-43337, Ahvaz, Iran

    Hamid Mohammad-Sedighi

  5. Drilling Center of Excellence and Research Center, Shahid Chamran University of Ahvaz, Ahvaz, Iran

    Hamid Mohammad-Sedighi

Authors
  1. A. E. Abouelregal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. W. Mohammed
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hamid Mohammad-Sedighi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. E. Abouelregal or Hamid Mohammad-Sedighi.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abouelregal, A.E., Mohammed, W.W. & Mohammad-Sedighi, H. Vibration analysis of functionally graded microbeam under initial stress via a generalized thermoelastic model with dual-phase lags. Arch Appl Mech 91, 2127–2142 (2021). https://doi.org/10.1007/s00419-020-01873-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-020-01873-2

Keywords

Search

Navigation