Smart Structures and Systems

  • 제19권3호
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  • Pages.243-257
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  • 2017
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  • 1738-1584(pISSN)
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  • 1738-1991(eISSN)
테크노프레스 (Techno-Press)
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DOI QR코드

DOI QR Code

Semi-analytical vibration analysis of functionally graded size-dependent nanobeams with various boundary conditions

  • 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 : 2015.04.02
  • 심사 : 2017.01.06
  • 발행 : 2017.03.25
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, free vibration of functionally graded (FG) size-dependent nanobeams is studied within the framework of nonlocal Timoshenko beam model. It is assumed that material properties of the FG nanobeam, vary continuously through the thickness according to a power-law form. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The non-classical governing differential equations of motion are derived through Hamilton's principle and they are solved utilizing both Navier-based analytical method and an efficient and semi-analytical technique called differential transformation method (DTM). Various types of boundary conditions such as simply-supported, clamped-clamped, clamped-simply and clamped-free are assumed for edge supports. The good agreement between the presented DTM and analytical results of this article and those available in the literature validated the presented approach. It is demonstrated that the DTM has high precision and computational efficiency in the vibration analysis of FG nanobeams. The obtained results show the significance of the material graduation, nonlocal effect, slenderness ratio and boundary conditions on the vibration characteristics of FG nanobeams.

