Skip to main content
SpringerLink
Log in

Nonlinear vibration and stability of functionally graded porous microbeam under electrostatic actuation

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

Abstract

In this work, a microbeam model based on the Euler–Bernoulli beam theory and the nonlocal strain gradient theory was developed to study nonlinear vibration and stability of a functionally graded porous microbeam under electrostatic actuation. The electrostatic force is the driving force per unit length, resulting from electrostatic excitation. The material properties of the microbeam change continuously along thickness direction according to simple power-law distribution in terms of the fractions of constituents. Two cases of even and uneven porosity distributions are considered. The Hamilton principle is used to obtain the governing equations of the motion for the microbeam. The equation of motion for the microbeams is reduced to a nonlinear ordinary differential equation with rational restoring force by applying Galerkin method. The frequency–amplitude relationship is given in the closed form by employing He’s Hamiltonian approach. The expression of the static pull-in voltage is also derived. Effects of the material length scale parameter, the nonlocal parameter, the power-law index, the porosity distribution factor, the applied voltage and the initial amplitude on nonlinear vibration and stability of the functionally graded porous microbeam are examined and discussed.

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
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

References

  1. Chuang, W.C., Lee, H.L., Chang, P.Z., Hu, Y.C.: Review on the modeling of electrostatic MEMS. Sensors 10, 6149–6171 (2010)

    Article  Google Scholar 

  2. Zhang, W.M., Yan, H., Peng, Z.K., Meng, G.: Electrostatic pull-in instability in MEMS/NEMS: a review. Sens. Actuators A Phys. 214, 187–218 (2014)

    Article  Google Scholar 

  3. Fu, Y., Du, H., Huang, W., Zhang, S., Hu, M.: TiNi-based thin films in MEMS applications: a review. Sens. Actuators A Phys. 112, 395–408 (2004)

    Article  Google Scholar 

  4. Clarke, D.R., Ma, Q., Clarke, D.R.: Size dependent hardness of silver single crystals. J. Mater. Res. 10(4), 853–863 (1995)

    Article  Google Scholar 

  5. Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory and experiment. Acta Metall. Mater. 42(2), 475–487 (1994)

    Article  Google Scholar 

  6. Stölken, J.S., Evans, A.G.: A microbend test method for measuring the plasticity length scale. Acta Mater. 46(14), 5109–5115 (1998)

    Article  Google Scholar 

  7. Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248 (1972)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  9. Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11(1), 415–448 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  10. Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  11. Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11(1), 385–414 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mindlin, R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1(1), 417–438 (1965)

    Article  Google Scholar 

  13. Aifantis, E.C.: On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci. 30(10), 1279–1299 (1992)

    Article  MATH  Google Scholar 

  14. Yang, F., Chong, A.C.M., Lam, D.C.C., et al.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39(10), 2731–2743 (2002)

    Article  MATH  Google Scholar 

  15. Lam, D.C.C., Yang, F., Chong, A.C.M., et al.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51(8), 1477–1508 (2003)

    Article  MATH  Google Scholar 

  16. Lim, C.W., Zhang, G., Reddy, J.N.: A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J. Mech. Phys. Solids 78, 298–313 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Reddy, J.N.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45, 288–307 (2007)

    Article  MATH  Google Scholar 

  18. Reddy, J.N.: Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. Int. J. Eng. Sci. 48(11), 1507–1518 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  20. Asghari, M., Kahrobaiyan, M.H., Ahmadian, M.T.: A nonlinear Timoshenko beam formulation based on the modified couple stress theory. Int. J. Eng. Sci. 48, 1749–1761 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Farokhi, H., Ghayesh, M.H.: Thermo-mechanical dynamics of perfect and imperfect Timoshenko microbeams. Int. J. Eng. Sci. 91, 12–33 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ghayes, M.H., Amabili, M., Farokhi, H.: Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory. Int. J. Eng. Sci. 63, 52–60 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Dang, V.H., Nguyen, D.A., Le, M.Q., Ninh, Q.H.: Nonlinear vibration of microbeams based on the nonlinear elastic foundation using the equivalent linearization method with a weighted averaging. Arch. Appl. Mech. 90, 87–106 (2020)

    Article  Google Scholar 

  24. Li, Li., Yujin, Hu.: Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory. Int. J. Eng. Sci. 97, 84–94 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lu, Lu., Guo, X., Zhao, J.: Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory. Int. J. Eng. Sci. 116, 12–24 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Arefi, M., Pourjamshidian, M., Arani, A.G.: Application of nonlocal strain gradient theory and various shear deformation theories to nonlinear vibration analysis of sandwich nano-beam with FG-CNTRCs face-sheets in electro-thermal environment. Appl. Phys. A 123, 323 (2017)

    Article  Google Scholar 

  27. Reddy, J.N., Pang, S.D.: Nonlocal continuum theories of beams for the analysis of carbon nanotubes. J. Appl. Phys. 103(2), 023511 (2008)

