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Nonlocal free vibration analysis of porous FG nanobeams using hyperbolic shear deformation beam theory

  • Hadji, Lazreg (Laboratory of Geomatics and Sustainable Development, University of Tiaret) ;
  • Avcar, Mehmet (Department of Civil Engineering, Suleyman Demirel University)
  • Received : 2020.09.05
  • Accepted : 2020.11.12
  • Published : 2021.03.25
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Abstract

This paper presents a new nonlocal Hyperbolic Shear Deformation Beam Theory (HSDBT) for the free vibration of porous Functionally Graded (FG) nanobeams. A new displacement field containing integrals is proposed which involves only three variables. The present model incorporates the length scale parameter (nonlocal parameter) which can capture the small scale effect and its account for shear deformation by a hyperbolic variation of all displacements through the thickness without using the shear correction factor. It has been observed that during the manufacture of Functionally Graded Materials (FGMs), micro-voids and porosities can occur inside the material. Thus, in this work, the investigation of the free vibration analysis of FG beams taking into account the influence of these imperfections is established. Four different porosity types are considered for FG nanobeam. Material characteristics of the FG beam are supposed to vary continuously within thickness direction according to a power-law scheme which is modified to approximate material characteristics for considering the influence of porosities. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion of the nanobeam are derived using Hamilton's principle. The effects of nonlocal parameter, aspect ratio, and the porosity types on the dynamic responses of the nanobeam are discussed.

Keywords

References

  1. Addou, F.Y., Meradjah, M., Bousahla, A.A., Benachour, A., Bourada, F., Tounsi, A. and Mahmoud, S.R. (2019), "Influences of porosity on dynamic response of FG plates resting on Winkler/Pasternak/Kerr foundation using quasi 3D HSDT", Comput. Concrete, Int. J., 24(4), 347-367. https://doi.org/10.12989/cac.2019.24.4.347
  2. Ait Atmane, H., Tounsi, A., Bernard, F. and Mahmoud, S.R. (2015), "A computational shear displacement model for Vibrational analysis of functionally graded beams with porosities", Steel Compos. Struct., Int. J., 19(2), 369-384. https://doi.org/10.12989/scs.2015.19.2.369.
  3. Akavci, S.S. (2010), "Two new hyperbolic shear displacement models for orthotropic laminated composite plates", Mech. Compos. Mater., 46(2), 215-2026. https://doi.org/10.1007/s11029-010-9140-3.
  4. Akbas, S.D. (2017a), "Thermal effects on the vibration of functionally graded deep beams with porosity", Int. J. Appl. Mech., 9(5), 1750076. https://doi.org/10.1142/S1758825117500764.
  5. Akbas, S.D. (2017b), "Free vibration of edge cracked functionally graded microscale beams based on the modified couple stress theory", Int. J. Struct. Stab. Dyn., 17(3), 1750033. https://doi.org/10.1142/S021945541750033X.
  6. Akgoz, B. and Civalek, O. (2012), "Free vibration analysis for single-layered graphene sheets in an elastic matrix via modified couple stress theory", Mater. Des., 42, 164-171. https://doi.org/10.1016/j.matdes.2012.06.002.
  7. Akgoz, B. and Civalek, O. (2013), "A size-dependent shear deformation beam model based on the strain gradient elasticity theory", Int. J. Eng. Sci., 70, 1-14. https://doi.org/10.1016/j.ijengsci.2013.04.004.
  8. Akgoz, B. and Civalek, O. (2017), "Effects of thermal and shear deformation on vibration response of functionally graded thick composite microbeams",Compos. B Eng., 129, 77-87. https://doi.org/10.1016/j.compositesb.2017.07.024.
  9. Al Rjoub, Y.S. and Hamad, A.G. (2017), "Free vibration of functionally Euler-Bernoulli and Timoshenko graded porous beams using the transfer matrix method", KSCE J. Civ. Eng., 21(3),792-806. https://doi.org/10.1007/s12205-016-0149-6.
