DOI QR코드

DOI QR Code

Natural frequency analysis of joined conical-cylindrical-conical shells made of graphene platelet reinforced composite resting on Winkler elastic foundation

  • Xiangling Wang (Department of Mining Engineering, Lyuliang University) ;
  • Xiaofeng Guo (College of information Science and Engineering, Shanxi Agricultural University) ;
  • Masoud Babaei (Department of Mechanical Engineering, University of Eyvanekey) ;
  • Rasoul Fili (Department of Engineering, Imam Ali University) ;
  • Hossein Farahani (Department of Civil engineering, Islamic Azad University)
  • 투고 : 2022.02.08
  • 심사 : 2023.05.02
  • 발행 : 2023.10.25

초록

Natural frequency behavior of graphene platelets reinforced composite (GPL-RC) joined truncated conical-cylindrical- conical shells resting on Winkler-type elastic foundation is presented in this paper for the first time. The rule of mixture and the modified Halpin-Tsai approach are applied to achieve the mechanical properties of the structure. Four different graphene platelets patterns are considered along the thickness of the structure such as GPLA, GPLO, GPLX, GPLUD. Finite element procedure according to Rayleigh-Ritz formulation has been used to solve 2D-axisymmetric elasticity equations. Application of 2D axisymmetric elasticity theory allows thickness stretching unlike simple shell theories, and this gives more accurate results, especially for thick shells. An efficient parametric investigation is also presented to show the effects of various geometric variables, three different boundary conditions, stiffness of elastic foundation, dispersion pattern and weight fraction of GPLs nanofillers on the natural frequencies of the joined shell. Results show that GPLO and BC3 provide the most rigidity that cause the most natural frequencies among different BCs and GPL patterns. Also, by increasing the weigh fraction of nanofillers, the natural frequencies will increase up to 200%.

키워드

과제정보

This work was supported by the National Science Fund for Distinguished Young Scholars (Grant number: 61525107).

참고문헌

  1. Abdelaziz, H.H., Meziane, M.A.A, Bousahla, A.A., Tounsi, A., Mahmoud, S.R. and Alwabli, A.S. (2017), "An efficient hyperbolic shear deformation theory for bending, buckling and free vibration of FGM sandwich plates with various boundary conditions", Steel Compos. Struct., 25(6), 693-704. http://doi.org/10.12989/scs.2017.25.6.693.
  2. Al-Furjan, M.S.H., Habibi, M., Ghabussi, A., Safarpour, H., Safarpour, M. and Tounsi, A. (2021), "Non-polynomial framework for stress and strain response of the FG-GPLRC disk using three-dimensional refined higher-order theory", Eng. Struct., 228, 111496. https://doi.org/10.1016/j.engstruct.2020.111496
  3. Al-Furjan, M. S. H., Habibi, M., Shan, L. and Tounsi, A. (2021), "On the vibrations of the imperfect sandwich higher-order disk with a lactic core using generalize differential quadrature method", Compos. Struct., 257, 113150. https://doi.org/10.1016/j.compstruct.2020.113150
  4. Al-Furjan, M.S.H., Habibi, M., Jung, D.W., Sadeghi, S., Safarpour, H., Tounsi, A. and Chen, G. (2022), "A computational framework for propagated waves in a sandwich doubly curved nanocomposite panel", Eng. Struct., 38(2), 1679-1696. https://doi.org/10.1007/s00366-020-01130-8
  5. Al-Furjan, M.S.H., Habibi, M., Ni, J., Jung, D.W. and Tounsi, A. (2022), "Frequency simulation of viscoelastic multi-phase reinforced fully symmetric systems", Eng. Struct., 38(5), 3725-3741. https://doi.org/10.1007/s00366-020-01200-x
  6. Alimirzaei, S., Mohammadimehr, M. and Tounsi, A. (2019), "Nonlinear analysis of viscoelastic micro-composite beam with geometrical imperfection using FEM: MSGT electro-magneto-elastic bending, buckling and vibration solutions", Struct. Eng. Mech., 71(5), 485-502. https://doi.org/10.12989/sem.2019.71.5.485
  7. 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., 19(2), 369-384. https://doi.org/10.12989/scs.2015.19.2.369.
