Acta Mechanica Solida Sinica

, Volume 26, Issue 4, pp 427–440 | Cite as

Stability analysis of a capacitive FGM micro-beam using modified couple stress theory

  • Behrokh Abbasnejad
  • Ghader RezazadehEmail author
  • Rasool Shabani


Based on the Modified Couple Stress Theory, a functionally graded micro-beam under electrostatic forces is studied. The FGM micro-beam is made of two materials and material properties vary continuously along the beam thickness according to a power-law. Dynamic and static pull-in voltages are obtained and it is shown that the static and dynamic pull-in voltages for some materials cannot be obtained using classic theories and components of couple stress must be taken into account. In addition, it is shown that the values of pull-in voltages depend on the variation through the thickness of the volume fractions of the two constituents.

Key Words

MEMS FGM micro-beam stability pull-in voltage electrostatic pressure modified couple stress theory 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Mohammadi, M., Saidi, A. and Jomehzadeh, E., Levy solution for buckling analysis of functionally graded rectangular plates. Applied Composite Materials, 2010, 17(2): 81–93.CrossRefGoogle Scholar
  2. 2.
    Kang, Y.-A. and Li, X.-F., Bending of functionally graded cantilever beam with power-law non-linearity subjected to an end force. International Journal of Non-Linear Mechanics , 2009, 44(6): 696–703.CrossRefGoogle Scholar
  3. 3.
    Simsek, M., Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories. Journal of Nuclear Engineering Design, 2010, 240(4): 697–705.CrossRefGoogle Scholar
  4. 4.
    Craciunescu, C.M. and Wuttig, M., New ferromagnetic and functionally grade shape memory alloys. J Optoelectron Advance Material, 2003, 5(1): 139–146.Google Scholar
  5. 5.
    Fu, Y.Q., Du, H. J. and Zhang, S., Functionally graded TiN/TiNi shape memory alloy films. Material Letters, 2003, 57(20): 2995–2999.CrossRefGoogle Scholar
  6. 6.
    Fu, Y.Q., Du, H.J., Huang, W.M., Zhang, S. and Hu, M., TiNi-based thin films in MEMS applications: a review. Sensors and Actuators A, 2004, 112(2–3): 395–408.CrossRefGoogle Scholar
  7. 7.
    Witvrouw, A. and Mehta, A., The use of functionally graded poly-SiGe layers for MEMS applications. Material Science Forum, 2005, 492–493: 255–260.CrossRefGoogle Scholar
  8. 8.
    Lee, Z., Ophus, C. and Fischer, L.M., et al., Metallic NEMS components fabricated from a nanocomposite Al-Mo films. Nanotechnology, 2006, 17(12): 3063–3070.CrossRefGoogle Scholar
  9. 9.
    Rahaeifard, M., Kahrobaiyan, M.H. and Ahmadian, M.T., Sensitivity analysis of atomic force microscope cantilever made of functionally graded materials. In: DETC2009-86254, 3rd International Conference on Micro- and Nanosystems (MNS3), August 30-September 2, 2009, San Diego, CA, USA.Google Scholar
  10. 10.
    Kong, S., Zhou, S., Nie, Z. and Wang, K., The size-dependent natural frequency of Bernoulli-Euler micro-beams. International Journal of Engineering Science, 2008, 46(5): 427–437.CrossRefGoogle Scholar
  11. 11.
    Nix, W.D., Mechanical properties of thin fillms. Metallurgical and Materials Transactions, 1989, 20A(11): 2217–2245.Google Scholar
  12. 12.
    Fleck, N.A., Muller G.M., Ashby, M.F. and Hutchinson, J.W., Strain gradient plasticity: theory and experiment. Acta Metallurgica et Materialia, 1994, 42(2): 475–487.CrossRefGoogle Scholar
  13. 13.
    Poole, W.J., Ashby, M.F. and Fleck, N.A., Micro-hardness of annealed and work-hardened copper polycrystals. Scripta Materialia, 1996, 34(4): 559–564.CrossRefGoogle Scholar
  14. 14.
    Lam, D.C.C. and Chong, A.C.M., Indentation model and strain gradient plasticity law for glassy polymers. Journal of Materials Research, 1999, 14(9): 3784–3788.CrossRefGoogle Scholar