  10. Alsaid-Alwan, H.H.S. and Avcar, M. (2020), "Analytical solution of free vibration of FG beam utilizing different types of beam theories: A comparative study", Comput. Concrete, Int. J., 26(3), 285-292. https://doi.org/10.12989/cac.2020.26.3.285.
  11. Ansari, R., Pourashraf, T. and Gholami, R. (2015), "An exact solution for the nonlinear forced vibration of functionally graded nanobeams in thermal environment based on surface elasticity theory", Thin-Wall. Struct., 93, 169-176. https://doi.org/10.1016/j.tws.2015.03.013.
  12. Apuzzo, A., Barretta, R., Faghidian, S.A., Luciano, R. and De Sciarra, F.M. (2018), "Free vibrations of elastic beams by modified nonlocal strain gradient theory", Int. J. Eng. Sci., 133, 99-108. https://doi.org/10.1016/j.ijengsci.2018.09.002.
  13. Arani, A.G., Abdollahian, M. and Kolahchi, R. (2015), "Nonlinear vibration of a nanobeam elastically bonded with a piezoelectric nanobeam via strain gradient theory", Int. J. Mech. Sci., 100, 32-40. https://doi.org/10.1016/j.ijmecsci.2015.06.002
  14. Arani, A.G., Pourjamshidian, M., Arefi, M. and Arani, M.R. (2019), "Application of nonlocal elasticity theory on the wave propagation of flexoelectric functionally graded (FG) timoshenko nano-beams considering surface effects and residual surface stress", Smart Struct. Syst., Int. J., 23(2), 141-153. http://dx.doi.org/10.12989/sss.2019.23.2.141.
  15. Arefi, M., Firouzeh, S., Bidgoli, E.M.R. and Civalek, O. (2020), "Analysis of porous micro-plates reinforced with FG-GNPS based on Reddy plate theory", Compos. Struct., 2020, 112391. https://doi.org/10.1016/j.compstruct.2020.112391.
  16. Aria, A.I., Rabczuk, T. and Friswell, M.I. (2019), "A finite element model for the thermo-elastic analysis of functionally graded porous nanobeams", Eur. J. Mech. A Solid, 77, 103767. https://doi.org/10.1016/j.euromechsol.2019.04.002.
  17. Asghar, S., Naeem, M.N., Hussain, M., Taj, M. and Tounsi, A. (2020), "Prediction and assessment of nonlocal natural frequencies of DWCNTs: Vibration analysis", Comput. Concrete, Int. J., 25(2), 133-144. https://doi.org/10.12989/cac.2020.25.2.133.
  18. Avcar, M. (2019), "Free vibration of imperfect sigmoid and power law functionally graded beams", Steel Compos. Struct., Int. J., 30(6), 603-615. https://doi.org/10.12989/scs.2019.30.6.603.
  19. Avcar, M. and Mohammed, W.K.M. (2018), "Free vibration of functionally graded beams resting on Winkler-Pasternak foundation", Arab. J. Geosci., 11(10), 232. https://doi.org/10.1007/s12517-018-3579-2.
  20. Aydogdu, M. (2009), "A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration", Physica E Low Dimens. Syst. Nanostruct., 41(9), 1651-1655. https://doi.org/10.1016/j.physe.2009.05.014.
  21. Barati, M.R. (2017), "Investigating dynamic response of porous inhomogeneous nanobeams on hybrid Kerr foundation under hygro-thermal loading", Appl. Phys. A, 123(5), 332. https://doi.org/10.1007/s00339-017-0908-3.
  22. Barretta, R., Feo, L., Luciano, R., De Sciarra, F.M. and Penna, R. (2016), "Functionally graded Timoshenko nanobeams: a novel nonlocal gradient formulation", Compos. Part B Eng., 100, 208-219. https://doi.org/10.1016/j.compositesb.2016.05.052.
