Skip to main content
SpringerLink
Log in

Investigation of size-dependent quasistatic response of electrically actuated nonlinear viscoelastic microcantilevers and microbridges

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

This study investigates the size-dependent quasistatic response of a nonlinear viscoelastic microelectromechanical system (MEMS) under an electric actuation. To have this problem in view, the deformable electrode of the MEMS is modelled using cantilever and doubly-clamped viscoelastic microbeams. The modified couple stress theory in conjunction with Bernoulli–Euler beam theory are used for mathematical modeling of the size-dependent instability of microsystems in the framework of linear viscoelastic theory. Simultaneous effect of electrostatic actuation including fringing field, residual stress, mid-plane stretching and Casimir and van der Waals intermolecular forces are considered in the theoretical model. A single element of the standard linear solid element is used to simulate the viscoelastic behavior. Based on the extended Hamilton’s variational principle, the nonlinear governing integro-differential equation and boundary conditions are derived. Thereafter, a new generalized differential-integral quadrature solution for the nonlinear quasistatic response of electrically actuated viscoelastic micro/nanobeams under two different boundary conditions; doubly-clamped microbridge and clamped-free microcantilever. The developed model is verified and a good agreement is obtained. Finally, a comprehensive study is conducted to investigate the effects of various parameters such as material relaxation time, durable modulus, material length scale parameter, Casimir force, van der Waals force, initial gap and beam length on the pull-in response of viscoelastic microbridges and microcantilevers in the framework of viscoelasticity.

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
Fig. 18
Fig. 19
Fig. 20

Similar content being viewed by others

References

  1. Tilmans HA, Legtenberg R (1994) Electrostatically driven vacuum-encapsulated polysilicon resonators: Part II. Theory and performance. Sens Actuators A 45(1):67–84

    Article  Google Scholar 

  2. Rezazadeh G, Tahmasebi A, Zubstov M (2006) Application of piezoelectric layers in electrostatic MEM actuators: controlling of pull-in voltage. Microsyst Technol 12(12):1163–1170

    Article  Google Scholar 

  3. Lin W-H, Zhao Y-P (2007) Stability and bifurcation behaviour of electrostatic torsional NEMS varactor influenced by dispersion forces. J Phys D Appl Phys 40(6):1649

    Article  ADS  Google Scholar 

  4. Das K, Batra R (2009) Symmetry breaking, snap-through and pull-in instabilities under dynamic loading of microelectromechanical shallow arches. Smart Mater Struct 18(11):115008

    Article  ADS  Google Scholar 

  5. Batra RC, Porfiri M, Spinello D (2006) Electromechanical model of electrically actuated narrow microbeams. J Microelectromech Syst 15(5):1175–1189

    Article  Google Scholar 

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

    Article  Google Scholar 

  7. Mokhtari J, Farrokhabadi A, Rach R, Abadyan M (2015) Theoretical modeling of the effect of Casimir attraction on the electrostatic instability of nanowire-fabricated actuators. Phys E 68:149–158

    Article  MATH  Google Scholar 

  8. Wineman AS, Rajagopal KR (2000) Mechanical response of polymers: an introduction. Cambridge University Press, Cambridge

    Google Scholar 

  9. Altenbach H, Eremeyev V (2011) Mechanics of viscoelastic plates made of FGMs. In: Computational modelling and advanced simulations. Springer, Berlin, pp 33–48

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  12. Kong S, Zhou S, Nie Z, Wang K (2008) The size-dependent natural frequency of Bernoulli–Euler micro-beams. Int J Eng Sci 46(5):427–437

    Article  MATH  Google Scholar 

  13. Arash B, Wang Q (2012) A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Comput Mater Sci 51(1):303–313

    Article  Google Scholar 

  14. Pamidighantam S, Puers R, Baert K, Tilmans HA (2002) Pull-in voltage analysis of electrostatically actuated beam structures with fixed–fixed and fixed–free end conditions. J Micromech Microeng 12(4):458

    Article  Google Scholar 

  15. Hung ES, Senturia SD (1999) Extending the travel range of analog-tuned electrostatic actuators. J Microelectromech Syst 8(4):497–505

    Article  Google Scholar 

  16. Xie W, Lee H, Lim S (2003) Nonlinear dynamic analysis of MEMS switches by nonlinear modal analysis. Nonlinear Dyn 31(3):243–256

