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Effects of Casimir Force and Thermal Stresses on the Buckling of Electrostatic Nanobridges Based on Couple Stress Theory

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Abstract

In this paper, the thermal and size effects on the buckling behavior of a nanobeam symmetrically located between two electrodes, subjected to the influence of the nonlinear external forces including electrostatic and Casimir attractions, have been investigated. Based on the modified couple stress theory and by using Hamilton’s principle, the governing equation is derived from Euler–Bernoulli beam model and the non-dimensional critical buckling load is presented. In order to achieve linearization the equations, instead of nonlinear forces, their corresponding Maclaurin series expansions are used. Also, the differential quadrature method is utilized to solve the buckling equations numerically. Finally, the material length scale parameters (the size effects), the electrostatic and Casimir nonlinear forces, the temperature change, as well as the effect of the initial gap between beam and fixed electrodes on the buckling load are studied. The results indicated that by increasing the forces dependent on the displacement such as Casimir and electrostatic, the buckling load decreases. However, increasing the material length scale parameters leads to an increase in the buckling load value. Meanwhile, the initial gap does not significantly affect the buckling load. Furthermore, the non-dimensional critical buckling load becomes higher with the temperature change increasing.

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References

  1. Poole, C.P.; Owens, F.J.:Introduction to Nanotechnology: Selected Topics. Wiley, New York (2003)

    Google Scholar 

  2. Rezazadeh, G.; Madinei, H.; Shabani, R.: Study of parametric oscillation of an electrostatically actuated microbeam using variational iteration method. Appl. Math. Model. 36, 430–443 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Tadi Beni, Y.: Use of augmented continuum theory for modeling the size dependent material behavior of nano-actuators. IJST Trans. Mech. Eng. 36(M1), 41–52 (2012)

    Google Scholar 

  4. Lu, P.; Lee, H.P.; Lu, C.; Zhang, P.Q.: Application of nonlocal beam models for carbon nanotubes. Int. J. Solids Struct. 44, 5289–5300 (2007)

    Article  MATH  Google Scholar 

  5. Darvishian, A.; Moeenfard, H.; Ahmadian, M.T.; Zohoor, H.: A coupled two degree of freedom pull-in model for micromirrors under capillary force. Acta Mech. 223, 387–394 (2012)

    Google Scholar 

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

    Google Scholar 

  7. Tadi B., Y.; Koochi, A.; Abadyan, M.: Theoretical study of the effect of Casimir force, elastic boundary conditions and size dependency on the pull-in instability of beam-type NEMS. Phys. E 43, 979–988 (2011)

  8. 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  MATH  MathSciNet  Google Scholar 

  9. Anthoine, A.: Effect of couple-stresses on the elastic bending of beams. Int. J. Solids Struct. 37, 1003–1018 (2000)

    Article  MATH  Google Scholar 

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

    Google Scholar 

  11. Park, S.K.; Gao, X.L.: Bernoulli–Euler beam model based on a modified couple stress theory. J. Micromech. Microeng. 16, 2355–2359 (2006)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  13. Park, S.K.; Gao, X.L.: Variational formulation of a modified couple stress theory and its application to a simple shear problem. Z. Angew. Math. Phys. 59, 904–917 (2008)

    Google Scholar 

  14. 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)

    Google Scholar 

  15. Akgoz, B.; Civalek, O.: Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams. Int. J. Eng. Sci. 49, 1268–1280 (2011)

    Google Scholar 

  16. Ma, H.M.; Gao, X.L.; Reddy J.N.: A non-classical Reddy–Levinson beam model based on a modified couple stress theory. Int. J. Multiscale Comput. Eng. 8, 167–180 (2010)

    Google Scholar 

  17. Salamat-talab, M.; Nateghi, A.; Torabi, J.: Static and dynamic analysis of third-order shear deformation FG micro beam based on modified couple stress theory. Int. J. Mech. Sci. 57, 63–73 (2012)

    Google Scholar 

  18. Reddy, J.N.: Microstructure-dependent couple stress theories of functionally graded beams. J. Mech. Phys. Solids 59, 2382–2399 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  19. 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 