  23. Bekki, H., Benferhat, R. and Hassaine Daouadji, T., (2019), "Influence of the distribution shape of porosity on the bending FGM new plate model resting on elastic foundations", Struct. Eng. Mech., Int. J., 72(1), 61-70. https://doi.org/10.12989/sem.2019.72.1.061
  24. Belmahi, S., Zidour, M. and Meradjah, M. (2019), "Small-scale effect on the forced vibration of a nano beam embedded an elastic medium using nonlocal elasticity theory", Adv. Aircr. Spacecr. Sci., Int. J., 6(1), 1-18. http://dx.doi.org/10.12989/aas.2019.6.1.001.
  25. Bendenia, N., Zidour, M., Bousahla, A.A., Bourada, F., Tounsi, A., Benrahou, K.H., Bedia, E.A., Mahmoud, S.R. and Tounsi, A. (2020), "Deflections, stresses and free vibration studies of FG-CNT reinforced sandwich plates resting on Pasternak elastic foundation", Comput. Concrete, Int. J., 26(3), 213-226. http://dx.doi.org/10.12989/cac.2020.26.3.213.
  26. Bensaid, I. (2017), "A refined nonlocal hyperbolic shear deformation beam model for bending and dynamic analysis of nanoscale beams", Adv. Nano Res., Int. J., 5(2), 113-126. https://doi.org/10.12989/anr.2017.5.2.113.
  27. Bensaid, I., Bekhadda, A. and Kerboua, B. (2018), "Dynamic analysis of higher order shear-deformable nanobeams resting on elastic foundation based on nonlocal strain gradient theory", Adv. Nano Res., Int. J., 6(3), 279-298. https://doi.org/10.12989/anr.2018.6.3.279.
  28. Berghouti, H., Adda Bedia, E.A., Benkhedda, A. and Tounsi, A. (2019), "Vibration analysis of nonlocal porous nanobeams made of functionally graded material", Adv. Nano Res., Int. J., 7(5), 351-364. https://doi.org/10.12989/anr.2019.7.5.351.
  29. Berrabah, H.M., Tounsi, A., Semmah, A. and Adda Bedia, E.A. (2013), "Comparison of various refined nonlocal beam theories for bending, vibration and buckling analysis of nanobeams", Struct. Eng. Mech., Int. J., 48(3), 351-365. https://doi.org/10.12989/sem.2013.48.3.351.
  30. Birman, V. and Byrd, L.W. (2007), "Modeling and analysis of functionally graded materials and structures", Appl. Mech. Rev., 60(5), 195-216. https://doi.org/10.1115/1.2777164.
  31. Bourada, F., Amara, K., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2018), "A novel refined plate theory for stability analysis of hybrid and symmetric S-FGM plates", Struct. Eng. Mech., Int. J., 68(6), 661-675. https://doi.org/10.12989/sem.2018.68.6.661.
  32. Bourada, F., Bousahla, A.A., Bourada, M., Azzaz, A., Zinata, A. and Tounsi, A. (2019), "Dynamic investigation of porous functionally graded beam using a sinusoidal shear deformation theory", Wind Struct., Int. J., 28(1), 19-30. https://doi.org/10.12989/was.2019.28.1.019.
  33. Boutaleb, S., Benrahou, K.H., Bakora, A., Algarni, A., Bousahla, A.A., Tounsi, A., Tounsi, A. and Mahmoud, S.R. (2019), "Dynamic analysis of nanosize FG rectangular plates based on simple nonlocal quasi 3D HSDT", Adv. Nano Res., Int. J., 7(3), 191-208. http://dx.doi.org/10.12989/anr.2019.7.3.191.
  34. Chaht, F.L., Kaci, A., Houari, M.S.A., Tounsi, A., Beg, O.A. and Mahmoud, S.R. (2015), "Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect", Steel Compos. Struct., Int. J., 18(2), 425-442. http://dx.doi.org/10.12989/scs.2015.18.2.425.
  35. Chakraverty, S. and Pradhan, K.K. (2016), Vibration of Functionally Graded Beams and Plates, Academic Press. London, UK. https://doi.org/10.1016/j.amc.2016.05.034
  36. Chikr, S.C., Kaci, A., Bousahla, A.A., Bourada, F., Tounsi, A., Adda Bedia, E.A., Mahmoud, S.R., Benrahou K.H. and Tounsi, A. (2019), "A novel four-unknown integral model for buckling response of FG sandwich plates resting on elastic foundations under various boundary conditions using Galerkin's approach", Geomech. Eng., Int. J., 21(5), 471-487. https://doi.org/10.12989/gae.2020.21.5.471.