  8. Arefi, M., Bidgoli, E.M.R., Dimitri, R. and Tornabene, F. (2018), "Free vibrations of functionally graded polymer composite nanoplates reinforced with graphene nano platelets", Aerosp. Sci. Technol., 81, 108-117. https://doi.org/10.1016/j.ast.2018.07.036
  9. Arefi, M., Bidgoli, E.M.R., Dimitri, R., Bacciocchi, M. and Tornabene, F. (2019a), "Nonlocal bending analysis of curved nanobeams reinforced by graphene nanoplatelets", Compos Part B: Eng. 166, 1-12. https://doi.org/10.1016/j.compositesb.2018.11.092
  10. Arefi, M., Bidgoli, E.M.R. and Rabczuk, T. (2019b), "Effect of various characteristics of graphene nanoplatelets on thermal buckling behavior of FGRC micro plate based onMCST", Eur. J. Mech. A Solids, 77, 103802. https://doi.org/10.1016/j.euromechsol.2019.103802
  11. Arshid, E., Khorasani, M., Soleimani-Javid, Z., Amir, S. and Tounsi, A. (2021), "Porosity-dependent vibration analysis of FG microplates embedded by polymeric nanocomposite patches considering hygrothermal effect via an innovative plate theory", Eng. Struct., 38, 1-22. https://doi.org/10.1007/s00366-021-01382-y
  12. Babaei, M., and Asemi, K. (2020a), "Stress analysis of functionally graded saturated porous rotating thick truncated cone", Mech. Base Des. Struct.,1-28. https://doi.org/10.1080/15397734.2020.1753536
  13. Babaei, M. and Asemi, K. (2020b), "Static, dynamic and natural frequency analyses of functionally graded carbon nanotube annular sector plates resting on viscoelastic foundation", SN Appl. Sci., 2(10), 1-21. https://doi.org/10.1007/s42452-020-03421-7
  14. Babaei, M., Kiarasi, F., Hossaeini Marashi, S.M., Ebadati, M., Masoumi, F., and Asemi, K. (2021), "Stress wave propagation and natural frequency analysis of functionally graded graphene platelet-reinforced porous joined conical-cylindrical-conical shell", Waves Random Complex Med., 1-33. https://doi.org/10.1080/17455030.2021.2003478
  15. Bagheri, H., Kiani, Y., Bagheri, N. and Eslami, M.R. (2020), "Free vibration of joined cylindrical-hemispherical FGM shells", Arch.Appl. Mech., 90, 2185-2199. https://doi.org/10.1007/s00419-020-01715-1
  16. Bagheri, H., Kiani, Y. and Eslami, M.R. (2018), "Free vibration of joined conical-cylindrical-conical shells", Acta Mechanica, 229(7), 2751-2764. https://doi.org/10.1007/s00707-018-2133-3
  17. 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. Concr., 26(3), 213-226. https://doi.org/10.12989/cac.2020.26.3.213
  18. Bidzard, A., Malekzadeh, P. and Mohebpour, S (2021), "Influences of pressure and thermal environment on nonlinear vibration characteristics of multilayer FG-GPLRC toroidal panels on nonlinear elastic foundation", Compos Struct., 259, 113503. https://doi.org/10.1016/j.compstruct.2020.113503
  19. Bouafia, H., Chikh, A., Bousahla, A.A., Bourada, F., Heireche, H., Tounsi, A., Benrahou, K.H., Tounsi, A., Al-Zahrani, M.M. and Hussain., M. (2021), "Natural frequencies of FGM nanoplates embedded in an elastic medium", Adv. Nano Res., 11(3), 239-249. https://doi.org/10.12989/anr.2021.11.3.239