  15. 15.
    Lam, D.C.C., Yang, F. and Chong, A.C.M., et al., Experiments and theory in strain gradient elasticity. Journal of Mechanics and Physics of Solids, 2003, 51(8): 1477–1508.CrossRefGoogle Scholar
  16. 16.
    McFarland, A.W. and Colton, J.S., Role of material microstructure in plate stiffness with relevance to micro cantilever sensors. Journal of Micromechanics and Microengineering, 2005, 15(5): 1060–1067.CrossRefGoogle Scholar
  17. 17.
    Chasiotis, I. and Knauss, W.G., The mechanical strength of polysilicon films: Part 2. Size effects associated with elliptical and circular perforations. Journal of Mechanics and Physics of Solids, 2003, 51(8): 1551–1572.CrossRefGoogle Scholar
  18. 18.
    Mindlin, R.D. and Tiersten, H.F., Effects of couple-stresses in linear elasticity. Archive for Rational Mechanics and Analysis, 1962, 11(1): 415–448.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Toupin, R.A., Elastic materials with couple-stresses. Archive for Rational Mechanics and Analysis, 1962, 11(1): 385–414.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mindlin, R.D., Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 1964, 16(1): 51–78.MathSciNetCrossRefGoogle Scholar
  21. 21.
    Koiter, W.T., Couple-stresses in the theory of elasticity: I and II. Proc. K. Ned. Akad. Wet. B, 1964, 67(1): 17–44.MathSciNetzbMATHGoogle Scholar
  22. 22.
    Mindlin, R.D., Stress functions for a Cosserat continuum. International Journal of Solids and Structures, 1965, 1(3): 265–271.CrossRefGoogle Scholar
  23. 23.
    Cosserat, E. and Cosserat, F., Theorie des corps deformables. Hermann et Fils, Paris, 1909.zbMATHGoogle Scholar
  24. 24.
    Yang, F., Chong, A.C.M., Lam, D.C.C. and Tong, P., Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 2002, 39(10): 2731–2743.CrossRefGoogle Scholar
  25. 25.
    Park, S.K., Gao, X.L., Bernoulli-Euler beam model based on a modified couple stress theory. J Micromechanics and Microengineering, 2006, 16(11): 2355–2359.CrossRefGoogle Scholar
  26. 26.
    Tsiatas, G.C., A new Kirchhoff plate model based on a modified couple stress theory. International Journal of Solids and Structures, 2009, 46(13): 2757–2764.CrossRefGoogle Scholar
  27. 27.
    Kahrobaiyan, M.H., Asghari, M., Rahaeifard, M. and Ahmadian, M.T., Investigation of the size-dependent dynamic characteristics of atomic force microscope microcantilevers based on the modified couple stress theory. International Journal of Engineering Science, 2010, 48(12): 1985–1994.CrossRefGoogle Scholar
  28. 28.
    Wang, L., Size-dependent vibration characteristics of fluid-conveying Microtubes. Journal of Fluids and Structures, 2010, 26(4): 675–684.CrossRefGoogle Scholar
  29. 29.
    Xia, W., Wang, L. and Yin, L., Nonlinear non-classical microscale beams: Static bending, postbuckling and free vibration. International Journal of Engineering Science, 2010, 48(12): 2044–2053.MathSciNetCrossRefGoogle Scholar
  30. 30.
    Asghari, M., Ahmadian, M.T., Kahrobaiyan, M.H. and Rahaeifard, M., On the size-dependent behavior of functionally graded micro-beams. Materials and Design, 2010, 31(5): 2324–2329.CrossRefGoogle Scholar
  31. 31.
    Ke, L.L. and Wang, Y.S., Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory. Composite Structures, 2011, 93(2): 342–350.CrossRefGoogle Scholar
  32. 32.
    Senturia, S., Microsystem Design. Norwell, MA: Kluwer, 2001.Google Scholar
  33. 33.
    Sadeghian, H., Rezazadeh, G. and Osterberg, P.M., Application of the generalized differential quadrature method to the study of pull-in phenomena of MEMS switches. J. Microelectromechanical System, 2007, 16(6): 1334–1340.CrossRefGoogle Scholar
  34. 34.
    Rezazadeh, G., Khatami, F. and Tahmasebi, A., Investigation of the torsion and bending effects on static stability of electrostatic torsional micromirrors. Microsystem Technologies, 2007, 13(7): 715–722.CrossRefGoogle Scholar
