Size-Dependent Pull-In Instability of Electrically Actuated Functionally Graded Nano-Beams Under Intermolecular Forces

  • Hossein Ataei
  • Yaghoub Tadi Beni
Research Paper


Pull-in instability of a cantilever beam-type nano-actuator made of functionally graded material is investigated using the strain gradient theory under the influence of electrostatic and intermolecular forces. Differential quadrature method is used to solve the nonlinear governing equation of nano-beam; the size effects, different materials and different volume fractions are examined. Results obtained from this model and numerical solution method are in good agreement with the experimental results mentioned in the references. Besides, the results demonstrate that size effect and amount of volume fraction have a substantial effect on the pull-in instability behavior of beam-type nano-actuator.


Modified strain gradient theory Intermolecular forces Functionally graded material Size effect 


  1. Abadyan M, Novinzadeh A, Kazemi A (2010) Approximating the effect of the Casimir force on the instability of electrostatic nano-cantilevers. Phys Scr 81:55–68CrossRefMATHGoogle Scholar
  2. Abbasnejad B, Rezazadeh G, Shabani R (2013) Stability analysis of a capacitive fgm micro-beam using modified couple stress theory. Acta Mech Solida Sin 26:427–440CrossRefGoogle Scholar
  3. Akgöz B, Civalek Ö (2013) Free vibration analysis of axially functionally graded tapered Bernoulli–Euler microbeams based on the modified couple stress theory. Compos Struct 98:314–322CrossRefGoogle Scholar
  4. Akgöz B, Civalek Ö (2015) Bending analysis of FG microbeams resting on Winkler elastic foundation via strain gradient elasticity. Compos Struct 134:294–301CrossRefGoogle Scholar
  5. Ansari R, Rouhi H, Arash B (2013) Vibrational analysis of double-walled carbon nanotubes based on the nonlocal Donnell shell theory via a new numerical approach. Iran J Sci Technol Trans Mech Eng 37:91–105Google Scholar
  6. Ansari R, Shojaei MF, Gholami R (2016) Size-dependent nonlinear mechanical behavior of third-order shear deformable functionally graded microbeams using the variational differential quadrature method. Compos Struct 136:669–683CrossRefGoogle Scholar
  7. Baghani M (2012) Analytical study on size-dependent static pull-in voltage of microcantilevers using the modified couple stress theory. Int J Eng Sci 54:99–105MathSciNetCrossRefGoogle Scholar
  8. Batra R, Porfiri M, Spinello D (2007) Effects of Casimir force on pull-in instability in micromembranes. Europhys Lett (EPL) 77:20010CrossRefGoogle Scholar
  9. Beni YT (2012) “Use of augmented continuum theory for modeling the size dependent material behavior of nano-actuators. Iran J Sci Technol Trans Mech Eng 36:41Google Scholar
  10. Beni YT, Abadyan M (2013) Size-dependent pull-in instability of torsional nano-actuator. Phys Scr 88:055801CrossRefMATHGoogle Scholar
  11. Beni YT, Abadyan M (2013b) Use of strain gradient theory for modeling the size-dependent pull-in of rotational nano-mirror in the presence of molecular force. Int J Mod Phys B 27:1350083.doi:
  12. Beni YT, Abadyan M, Noghrehabadi A (2011) Investigation of size effect on the pull-in instability of beam-type NEMS under van der Waals attraction. Procedia Eng 10:1718–1723CrossRefGoogle Scholar
  13. 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 39:2633–2648MathSciNetCrossRefGoogle Scholar
  14. Bordag M, Mohideen U, Mostepanenko VM (2001) New developments in the Casimir effect. Phys Rep 353:1–205MathSciNetCrossRefMATHGoogle Scholar
  15. Civalek Ö (2004) Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns. Eng Struct 26:171–186CrossRefGoogle Scholar
  16. Dashtaki PM, Beni YT (2014) Effects of Casimir force and thermal stresses on the buckling of electrostatic nanobridges based on couple stress theory. Arab J Sci Eng 39:5753–5763CrossRefGoogle Scholar