  20. Bourada, F., Bousahla, A.A., Tounsi, A., Bedia, E.A., Mahmoud, S.R., Benrahou, K.H. and Tounsi, A. (2020), "Stability and dynamic analyses of SW-CNT reinforced concrete beam resting on elastic-foundation", Comput. Concr., 25(6), 485-495. https://doi.org/10.12989/cac.2020.25.6.485
  21. Chaubey, A.K., A., Kumar and A., Chakrabarti (2018), "Novel shear deformation model for moderately thick and deep laminated composite conoidal shell", Mech. Base Des. Struct., 46(5), 50668. https://doi.org/10.1080/15397734.2017.1422433
  22. Chen, C.S., Liu, F.H. and Chen, W.R. (2017), "Vibration and stability of initially stressed sandwich plates with FGM face sheets in thermal environments", Steel Compos. Struct., 23(3), 251-261. https://doi.org/10.12989/scs.2017.23.3.251
  23. Civalek, O., Dastjerdi, S., Akbas, S.D. and Akgoz, B. (2020a), "Vibration analysis of carbon nanotube-reinforced composite microbeams", Math. Meth. Appl. Sci., Special Issue Paper. https://doi.org/10.1002/mma.7069.
  24. Civalek, O., Uzun, B., Yayli, M.O. and Akgoz, B. (2020b), "Size-dependent transverse and longitudinal vibrations of embedded carbon and silica carbide nanotubes by nonlocal finite element method", Eur. Phys. J. Plus, 135, 381. https://doi.org/10.1140/epjp/s13360-020-00385-w.
  25. Cuong-Le, T., Nguyen, K.D., Le-Minh, H., Phan-Vu, P., Nguyen-Trong, P. and Tounsi, A. (2022), "Nonlinear bending analysis of porous sigmoid FGM nanoplate via IGA and nonlocal strain gradient theory", Adv. Nano Res., 12(5), 441-455. https://doi.org/10.12989/anr.2022.12.5.441
  26. Dastjerdi, S. and Beni, Y.T. (2019), "Anovel approach for nonlinear bending response of macro and nanoplates with irregular variable thickness under nonuniform loading in thermal environment", Mech. Base Des. Struct., 47(4), 453-478. https://doi.org/10.1080/15397734.2018.1557529
  27. Djilali, N., A.A., Bousahla, A., Kaci, M.M., Selim, F., Bourada, A., Tounsi, A., Tounsi, K.H. Benrahou and S.R., Mahmoud. (2022), "Large cylindrical deflection analysis of FG carbon nanotube-reinforced plates in thermal environment using a simple integral HSDT", Steel Compos. Struct., 42(6), 779-789. https://doi.org/10.12989/scs.2022.42.6.779
  28. Dong, Y.H., B., Zhu, Y., Wang, Y.H., Li, and J., Yang (2018), "Nonlinear free vibration of graded grapheme reinforced cylindrical shells: Effects of spinning motion and axial load", J Sound Vib. 437, 79-96. https://doi.org/10.1016/j.jsv.2018.08.036
  29. Ebrahimi, F. and Jafari, A. (2016), "Thermo-mechanical vibration analysis of temperature-dependent porous FG beams based on Timoshenko beam theory", Struct. Eng. Mech., 59(2), 343-371. https://doi.org/10.12989/sem.2016.59.2.343
  30. 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, 61(6), 721-736. https://doi.org/10.12989/sem.2017.61.6.721.
  31. Ebrahimi, F., Karimiasl, M. and Selvamani, R. (2020), "Bending analysis of magneto-electro piezoelectric nanobeams system under hygro-thermal loading", Adv. Nano Res., 8(3), 203-214. https://doi.org/10.12989/anr.2020.8.3.203.
