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Size-dependent dynamic pull-in instability of hydrostatically and electrostatically actuated circular microplates

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Abstract

In the present study, the dynamic pull-in instability and free vibration of circular microplates subjected to combined hydrostatic and electrostatic forces are investigated. To take size effects into account, the strain gradient elasticity theory is incorporated into the Kirchhoff plate theory to develop a nonclassical plate model including three internal material length scale parameters. By using Hamilton’s principle, the higher-order governing equation and the corresponding boundary conditions are obtained. Afterward, a generalized differential quadrature (GDQ) method is employed to discritize the governing differential equations along with simply supported and clamped edge supports. To evaluate the pull-in voltage and vibration frequencies of actuated microplates, the hydrostatic-electrostatic actuation is assumed to be calculated by neglecting the fringing field effects and utilizing the parallel plate approximation. Also, a comparison between the pull-in voltages predicted by the strain gradient theory and the degenerated ones is presented. It is revealed that increasing the dimensionless internal length scale parameter or decreasing the applied hydrostatic pressures leads to higher values of the pull-in voltage. Moreover, it is found that the value of pull-in hydrostatic pressure decreases corresponding to higher dimensionless internal length scale parameters and applied voltages.

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References

  1. Saif, M.T.A., Alaca, B.E., Sehitoglu, H.: Analytical modeling of electrostatic membrane actuator micro pumps. J. Microelectromech. Syst. 81, 335–345 (1999)

    Article  Google Scholar 

  2. Soleymani, P., Sadeghian, H., Tahmasebi, A., Rezazadeh, G.: Pull-in instability investigation of circular micro pump subjected to nonlinear electrostatic force. Sens. Transducers J. 69, 622–628 (2006)

    Google Scholar 

  3. Sallese, J.M., Grabinski, W., Meyer, V., Bassin, C., Fazan, P.: Electrical modeling of a pressure sensor MOSFET. Sens. Actuators A, Phys. 94, 53–58 (2001)

    Article  Google Scholar 

  4. Nabian, A., Rezazadeh, G., Haddad-derafshi, M., Tahmasebi, A.: Mechanical behavior of a circular microplate subjected to uniform hydrostatic and non-uniform electrostatic pressure. J. Microsystems Technol. 14, 235–240 (2008)

    Article  Google Scholar 

  5. Bao, M., Wang, W.: Future of microelectromechanical systems (MEMS). Sens. Actuators A, Phys. 56, 135–141 (1996)

    Article  Google Scholar 

  6. Younis, M.I., Abdel-Rahman, E.M., Nayfeh, A.: A reduced-order model for electrically actuated microbeam-based MEMS. J. Microelectromech. Syst. 12, 672–680 (2003)

    Article  Google Scholar 

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

    Article  Google Scholar 

  8. Nayfeh, A.H., Younis, M.I., Abdel-Rahman, E.M.: Dynamic pull-in phenomenon in MEMS resonators. Nonlinear Dyn. 48, 153–163 (2007)

    Article  MATH  Google Scholar 

  9. Nayfeh, A.H., Younis, M.I.: Modeling and simulations of thermoelastic damping in microplates. J. Micromech. Microeng. 14, 1711–1717 (2004)

    Article  Google Scholar 

  10. Zhao, X.P., Abdel-Rahman, E.M., Nayfeh, A.H.: A reduced-order model for electrically actuated microplates. J. Micromech. Microeng. 14, 900–906 (2004)

    Article  Google Scholar 

  11. Machauf, A., Nemirovsky, Y., Dinnar, U.: A membrane micropump electrostatically actuated across the working fluid. J. Micromech. Microeng. 15, 2309–2316 (2005)

    Article  Google Scholar 

  12. Mukherjee, S., Bao, Z.P., Roman, M., Aubry, N.: Nonlinear mechanics of MEMS plates with a total Lagrangian approach. Comput. Struct. 83, 758–768 (2005)

    Article  Google Scholar 

  13. Batra, R.C., Porfiri, M., Spinello, D.: Reduced-order models for microelectromechanical rectangular and circular plates incorporating the Casimir force. Int. J. Solids Struct. 45, 3558–3583 (2008)

    Article  MATH  Google Scholar 

  14. Chao, P.C.P., Chiu, C.W., Tsai, C.Y.: A novel method to predict the pull-in voltage in a closed form for micro-plates actuated by a distributed electrostatic force. J. Micromech. Microeng. 16, 986–998 (2006)

    Article  Google Scholar 

  15. Lam, D.C.C., Chong, A.C.M.: Indentation model and strain gradient plasticity law for glassy polymers. Int. J. Mater. Res. 14, 3784–3788 (1999)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  17. McFarland, A.W., Colton, J.S.: Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J. Micromech. Microeng. 15, 1060–1067 (2005)

    Article  Google Scholar 

  18. Nix, W.D.: Mechanical properties of thin films. Metall. Mater. Trans. A, Phys. Metall. Mater. Sci. 20, 2217–2245 (1989)

    Article  Google Scholar 

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

    Article  Google Scholar 

  20. Poole, W.J., Ashby, M.F., Fleck, N.A.: Micro-hardness of annealed and work-hardened copper polycrystals. Scr. Mater. 34, 559–564 (1996)

