Advances in nano research

  • 제4권2호
  • /
  • Pages.85-111
  • /
  • 2016
  • /
  • 2287-237X(pISSN)
  • /
  • 2287-2388(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Nonlocal vibration analysis of FG nano beams with different boundary conditions

  • Ehyaei, Javad (Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University) ;
  • Ebrahimi, Farzad (Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University) ;
  • Salari, Erfan (Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University)
  • 투고 : 2015.12.24
  • 심사 : 2016.05.21
  • 발행 : 2016.06.25
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, the classical and non-classical boundary conditions effect on free vibration characteristics of functionally graded (FG) size-dependent nanobeams are investigated by presenting a semi analytical differential transform method (DTM) for the first time. Three kinds of mathematical models, namely; power law (P-FGM), sigmoid (S-FGM) and Mori-Tanaka (MT-FGM) distribution are considered to describe the material properties in the thickness direction. The nonlocal Eringen theory takes into account the effect of small size, which enables the present model to become effective in the analysis and design of nanosensors and nanoactuators. Governing equations are derived through Hamilton's principle and they are solved applying semi analytical differential transform method. The good agreement between the results of this article and those available in literature validated the presented approach. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as small scale effects, spring constant factors, various material compositions and mode number on the normalized natural frequencies of the FG nanobeams in detail. It is explicitly shown that the vibration of FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.

키워드

참고문헌

  1. Alshorbagy, A.E., Eltaher, M.A. and Mahmoud, F.F. (2011), "Free vibration characteristics of afunctionally graded beam by finite element method", Appl. Math. Model., 35(1), 412-425. https://doi.org/10.1016/j.apm.2010.07.006
  2. Ansari, R., Gholami, R. and Sahmani, S. (2011), "Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory", Compos. Struct., 94(1), 221-228. https://doi.org/10.1016/j.compstruct.2011.06.024
  3. Asghari, M., Ahmadian, M.T., Kahrobaiyan, M.H.and Rahaeifard, M. (2010), "On the size-dependent behavior of functionally graded micro-beams", Mater. Des., 31(5), 2324-2329. https://doi.org/10.1016/j.matdes.2009.12.006
  4. Asghari, M., Rahaeifard, M., Kahrobaiyan, M.H. and Ahmadian, M.T. (2011), "The modified couple stressfunctionally graded Timoshenko beam formulation", Mater. Des., 32(3) 1435-1443. https://doi.org/10.1016/j.matdes.2010.08.046
  5. Aydogdu, M. (2009), "A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration", Physica E: Low-dimen. Syst. Nanostr., 41(9), 1651-1655. https://doi.org/10.1016/j.physe.2009.05.014
  6. Belabed, Z., Houari, M.S.A., Tounsi, A., Mahmoud, S. and Beg, O.A. (2014), "An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates", Compos. Part B: Eng., 60, 274-283. https://doi.org/10.1016/j.compositesb.2013.12.057
  7. Bennoun, M., Houari, M.S.A. and Tounsi, A. (2016), "A novel five-variable refined plate theory for vibration analysis of functionally graded sandwich plates", Mech. Adv. Mater. Struct., 23(4), 423-431. https://doi.org/10.1080/15376494.2014.984088
  8. Besseghier, A., Heireche, H., Bousahla, A.A., Tounsi, A. and Benzair, A. (2015), "Nonlinear vibration properties of a zigzag single-walled carbon nanotube embedded in a polymer matrix", Adv. Nano Res., 3(1), 029. https://doi.org/10.12989/anr.2015.3.1.029
  9. 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., 20(2), 227-249. https://doi.org/10.12989/scs.2016.20.2.227
  10. Chemi, A., Heireche, H., Zidour, M., Rakrak, K. and Bousahla, A.A. (2015), "Critical buckling load of chiral double-walled carbon nanotube using non-local theory elasticity", Adv. Nano Res., 3(4), 193-206. https://doi.org/10.12989/anr.2015.3.4.193
  11. Civalek, Ö., Cigdem, D., and Bekir, A. (2010), "Free vibration and bending analyses of cantilever microtubules based on nonlocal continuum model", Math. Comput. Appl., 15(2), 289-298.
  12. Eltaher, M.A., Alshorbagy, A.E. and Mahmoud, F.F. (2013), "Determination of neutral axis position andits effect on natural frequencies of functionally graded macro/nanobeams", Compos. Struct., 99, 193-201. https://doi.org/10.1016/j.compstruct.2012.11.039
  13. Eltaher, M.A., Emam, S.A. and Mahmoud, F.F. (2012), "Free vibration analysis of functionally graded size-dependent nanobeams", Appl. Math. Comput., 218(14), 7406-7420. https://doi.org/10.1016/j.amc.2011.12.090
  14. Eltaher, M.A., Emam, S.A. and Mahmoud, F.F. (2013), "Static and stability analysis of nonlocal functionally graded nanobeams", Compos. Struct., 96, 82-88. https://doi.org/10.1016/j.compstruct.2012.09.030
  15. 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
  16. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screwdislocation and surface waves", J. Appl. Phys., 54(9) 4703-4710. https://doi.org/10.1063/1.332803
  17. Hamidi, A., Houari, M.S.A., Mahmoud, S. and Tounsi, A. (2015), "A sinusoidal plate theory with 5-unknowns and stretching effect for thermomechanical bending of functionally graded sandwich plates", Steel Compos. Struct., 18(1), 235-253. https://doi.org/10.12989/scs.2015.18.1.235
  18. Hebali, H., Tounsi, A., Houari, M.S.A., Bessaim, A. and Bedia, E.A.A. (2014), "New quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates", J. Eng. Mech., 140(2), 374-383. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000665
  19. Hosseini-Hashemi, S. and Nazemnezhad, R. (2013), "An analytical study on the nonlinear freevibration of functionally graded nanobeams incorporating surface effects", Compos. Part B: Eng., 52, 199-206. https://doi.org/10.1016/j.compositesb.2013.04.023
  20. Ke, L.L. and Wang, Y.S. (2011), "Size effect on dynamic stability of functionally gradedmicrobeams based on a modified couple stress theory", Compos. Struct., 93(2), 342-350. https://doi.org/10.1016/j.compstruct.2010.09.008
  21. Ke, L.L., Wang, Y.S., Yang, J. and Kitipornchai, S. (2012), "Nonlinear free vibration ofsize-dependent functionally graded microbeams", Int. J. Eng. Sci., 50(1), 256-267. https://doi.org/10.1016/j.ijengsci.2010.12.008
  22. Ma'en, S.S. and Butcher, E.A. (2012), "Free vibration analysis of non-rotating and rotatingTimoshenko beams with damaged boundaries using the Chebyshev collocation method", Int. J. Mech. Sci., 60(1), 1-11. https://doi.org/10.1016/j.ijmecsci.2012.03.008
  23. Mahmoud, S., Chaht, F.L., Kaci, A., Houari, M.S.A., Tounsi, A. and Beg, O.A. (2015), "Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect", Steel Compos. Struct., 18(2), 425. https://doi.org/10.12989/scs.2015.18.2.425
  24. Pradhan, S.C. and Murmu, T. (2010), "Application of nonlocal elasticity and DQM in the flap wise bending vibration of a rotating nano cantilever", Physica E: Low-dimen. Syst. Nanostr., 42(7), 1944-1949. https://doi.org/10.1016/j.physe.2010.03.004
  25. Shahba, A., Attarnejad, R., Marvi, M.T. and Hajilar, S. (2011), "Free vibration and stability analysisof axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions", Compos. Part B: Eng., 42(4), 801-808. https://doi.org/10.1016/j.compositesb.2011.01.017
  26. Sharabiani, P.A. and Yazdi, M.R.H. (2013). (2013), "Nonlinear free vibrations of functionally graded nanobeams with surface effects", Compos. Part B: Eng., 45(1), 581-586. https://doi.org/10.1016/j.compositesb.2012.04.064
  27. Simsek, M. (2010), "Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories", Nucl. Eng. Des., 240(4), 697-705. https://doi.org/10.1016/j.nucengdes.2009.12.013
  28. Tounsi, A., Benguediab, S., Adda, B., Semmah, A. and Zidour, M. (2013), "Nonlocal effects on thermal buckling properties of double-walled carbon nanotubes", Adv. Nano Res., 1(1), 1-11. https://doi.org/10.12989/anr.2013.1.1.001
  29. Tounsi, A., Bourada, M., Kaci, A. and Houari, M.S.A. (2015), "A new simple shear and normal deformations theory for functionally graded beams", Steel Compos. Struct., 18(2), 409. https://doi.org/10.12989/scs.2015.18.2.409
  30. Wang, Q. (2005), "Wave propagation in carbon nanotubes via nonlocal continuum mechanics", J. Appl. Phys., 98(12), 124301. https://doi.org/10.1063/1.2141648
  31. Wang, Q., and Varadan, V.K. (2006), "Vibration of carbon nanotubes studied using nonlocal continuum mechanics", Smart Mater. Struct., 15(2), 659. https://doi.org/10.1088/0964-1726/15/2/050
  32. Wattanasakulpong, N. and Variddhi, U. (2014), "Linear and nonlinear vibration analysis ofelastically restrained ends FGM beams with porosities", Aerosp. Sci. Tech., 32(1), 111-120. https://doi.org/10.1016/j.ast.2013.12.002
  33. Witvrouw, A. and Mehta, A. (2005), "The use of functionally graded poly-SiGe layers for MEMS applications", Materials science forum, 492, Trans Tech Publications.
  34. Zemri, A., Houari, M.S.A., Bousahla, A.A. and Tounsi, A. (2015), "A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory", Struct. Eng. Mech., 54(4), 693-710. https://doi.org/10.12989/sem.2015.54.4.693
  35. Zhang, Y.Q., Liu, G.R. and Wang, J.S. (2004), "Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression", Phys.Rrev. B, 70(20), 205430. https://doi.org/10.1103/PhysRevB.70.205430