  37. Civalek, O. (2017), "Free vibration of carbon nanotubes reinforced (CNTR) and functionally graded shells and plates based on FSDT via discrete singular convolution method", Compos. B Eng., 111, 45-59. https://doi.org/10.1016/j.compositesb.2016.11.030.
  38. Civalek, O. and Avcar, M (2020), "Free vibration and buckling analyses of CNT reinforced laminated non-rectangular plates by discrete singular convolution method", Eng. Comput., 2020, 1-33. https://doi.org/10.1007/s00366-020-01168-8.
  39. Civalek, O., Ersoy, H., Numanoglu, H.M. and Akgoz, B. (2018), "Small size and rotary inertia effects on the natural frequencies of carbon nanotubes", Curved Layer. Struct., 5(1), 273-279. https://doi.org/10.1515/cls-2018-0020.
  40. Civalek, O., Uzun, B. and Yayli, M.O. (2020), "Frequency, bending and buckling loads of nanobeams with different cross sections", Adv. Nano Res., Int. J., 9(2), 91-104. https://doi.org/10.12989/anr.2020.9.2.091.
  41. Danesh, M., Farajpour, A. and Mohammadi, M. (2012), "Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method", Mech. Res. Commun., 39(1), 23-27. https://doi.org/10.1016/j.mechrescom.2011.09.004.
  42. Demir, C. and Civalek, O. (2013), "Torsional and longitudinal frequency and wave response of microtubules based on the nonlocal continuum and nonlocal discrete models", Appl. Math. Model., 37(22), 9355-9367. https://doi.org/10.1016/j.apm.2013.04.050.
  43. Demir, C. and Civalek, O. (2017), "On the analysis of microbeams", Int. J. Eng. Sci., 121, 14-33. https://doi.org/10.1016/j.ijengsci.2017.08.016.
  44. Demir, C., Mercan, K., Numanoglu, H.M. and Civalek, O. (2018), "Bending response of nanobeams resting on elastic foundation", J. Appl. Comput. Mech., 4(2), 105-114. https://doi.org/10.22055/JACM.2017.22594.1137.
  45. Ebrahimi-Nejad, S., Shaghaghi, G.R., Miraskari, F. and Kheybari, M. (2019), "Size-dependent vibration in two-directional functionally graded porous nanobeams under hygro-thermo-mechanical loading", Eur. Phys. J. Plus, 134(9), 465. https://doi.org/10.1140/epjp/i2019-12795-6.
  46. Ebrahimi, F. and Barati, M.R. (2016), "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
  47. Ebrahimi, F. and Barati, M.R. (2017). "Vibration analysis of embedded size dependent FG nanobeams based on third-order shear deformation beam theory", Struct. Eng. Mech., Int. J., 61(6), 721-736. http://dx.doi.org/10.12989/sem.2017.61.6.721.
  48. Ebrahimi, F. and Daman, M. (2017), "Nonlocal thermo-electromechanical vibration analysis of smart curved FG piezoelectric Timoshenko nanobeam", Smart Struct. Syst., Int. J., 20(3), 351-368. http://dx.doi.org/10.12989/sss.2017.20.3.351.
  49. Ebrahimi, F., Jafari, A. and Reza Barati, M. (2017), "Dynamic modeling of porous heterogeneous micro/nanobeams", Eur. Phys. J. Plus, 132(12), 521. https://doi.org/10.1140/epjp/i2017-11754-7.
  50. Ebrahimi, F., Karimiasl, M., Civalek, O. and Vinyas, M. (2019), "Surface effects on scale-dependent vibration behavior of flexoelectric sandwich nanobeams", Adv. Nano Res., Int. J., 7(2), 77-88. https://doi.org/10.12989/anr.2019.7.2.077.