  35. 35.
    Sazonova, V., A Tunable Carbon Nanotube Resonator, Ph.D. Thesis, Cornell University, 2006.Google Scholar
  36. 36.
    Rezazadeh, G., Tahmasebi, A. and Zubtsov, M., Application of piezoelectric layers in electrostatic MEM actuators: Controlling of pull-in voltage. Microsystem Technologies, 2006, 12(12): 1163–1170.CrossRefGoogle Scholar
  37. 37.
    Bao, M. and Wang, W., Future of microelectromechanical systems (MEMS). Sensors and Actuators A: Physical, 1996, 56(1–2): 135–141.CrossRefGoogle Scholar
  38. 38.
    Mehdaoui, Pisani, M.B., Tsamados, D., Casset, F., Ancey, P. and Ionescu, A.M., MEMS tunable capacitors with fragmented electrodes and rotational electro-thermal drive. Microsystem Technologies, 2007, 13(11): 1589–1594.CrossRefGoogle Scholar
  39. 39.
    Zhang, Y. and Zhao, Y., Numerical and analytical study on the pull-in instability of micro-structure under electrostatic loading. Sensors and Actuators A, 2006, 127(2): 366–367.CrossRefGoogle Scholar
  40. 40.
    Nguyen, C.T.C., Katehi, L.P.B. and Rebeiz, G.M., Micromachined devices for wireless communications. Proc. IEEE, 1998, 86(8): 1756–1768.CrossRefGoogle Scholar
  41. 41.
    Hasanyan, D.J., Batra, R.C. and Harutyunyan, S., Pull-in Instabilities in functionally graded microthermo-electromechanical systems. Journal of Thermal Stresses, 2008, 31(10): 1006–1021.CrossRefGoogle Scholar
  42. 42.
    Jia, X.L., Yang, J. and Kitipornchai, S., Characterization of FGM micro-switches under electrostatic and Casimir forces. IOP Conference Series, Materials Science and Engineering, 2010, 10, 012178.CrossRefGoogle Scholar
  43. 43.
    Ballestra, A., Brusa, E., De Pasquale, G., Munteanu, M.G. and Soma, A., FEM modelling and experimental characterization of microbeams in presence of residual stress. Analog Integrated Circuits and Signal Processing, 2010, 63(3): 477–488.CrossRefGoogle Scholar
  44. 44.
    Sadeghian, H., Goosen, H., Bossche, A., Thijsse, B. and van Keulen, F., On the size-dependent elasticity of silicon nanocantilevers: impact of defects. Journal of Physics D: Applied Physics, 2011, 44(7): 072001.CrossRefGoogle Scholar
  45. 45.
    Rezazadeh, G., Fathalilou, M., Shabani, R., Tarverdilou, S. and Talebian, S., Dynamic characteristics and forced response of an electrostatically-actuated micro-beam subjected to fluid loading. Microsystem Technologies, 2009, 15(9): 1355–1363.CrossRefGoogle Scholar
  46. 46.
    Pacheco, S.P., Katehi, L.P.B. and Nguyen, C.T.C., Design of low actuation voltage RF MEMS switch. Microwave Symposium Digest., 2000 IEEE MTT-S International, 2000, 1(11–16): 165–168.Google Scholar
  47. 47.
    Son, D., Kim, J.J., Kim, J.Y. and Kwon, D., Tensile properties and fatigue crack growth in LIGA nickel MEMS structures. Materials Science and Engineering A, 2005, 406(1–2): 274–278.CrossRefGoogle Scholar
  48. 48.
    Lin, W.H. and Zhao, Y.P., Stability and bifurcation behavior of electrostatic torsional NEMS varactor influenced by dispersion forces. Journal of Physics D: Applied Physics, 2007, 40(6): 1649–1654.CrossRefGoogle Scholar
  49. 49.
    Seydel, R., Practical Bifurcation and Stability Analysis, Third Edition, Springer, aDOI 10.1007/978-1-4419-1740-9.Google Scholar
  50. 50.
    Azizi, S., Design of micro accelerometer to use as airbag activator, MSc thesis, Mechanical Engineering Department, Tarbiat Modares University, Tehran, Iran, 2008: 53–4.Google Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2013

Authors and Affiliations

  • Behrokh Abbasnejad
    • 1
  • Ghader Rezazadeh
    • 1
    Email author
  • Rasool Shabani
    • 1
  1. 1.Mechanical Engineering DepartmentUrmia UniversityUrmiaIran

Personalised recommendations