  32. Ebrahimi, F., Nouraei, M. and Dabbagh, A. (2020), "Thermal vibration analysis of embedded graphene oxide powder-reinforced nanocomposite plates", Eng Comput., 36(3), 879-895. https://doi.org/10.1016/j.jsv.2018.08.036
  33. Ebrahimi, F. and Seyfi, A. (2020), "Studying propagation of wave in metal foam cylindrical shells with graded porosities resting on variable elastic substrate", Eng Comput., 36, 1-17. https://doi.org/10.1007/s00366-020-01069-w
  34. Faghidian, A. and Tounsi, A. (2022), "Dynamic characteristics of mixture unified gradient elastic nanobeams", Facta Univ. Series Mech. Eng., 20(3), 539-552. https://doi.org/10.22190/FUME220703035F
  35. Feng, C., Kitipornchai, S. and Yang, J. (2017), "Nonlinear bending of polymer nanocomposite beams reinforced with non-uniformly distributed graphene platelets (GPLs)", Compos. Part B Eng., 110, 132-140. https://doi.org/10.1016/j.compositesb.2016.11.024
  36. Garg, A., Belarbi, M.O., Tounsi, A., Li, L., Singh, A. and Mukhopadhyay, T. (2022a), "Predicting elemental stiffness matrix of FG nanoplates using Gaussian Process Regression based surrogate model in framework of layerwise model", Eng. Anal. Bound. Elem., 143, 779-795. https://doi.org/10.1016/j.enganabound.2022.08.001
  37. Garg, A., Aggarwal, P., Aggarwal, Y., Belarbi, M.O., Chalak, H. D., Tounsi, A. and Gulia, R. (2022b), "Machine learning models for predicting the compressive strength of concrete containing nano silica", Comput. Concr., 30(1), 33-42. https://doi.org/10.12989/cac.2022.30.1.033
  38. Ghahfarokhi, D.S., Safarpour, M. and Rahimi, A.R. (2019), "Torsional buckling analyses of functionally graded porous nanocomposite cylindrical shells reinforced with graphene platelets (GPLs)", Mech. Base Des. Struct., 81-102. https://doi.org/10.1080/15397734.2019.1666723
  39. Guo, H., Zhuang, X. and Rabczuk, T. (2021), "A deep collocation method for the bending analysisof Kirchhoff plate", Comput. Mater. Continua, 59(2), 433-456. https://doi.org/10.32604/cmc.2019.06660
  40. Gupta, A. and M., Talha (2018), "Influence of initial geometric imperfections and porosity on the stability of functionally graded material plates", Mech. Base Des Struc., 46(6), 693-711. https://doi.org/10.1080/15397734.2018.1449656
  41. Hachemi, M. and Hamza-Cherif, S.M. (2020), "Free vibration of composite laminated plate with complicated cutout", Mech. Base Des. Struct., 48(2), 192-216. https://doi.org/10.1080/15397734.2019.1633341
  42. He, Q., Dai, H.L., Gui, Q.F. and Li, J.J. (2020), "Analysis of vibration characteristics of joined cylindrical-spherical shells", Eng Struct., 218, 110767. https://doi.org/10.1016/j.engstruct.2020.110767
  43. Heidari, F., Taheri, K., Sheybani, M., Janghorban, M. and Tounsi, A. (2021), "On the mechanics of nanocomposites reinforced by wavy/defected/aggregated nanotubes", Steel Compos. Struct., 38(5), 533-545. https://doi.org/10.12989/scs.2021.38.5.533
  44. Hosseini, S.M. and Zhang, C. (2018), "Elastodynamic and wave propagation analysis in a FG Graphene platelets-reinforced nanocomposite cylinder using a modified nonlinear micromechanical model", Steel Compos. Struct., 27(3), 255-271. https://doi.org/10.12989/scs.2018.27.3.255.