  51. Ehyaei, J., Akbarshahi, A. and Shafiei, N. (2017), "Influence of porosity and axial preload on vibration behavior of rotating FG nanobeam", Adv. Nano Res., Int. J., 5(2), 141-169. http://dx.doi.org/10.12989/anr.2017.5.2.141.
  52. Ehyaei, J., Ebrahimi, F. and Salari, E. (2016), "Nonlocal vibration analysis of FG nano beams with different boundary conditions", Adv. Nano Res., Int. J., 4(2), 85-111. http://dx.doi.org/10.12989/anr.2016.4.2.085.
  53. Elishakoff, I. and Pentaras, D. (2009), "Fundamental natural frequencies of double-walled carbon nanotubes", J. Sound Vib., 322(4-5), 652-664. https://doi.org/10.1016/j.jsv.2009.02.037.
  54. 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.
  55. Eltaher, M.A., Fouda, N., El-midany, T. and Sadoun, A.M. (2018), "Modified porosity model in analysis of functionally graded porous nanobeams", J. Braz. Soc. Mech. Sci. Eng., 40(3), 141. https://doi.org/10.1007/s40430-018-1065-0.
  56. Eltaher, M.A., Omar, F.A., Abdalla, W.S. and Gad, E.H. (2019), "Bending and vibrational behaviors of piezoelectric nonlocal nanobeam including surface elasticity", Wave Random Complex Media, 29(2), 264-280. https://doi.org/10.1080/17455030.2018.1429693.
  57. Eringen, A.C. (1972), "Linear theory of nonlocal elasticity and dispersion of plane waves", Int. J. Eng. Sci., 10(5), 425-435. https://doi.org/10.1016/0020-7225(72)90050-X.
  58. 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.
  59. Gafour, Y., Hamidi, A., Benahmed, A., Zidour, M. and Bensattalah, T. (2020), "Porosity-dependent free vibration analysis of FG nanobeam using non-local shear deformation and energy principle", Adv. Nano Res., Int. J., 8(1), 37-47. https://doi.org/10.12989/anr.2020.8.1.037.
  60. Gheshlaghi, B. and Hasheminejad, S.M. (2011), "Surface effects on nonlinear free vibration of nanobeams", Compos. Part B. Eng., 42(4), 934-937. https://doi.org/10.1016/j.compositesb.2010.12.026.
  61. Hadji, L. and Adda Bedia, E.A. (2015), "Influence of the porosities on the free vibration of FGM beams", Wind Struct., 21(3), 273-287. https://doi.org/10.12989/was.2015.21.3.273.
  62. Hadi, A., Nejad, M.Z. and Hosseini, M. (2018), "Vibrations of three-dimensionally graded nanobeams", Int. J. Eng. Sci., 128, 12-23. https://doi.org/10.1016/j.ijengsci.2018.03.004.
  63. Iijima, S. (1991), "Helical microtubules of graphitic carbon", Nature, 354(6348), 56-58. https://doi.org/10.1038/354056a0.
  64. Jalaei, M.H. and Civalek, O. (2019), "On dynamic instability of magnetically embedded viscoelastic porous FG nanobeam", Int. J. Eng. Sci., 143, 14-32. https://doi.org/10.1016/j.ijengsci.2019.06.013.
  65. Jouneghani, F.Z., Dimitri, R. and Tornabene, F. (2018), "Structural response of porous FG nanobeams under hygro-thermo-mechanical loadings", Compos. Part B Eng., 152, 71-78. https://doi.org/10.1016/j.compositesb.2018.06.023.
  66. Kaddari, M., Kaci, A., Bousahla, A.A., Tounsi, A., Bourada, F., Tounsi, A., Bedia, E.A. and Al-Osta, M.A., (2020), "A study on the structural behaviour of functionally graded porous plates on elastic foundation using a new quasi-3D model: bending and free vibration analysis", Comput. Concrete, Int. J., 25(1), 37-57. https://doi.org/10.12989/cac.2020.25.1.037.
  67. Karami, B., Janghorban, M. and Tounsi, A. (2019), "On pre-stressed functionally graded anisotropic nanoshell in magnetic field", J. Braz. Soc. Mech. Sci. Eng., 41, 495. https://doi.org/10.1007/s40430-019-1996-0.