  17. Fleck N, Hutchinson J (2001) A reformulation of strain gradient plasticity. J Mech Phys Solids 49:2245–2271CrossRefMATHGoogle Scholar
  18. Fleck NA, Muller GM, Ashby MF, Hutchinson JW (1994) Strain gradient plasticity: theory and experiment. Acta Metall Mater 42:475–487CrossRefGoogle Scholar
  19. Gholami R, Ansari R, Rouhi H (2015) Studying the effects of small scale and Casimir force on the non-linear pull-in instability and vibrations of FGM microswitches under electrostatic actuation. Int J Non-linear Mech 77:193–207CrossRefGoogle Scholar
  20. Gusso A, Delben GJ (2008) Dispersion force for materials relevant for micro-and nanodevices fabrication. J Phys D Appl Phys 41:175405CrossRefGoogle Scholar
  21. Hu Y-C (2006) Closed form solutions for the pull-in voltage of micro curled beams subjected to electrostatic loads. J Micromech Microeng 16:648CrossRefGoogle Scholar
  22. Jia XL, Yang J, Kitipornchai S, Lim CW (2012) Pull-in instability and free vibration of electrically actuated poly-SiGe graded micro-beams with a curved ground electrode. Appl Math Model 36:1875–1884MathSciNetCrossRefMATHGoogle Scholar
  23. Jia XL, Zhang SM, Ke LL, Yang J, Kitipornchai S (2014) Thermal effect on the pull-in instability of functionally graded micro-beams subjected to electrical actuation. Compos Struct 116:136–146CrossRefGoogle Scholar
  24. Jia XL, Ke LL, Feng CB, Yang J, Kitipornchai S (2015) Size effect on the free vibration of geometrically nonlinear functionally graded micro-beams under electrical actuation and temperature change. Compos Struct 133:1137–1148CrossRefGoogle Scholar
  25. Koizumi M (1993) Concept of FGM. Ceram Trans Func Grad Mater 34:3–10Google Scholar
  26. Kong S, Zhou S, Nie Z, Wang K (2009) Static and dynamic analysis of micro beams based on strain gradient elasticity theory. Int J Eng Sci 47:487–498MathSciNetCrossRefMATHGoogle Scholar
  27. Lam DC, Chong A (1999) Indentation model and strain gradient plasticity law for glassy polymers. J Mater Res 14:3784–3788CrossRefGoogle Scholar
  28. Lam DCC, Yang F, Chong ACM, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51:1477–1508CrossRefMATHGoogle Scholar
  29. Lamoreaux SK (2005) The Casimir force: background, experiments, and applications. Rep Prog Phys 68:201CrossRefGoogle Scholar
  30. Malihi S, Beni YT, Golestanian H (2016) Analytical modeling of dynamic pull-in instability behavior of torsional nano/micromirrors under the effect of Casimir force. Opt Int J Light Electron Opt 127:4426–4437CrossRefGoogle Scholar
  31. Miandoab EM, Yousefi-Koma A, Pishkenari HN (2014) Poly silicon nanobeam model based on strain gradient theory. Mech Res Commun 62:83–88CrossRefGoogle Scholar
  32. Mindlin RD, Tiersten HF (1962) Effects of couple-stresses in linear elasticity. Arch Ration Mech Anal 11:415–448MathSciNetCrossRefMATHGoogle Scholar
  33. Mohammadi H, Mahzoon M (2014) Investigating thermal effects in nonlinear buckling analysis of micro beams using modified strain gradient theory. Iran J Sci Technol Trans Mech Eng 38:303–320Google Scholar
  34. Mohammadi-Alasti B, Rezazadeh G, Borgheei A-M, Minaei S, Habibifar R (2011) On the mechanical behavior of a functionally graded micro-beam subjected to a thermal moment and nonlinear electrostatic pressure. Compos Struct 93:1516–1525CrossRefGoogle Scholar
  35. Mousavi T, Bornassi S, Haddadpour H (2013) The effect of small scale on the pull-in instability of nano-switches using DQM. Int J Solids Struct 50:1193–1202CrossRefGoogle Scholar
  36. Nami M, Janghorban M (2015) Free vibration of functionally graded size dependent nanoplates based on second order shear deformation theory using nonlocal elasticity theory. Iran J Sci Technol Trans Mech Eng 39:15–28Google Scholar
  37. Nayfeh AH, Younis MI, Abdel-Rahman EM (2007) Dynamic pull-in phenomenon in MEMS resonators. Nonlinear Dyn 48:153–163CrossRefMATHGoogle Scholar