    Article  Google Scholar 

  28. De Rosa, M.A., Lippiello, M.: Nonlocal frequency analysis of embedded single-walled carbon nanotube using the differential quadrature method. Compos. B 84, 41–51 (2016)

    Article  Google Scholar 

  29. Zhen, Y.X., Wen, S.L., Tang, Y.: Free vibration analysis of viscoelastic nanotubes under longitudinal magnetic field based on nonlocal strain gradient Timoshenko beam model. Physica E 105, 116–124 (2019)

    Article  Google Scholar 

  30. Atashafrooz, M., Bahaadini, R., Sheibani, H.R.: Nonlocal, strain gradient and surface effects on vibration and instability of nanotubes conveying nanoflow. Mech. Adv. Mater. Struct. 27(7), 586–598 (2020)

    Article  Google Scholar 

  31. Wang, B.L., Shen, S.J., Zhao, J.F., Chen, X.: A size-dependent Kirchhoff micro-plate model based on strain gradient elasticity theory. Eur. J. Mech. A. Solids 30, 517–524 (2011)

    Article  MATH  Google Scholar 

  32. Thanh, C.L., Loc, V.T., Huu, T.V., Abdel-Wahab, M.: The size-dependent thermal bending and buckling analyses of composite laminate microplate based on new modified couple stress theory and isogeometric analysis. Comput. Methods Appl. Mech. Eng. 350(15), 337–361 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Shahsavari, D., Karami, B., Mansouri, S.: Shear buckling of single layer graphene sheets in hygrothermal environment resting on elastic foundation based on different nonlocal strain gradient theories. Eur. J. Mech. A. Solids 67, 200–214 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  34. Thai, H.T., Vo, T.P., Nguyen, T.K., Kim, S.-E.: A review of continuum mechanics models for size-dependent analysis of beams and plates. Compos. Struct. 177, 196–219 (2017)

    Article  Google Scholar 

  35. Farajpour, A., Ghayesh, M.H., Farokhi, H.: A review on the mechanics of nanostructures. Int. J. Eng. Sci. 133, 231–263 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  36. Li, Li., Yujin, Hu.: Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material. Int. J. Eng. Sci. 107, 77–97 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. Şimşek, M.: Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach. Int. J. Eng. Sci. 105, 12–27 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  38. Tounsi, A., Al-Basyouni, K.S., Mahmoud, S.R.: Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position. Compos. Struct. 125, 621–630 (2015)

    Article  Google Scholar 

  39. Allam, M.N.M., Radwan, A.F.: Nonlocal strain gradient theory for bending, buckling, and vibration of viscoelastic functionally graded curved nanobeam embedded in an elastic medium. Adv. Mech. Eng. 11, 1–15 (2019)

    Article  Google Scholar 

  40. Arefi M, Pourjamshidian M, Arani AG, Rabczuk T (2019) Influence of flexoelectric, small-scale, surface and residual stress on the nonlinear vibration of sigmoid exponential and power-law FG Timoshenko nano-beams. J. Low Freq. Noise Vib. Act. Control; 38(1):122–142.

  41. Ansari, R., Shojaei, M.F., Mohammadi, V., Gholami, R., Darabi, M.A.: Nonlinear vibrations of functionally graded Mindlin microplates based on the modified couple stress theory. Compos. Struct. 114, 124–134 (2014)

    Article  Google Scholar 

  42. Ke, L.L., Yang, J., Kitipornchai, S., Bradford, M.A.: Bending, buckling and vibration of size-dependent functionally graded annular microplates. Compos. Struct. 94(1), 3250–3257 (2012)

    Article  Google Scholar 

  43. Motezaker, M., Jamali, M., Kolahchi, R.: Application of differential cubature method for nonlocal vibration, buckling and bending response of annular nanoplates integrated by piezoelectric layers based on surface-higher order nonlocal-piezoelasticity theory. J. Comput. Appl. Math. 369, 112625 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  44. Wattanasakulpong, N., Chaikittiratana, A.: Flexural vibration of imperfect functionally graded beams based on Timoshenko beam theory: Chebyshev collocation method. Meccanica 50, 1331–1342 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  45. Liu, Hu., Liu, H., Yang, J.: Vibration of FG magneto-electro-viscoelastic porous nanobeams on visco-Pasternak foundation. Compos. B 155, 244–256 (2018)

    Article  Google Scholar 

  46. Akbaş, S.D.: Forced vibration analysis of functionally graded porous deep beams. Compos. Struct. 186, 293–302 (2018)

    Article  Google Scholar 

  47. Karami, B., Janghorban, M., Rabczuk, T.: Dynamics of two-dimensional functionally graded tapered Timoshenko nanobeam in thermal environment using nonlocal strain gradient theory. Compos. B Eng. 182, 107622 (2020)

