Advertisement

Pull-in behavior of functionally graded micro/nano-beams for MEMS and NEMS switches

  • Hamid Haghshenas Gorgani
  • Mohsen Mahdavi Adeli
  • Mohammad Hosseini
Technical Paper
  • 29 Downloads

Abstract

In this paper, pull-in behavior of cantilever micro/nano-beams made of functionally graded materials (FGM) with small-scale effects under electrostatic force is investigated. Consistent couple stress theory is employed to study the influence of small-scale on pull-in behavior. According to this theory, the couple tensor is skew-symmetric by adopting the skew-symmetric part of the rotation gradients. The material properties except Poisson’s ratio obey the power law distribution in the thickness direction. The approximate analytical solutions for the pull-in voltage and pull-in displacement of the microbeams are derived using the Rayleigh–Ritz method. Comparison between the results of the present work with Osterberg and Senturia’s article for pull-in behavior of microbeams made of isotropic material reveals the accuracy of this study. Numerical results explored the effects of material length scale parameter, inhomogeneity constant, gap distance and dimensionless thickness. Presented model has the ability to turn into the classical model if the material length scale parameter is taken to be zero. A comparison between classical and consistent couple stress theories is done which reveals the application of the consistent couple stress theory. As an important result of this study can be stated that a micro/nano-beams model based on the couple stress theory behaves stiffer and has larger pull-in voltages.