  38. Nix W (1989) Mechanical properties of thin films. Metall Trans A 20:2217–2245CrossRefGoogle Scholar
  39. Osterberg PM, Senturia SD (1997) M-TEST: a test chip for MEMS material property measurement using electrostatically actuated test structures. J Microelectromech Syst 6:107–118CrossRefGoogle Scholar
  40. Pelesko JA, Bernstein DH (2002) Modeling Mems and Nems. CRC Press, Boca RatonCrossRefMATHGoogle Scholar
  41. Rafieipour H, Lotfavar A, Masroori A (2013) Ananlytical approximate solution for nonlinear vibration of microelecromechanical system using He’s frequency amplitude formulation. Iran J Sci Technol Trans Mech Eng 37:83Google Scholar
  42. 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 171:370–374CrossRefMATHGoogle Scholar
  43. Rezazadeh G, Tahmasebi A, Zubstov M (2006) Application of piezoelectric layers in electrostatic MEM actuators: controlling of pull-in voltage. Microsyst Technol 12:1163–1170CrossRefGoogle Scholar
  44. 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 190:32–43CrossRefGoogle Scholar
  45. Sedighi HM, Daneshmand F, Abadyan M (2015) Modeling the effects of material properties on the pull‐in instability of nonlocal functionally graded nano-actuators. ZAMM J Appl Math Mech 95:385–400Google Scholar
  46. Shojaeian M, Beni YT (2015) Size-dependent electromechanical buckling of functionally graded electrostatic nano-bridges. Sens Actuators A 232:49–62CrossRefGoogle Scholar
  47. Shojaeian M, Beni YT, Ataei H (2016) Electromechanical buckling of functionally graded electrostatic nanobridges using strain gradient theory. Acta Astronaut 118:62–71CrossRefGoogle Scholar
  48. Shu C (2000) Differential quadrature and its application in engineering. Springer, LondonCrossRefMATHGoogle Scholar
  49. Shu C, Du H (1997) A generalized approach for implementing general boundary conditions in the GDQ free vibration analysis of plates. Int J Solids Struct 34:837–846CrossRefMATHGoogle Scholar
  50. Şimşek M, Kocatürk T, Akbaş ŞD (2013) Static bending of a functionally graded microscale Timoshenko beam based on the modified couple stress theory. Compos Struct 95:740–747CrossRefGoogle Scholar
  51. Taylor G (1968) The coalescence of closely spaced drops when they are at different electric potentials. Proc R Soc Lond A 306:423–434CrossRefGoogle Scholar
  52. Witvrouw A, Mehta A (2005) The use of functionally graded poly-SiGe layers for MEMS applications. In: Materials science forum, pp 255–260Google Scholar
  53. Yamanouti M (1990) Functionally gradient materials forum. In: Proceedings of the first international symposium on functionally gradient materials (FGM ‘90), October 8–9, Hotel Sendai Plaza, Sendai, JapanGoogle Scholar
  54. Yang F, Chong ACM, Lam DCC, Tong P (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39:2731–2743CrossRefMATHGoogle Scholar
  55. 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 37:6964–6978MathSciNetCrossRefGoogle Scholar
  56. 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 224:2037–2047CrossRefGoogle Scholar
  57. Zare J (2014) Pull-in behavior analysis of vibrating functionally graded micro-cantilevers under suddenly DC voltage. J Appl Comput Mech 1:17–25Google Scholar
  58. Zeighampour H, Beni YT (2014a) Size-dependent vibration of fluid-conveying double-walled carbon nanotubes using couple stress shell theory. Phys E 61:28–39CrossRefGoogle Scholar
  59. Zeighampour H, Beni YT (2014b) Analysis of conical shells in the framework of coupled stresses theory. Int J Eng Sci 81:107–122MathSciNetCrossRefGoogle Scholar
  60. Zeverdejani MK, Beni YT (2013) The nano scale vibration of protein microtubules based on modified strain gradient theory. Curr Appl Phys 13:1566–1576CrossRefGoogle Scholar

Copyright information

© Shiraz University 2016

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentShahrekord UniversityShahrekordIran
  2. 2.Faculty of EngineeringShahrekord UniversityShahrekordIran

Personalised recommendations