  68. Karlicic, D., Murmu, T., Adhikari, S. and McCarthy, M. (2015), Non-Local Structural Mechanics, Wiley, New York, USA.
  69. Khadimallah, M.A., Hussain, M., Khedher, K.M., Naeem, M.N. and Tounsi, A. (2020), "Backward and forward rotating of FG ring support cylindrical shells", Steel Compos. Struct., Int. J., 37(2), 137-150. http://dx.doi.org/10.12989/scs.2020.37.2.137.
  70. Kheroubi, B., Benzair, A., Tounsi, A. and Semmah, A. (2016), "A new refined nonlocal beam theory accounting for effect of thickness stretching in nanoscale beams", Adv. Nano Res., Int. J., 4(4), 251-264. https://doi.org/10.12989/anr.2016.4.4.251.
  71. Khorshidi, M.A. and Shariati, M. (2016), "Free vibration analysis of sigmoid functionally graded nanobeams based on a modified couple stress theory with general shear deformation theory", J. Braz. Soc. Mech. Sci. Eng., 38(8), 2607-2619. https://doi.org/10.1007/s40430-015-0388-3.
  72. Khosravi, F., Simyari, M., Hosseini, S.A. and Tounsi, A. (2020), "Size dependent axial free and forced vibration of carbon nanotube via different rod models", Adv. Nano Res., Int. J., 9(3), 157-172. http://dx.doi.org/10.12989/anr.2020.9.3.157.
  73. Koizumi, M. (1993), "The concept of FGM", Ceram. Trans. Funct. Graded Mater., 34, 3-10. https://doi.org/10.1080/10426919508935030.
  74. Koizumi, M. (1997), "FGM activities in Japan", Compos. Part B Eng., 28(1-2), 1-4. https://doi.org/10.1016/S1359-8368(96)00016-9.
  75. Liu, H., Lv, Z. and Wu, H. (2019), "Nonlinear free vibration of geometrically imperfect functionally graded sandwich nanobeams based on nonlocal strain gradient theory", Compos. Struct., 214, 47-61. https://doi.org/10.1016/j.compstruct.2019.01.090.
  76. Mahi, A., Adda Bedia, E.A. and Tounsi, A. (2015), "A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates", Appl. Math. Model., 39(9), 2489-2508. https://doi.org/10.1016/j.apm.2014.10.045.
  77. Matouk, H., Bousahla, A.A., Heireche, H., Bourada, F., Bedia, E.A., Tounsi, A., Mahmood S.R., Tounsi, A. and Benrahou, K.H. (2020), "Investigation on hygro-thermal vibration of P-FG and symmetric S-FG nanobeam using integral Timoshenko beam theory", Adv. Nano Res., Int. J., 8(4), 293-305. http://dx.doi.org/10.12989/anr.2020.8.4.293.
  78. Menasria, A., Kaci, A., Bousahla, A.A., Bourada, F., Tounsi, A., Benrahou, K.H., Tounsi, A., Adda Bedia, E.A. and Mahmoud, S.R. (2020), "A four-unknown refined plate theory for dynamic analysis of FG-sandwich plates under various boundary conditions", Steel Compos. Struct., Int. J., 36(3), 355-367. https://doi.org/10.12989/scs.2020.36.3.355.
  79. Mirjavadi, S.S., Afshari, B.M., Shafiei, N., Hamouda, A.M.S. and Kazemi, M. (2017), "Thermal vibration of two-dimensional functionally graded (2D-FG) porous Timoshenko nanobeams", Steel Compos. Struct., Int. J., 25(4), 415-426. http://dx.doi.org/10.12989/scs.2017.25.4.415.
  80. Murmu, T. and Adhikari, S. (2010), "Nonlocal transverse vibration of double-nanobeam-systems", J. Appl. Phys., 108(8), 083514. https://doi.org/10.1063/1.3496627.