  45. Huang, Y., Karami, B., Shahsavari, D. and Tounsi, A. (2021), "Static stability analysis of carbon nanotube reinforced polymeric composite doubly curved micro-shell panels", Arch. Civil Mech. Eng., 21(4), 139. https://doi.org/10.1007/s43452-021-00291-7
  46. Irie, T., Yamada, G. and Muramoto, Y. (1984), "Free vibration of joined conical-cylindrical shells", J Sound Vib., 95(1), 31-39. https://doi.org/10.1007/s00707-018-2133-3
  47. Izadi, M.H., Hosseini-Hashemi, S. and Korayem, M.H. (2018), "Analytical and FEM solutions for free vibration of joined cross-ply laminated thick conical shells using shear deformation theory", Arch Appl Mech., 88(12), 2231-2246. https://doi.org/10.1007/s00419-018-1446-y
  48. Jalaei, M. 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.
  49. Jamalabadi, M.Y.A., Borji, P., Habibi, M. and Pelalak, R. (2021), "Nonlinear vibration analysis of functionally graded GPL-RC conical panels resting on elastic medium", Thin Wall. Struct., 160, 107370. https://doi.org/10.1016/j.tws.2020.107370
  50. Javani, M., Kiani, Y. and Eslami, M.R. (2020), "Thermal buckling of FG graphene platelet reinforced composite annular sector plates", Thin Wall. Struct., 148, 106589. https://doi.org/10.1016/j.tws.2019.106589
  51. Javani, M., Kiani, Y. and Eslami, M.R. (2021), "Geometrically nonlinear free vibration of FG-GPLRC circular plate on the nonlinear elastic foundation", Compos. Struct. 261, 113515. https://doi.org/10.1016/j.compstruct.2020.113515
  52. Jrad, H., Mars, J., Wali, M. and Dammak, F. (2019), "Geometrically nonlinear analysis of elastoplastic behavior of functionally graded shells", Eng Comput., 35(3), 833-847. https://doi.org/10.1007/s00366-018-0633-3
  53. Kaddari, M., Kaci, A., Bousahla, A.A., Tounsi, A., Bourada, F., Tounsi, A., Bourada, F., Tounsi, A., Bedia, E.A.A. and Al-Osta, M.A. (2020), "Astudy on the structural behaviour of functionally graded porous plates on elastic foundation using a new quasi-3Dmodel: Bending and free vibration analysis", Comput. Concr. Int.J., 25(1), 37-57. https://doi.org/10.12989/cac.2020.25.1.037
  54. Kaghazian, A., Hajnayeb, A. and Foruzande, H. (2017), "Free vibration analysis of a piezoelectric nanobeam using nonlocal elasticity theory", Struct. Eng. Mech., 61(5), 617-624. https://doi.org/10.12989/sem.2017.61.5.617.
  55. Karami, B., Janghorban, M., Shahsavari, D. and Tounsi, A. (2018), "A size-dependent quasi-3D model for wave dispersion analysis of FG nanoplates", Steel Compos. Struct., 28(1), 99-110. https://doi.org/10.12989/sem.2019.69.5.487.
  56. Katiyar, V., Gupta, A., and Tounsi, A. (2022), "Microstructural/geometric imperfection sensitivity on the vibration response of geometrically discontinuous bi-directional functionally graded plates (2D-FGPs) with partial supports by using FEM", Steel Compos. Struct., 45(5), 621-640. https://doi.org/10.12989/scs.2022.45.5.621
  57. Kiani, Y. (2018a), "Isogeometric large amplitude free vibration of graphene reinforced laminated plates in thermal environment using NURBS formulation", Comp. Meth. Appl. Mech. Eng., 332, 86-101. https://doi.org/10.1016/j.cma.2017.12.015
  58. Kiani, Y. (2018b), "NURBS-based isogeometric thermal postbuckling analysis of temperature dependent graphenereinforced composite laminated plates", Thin Wall. Struct., 125, 211-219. https://doi.org/10.1016/j.tws.2018.01.024
  59. Kiani, Y. (2019), "Buckling of functionally graded graphene reinforced conical shells under external pressure in thermal environment", Compos. Part B Eng., 156, 128-137. https://doi.org/10.1016/j.compositesb.2018.08.052
  60. Kiani, Y. (2020), "Influence of graphene platelets on the response of composite plates subjected to a moving load", Mech. Base Des. Struct., 1-14.