    Article  MATH  Google Scholar 

  17. Batra R, Porfiri M, Spinello D (2007) Effects of Casimir force on pull-in instability in micromembranes. Europhys Lett 77(2):20010

    Article  ADS  Google Scholar 

  18. Ramezani A, Alasty A, Akbari J (2007) Closed-form solutions of the pull-in instability in nano-cantilevers under electrostatic and intermolecular surface forces. Int J Solids Struct 44(14):4925–4941

    Article  MATH  Google Scholar 

  19. Zand MM, Ahmadian M (2010) Dynamic pull-in instability of electrostatically actuated beams incorporating Casimir and van der Waals forces. Proc Inst Mech Eng Part C: J Mech Eng Sci 224(9):2037–2047

    Article  Google Scholar 

  20. Eringen AC (1972) Nonlocal polar elastic continua. Int J Eng Sci 10(1):1–16

    Article  MathSciNet  MATH  Google Scholar 

  21. Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54(9):4703–4710

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Mindlin R, Eshel N (1968) On first strain-gradient theories in linear elasticity. Int J Solids Struct 4(1):109–124

    Article  MATH  Google Scholar 

  24. Yang F, Chong A, Lam DCC, Tong P (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39(10):2731–2743

    Article  MATH  Google Scholar 

  25. Lam DCC, Yang F, Chong A, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51(8):1477–1508

    Article  ADS  MATH  Google Scholar 

  26. Rahaeifard M, Kahrobaiyan M, Asghari M, Ahmadian M (2011) Static pull-in analysis of microcantilevers based on the modified couple stress theory. Sens Actuators, A 171(2):370–374

    Article  MATH  Google Scholar 

  27. Yin L, Qian Q, Wang L (2011) Size effect on the static behavior of electrostatically actuated microbeams. Acta Mech Sin 27(3):445–451

    Article  ADS  MATH  Google Scholar 

  28. Kong S (2013) Size effect on pull-in behavior of electrostatically actuated microbeams based on a modified couple stress theory. Appl Math Model 37(12):7481–7488

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  30. Rokni H, Seethaler RJ, Milani AS, Hosseini-Hashemi S, Li X-F (2013) Analytical closed-form solutions for size-dependent static pull-in behavior in electrostatic micro-actuators via Fredholm integral equation. Sens Actuators, A 190:32–43

    Article  Google Scholar 

  31. Zamanzadeh M, Rezazadeh G, Jafarsadeghi-Poornaki I, Shabani R (2013) Static and dynamic stability modeling of a capacitive FGM micro-beam in presence of temperature changes. Appl Math Model 37(10):6964–6978

    Article  MathSciNet  Google Scholar 

  32. Shaat M, Mohamed SA (2014) Nonlinear-electrostatic analysis of micro-actuated beams based on couple stress and surface elasticity theories. Int J Mech Sci 84:208–217

    Article  Google Scholar 

  33. Shaat M, Abdelkefi A (2015) Modeling the material structure and couple stress effects of nanocrystalline silicon beams for pull-in and bio-mass sensing applications. Int J Mech Sci 101:280–291

    Article  Google Scholar 

  34. Wang Q (2006) Axi-symmetric wave propagation of carbon nanotubes with non-local elastic shell model. Int J Struct Stab Dyn 6(02):285–296

    Article  Google Scholar 

  35. Liang B, Zhang L, Wang B, Zhou S (2015) A variational size-dependent model for electrostatically actuated NEMS incorporating nonlinearities and Casimir force. Phys E 71:21–30

    Article  ADS  Google Scholar 

  36. Beni YT, Karimipour I, Abadyan M (2015) Modeling the instability of electrostatic nano-bridges and nano-cantilevers using modified strain gradient theory. Appl Math Model 39(9):2633–2648

    Article  MathSciNet  Google Scholar 

  37. Noghrehabadi A, Eslami M (2016) Analytical study on size-dependent static pull-in analysis of clamped–clamped nano-actuators in liquid electrolytes. Appl Math Model 40(4):3011–3028

    Article  MathSciNet  Google Scholar 

  38. Bethe K, Baumgarten D, Frank J (1990) Creep of sensor’s elastic elements: metals versus non-metals. Sens Actuators, A 23(1):844–849

    Article  Google Scholar 

  39. Fu Y, Zhang J, Bi R (2009) Analysis of the nonlinear dynamic stability for an electrically actuated viscoelastic microbeam. Microsyst Technol 15(5):763–769