  81. Noori, A.R., Aslan, T.A. and Temel, B. (2018), "An efficient approach for in-plane free and forced vibrations of axially functionally graded parabolic arches with nonuniform cross section", Compos. Struct., 200, 701-710. https://doi.org/10.1016/j.compstruct.2018.05.077.
  82. Noori, A.R., Aslan, T.A. and Temel, B. (2021), "Dynamic analysis of functionally graded porous beams using complementary functions method in the Laplace domain", Compos. Struct., 256, 113094. https://doi.org/10.1016/j.compstruct.2020.113094.
  83. Numanoglu, H.M., Akgoz, B. and Civalek, O. (2018), "On dynamic analysis of nanorods", Int. J. Eng. Sci., 130, 33-50. https://doi.org/10.1016/j.ijengsci.2018.05.001.
  84. Rahmani, O., Hosseini, S.A.H., Ghoytasi, I. and Golmohammadi, H. (2018), "Free vibration of deep curved FG nano-beam based on modified couple stress theory", Steel Compos. Struct., Int. J., 26(5), 607-620. http://dx.doi.org/10.12989/scs.2018.26.5.607.
  85. Rahmani, A., Faroughi, S. and Friswell, M.I. (2020), "The vibration of two-dimensional imperfect functionally graded (2D-FG) porous rotating nanobeams based on general nonlocal theory", Mech. Syst. Signal Process, 144, 106854. https://doi.org/10.1016/j.ymssp.2020.106854.
  86. Reddy, J.N. (2002), Energy Principles and Variational Methods in Applied Mechanics, Wiley, New York, USA.
  87. Reddy, J.N. (2007), "Nonlocal theories for bending, buckling and vibration of beams", Int. J. Eng. Sci., 45(2-8), 288-307. https://doi.org/10.1016/j.ijengsci.2007.04.004.
  88. Semmah, A., Tounsi, A., Zidour, M., Heireche, H. and Naceri, M. (2014), "Effect of chirality on critical buckling temperature of a zigzag single-walled carbon nanotubes using nonlocal continuum theory", Full. Nanotub. Carbon Nanostr., 23, 518-522. https://doi.org/10.1080/1536383X.2012.749457.
  89. Shafiei, N., Mirjavadi, S.S., MohaselAfshari, B., Rabby, S. and Kazemi, M. (2017), "Vibration of two-dimensional imperfect functionally graded (2D-FG) porous nano-/microbeams", Comput. Method Appl. M., 322, 615-632. https://doi.org/10.1016/j.cma.2017.05.007.
  90. Shanab, R.A., Mohamed, S.A., Mohamed, N.A. and Attia, M.A. (2020), "Comprehensive investigation of vibration of sigmoid and power law FG nanobeams based on surface elasticity and modified couple stress theories", Acta Mech., 231(5), 1977-2010. https://doi.org/10.1007/s00707-020-02623-9.
  91. She, G.L., Liu, H.B. and Karami, B. (2020), "On resonance behavior of porous FG curved nanobeams", Steel Compos. Struct., Int. J., 36(2), 179-186. http://dx.doi.org/10.12989/scs.2020.36.2.179.
  92. Simsek, M. (2019), "Some closed-form solutions for static, buckling, free and forced vibration of functionally graded (FG) nanobeams using nonlocal strain gradient theory", Compos. Struct., 224, 111041. https://doi.org/10.1016/j.compstruct.2019.111041.
  93. Tagrara, S.H., Benachour, A., Bachir Bouiadjra, M. and Tounsi, A. (2015), "On bending, buckling and vibration responses of functionally graded carbon nanotube-reinforced composite beams", Steel Compos. Struct., Int. J., 19(5), 1259-1277. https://doi.org/10.12989/scs.2015.19.5.1259.
  94. Temel, B. and Noori, A.R. (2019), "Out-of-plane vibrations of shear-deformable AFG cycloidal beams with variable cross section", Appl. Acoust., 155, 84-96. https://doi.org/10.1016/j.apacoust.2019.05.010.
  95. Temel, B. and Noori, A.R. (2020), "A unified solution for the vibration analysis of two-directional functionally graded axisymmetric Mindlin-Reissner plates with variable thickness", Int. J. Mech. Sci., 174, 105471. https://doi.org/10.1016/j.ijmecsci.2020.105471.
