Structural Engineering and Mechanics

  • 제61권2호
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  • Pages.209-220
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  • 2017
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  • 1225-4568(pISSN)
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  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
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DOI QR코드

DOI QR Code

Large amplitude free vibration analysis of functionally graded nano/micro beams on nonlinear elastic foundation

  • Setoodeh, AliReza (Department of Mechanical and Aerospace Engineering, Shiraz University of Technology) ;
  • Rezaei, Mohammad (Department of Mechanical and Aerospace Engineering, Shiraz University of Technology)
  • 투고 : 2016.06.25
  • 심사 : 2016.08.15
  • 발행 : 2017.01.25
⟨ 이전 논문 다음 논문 ⟩

초록

The purpose of this paper is to study the geometrically nonlinear free vibration of functionally graded nano/micro beams (FGNBs) based on the modified couple stress theory. For practical applications, some analytical expressions of nonlinear frequencies for FGNBs on a nonlinear Pasternak foundation are developed. Hamilton's principle is employed to obtain nonlinear governing differential equations in the context of both Euler-Bernoulli and Timoshenko beam theories for a comprehensive investigation. The modified continuum theory contains one material length scale parameter to capture the size effect. The variation of two-constituent material along the thickness is modeled using Reddy's power-law. Also, the Mori-Tanaka method as an accurate homogenization technique is implemented to estimate the effective material properties of the FGNBs. The results are presented for both hinged-hinged and clamped-clamped boundary conditions. The nonlinear partial differential equations are reduced to ordinary differential equations using Galerkin method and then the powerful method of homotopy analysis is utilized to obtain the semi-analytical solutions. Eventually, the presented analytical expressions are used to examine the influences of the length scale parameter, material gradient index, and elastic foundation on the nonlinear free vibration of FGNBs.