Notes

References

  1. Adeli MM, Hadi A, Hosseini M, Gorgani HH (2017) Torsional vibration of nano-cone based on nonlocal strain gradient elasticity theory. Eur Phys J Plus 132:393.  https://doi.org/10.1140/epjp/i2017-11688-0 CrossRefGoogle Scholar
  2. Aifantis EC (1999) Strain gradient interpretation of size effects. In: Bažant ZP, Rajapakse YDS (eds) Fracture scaling. Springer, Dordrecht, pp 299–314.  https://doi.org/10.1007/978-94-011-4659-3_16 CrossRefGoogle Scholar
  3. Akgöz B, Civalek Ö (2014) Thermo-mechanical buckling behavior of functionally graded microbeams embedded in elastic medium. Int J Eng Sci 85:90–104.  https://doi.org/10.1016/j.ijengsci.2014.08.011 CrossRefGoogle Scholar
  4. Asghari M (2012) Geometrically nonlinear micro-plate formulation based on the modified couple stress theory. Int J Eng Sci 51:292–309.  https://doi.org/10.1016/j.ijengsci.2011.08.013 MathSciNetCrossRefzbMATHGoogle Scholar
  5. Asghari M, Kahrobaiyan MH, Ahmadian MT (2010) A nonlinear Timoshenko beam formulation based on the modified couple stress theory. Int J Eng Sci 48:1749–1761.  https://doi.org/10.1016/j.ijengsci.2010.09.025 MathSciNetCrossRefzbMATHGoogle Scholar
  6. Beni YT, Abadyan MR, Noghrehabadi A (2011) Investigation of size effect on the pull-in instability of beam-type NEMS under van der waals attraction. Proc Eng 10:1718–1723.  https://doi.org/10.1016/j.proeng.2011.04.286 CrossRefGoogle Scholar
  7. Demir Ç, Civalek Ö (2017) On the analysis of microbeams. Int J Eng Sci 121:14–33.  https://doi.org/10.1016/j.ijengsci.2017.08.016 MathSciNetCrossRefGoogle Scholar
  8. Ebrahimi F, Barati MR (2016) A nonlocal higher-order refined magneto-electro-viscoelastic beam model for dynamic analysis of smart nanostructures. Int J Eng Sci 107:183–196.  https://doi.org/10.1016/j.ijengsci.2016.08.001 CrossRefGoogle Scholar
  9. Ebrahimi F, Barati MR, Dabbagh A (2016) A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates. Int J Eng Sci 107:169–182.  https://doi.org/10.1016/j.ijengsci.2016.07.008 CrossRefGoogle Scholar
  10. Eringen AC (1972a) Nonlocal polar elastic continua. Int J Eng Sci 10:1–16.  https://doi.org/10.1016/0020-7225(72)90070-5 MathSciNetCrossRefzbMATHGoogle Scholar
  11. Eringen AC (1972b) Theory of micromorphic materials with memory. Int J Eng Sci 10:623–641.  https://doi.org/10.1016/0020-7225(72)90089-4 CrossRefzbMATHGoogle Scholar
  12. Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54:4703–4710.  https://doi.org/10.1063/1.332803 CrossRefGoogle Scholar
  13. Eringen AC (2002) Nonlocal continuum field theories. Springer Science & Business Media, Berlin, HeidelbergzbMATHGoogle Scholar
  14. Goodarzi M, Nikkhah Bahrami M, Tavaf V (2017) Refined plate theory for free vibration analysis of FG nanoplates using the nonlocal continuum plate model. J Comput Appl Mech 48:123–136.  https://doi.org/10.22059/jcamech.2017.236217.155 CrossRefGoogle Scholar
  15. Hadjesfandiari AR, Dargush GF (2011) Couple stress theory for solids. Int J Solids Struct 48:2496–2510.  https://doi.org/10.1016/j.ijsolstr.2011.05.002 CrossRefGoogle Scholar
  16. Hakamiha S, Mojahedi M (2017) Nonlinear analysis of microswitches considering nonclassical theory. Int J Appl Mech 09:1750113.  https://doi.org/10.1142/s1758825117501137 CrossRefGoogle Scholar
  17. Hossein Bakhshi K, Sundaramoorthy R (2018) Mechanical analysis of non-uniform bi-directional functionally graded intelligent micro-beams using modified couple stress theory. Mater Res Exp 5:055703.  https://doi.org/10.1088/2053-1591/aabe62 CrossRefGoogle Scholar
  18. Hosseini M, Shishesaz M, Naderan-Tahan K, Hadi A (2016) Stress analysis of rotating nano-disks of variable thickness made of functionally graded materials. Int J Eng Sci 109:29–53.  https://doi.org/10.1016/j.ijengsci.2016.09.002 MathSciNetCrossRefGoogle Scholar
  19. Hosseini M, Haghshenas Gorgani H, Shishesaz M, Hadi A (2017) Size-dependent stress analysis of single-wall carbon nanotube based on strain gradient theory. Int J Appl Mech 09:1750087.  https://doi.org/10.1142/s1758825117500879 CrossRefGoogle Scholar
  20. Jomehzadeh E, Noori HR, Saidi AR (2011) The size-dependent vibration analysis of micro-plates based on a modified couple stress theory. Physica E 43:877–883.  https://doi.org/10.1016/j.physe.2010.11.005 CrossRefGoogle Scholar
  21. Kahrobaiyan MH, Rahaeifard M, Tajalli SA, Ahmadian MT (2012) A strain gradient functionally graded Euler-Bernoulli beam formulation. Int J Eng Sci 52:65–76.  https://doi.org/10.1016/j.ijengsci.2011.11.010 MathSciNetCrossRefzbMATHGoogle Scholar
  22. Ke L-L, Wang Y-S, Yang J, Kitipornchai S (2012) Nonlinear free vibration of size-dependent functionally graded microbeams. Int J Eng Sci 50:256–267.  https://doi.org/10.1016/j.ijengsci.2010.12.008 MathSciNetCrossRefzbMATHGoogle Scholar
  23. Koiter W (1964) Couple stresses in the theory of elasticity. In: I and II Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 67:17-44Google Scholar
  24. Kong S (2013) Size effect on pull-in behavior of electrostatically actuated microbeams based on a modified couple stress theory. Appl Math Model 37:7481–7488.  https://doi.org/10.1016/j.apm.2013.02.024 MathSciNetCrossRefzbMATHGoogle Scholar
  25. Koochi A, Kazemi AS, Noghrehabadi A, Yekrangi A, Abadyan M (2011) New approach to model the buckling and stable length of multi walled carbon nanotube probes near graphite sheets. Mater Des 32:2949–2955.  https://doi.org/10.1016/j.matdes.2010.08.002 CrossRefGoogle Scholar
  26. Lam DCC, Yang F, Chong ACM, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51:1477–1508.  https://doi.org/10.1016/S0022-5096(03)00053-X CrossRefzbMATHGoogle Scholar