  96. 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.
  97. Thai, H.T., Vo, T.P., Nguyen, T.K. and Kim, S.E. (2017), "A review of continuum mechanics models for size-dependent analysis of beams and plates", Compos. Struct., 177, 196-219. https://doi.org/10.1016/j.compstruct.2017.06.040.
  98. Togun, N. and Bagdatli, S.M. (2016), "Size dependent nonlinear vibration of the tensioned nanobeam based on the modified couple stress theory", Compos. Part B Eng., 97, 255-262. https://doi.org/10.1016/j.compositesb.2016.04.074.
  99. Tounsi, A., Semmah, A. and Bousahla, A.A. (2013a), "Thermal buckling behavior of nanobeams using an efficient higher-order nonlocal beam theory", ASCE J. Nanomech. Micromech., 3(3), 37-42. https://doi.org/10.1061/(ASCE)NM.2153-5477.0000057.
  100. Tounsi, A., Benguediab, S., Houari, M.S.A. and Semmah, A. (2013b), "A new nonlocal beam theory with thickness stretching effect for nanobeams", Int. J. Nanosci., 12(4), 1350025. https://doi.org/10.1142/S0219581X13500257.
  101. Uzun, B. and Civalek, O. (2019), "Nonlocal FEM formulation for vibration analysis of nanowires on elastic matrix with different materials", Math. Comput. Appl., 24(2), 38. https://doi.org/10.3390/mca24020038.
  102. Wang, Q. (2005), "Wave propagation in carbon nanotubes via nonlocal continuum mechanics", J. Appl. Phys., 98, 124301. https://doi.org/10.1063/1.2141648.
  103. Wang, G.F. and Feng, X.Q. (2007), "Effects of surface elasticity and residual surface tension on the natural frequency of microbeams", Appl. Phys. Lett., 90(23), 231904. https://doi.org/10.1063/1.2746950.
  104. Wattanasakulpong, N. and Ungbhakorn, V. (2014), "Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities", Aerosp. Sci. Technol., 32(1), 111-120. https://doi.org/10.1016/j.ast.2013.12.002.
  105. Wattanasakulpong, N., Gangadhara Prusty, B., Kelly, D.W. and Hoffman, M. (2012), "Free vibration analysis of layered functionally graded beams with experimental validation", Mater. Des., 36, 182-190. https://doi.org/10.1016/j.matdes.2011.10.049.
  106. Xu, X.J., Zheng, M.L. and Wang, X.C. (2017), "On vibrations of nonlocal rods: Boundary conditions, exact solutions and their asymptotics", Int. J. Eng. Sci., 119, 217-231. https://doi.org/10.1016/j.ijengsci.2017.06.025.
  107. Yan, Z. and Jiang, L.Y. (2011), "The vibrational and buckling behaviors of piezoelectric nanobeams with surface effects", Nanotechnology, 22(24), 245703. https://doi.org/10.1088/0957-4484/22/24/245703.
  108. Zaoui, F.Z., Ouinas, D. and Tounsi, A. (2019), "New 2D and quasi-3D shear deformation theories for free vibration of functionally graded plates on elastic foundations", Compos. B Eng., 159, 231-247. https://doi.org/10.1016/j.compositesb.2018.09.051.
  109. Zhu, J., Lai, Z., Yin, Z., Jeon, J. and Lee, S. (2001), "Fabrication of ZrO2-NiCr functionally graded material by powder metallurgy", Mater. Chem. Phys., 68(1-3), 130-135. https://doi.org/10.1016/S0254-0584(00)00355-2.
  110. Zine, A., Bousahla, A.A., Bourada, F., Benrahou, K.H., Tounsi, A., Adda Bedia, E.A., Mahmoud, S.R. and Tounsi, A. (2020), "Bending analysis of functionally graded porous plates via a refined shear deformation theory", Comput. Concrete, Int. J., 26(1), 63-74. http://dx.doi.org/10.12989/cac.2020.26.1.063.

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