  61. Kiani, Y. and Zur, K.K. (2022), "Free vibrations of graphene platelet reinforced composite skew plates resting on point supports", Thin Wall. Struct., 176, 109363. https://doi.org/10.1016/j.tws.2022.109363
  62. Kumar, Y., Gupta, A. and Tounsi, A. (2021), "Size-dependent vibration response of porous graded nanostructure with FEM and nonlocal continuum model", Adv. Nano Res., 11(1), 1-17. https://doi.org/10.12989/anr.2021.11.1.001
  63. Lee, Y.S., Yang, M.S., Kim, H.S. and Kim, J.H. (2002), "A study on the free vibration of the joined cylindrical-spherical shell structures", Comput. Struct., 80(27-30), 2405-2414. https://doi.org/10.1016/S0045-7949(02)00243-2
  64. Lee, J. (2018), "Free vibration analysis of joined conical-cylindrical shells by matched Fourier-Chebyshev collocation method", J. Mech. Sci. Tech., 32(10), 4601-4612. https://doi.org/10.1007/s12206-018-0907-0
  65. Leissa, A.W. (1993), Vibration of Shells, American Institute of Physics, New York, U.S.A.
  66. Liu, J., Yan, H., and Jiang, K. (2013), "Mechanical properties of graphene platelet-reinforced alumina ceramic composites", Ceram. Int., 39(6), 6215-6221. https://doi.org/10.1016/j.ceramint.2013.01.041
  67. Liu, G., Wu, S., Shahsavari, D., Karami, B. and Tounsi, A. (2022), "Dynamics of imperfect inhomogeneous nanoplate with exponentially-varying properties resting on viscoelastic foundation", Eur. J. Mech. A Solids, 95, 104649. https://doi.org/10.1016/j.euromechsol.2022.104649
  68. Mangalasseri, A.S., Mahesh, V., Mukunda, S., Mahesh, V., Ponnusami, S.A., Harursampath, D. and Tounsi, A. (2023), "Vibration based energy harvesting performance of magneto-electro-elastic beams reinforced with carbon nanotubes", Adv. Nano Res., 14(1), 27-43. https://doi.org/10.12989/anr.2023.14.1.027
  69. Moradi, S. and Mansouri, M.H. (2012), "Thermal buckling analysis of shear deformable laminated orthotopic plates by differential quadrature", Steel Compos. Struct., 12(2), 129-147. https://doi.org/10.12989/scs.2012.12.2.129
  70. Nguyen, L.B., Thai, C.H., and Nguyen-Xuan, H. (2016), "A generalized unconstrained theory and isogeometric finite element analysis based on Bezier extraction for laminated composite plates", Eng Comput., 32(3), 457-475. https://doi.org/10.1007/s00366-015-0426-x
  71. Nguyen, P.C., Pham, Q.H., Tran, T.T. and Nguyen-Thoi, T. (2022), "Effects of partially supported elastic foundation on free vibration of FGP plates using ES-MITC3 elements", Ain Shams Eng. J., 13(3), 101615. https://doi.org/10.1016/j.asej.2021.10.010
  72. Patel, B.P., Ganapathi, M. and Kamat, S. (2000), "Free vibration characteristics of laminated composite joined conical-cylindrical shells", J Sound Vib., 237(5), 920-930. https://doi.org/10.1006/jsvi.2000.3018
  73. Pham, Q.H., Nguyen, P.C., Tran, V.K. and Nguyen-Thoi, T. (2021), "Finite element analysis for functionally graded porous nano-plates resting on elastic foundation", Steel Compos. Struct., 41(2), 149-166. https://doi.org/10.12989/scs.2021.41.2.149
  74. Pham, Q.H. and Nguyen, P.C. (2022), "Effects of size-dependence on static and free vibration of FGP nanobeams using finite element method based on nonlocal strain gradient theory", Steel Compos. Struct., 45(3), 331-348. https://doi.org/10.12989/scs.2022.45.3.331
  75. Pham, Q.H., Nguyen, P.C., Tran, V.K. and Nguyen-Thoi, T. (2022a), "Isogeometric analysis for free vibration of bidirectional functionally graded plates in the fluid medium", Defence Technol., 18(8), 1311-1329. https://doi.org/10.1016/j.dt.2021.09.006
