Steel and Composite Structures

  • 제28권6호
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  • Pages.749-758
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  • 2018
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  • 1229-9367(pISSN)
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  • 1598-6233(eISSN)
테크노프레스 (Techno-Press)
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DOI QR코드

DOI QR Code

Free axial vibration analysis of axially functionally graded thick nanorods using nonlocal Bishop's theory

  • Nazemnezhad, Reza (School of Engineering, Damghan University) ;
  • Kamali, Kamran (Impact Research Laboratory, Department of Mechanical Engineering, Iran University of Science and Technology)
  • 투고 : 2017.11.16
  • 심사 : 2018.07.06
  • 발행 : 2018.09.25
⟨ 이전 논문 다음 논문 ⟩

초록

Free axial vibration of axially functionally graded (AFG) nanorods is studied by focusing on the inertia of lateral motions and shear stiffness effects. To this end, Bishop's theory considering the inertia of the lateral motions and shear stiffness effects and the nonlocal theory considering the small scale effect are used. The material properties are assumed to change continuously through the length of the AFG nanorod according to a power-law distribution. Then, nonlocal governing equation of motion and boundary conditions are derived by implementing the Hamilton's principle. The governing equation is solved using the harmonic differential quadrature method (HDQM), After that, the first five axial natural frequencies of the AFG nanorod with clamped-clamped end condition are obtained. In the next step, effects of various parameters like the length of the AFG nanorod, the diameter of the AFG nanorod, material properties, and the nonlocal parameter value on natural frequencies are investigated. Results of the present study can be useful in more accurate design of nano-electro-mechanical systems in which nanotubes are used.