  20. Ma, H.M.; Gao, X.L.; Reddy, J.N.: A non-classical Mindlin plate model based on a modified couple stress theory. Acta Mech. 220, 217–235 (2011)

    Article  MATH  Google Scholar 

  21. Gao, X.L.; Huang, J.X.; Reddy J.N.: A non-classical third-order shear deformation plate model based on a modified couple stress theory. Acta Mech. 224, 2699–2718 (2013)

    Google Scholar 

  22. Yin, L.; Qian, Q.; Wang, L.; Xia, W.: Vibration analysis of microscale plates based on modified couple stress theory. Acta Mech. Solida Sin. 23, 386–393 (2010)

    Article  Google Scholar 

  23. Tadi Beni, Y.; Abadyan, M.; Koochi, A.: Effect of the Casimir attraction on the torsion/bending coupled instability of electrostatic nano-actuators. Phys. Scr. 84, 065801 (2011)

    Google Scholar 

  24. Tadi Beni, Y.; Vahdati, A.R.; Abadyan, M.: Using ALE-FEM to simulate the instability of beam-type nano-actuator in the presence of electrostatic field and dispersion forces. IJST Trans. Mech. Eng. 37(M1), 1–9 (2013)

    Google Scholar 

  25. Tadi Beni, Y.; Abadyan, M.: Size-dependent pull-in instability of torsional nanoactuator. Phys. Scr. 88, 055801 (2013)

    Article  Google Scholar 

  26. Lin, W.H.; Zhao, Y.P.: Nonlinear behavior for nanoscale electrostatic actuators with Casimir force. Chaos Solitons Fractals 23, 1777–1785 (2005)

    Article  MATH  Google Scholar 

  27. Ghalambaz, M.; Noghrehabadi, A.; Abadyan, M.; Tadi Beni, Y.; Noghrehabadi, A.R.; Noghrehabadi, M.: A deflection of nano cantilevers using monotone solution. Procedia Eng. 10, 3717–3724 (2011)

    Google Scholar 

  28. Jia, X.L.; Yang, J.; Kitipornchai, S.: Pull-in instability of geometrically nonlinear micro-switches under electrostatic and Casimir forces. Acta Mech. 218, 161–174 (2011)

    Google Scholar 

  29. Koochi, A.; Kazemi, A.S.; Tadi Beni, Y.; Yekrangi, A.; Abadyan, M.: Theoretical study of the effect of Casimir attraction on the pull-in behavior of beam-type NEMS using modified Adomian method. Physica E 43, 625–632 (2010)

    Google Scholar 

  30. Kacem, N.; Baguet, S.; Hentz, S.; Dufour, R.: Computational and quasi-analytical models for non-linear vibrations of resonant MEMS and NEMS sensors. Int. J. Nonlin. Mech. 46, 532–542 (2011)

    Google Scholar 

  31. Shu, C.: Differential Quadrature and its Application in Engineering: Engineering Applications. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  32. Bellman, R.; Casti, J.: Differential quadrature and long term integration. J. Math. Anal. Appl. 34, 235–238 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  33. Murmu, T.; Pradhan, S.C.: Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory. Comput. Mater. Sci. 46, 854–859 (2009)

    Article  Google Scholar 

  34. Murmu, T.; Pradhan, S.C.: Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM. Physica E 41, 1232–1239 (2009)

    Google Scholar 

  35. 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 

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

    Article  Google Scholar 

  37. Lifshitz, E.M.: The theory of molecular attractive forces between solids. Soviet Physics JETP 2, 73–83 (1956)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Engineering, Shahrekord University, Shahrekord, Iran

    P. Mohammadi Dashtaki & Y. Tadi Beni

  2. Nanotechnology Research Center, Shahrekord University, Shahrekord, Iran

    Y. Tadi Beni

Authors
  1. P. Mohammadi Dashtaki
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  2. Y. Tadi Beni
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Correspondence to Y. Tadi Beni.

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Dashtaki, P.M., Beni, Y.T. Effects of Casimir Force and Thermal Stresses on the Buckling of Electrostatic Nanobridges Based on Couple Stress Theory. Arab J Sci Eng 39, 5753–5763 (2014). https://doi.org/10.1007/s13369-014-1107-6

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  • DOI: https://doi.org/10.1007/s13369-014-1107-6

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