키워드

참고문헌

  1. Asghari, M., M.T. Ahmadian, M.H. Kahrobaiyan, M. Rahaeifard (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
  2. Alshorbagy, A.E., M.A. Eltaher, F.F. Mahmoud (2011), "Free vibration characteristics of a functionally graded beam by finite element method", Appl. Math. Model., 35(1), 412-425. https://doi.org/10.1016/j.apm.2010.07.006
  3. Aghelinejad, M., Zare, K., Ebrahimi, F. and Rastgoo, A. (2011), "Nonlinear thermomechanical post-buckling analysis of thin functionally graded annular plates based on Von-Karman's Plate Theory", Mech. Adv. Mater. Struct., 18(5), 319-326. https://doi.org/10.1080/15376494.2010.516880
  4. Aifantis, E.C. (1984), "On the microstructural origin of certain inelastic models", J. Eng. Mater. Technol., 106(4), 326-330. https://doi.org/10.1115/1.3225725
  5. Baughman, R.H., Zakhidov, A.A. and de Heer, W.A. (2002), "Carbon nanotubes--the route toward applications", Science, 297(5582), 787-792. https://doi.org/10.1126/science.1060928
  6. Civalek, O., C. Demir, B. Akgoz (2010), "Free vibration and bending analyses of cantilever microtubules based on nonlocal continuum model", Math. Comput. Appl., 15(2), 289-298.
  7. Ebrahimi, F. (2013), "Analytical investigation on vibrations and dynamic response of functionally graded plate integrated with piezoelectric layers in thermal environment", Mech. Adv. Mater. Struct., 20(10), 854-870. https://doi.org/10.1080/15376494.2012.677098
  8. Ebrahimi, F. and A. Rastgoo (2008a), "Free vibration analysis of smart annular FGM plates integrated with piezoelectric layers", Smart Mater. Struct., 17(1), 015044. https://doi.org/10.1088/0964-1726/17/1/015044
  9. Ebrahimi, F. and A. Rastgoo (2008b), "An analytical study on the free vibration of smart circular thin FGM plate based on classical plate theory", Thin-Wall. Struct., 46(12), 1402-1408. https://doi.org/10.1016/j.tws.2008.03.008
  10. Ebrahimi, F., Naei, M.H. and Rastgoo, A. (2009), "Geometrically nonlinear vibration analysis of piezoelectrically actuated FGM plate with an initial large deformation", J. Mech. Sci. Technol., 23(8), 2107-2124. https://doi.org/10.1007/s12206-009-0358-8
  11. Ebrahimi, F. and Rastgoo, A. (2011), "Nonlinear vibration analysis of piezo-thermo-electrically actuated functionally graded circular plates", Arch. Appl. Mech., 81(3), 361-383. https://doi.org/10.1007/s00419-010-0415-x
  12. Ebrahimi, F. and Rastgoo, A. (2009), "Nonlinear vibration of smart circular functionally graded plates coupled with piezoelectric layers", Int. J. Mech. Mater. Des., 5(2), 157-165. https://doi.org/10.1007/s10999-008-9091-1
  13. Ebrahimi, F. and Barati, M.R. (2016a), "Temperature distribution effects on buckling behavior of smart heterogeneous nanosize plates based on nonlocal four-variable refined plate theory", Int. J. Smart Nano Mater., 7(3) 1-25. https://doi.org/10.1080/19475411.2016.1148077
  14. Ebrahimi, F. and Barati, M.R. (2016b), "Vibration analysis of smart piezoelectrically actuated nanobeams subjected to magneto-electrical field in thermal environment", J. Vib. Control, 1077546316646239.
  15. Ebrahimi, F. and Barati, M.R. (2016c), "Size-dependent thermal stability analysis of graded piezomagnetic nanoplates on elastic medium subjected to various thermal environments", Appl. Phys. A, 122(10), 910. https://doi.org/10.1007/s00339-016-0441-9
  16. Ebrahimi, F. and Barati, M.R. (2016d), "Static stability analysis of smart magneto-electro-elastic heterogeneous nanoplates embedded in an elastic medium based on a four-variable refined plate theory", Smart Mater. Struct., 25(10), 105014. https://doi.org/10.1088/0964-1726/25/10/105014
  17. Ebrahimi, F. and Barati, M.R. (2016e), "Buckling analysis of piezoelectrically actuated smart nanoscale plates subjected to magnetic field", J. Intel. Mater. Syst. Struct., 1045389X16672569.
  18. Ebrahimi, F., Barati, M.R. and Dabbagh, A. (2016), "A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates", Int. J. Eng. Sci., 107, 169-182. https://doi.org/10.1016/j.ijengsci.2016.07.008
  19. Ebrahimi, F. and Dabbagh, A. (2016), "On flexural wave propagation responses of smart FG magneto-electro-elastic nanoplates via nonlocal strain gradient theory", Compos. Struct., 162, 281-293.