    Article  Google Scholar 

  48. Li, Q., Wu, D., Chen, X., Liu, L., Yu, Y., Gao, W.: Nonlinear vibration and dynamic buckling analyses of sandwich functionally graded porous plate with graphene platelet reinforcement resting on Winkler-Pasternak elastic foundation. Int. J. Mech. Sci. 148, 596–610 (2018)

    Article  Google Scholar 

  49. Younis, M.I., Nayfeh, A.H.: A study of the nonlinear response of a resonant microbeam to an electric actuation. Nonlinear Dyn. 31, 91–117 (2003)

    Article  MATH  Google Scholar 

  50. Fu, Y.M., Zhang, J., Wan, L.J.: Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS). Curr. Appl. Phys. 11, 482–485 (2011)

    Article  Google Scholar 

  51. Dang, V.H., Nguyen, D.A., Le, M.Q., Duong, T.H.: Nonlinear vibration of nanobeams under electrostatic force based on the nonlocal strain gradient theory. Int. J. Mech. Mater. Des. 16, 289–308 (2020)

    Article  Google Scholar 

  52. Mojahedi, M., Zand, M.M., Ahmadian, M.T.: Static pull-in analysis of electrostatically actuated microbeams using homotopy perturbation method. Appl. Math. Model. 34, 1032–1041 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  53. Fu, Y., Zhang, J.: Size-dependent pull-in phenomena in electrically actuated nanobeams incorporating surface energies. Appl. Math. Model. 35, 941–951 (2011)

    Article  MathSciNet  Google Scholar 

  54. Rahaeifard, M., Kahrobaiyan, M.H., Asghari, M., Ahmadian, M.T.: Static pull-in analysis of microcantilevers based on the modified couple stress theory. Sens. Actuators A 171, 370–374 (2011)

    Article  MATH  Google Scholar 

  55. Baghani, M.: Analytical study on size-dependent static pull-in voltage of microcantilevers using the modified couple stress theory. Int. J. Eng. Sci. 54, 99–105 (2012)

    Article  MathSciNet  Google Scholar 

  56. Ouakada, H.M., Hasan, M.H., Jaber, N.R., Hafiz, M.A.A., Alsaleem, F., Younis, M.: On the double resonance activation of electrostatically actuated microbeam based resonators. Int. J. Non-Linear Mech. 121, 103437 (2020)

    Article  Google Scholar 

  57. Li, L., Tang, H., Hu, Y.: The effect of thickness on the mechanics of nanobeams. Int. J. Eng. Sci. 123, 81–91 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  58. Tang, H., Li, L., Hu, Y.: Coupling effect of thickness and shear deformation on size-dependent bending of micro/nano-scale porous beams. Appl. Math. Model. 66, 527–547 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  59. Chen, W., Wang, L., Dai, H.: Stability and nonlinear vibration analysis of an axially loaded nanobeam based on nonlocal strain gradient theory. Int. J. Appl. Mech. 11(07), 1950069 (2019)

    Article  Google Scholar 

  60. Chen, W., Wang, L., Dai, H.: Nonlinear free vibration of nanobeams based on nonlocal strain gradient theory with the consideration of thickness-dependent size effect. J. Mech. Mater. Struct. 14(1), 119–137 (2019)

    Article  MathSciNet  Google Scholar 

  61. Hieu, D. V., Duong, T. H., Bui, G. P.: Nonlinear vibration of a functionally graded nanobeam based on the nonlocal strain gradient theory considering thickness effect. Adv. Civ. Eng. Vol. 2020, Article ID 9407673

  62. Tang, H., Li, L., Hu, Y., Meng, W., Duan, K.: Vibration of nonlocal strain gradient beams incorporating Poisson’s ratio and thickness effects. Thin-Walled Struct. 137, 377–391 (2019)

    Article  Google Scholar 

  63. He, J.H.: Hamiltonian approach to nonlinear approach. Phys. Lett. A 374, 2312–2314 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  64. He, J.H.: Preliminary report on the energy balance for nonlinear oscillations. Mech. Res. Commun. 29, 107–111 (2002)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 107.02-2020.03. The authors are grateful for this support.

Author information

Authors and Affiliations

  1. TNU, Thai Nguyen University of Technology (TNUT), Thainguyen, Vietnam

    Van-Hieu Dang

  2. University of Transport Technology, Hanoi, Vietnam

    Quang-Chan Do

Authors
  1. Van-Hieu Dang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Quang-Chan Do
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Van-Hieu Dang.

Ethics declarations

Conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

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

Dang, VH., Do, QC. Nonlinear vibration and stability of functionally graded porous microbeam under electrostatic actuation. Arch Appl Mech 91, 2301–2329 (2021). https://doi.org/10.1007/s00419-021-01884-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-021-01884-7

Keywords

Search

Navigation