  76. Pham, Q.H., Tran, V.K., Tran, T.T., Nguyen, P.C. and Malekzadeh, P. (2022b), "Dynamic instability of magnetically embedded functionally graded porous nanobeams using the strain gradient theory", Alexandria Eng. J., 61(12), 10025-10044. https://doi.org/10.1016/j.aej.2022.03.007
  77. Phung-Van, P., Thai, C.H., Ferreira, A.J.M. and Rabczuk, T. (2020), "Isogeometric nonlinear transient analysis of porous FGM plates subjected to hygro-thermo-mechanical loads", Thin-Walled Struct., 148, 106497. https://doi.org/10.1016/j.tws.2019.106497
  78. Qatu, M.S. (2004), Vibration of Laminated Shells and Plates, Elsevier, New York, U.S.A.
  79. Qu, Y., Chen, Y., Long, X., Hua, H. and Meng, G. (2013), "A variational method for free vibration analysis of joined cylindrical-conical shells", J. Vib. Control, 19(16), 2319-2334. https://doi.org/10.1177%2F1077546312456227 https://doi.org/10.1177%2F1077546312456227
  80. Rafiee, M.A., J.Z., Wang, H., Song, Z.Z., Yu, and N., Koratkar (2009), "Enhanced mechanical properties of nanocomposites at low graphene content", ACS Nano, 3, 3884-3990. https://doi.org/10.1021/nn9010472
  81. Reddy, K.R., El-Zein, A., Airey, D.W., Alonso-Marroquin, F., Schubel, P. and Manalo, A. (2020), "Self-healing polymers: Synthesis methods and applications", Nano Struct. Nano Objects, 23, 100500. https://doi.org/10.1016/j.nanoso.2020.100500
  82. Rouabhia, A., Chikh, A., Bousahla, A.A., Bourada, F., Heireche, H., Tounsi, A., Kouider H. and Tounsi, A. (2020), "Physical stability response of a SLGS resting on viscoelastic medium using nonlocal integral first-order theory", Steel Compos. Struct., 37(6), 695-709. https://doi.org/10.12989/scs.2020.37.6.695
  83. Safarpour M., Rahimi, A.R. and Alibeigloo, A. (2020), "Static and free vibration analysis of graphene platelets reinforced composite truncated conical shell, cylindrical shell, and annular plate using theory of elasticity and DQM", Mech. Base Des. Struct., 48, 496-524. https://doi.org/10.1080/15397734.2019.1646137
  84. Sarkheil, S., Foumani, M.S. and Navazi, H.M. (2017), "Free vibrations of a rotating shell made of p joined cones", Int. J. Mech.Sci., 124, 83-94. https://doi.org/10.1016/j.ijmecsci.2017.02.003
  85. Shakouri, M. and M.A., Kouchakzadeh (2014), "Free vibration analysis of joined conical shells: Analytical and experimental study", Thin Wall. Struct., 85, 350-358. https://doi.org/10.1016/j.tws.2014.08.022
  86. Shojaei, A., Galvanetto, U., Rabczuk, T., Jenabi, A. and Zaccariotto, M. (2019), "A generalized finite difference method based on the Peridynamic differential operator for the solution of problemsin bounded and unbounded domains", Comp Meth Appl Mech Eng. 343,100-126. https://doi.org/10.1016/j.cma.2018.08.033
  87. Soedel, W. (2004), Vibrations of Shells and Plates, Marcel Dekker, New York., U.S.A. https://doi.org/10.3233/SAV-1995-2209
  88. Song, M., Kitipornchai, S. and Yang, J. (2017), "Free and forced vibrations of functionally graded polymer composite plates reinforced with graphene nanoplatelets", Compos Struct., 159, 579-588. https://doi.org/10.1016/j.compstruct.2016.09.070
  89. Soureshjani A.H., Talebitooti R. and Talebitooti, M. (2020), "Thermal effects on the free vibration of joined FG-CNTRC conical-conical shells", Thin Wall. Struct., 156, 106960. https://doi.org/10.1016/j.tws.2020.106960
  90. Tahouneh, V. (2014), "Free vibration analysis of bidirectional functionally graded annular plates resting on elastic foundations using differential quadrature method", Struct. Eng. Mech., 52(4), 663-686. https://doi.org/10.12989/sem.2014.52.4.663.