  20. Ebrahimi, F. and Hosseini, S.H.S. (2016a), "Thermal effects on nonlinear vibration behavior of viscoelastic nanosize plates", J. Therm. Stress., 39(5), 606-625. https://doi.org/10.1080/01495739.2016.1160684
  21. Ebrahimi, F. and Hosseini, S.H.S. (2016b), "Double nanoplatebased NEMS under hydrostatic and electrostatic actuations", Eur. Phys. J. Plus, 131(5), 1-19. https://doi.org/10.1140/epjp/i2016-16001-3
  22. Ebrahimi, F. and Barati, M.R. (2016f), "A nonlocal higher-order shear deformation beam theory for vibration analysis of sizedependent functionally graded nanobeams", Arab. J. Sci. Eng., 41(5), 1679-1690. https://doi.org/10.1007/s13369-015-1930-4
  23. Ebrahimi, F. and Barati, M.R. (2016g), "Vibration analysis of nonlocal beams made of functionally graded material in thermal environment", Eur. Phys. J. Plus, 131(8), 279. https://doi.org/10.1140/epjp/i2016-16279-y
  24. Ebrahimi, F. and Barati, M.R. (2016h), "Dynamic modeling of a thermos-piezo-electrically actuated nanosize beam subjected to a magnetic field", Appl. Phys. A, 122(4), 1-18.
  25. Ebrahimi, F. and Barati, M.R. (2016i), "A unified formulation for dynamic analysis of nonlocal heterogeneous nanobeams in hygro-thermal environment", Appl. Phys. A, 122(9), 792. https://doi.org/10.1007/s00339-016-0322-2
  26. Ebrahimi, F. and Barati, M.R. (2016j), "A nonlocal higher-order refined magneto-electro-viscoelastic beam model for dynamic analysis of smart nanostructures", Int. J. Eng. Sci., 107, 183-196. https://doi.org/10.1016/j.ijengsci.2016.08.001
  27. Ebrahimi, F. and Barati, M.R. (2016k), "Hygrothermal effects on vibration characteristics of viscoelastic FG nanobeams based on nonlocal strain gradient theory", Compos. Struct., 159, 433-444.
  28. Ebrahimi, F. and Barati, M.R. (2016l), "Buckling analysis of nonlocal third-order shear deformable functionally graded piezoelectric nanobeams embedded in elastic medium", J. Brazil. Soc. Mech. Sci. Eng., 1-16.
  29. Ebrahimi, F. and Barati, M.R. (2016m), "Magnetic field effects on buckling behavior of smart size-dependent graded nanoscale beams", Eur. Phys. J. Plus, 131(7), 1-14. https://doi.org/10.1140/epjp/i2016-16001-3
  30. Ebrahimi, F. and Barati, M.R. (2016n), "Buckling analysis of smart size-dependent higher order magneto-electro-thermoelastic functionally graded nanosize beams", J. Mech., 1-11.
  31. Ebrahimi, F. and Barati, M.R. (2017), "A nonlocal strain gradient refined beam model for buckling analysis of size-dependent shear-deformable curved FG nanobeams", Compos. Struct., 159, 174-182. https://doi.org/10.1016/j.compstruct.2016.09.058
  32. Ebrahimi, F., Ghadiri, M., Salari, E., Hoseini, S. A.H. and Shaghaghi, G.R. (2015a), "Application of the differential transformation method for nonlocal vibration analysis of functionally graded nanobeams", J. Mech. Sci. Technol., 29(3), 1207-1215. https://doi.org/10.1007/s12206-015-0234-7
  33. Ebrahimi, F. and Salari, E. (2015a), "Thermal buckling and free vibration analysis of size dependent Timoshenko FG nanobeams in thermal environments", Compos. Struct., 128, 363-380. https://doi.org/10.1016/j.compstruct.2015.03.023
  34. Ebrahimi, F. and Salari, E. (2015b), "Size-dependent free flexural vibrational behavior of functionally graded nanobeams using semi-analytical differential transform method", Compos. Part B: Eng., 79, 156-169. https://doi.org/10.1016/j.compositesb.2015.04.010
  35. Ebrahimi, F. and Salari, E. (2015c), "Nonlocal thermo-mechanical vibration analysis of functionally graded nanobeams in thermal environment", Acta Astronautica, 113, 29-50. https://doi.org/10.1016/j.actaastro.2015.03.031
  36. Ebrahimi, F., Salari, E. and Hosseini, S.A.H. (2015b), "Thermomechanical vibration behavior of FG nanobeams subjected to linear and nonlinear temperature distributions", J. Therm. Stress., 38(12), 1360-1386. https://doi.org/10.1080/01495739.2015.1073980
  37. Ebrahimi, F. and Barati, M.R. (2016o), "An exact solution for buckling analysis of embedded piezoelectro-magnetically actuated nanoscale beams", Adv. Nano Res., 4(2), 65-84. https://doi.org/10.12989/anr.2016.4.2.065