    Article  Google Scholar 

  40. Fu Y, Zhang J (2009) Nonlinear static and dynamic responses of an electrically actuated viscoelastic microbeam. Acta Mech Sin 25(2):211–218

    Article  ADS  MATH  Google Scholar 

  41. Zhang J, Fu Y (2012) Pull-in analysis of electrically actuated viscoelastic microbeams based on a modified couple stress theory. Meccanica 47(7):1649–1658

    Article  MathSciNet  MATH  Google Scholar 

  42. Chen C, Li S, Dai L, Qian C (2014) Buckling and stability analysis of a piezoelectric viscoelastic nanobeam subjected to van der Waals forces. Commun Nonlinear Sci Numer Simul 19(5):1626–1637

    Article  ADS  MathSciNet  Google Scholar 

  43. Phan-Thien N (2012) Understanding viscoelasticity: an introduction to rheology. Springer, Berlin

    Google Scholar 

  44. Osterberg PM, Senturia SD (1997) M-TEST: a test chip for MEMS material property measurement using electrostatically actuated test structures. J Microelectromech Syst 6(2):107–118

    Article  Google Scholar 

  45. Schapery RA (1974) Viscoelastic behavior and analysis of composite materials. Mechanics of composite materials (A 75-24868 10-39). Academic Press Inc., New York, pp 85–168

    Google Scholar 

  46. Chen L, Zhang W, Liu Y (2007) Modeling of nonlinear oscillations for viscoelastic moving belt using generalized Hamilton’s principle. J Vib Acoust 129(1):128–132

    Article  Google Scholar 

  47. Nayfeh AH, Emam SA (2008) Exact solution and stability of postbuckling configurations of beams. Nonlinear Dyn 54(4):395–408

    Article  MathSciNet  MATH  Google Scholar 

  48. Huang J-M, Liu A, Lu C, Ahn J (2003) Mechanical characterization of micromachined capacitive switches: design consideration and experimental verification. Sens Actuators, A 108(1):36–48

    Article  Google Scholar 

  49. Gusso A, Delben GJ (2008) Dispersion force for materials relevant for micro-and nanodevices fabrication. J Phys D Appl Phys 41(17):175405

    Article  ADS  Google Scholar 

  50. Batra R, Porfiri M, Spinello D (2008) Vibrations of narrow microbeams predeformed by an electric field. J Sound Vib 309(3):600–612

    Article  ADS  Google Scholar 

  51. Attia MA, Mahmoud FF (2016) Analysis of viscoelastic Bernoulli-Euler nanobeams incorporating nonlocal and microstructure effects. Int J Mech Mater Des 1–12. doi:10.1007/s10999-016-9343-4

  52. Shu C (2012) Differential quadrature and its application in engineering. Springer, Berlin

    Google Scholar 

  53. Attia MA, Mohamed SA (2016) Nonlinear modeling and analysis of electrically actuated viscoelastic microbeams based on the modified couple stress theory. Appl Math Model. doi:10.1016/j.apm.2016.08.036

    MathSciNet  Google Scholar 

  54. Schapery RA (1965) A method of viscoelastic stress analysis using elastic solutions. J Franklin Inst 279(4):268–289

    Article  MathSciNet  MATH  Google Scholar 

  55. Nayfeh AH, Mook DT (2008) Nonlinear oscillations. Wiley, London

    MATH  Google Scholar 

  56. Kahn H, Heuer AH (2002) Thermal stability of residual stresses and residual stress gradients in multilayer LPCVD polysilicon films. J Ceram Process Res 3(1):22–24

    Google Scholar 

Download references

Acknowledgements

The author would like to thank the reviewers for their insightful comments and insight suggestions to improve the clarity of this article. The author would also like to thank Prof. Fatin F. Mahmoud and Prof. Salwa A. Mohamed (Zagazig University-Egypt) for their help in this research and for the useful discussions and advises.

Author information

Authors and Affiliations

  1. Mechanical Design and Production Engineering Department, Zagazig University, Zagazig, 44511, Egypt

    Mohamed A. Attia

Authors
  1. Mohamed A. Attia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohamed A. Attia.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Attia, M.A. Investigation of size-dependent quasistatic response of electrically actuated nonlinear viscoelastic microcantilevers and microbridges. Meccanica 52, 2391–2420 (2017). https://doi.org/10.1007/s11012-016-0595-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-016-0595-8

Keywords

Search

Navigation