피인용 문헌

  1. On thermal stability of plates with functionally graded coefficient of thermal expansion vol.60, pp.2, 2016, https://doi.org/10.12989/sem.2016.60.2.313
  2. Forced Vibration Analysis of Functionally Graded Nanobeams vol.09, pp.07, 2017, https://doi.org/10.1142/S1758825117501009
  3. Large amplitude free vibration analysis of functionally graded nano/micro beams on nonlinear elastic foundation vol.61, pp.2, 2017, https://doi.org/10.12989/sem.2017.61.2.209
  4. A nonlocal quasi-3D theory for bending and free flexural vibration behaviors of functionally graded nanobeams vol.19, pp.2, 2017, https://doi.org/10.12989/sss.2017.19.2.115
  5. Size dependent vibration of embedded functionally graded nanoplate in hygrothermal environment by Rayleigh-Ritz method vol.10, pp.1, 2016, https://doi.org/10.12989/anr.2021.10.1.025
  6. A compressive study for porous FG curved nanobeam under various boundary conditions via a nonlocal strain gradient theory vol.136, pp.2, 2016, https://doi.org/10.1140/epjp/s13360-021-01238-w
  7. Nonlocal free vibration analysis of porous FG nanobeams using hyperbolic shear deformation beam theory vol.10, pp.3, 2016, https://doi.org/10.12989/anr.2021.10.3.281