  38. Ebrahimi, F. and Barati, M.R. (2016p), "Electromechanical buckling behavior of smart piezoelectrically actuated higherorder size-dependent graded nanoscale beams in thermal environment", Int. J. Smart Nano Mater., 7(2) 1-22. https://doi.org/10.1080/19475411.2016.1148077
  39. Ebrahimi, F. and Barati, M.R. (2016q), "Small scale effects on hygro-thermo-mechanical vibration of temperature dependent nonhomogeneous nanoscale beams", Mech. Adv. Mater. Struct., 1-13.
  40. Ebrahimi, F., Salari, E. and Hosseini, S.A.H. (2016), "In-plane thermal loading effects on vibrational characteristics of functionally graded nanobeams", Meccanica, 51(4), 951-977. https://doi.org/10.1007/s11012-015-0248-3
  41. Eltaher M.A., S.A. Emam, F.F. Mahmoud (2012), "Free vibration analysis of functionally graded size-dependent nanobeams", Appl. Math. Comput., 218, 7406-7420.
  42. Eltaher M.A., A.E. Alshorbagy, F.F. Mahmoud (2013), "Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/ nanobeams", Compos. Struct., 99, 193-201. https://doi.org/10.1016/j.compstruct.2012.11.039
  43. Eltaher M.A., S.A. Emam, F.F. Mahmoud (2013b), "Static and stability analysis of nonlocal functionally graded nanobeams", Compos. Struct., 96, 82-88. https://doi.org/10.1016/j.compstruct.2012.09.030
  44. Eringen, A.C. (1972b), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10(1), 1-16. https://doi.org/10.1016/0020-7225(72)90070-5
  45. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54(9), 4703-4710. https://doi.org/10.1063/1.332803
  46. Iijima, S. (1991), "Helical microtubules of graphitic carbon", Nature, 354(6348), 56-58. https://doi.org/10.1038/354056a0
  47. Ke, L.L. and Y.S. Wang (2011), "Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory", Compos. Struct., 93(2), 342-350. https://doi.org/10.1016/j.compstruct.2010.09.008
  48. Lee, Z., Ophus, C., Fischer, L.M., Nelson-Fitzpatrick, N., Westra, K.L., Evoy, S. and Mitlin, D. (2006), "Metallic NEMS components fabricated from nanocomposite Al-Mo films", Nanotechnology, 17(12), 3063-3070. https://doi.org/10.1088/0957-4484/17/12/042
  49. Maranganti, R. and Sharma, P. (2007), "Length scales at which classical elasticity breaks down for various materials", Phys. Rev. Lett., 98(19), 195504. https://doi.org/10.1103/PhysRevLett.98.195504
  50. Mindlin, R.D. (1964), "Micro-structure in linear elasticity", Arch. Ration. Mech. Anal., 16(1), 51-78. https://doi.org/10.1007/BF00248490
  51. Reddy, J.N. (2007), "Nonlocal theories for bending, buckling and vibration of beams", Int. J. Eng. Sci., 45(2), 288-307. https://doi.org/10.1016/j.ijengsci.2007.04.004
  52. Simsek, M. and H.H. Yurtcu (2013), "Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory", Compos. Struct., 97, 378-386. https://doi.org/10.1016/j.compstruct.2012.10.038
  53. Tauchert, T.R. (1974), Energy principles in structural mechanics, New York : McGraw-Hill.
  54. Thai, H.T. (2012), "A nonlocal beam theory for bending, buckling, and vibration of nanobeams", Int. J. Eng. Sci., 52, 56-64. https://doi.org/10.1016/j.ijengsci.2011.11.011
  55. Wang, X. and Cai, H. (2006), "Effects of initial stress on noncoaxial resonance of multi-wall carbon nanotubes", Acta Materialia, 54(8), 2067-2074. https://doi.org/10.1016/j.actamat.2005.12.039
  56. 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
  57. 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
  58. Witvrouw, A. and Mehta, A. (2005), "The use of functionally graded poly-SiGe layers for MEMS applications", Funct. Graded Mater. VIII, 492-493, 255-260.
  59. Zhang, Y.Q., Liu, G.R. and Wang, J.S. (2004), "Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression", Phys. Rev. B, 70(20), 205430. https://doi.org/10.1103/PhysRevB.70.205430
  60. Zhu, H., Wang, J. and Karihaloo, B. (2009), "Effects of surface and initial stresses on the bending stiffness of trilayer plates and nanofilms", J. Mech. Mater. Struct., 4(3), 589-604. https://doi.org/10.2140/jomms.2009.4.589
  61. Zhou, J.K. (1986), "Differential transformation and its applications for electrical circuits", Huazhong University Press, Wuhan, China.

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