키워드

참고문헌

  1. Adhikari, S., Murmu, T. and McCarthy, M.A. (2013), "Dynamic finite element analysis of axially vibrating nonlocal rods", Finite Elem. Anal. Des., 63, 42-50. https://doi.org/10.1016/j.finel.2012.08.001
  2. Adhikari, S., Murmu, T. and McCarthy, M.A. (2014), "Frequency domain analysis of nonlocal rods embedded in an elastic medium", Physica E: Low-dimen. Syst. Nanostruct., 59, 33-40. https://doi.org/10.1016/j.physe.2013.11.001
  3. Akgoz, B. and Civalek, O. (2011), "Buckling analysis of cantilever carbon nanotubes using the strain gradient elasticity and modified couple stress theories", J. Computat. Theor. Nanosci., 8, 1821-1827. https://doi.org/10.1166/jctn.2011.1888
  4. Akgoz, B. and Civalek, O. (2013), "Longitudinal vibration analysis of strain gradient bars made of functionally graded materials (FGM)", Compos. Part B: Eng., 55, 263-268. https://doi.org/10.1016/j.compositesb.2013.06.035
  5. Akgoz, B. and Civalek, O. (2017), "Effects of thermal and shear deformation on vibration response of functionally graded thick composite microbeams", Compos. Part B: Eng., 129, 77-87. https://doi.org/10.1016/j.compositesb.2017.07.024
  6. Allahkarami, F., Nikkhah-Bahrami, M. and Saryazdi, M.G. (2017), "Damping and vibration analysis of viscoelastic curved microbeam reinforced with FG-CNTs resting on viscoelastic medium using strain gradient theory and DQM", Steel Compos. Struct., Int. J., 25(2), 141-155.
  7. Amar, L.H.H., Kaci, A., Yeghnem, R. and Tounsi, A. (2018), "A new four-unknown refined theory based on modified couple stress theory for size-dependent bending and vibration analysis of functionally graded micro-plate", Steel Compos. Struct., Int. J., 26(1), 89-102.
  8. Assadi, A. and Farshi, B. (2011), "Size-dependent longitudinal and transverse wave propagation in embedded nanotubes with consideration of surface effects", Acta Mech., 222(1-2), 27-39. https://doi.org/10.1007/s00707-011-0521-z
  9. Aydogdu, M. (2009), "Axial vibration of the nanorods with the nonlocal continuum rod model", Physica E: Low-dimen. Syst. Nanostruct., 41(5), 861-864. https://doi.org/10.1016/j.physe.2009.01.007
  10. Aydogdu, M. (2014), "A nonlocal rod model for axial vibration of double-walled carbon nanotubes including axial Van der Waals force effects", J. Vib. Control, 21(16), 3132-3154. https://doi.org/10.1177/1077546313518954
  11. Belkorissat, I., Houari, M.S.A., Tounsi, A., Bedia, E.A. and Mahmoud, S.R. (2015), "On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable mode", Steel Compos. Struct., Int. J., 18(4), 1063-1081. https://doi.org/10.12989/scs.2015.18.4.1063
  12. Bert, C.W. and Malik, M. (1996), "Differential quadrature method in computational mechanics: a review", Appl. Mech. Rev., 49(1), 1-28. https://doi.org/10.1115/1.3101882
  13. Bounouara, F., Benrahou, K.H., Belkorissat, I. and Tounsi, A. (2016), "A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation", Steel Compos. Struct., Int. J., 20(2), 227-249. https://doi.org/10.12989/scs.2016.20.2.227
  14. Civalek, O. (2004), "Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns", Eng. Struct., 26(2), 171-186. https://doi.org/10.1016/j.engstruct.2003.09.005
  15. Faroughi, S. and Shaat, M. (2017), "Poisson ratio effects on the mechanics of auxetic nanobeams", Eur. J. Mech. / A Solids, 70, 8-14.
  16. Faroughi, S., Goushegir, S.M.H., Khodaparast, H.H. and Friswell, M.I. (2017), "Nonlocal elasticity in plates using novel trial functions", Int. J. Mech. Sci., 130, 221-233. https://doi.org/10.1016/j.ijmecsci.2017.05.034
  17. Fernandez-Saez, J., Zaera, R., Loya, J. and Reddy, J. (2016), "Bending of Euler-Bernoulli beams using Eringen's integral formulation: a paradox resolved", Int. J. Eng. Sci., 99, 107-116. https://doi.org/10.1016/j.ijengsci.2015.10.013
  18. Goushegir, S.M.H. and Faroughi, S. (2016), "Analysis of axial vibration of non-uniform nanorods using boundary characteristic orthogonal polynomials", Modares Mech. Eng., 16(1), 203-212.
  19. Goushegir, S.M.H. and Faroughi, S. (2017), "Free vibration and wave propagation of thick plates using the generalized nonlocal strain gradient theory", J. Theor. Appl. Vib. Acoust., 3(2), 165-198.
  20. Guven, U. (2014), "Love-Bishop rod solution based on strain gradient elasticity theory", Comptes Rendus Mecanique, 342, 8-16. https://doi.org/10.1016/j.crme.2013.10.011
  21. Hsu, J.-C., Lee, H.-L. and Chang, W.-J. (2011), "Longitudinal vibration of cracked nanobeams using nonlocal elasticity theory", Current Appl. Phys., 11(6), 1384-1388. https://doi.org/10.1016/j.cap.2011.04.026
  22. Karlicic, D., Cajic, M., Murmu, T. and Adhikari, S. (2015), "Nonlocal longitudinal vibration of viscoelastic coupled doublenanorod systems", Eur. J. Mech.-A/Solids, 49, 183-196. https://doi.org/10.1016/j.euromechsol.2014.07.005
  23. Kiani, K. (2010), "Free longitudinal vibration of tapered nanowires in the context of nonlocal continuum theory via a perturbation technique", Physica E: Low-dimen. Syst. Nanostruct., 43(1), 387-397. https://doi.org/10.1016/j.physe.2010.08.022
  24. Li, X.F., Shen, Z.B., and Lee, K.Y. (2017), "Axial wave propagation and vibration of nonlocal nanorods with radial deformation and inertia", ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift fur Angewandte Mathematik und Mechanik, 97, 602-616. https://doi.org/10.1002/zamm.201500186
  25. Malekzadeh, P. and Karami, G. (2005), "Polynomial and harmonic differential quadrature methods for free vibration of variable thickness thick skew plates", Eng. Struct., 27(10), 1563-1574. https://doi.org/10.1016/j.engstruct.2005.03.017
  26. Murmu, T. and Adhikari, S. (2010), "Nonlocal effects in the longitudinal vibration of double-nanorod systems", Physica E: Low-dimen. Syst. Nanostruct., 43(1), 415-422. https://doi.org/10.1016/j.physe.2010.08.023
  27. Narendar, S. and Gopalakrishnan, S. (2011), "Axial wave propagation in coupled nanorod system with nonlocal small scale effects", Compos. Part B: Eng., 42(7), 2013-2023. https://doi.org/10.1016/j.compositesb.2011.05.021
  28. Nazemnezhad, R. and Hosseini-Hashemi, S. (2014), "Nonlocal nonlinear free vibration of functionally graded nanobeams", Compos. Struct., 110, 192-199. https://doi.org/10.1016/j.compstruct.2013.12.006
  29. Nazemnezhad, R. and Kamali, K. (2016), "Investigation of the inertia of the lateral motions effect on free axial vibration of nanorods using nonlocal Rayleigh theory", Modares Mech. Eng., 16(5), 19-28. [In Persian]
  30. Nguyen, N.-T., Kim, N.-I. and Lee, J. (2014), "Analytical solutions for bending of transversely or axially FG nonlocal beams", Steel Compos. Struct., Int. J., 17(5), 641-665. https://doi.org/10.12989/scs.2014.17.5.641
  31. Rahmani, O., Refaeinejad, V. and Hosseini, S. (2017), "Assessment of various nonlocal higher order theories for the bending and buckling behavior of functionally graded nanobeams", Steel Compos. Struct., Int. J., 23(3), 339-350. https://doi.org/10.12989/scs.2017.23.3.339
  32. Rao, S.S. (2007), Vibration of Continuous Systems, John Wiley & Sons.
  33. Romano, G., Barretta, R., Diaco, M. and de Sciarra, F.M. (2017), "Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams", Int. J. Mech. Sci., 121, 151-156. https://doi.org/10.1016/j.ijmecsci.2016.10.036
  34. Shaat, M. and Faroughi, S. (2018), "Influence of surface integrity on vibration characteristics of microbeams", Eur. J. Mech. -A/Solids, 71, 365-377. https://doi.org/10.1016/j.euromechsol.2018.04.009
  35. Simsek, M. (2011), "Forced vibration of an embedded singlewalled carbon nanotube traversed by a moving load using nonlocal Timoshenko beam theory", Steel Compos. Struct., Int. J., 11(1), 59-76. https://doi.org/10.12989/scs.2011.11.1.059
  36. Simsek, M. (2012), "Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nanorods", Computat. Mater. Sci., 61, 257-265. https://doi.org/10.1016/j.commatsci.2012.04.001
  37. Striz, A., Wang, X. and Bert, C. (1995), "Harmonic differential quadrature method and applications to analysis of structural components", Acta Mech., 111(1-2), 85-94. https://doi.org/10.1007/BF01187729

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