    Article  Google Scholar 

  21. Chasiotis, I., Knauss, W.G.: The mechanical strength of polysilicon films: Part 2. Size effects associated with elliptical and circular perforations. J. Mech. Phys. Solids 51, 1551–1572 (2003)

    Article  Google Scholar 

  22. Aifantis, E.C.: Exploring the applicability of gradient elasticity to certain micro/nano reliability problems, microsystem technologies-micro-and nanosystems. J. Inf. Storage Process. Syst. 15, 109–115 (2009)

    Google Scholar 

  23. Yang, J., Jia, X.L., Kitipornchai, S.: Pull-in instability of nano-switches using nonlocal elasticity theory. J. Phys. D, Appl. Phys. 41, 035103 (2008)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Koiter, W.T.: Couple stresses in the theory of elasticity I and II. Proc. K. Ned. Akad. Wet. 67, 17–44 (1964)

    MATH  Google Scholar 

  26. Eringen, A.C., Suhubi, E.S.: Nonlinear theory of simple microelastic solid-I. Int. J. Eng. Sci. 2, 189–203 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  27. Eringen, A.C., Suhubi, E.S.: Nonlinear theory of simple microelastic solid-II. Int. J. Eng. Sci. 2, 389–404 (1964)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Toupin, R.A.: Theory of elasticity with couple stresses. Arch. Ration. Mech. Anal. 17, 85–112 (1964)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  32. 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 

  33. Vardoulaksi, I., Exadaktylos, G., Kourkoulis, S.K.: Bending of marble with intrinsic length scales: a gradient theory with surface energy and size effects. J. Phys. IV 8, 399–406 (1998)

    Google Scholar 

  34. 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, 2731–2743 (2002)

    Article  MATH  Google Scholar 

  35. Tsiatas, G.C.: A new Kirchhoff plate model based on a modified couple stress theory. Int. J. Solids Struct. 46, 2757–2764 (2009)

    Article  MATH  Google Scholar 

  36. Xia, W., Wang, L., Yin, L.: Nonlinear non-classical microscale beams: static bending, postbuckling and free vibration. Int. J. Eng. Sci. 48, 2044–2053 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Asghari, M., Rahaeifard, M., Kahrobaiyan, M.H., Ahmadian, M.T.: The modified couple stress functionally graded Timoshenko beam formulation. Mater. Des. 32, 1435–1443 (2011)

    Article  Google Scholar 

  38. Ke, L.L., Wang, Y.S.: Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory. Compos. Struct. 93, 342–350 (2011)

    Article  Google Scholar 

  39. Ke, L.L., Wang, Y.S., Yang, J., Kitipornchai, S.: Nonlinear free vibration of size-dependent functionally graded microbeams. Int. J. Eng. Sci. 50, 256–267 (2012)

    Article  MathSciNet  Google Scholar 

  40. Ke, L.L., Wang, Y.S., Wang, Z.D.: Thermal effect on free vibration and buckling of size-dependent microbeams. Physica E 43, 1387–1393 (2011)

    Article  Google Scholar 

  41. Fleck, N.A., Hutchinson, J.W.: Phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids 41, 1825–1857 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  42. Kong, S.L., Zhou, S.J., Nie, Z.F., Wang, K.: Static and dynamic analysis of microbeams based on strain gradient theory. Int. J. Eng. Sci. 47, 487–498 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wang, B., Zhao, J., Zhou, S.: A micro scale Timoshenko beam model based on strain gradient elasticity theory. Eur. J. Mech. A, Solids 29, 591–599 (2010)

    Article  Google Scholar 

  44. Ansari, R., Gholami, R., Sahmani, S.: Free vibration of size-dependent functionally graded microbeams based on a strain gradient theory. Compos. Struct. 94, 221–228 (2011)

    Article  Google Scholar 

  45. Ansari, R., Gholami, R., Sahmani, S.: Study of small scale effects on the nonlinear vibration response of functionally graded Timoshenko microbeams based on the strain gradient theory. ASME J. Comput. Nonlinear Dyn. 7, 031010 (2012)

    Article  Google Scholar 

  46. Sahmani, S., Ansari, R.: On the free vibration response of functionally graded higher-order shear deformable microplates based on the strain gradient elasticity theory. Compos. Struct. 95, 430–442 (2013)

    Article  Google Scholar 

  47. Ghayesh, 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  Google Scholar 

  48. Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity, 3rd edn. McGraw-Hill, New York (1970)

    MATH  Google Scholar 

  49. Shu, C.: Differential Qquadrature and Its Application in Engineering. Springer, London (2000)

    Book  Google Scholar 

  50. Ma, H.M., Gao, X.L., Reddy, J.N.: A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids 56, 3379–3391 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran

    V. Mohammadi, R. Ansari, M. Faghih Shojaei, R. Gholami & S. Sahmani

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  1. V. Mohammadi
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  2. R. Ansari
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  3. M. Faghih Shojaei
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Correspondence to R. Ansari.

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Mohammadi, V., Ansari, R., Faghih Shojaei, M. et al. Size-dependent dynamic pull-in instability of hydrostatically and electrostatically actuated circular microplates. Nonlinear Dyn 73, 1515–1526 (2013). https://doi.org/10.1007/s11071-013-0882-z

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  • DOI: https://doi.org/10.1007/s11071-013-0882-z

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