키워드

참고문헌

  1. Ansari, R., Gholami, R., Faghih Shojaei, M., Mohammadi, V. and Darabi, M.A. (2016), "Coupled longitudinal-transverserotational free vibration of post-buckled functionally graded first-order shear deformable micro- and nano-beams based on the Mindlin's strain gradient Theory", Appl. Math. Model., 40(23), 9872-9891. https://doi.org/10.1016/j.apm.2016.06.042
  2. Asghari, M., Rahaeifard, M., Kahrobaiyan, M.H. and Ahmadian, M.T. (2011), "The modified couple stress functionally graded Timoshenko beam formulation", Mater. Des., 32(3), 1435-1443. https://doi.org/10.1016/j.matdes.2010.08.046
  3. Bagdatli, S.M. (2015), "Non-linear transverse vibrations of tensioned nanobeams using nonlocal beam theory", Struct. Eng. Mech., 55(2), 281-298. https://doi.org/10.12989/sem.2015.55.2.281
  4. Bayat, M., Pakar, I. and Emadi, A. (2013), "Vibration of electrostatically actuated microbeam by means of homotopy perturbation method", Struct. Eng. Mech., 48(6), 823-831. https://doi.org/10.12989/sem.2013.48.6.823
  5. Ebrahimi, F. and Shafiei, N. (2016), "Application of Eringen's nonlocal elasticity theory for vibration analysis of rotating functionally graded nanobeams", Smart. Struct. Syst., 17(5), 837-857. https://doi.org/10.12989/sss.2016.17.5.837
  6. Ehyaei, J., Ebrahimi, F. and Salari, E. (2016), "Nonlocal vibration analysis of FG nano beams with different boundary conditions", Adv. Nano Res., 4(2), 85-111. https://doi.org/10.12989/anr.2016.4.2.085
  7. Eringen, A.C. (1972), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10(1), 1-16. https://doi.org/10.1016/0020-7225(72)90070-5
  8. Jafari-Talookolaei, R.A., Salarieh, H. and Kargarnovin, M.H. (2011), "Analysis of large amplitude free vibrations of unsymmetrically laminated composite beams on a nonlinear elastic foundation", Acta Mech., 219, 65-75. https://doi.org/10.1007/s00707-010-0439-x
  9. Janghorban, M. and Zare, A. (2011), "Free vibration analysis of functionally graded carbon nanotubes with variable thickness by differential quadrature method", Physica. E Low Dimens. Syst. Nanostruct., 43(9), 1602-1604. https://doi.org/10.1016/j.physe.2011.05.002
  10. Jia, X.L., Ke, L.L., Feng, C.B., Yang, J. and 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, 1137-1148. https://doi.org/10.1016/j.compstruct.2015.08.044
  11. Kahrobaiyan, M.H., Asghari, M., Rahaeifard, M. and Ahmadian, M.T. (2010), "Investigation of the size-dependent dynamic characteristics of atomic force microscope microcantilevers based on the modified couple stress theory", Int. J. Eng. Sci., 48(12), 1985-1994. https://doi.org/10.1016/j.ijengsci.2010.06.003
  12. Ke, L.L., Wang, Y.S., Yang, J. and Kitipornchai, S. (2012), "Nonlinear free vibration of size-dependent functionally graded microbeams", Int. J. Eng. Sci., 50(1), 256-267. https://doi.org/10.1016/j.ijengsci.2010.12.008
  13. Ke, L.L., Yang, J., Kitipornchai, S. and Xiang, Y. (2009), "Flexural vibration and elastic buckling of a cracked Timoshenko beam made of functionally graded materials", Mech. Adv. Mater. Struct., 16(6), 488-502. https://doi.org/10.1080/15376490902781175
  14. Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solid., 51(8), 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X
  15. Liao, S.J. (2004), Beyond perturbation: introduction to homotopy analysis method, CRC Press, Boca Raton.
  16. Ma, H.M., Gao, X.L. and Reddy, J.N. (2008), "A microstructuredependent Timoshenko beam model based on a modified couple stress theory", J. Mech. Phys. Solid., 56(12), 3379-3391. https://doi.org/10.1016/j.jmps.2008.09.007
  17. Malekzadeh, P. and Shojaee, M. (2013), "Surface and nonlocal effects on the nonlinear free vibration of non-uniform nanobeams", Compos. Part B-Eng., 52, 84-92. https://doi.org/10.1016/j.compositesb.2013.03.046
  18. Malekzadeh, P. and Shojaee, M. (2015), "A two-variable firstorder shear deformation theory coupled with surface and nonlocal effects for free vibration of nanoplates", J. Vib.. Control., 21, 2755-2772. https://doi.org/10.1177/1077546313516667
  19. Mindlin, R.D. and Tiersten, H.F. (1962), "Effects of couplestresses in linear elasticity", Arch. Ration. Mech. Anal., 11(1), 415-448. https://doi.org/10.1007/BF00253946
  20. Nateghi, A. and Salamat-talab, M. (2013), "Thermal effect on size dependent behavior of functionally graded micro beams based on modified couple stress theory", Compos. Struct., 96, 97-110. https://doi.org/10.1016/j.compstruct.2012.08.048
  21. Rao, S.S. (2007), Vibration of continuous systems. John Wiley & Sons, New Jersi.
  22. Schmid, S., Wagli, P. and Hierold, C. (2009, January), "Biosensor based on all-polymer resonant microbeams", Micro Electro Mechanical Systems, IEEE 22nd International Conference, 300-303.
  23. Sedighi, H.M., Chan-Gizian, M. and Noghreha-Badi, A. (2014), "Dynamic pull-in instability of geometrically nonlinear actuated micro-beams based on the modified couple stress theory", Lat. Am. J. Solid. Struct., 11(5), 810-825. https://doi.org/10.1590/S1679-78252014000500005
  24. Setoodeh, A.R. and Afrahim, S. (2014), "Nonlinear dynamic analysis of FG micro-pipes conveying fluid Based on strain gradient theory", Compos. Struct., 116, 128-135. https://doi.org/10.1016/j.compstruct.2014.05.013
  25. Setoodeh, A.R., Derahaki, M. and Bavi, N. (2015), "DQ thermal buckling analysis of embedded curved carbon nanotubes based on nonlocal elasticity theory", Lat. Am. J. Solid. Struct., 12(10), 1901-1917. https://doi.org/10.1590/1679-78251894
  26. Setoodeh, A.R., Rezaei, M. and Zendehdel Shahri, M.R. (2016), "Linear and nonlinear torsional free vibration of functionally graded micro/nano-tubes based on modified couple stress theory", Appl. Math. Mech., English Edition, 37, 1-16. https://doi.org/10.1007/s10483-016-2051-9
  27. Shenas, A.G. and Malekzadeh, P. (2016), "Free vibration of functionally graded quadrilateral microplates in thermal environment", Thin. Wall. Struct., 106, 294-315. https://doi.org/10.1016/j.tws.2016.05.001
  28. Taeprasartsit, S. (2013), "Nonlinear free vibration of thin functionally graded beams using the finite element method", J. Vib. Control., 1077546313484506. https://doi.org/10.1177/1077546313484506
  29. Thai, H.T. and Choi, D.H. (2013), "Size-dependent functionally graded Kirchhoff and Mindlin plate models based on a modified couple stress theory", Compos. Struct., 95, 142-153. https://doi.org/10.1016/j.compstruct.2012.08.023
  30. Younis, M.I., Abdel-Rahman, E.M. and Nayfeh, A. (2003), "A reduced-order model for electrically actuated microbeam-based MEMS", J. Microelectromech. Syst., 12(5), 672-680. https://doi.org/10.1109/JMEMS.2003.818069

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