  27. Lazar M, Maugin GA, Aifantis EC (2006) Dislocations in second strain gradient elasticity. Int J Solids Struct 43:1787–1817.  https://doi.org/10.1016/j.ijsolstr.2005.07.005 CrossRefzbMATHGoogle Scholar
  28. Ma HM, Gao XL, Reddy JN (2008) A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J Mech Phys Solids 56:3379–3391.  https://doi.org/10.1016/j.jmps.2008.09.007 MathSciNetCrossRefzbMATHGoogle Scholar
  29. Mehar K, Panda SK, Mahapatra TR (2018) Nonlinear frequency responses of functionally graded carbon nanotube-reinforced sandwich curved panel under uniform temperature field. Int J Appl Mech 10:1850028.  https://doi.org/10.1142/s175882511850028x CrossRefGoogle Scholar
  30. Mindlin RD, Tiersten HF (1962) Effects of couple-stresses in linear elasticity. Arch Ration Mech Anal 11:415–448.  https://doi.org/10.1007/bf00253946 MathSciNetCrossRefzbMATHGoogle Scholar
  31. Mohammad-Abadi M, Daneshmehr AR (2014) Size dependent buckling analysis of microbeams based on modified couple stress theory with high order theories and general boundary conditions. Int J Eng Sci 74:1–14.  https://doi.org/10.1016/j.ijengsci.2013.08.010 MathSciNetCrossRefGoogle Scholar
  32. Mohammad-Abadi M, Daneshmehr AR (2015) Modified couple stress theory applied to dynamic analysis of composite laminated beams by considering different beam theories. Int J Eng Sci 87:83–102.  https://doi.org/10.1016/j.ijengsci.2014.11.003 CrossRefGoogle Scholar
  33. Mohammad-Sedighi H, Koochi A, Keivani M, Abadyan M (2017) Microstructure-dependent dynamic behavior of torsional nano-varactor. Measurement 111:114–121.  https://doi.org/10.1016/j.measurement.2017.07.011 CrossRefGoogle Scholar
  34. Nejad MZ, Rastgoo A, Hadi A (2014) Exact elasto-plastic analysis of rotating disks made of functionally graded materials. Int J Eng Sci 85:47–57.  https://doi.org/10.1016/j.ijengsci.2014.07.009 MathSciNetCrossRefGoogle Scholar
  35. Osterberg PM, Senturia SD (1997) M-TEST: a test chip for MEMS material property measurement using electrostatically actuated test structures. J Microelectromech Syst 6:107–118.  https://doi.org/10.1109/84.585788 CrossRefGoogle Scholar
  36. Park S, Gao X (2006) Bernoulli-Euler beam model based on a modified couple stress theory. J Micromech Microeng 16:2355.  https://doi.org/10.1088/0960-1317/16/11/015 CrossRefGoogle Scholar
  37. Pasoodeh B, Parvizi A, Akbari H (2018) Effect of the asymmetrical rolling process on the micro hardness and microstructure of brass wire. J Comput Appl Mech 49:143–148.  https://doi.org/10.22059/jcamech.2018.246027.208 CrossRefGoogle Scholar
  38. Rafiee M, Nitzsche F, Labrosse MR (2018) Cross-sectional design and analysis of multiscale carbon nanotubes-reinforced composite beams and blades. Int J Appl Mech 10:1850032.  https://doi.org/10.1142/s1758825118500321 CrossRefGoogle Scholar
  39. Rahaeifard M, Kahrobaiyan MH, Asghari M, Ahmadian MT (2011) Static pull-in analysis of microcantilevers based on the modified couple stress theory. Sens Actuators A 171:370–374.  https://doi.org/10.1016/j.sna.2011.08.025 CrossRefzbMATHGoogle Scholar
  40. Shishesaz M, Hosseini M, Naderan-Tahan K, Hadi A (2017) Analysis of functionally graded nanodisks under thermoelastic loading based on the strain gradient theory. Acta Mech.  https://doi.org/10.1007/s00707-017-1939-8 MathSciNetCrossRefzbMATHGoogle Scholar
  41. Şimşek M, Reddy JN (2013) Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory. Int J Eng Sci 64:37–53.  https://doi.org/10.1016/j.ijengsci.2012.12.002 MathSciNetCrossRefGoogle Scholar
  42. Toupin RA (1962) Elastic materials with couple-stresses. Arch Ration Mech Anal 11:385–414.  https://doi.org/10.1007/bf00253945 MathSciNetCrossRefzbMATHGoogle Scholar
  43. Xia W, Wang L, Yin L (2010) Nonlinear non-classical microscale beams: static bending, postbuckling and free vibration. Int J Eng Sci 48:2044–2053.  https://doi.org/10.1016/j.ijengsci.2010.04.010 MathSciNetCrossRefzbMATHGoogle Scholar
  44. Xue C-X, Pan E (2013) On the longitudinal wave along a functionally graded magneto-electro-elastic rod. Int J Eng Sci 62:48–55.  https://doi.org/10.1016/j.ijengsci.2012.08.004 CrossRefzbMATHGoogle Scholar
  45. Yang F, Chong ACM, Lam DCC, Tong P (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39:2731–2743.  https://doi.org/10.1016/S0020-7683(02)00152-X CrossRefzbMATHGoogle Scholar
  46. Yang J, Jia X, Kitipornchai S (2008) Pull-in instability of nano-switches using nonlocal elasticity theory. J Phys D Appl Phys 41:035103.  https://doi.org/10.1088/0022-3727/41/3/035103 CrossRefGoogle Scholar
  47. Yin L, Qian Q, Wang L, Xia W (2010) Vibration analysis of microscale plates based on modified couple stress theory. Acta Mech Solida Sin 23:386–393.  https://doi.org/10.1016/S0894-9166(10)60040-7 CrossRefGoogle Scholar
  48. Zargaripoor A, Daneshmehr A, Isaac-Hosseini I, Rajabpoor A (2018) Free vibration analysis of nanoplates made of functionally graded materials based on nonlocal elasticity theory using finite element method. J Comput Appl Mech 49:86–101.  https://doi.org/10.22059/jcamech.2018.248906.223 CrossRefGoogle Scholar
  49. Zenkour AM (2013) Bending of fgm plates by a simplified four-unknown shear and normal deformations theory. Int J Appl Mech 05:1350020.  https://doi.org/10.1142/s1758825113500208 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Engineering Graphics CenterSharif University of TechnologyTehranIran
  2. 2.Department of Mechanical EngineeringSousangerd Branch, Islamic Azad UniversitySousangerdIran
  3. 3.Department of Mechanical EngineeringShahid Chamran University of AhvazAhvazIran

Personalised recommendations