  91. Tahouneh, V. (2016), "Using an equivalent continuum model for 3D dynamic analysis of nanocomposite plates", Steel Compos. Struct., 20(3), 623-649. https://doi.org/10.12989/scs.2016.20.3.623.
  92. Tran, T.V., Tran, T.D., Hoa Pham, Q., Nguyen-Thoi, T. and Tran, V.K. (2020), "An ES-MITC3 finite element method based on higher-order shear deformation theory for static and free vibration analyses of FG porous plates reinforced by GPLs", Math. Probl. Eng., 1-18. https://doi.org/10.1155/2020/7520209
  93. Van Vinh, P. and Tounsi, A. (2022a), "The role of spatial variation of the nonlocal parameter on the free vibration of functionally graded sandwich nanoplates", Eng. Struct., 38(5), 4301-4319. https://doi.org/10.1007/s00366-021-01475-8
  94. Van Vinh, P. and Tounsi, A. (2022b), "Free vibration analysis of functionally graded doubly curved nanoshells using nonlocal first-order shear deformation theory with variable nonlocal parameters", Thin Wall. Struct., 174, 109084. https://doi.org/10.1016/j.tws.2022.109084
  95. Van Vinh, P., Van Chinh, N. and Tounsi, A. (2022), "Static bending and buckling analysis of bi-directional functionally graded porous plates using an improved first-order shear deformation theory and FEM", Eur. J. Mech. A Solids, 96, 104743. https://doi.org/10.1016/j.euromechsol.2022.104743
  96. Wang, Y., Feng, C., Wang, X., Zhao, Z., Romero, C. S. and Yang, J. (2019), "Nonlinear free vibration of graphene platelets (GPLs)/polymer dielectric beam", Smart Mater. Struct., 28(5), 055013. https://doi.org/10.1088/1361-665X/ab0b51
  97. Wu, H., Kitipornchai, S. and Yang, J. (2017), "Thermal buckling and postbuckling of functionally graded graphene nanocomposite plates", Mater. Des., 132, 430-441. https://doi.org/10.1016/j.tws.2017.05.006
  98. Wu, C.P. and Liu, Y.C. (2016), "A state space meshless method for the 3D analysis of FGM axisymmetric circular plates", Steel Compos. Struct., 22(1), 161-182. https://doi.org/10.12989/scs.2016.22.1.161.
  99. Yang, J., Wu, H. and Kitipornchai, S. (2017), "Buckling and postbuckling of functionally graded multilayer graphene platelet-reinforced composite beams", Compos.Struct., 161, 111-118. https://doi.org/10.1016/j.compstruct.2016.11.048
  100. 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
  101. Zeighampour, H. and Beni, Y.T., (2014), "Cylindrical thin-shell model based on modified strain gradient theory", Int. J. Eng. Sci., 78, 27-47. https://doi.org/10.1016/j.ijengsci.2014.01.004
  102. Zeighampour, H., Beni Y.T. and Dehkordi, M.B. (2018), "Wave propagation in viscoelastic thin cylindrical nanoshell resting on a visco-Pasternak foundation based on nonlocal strain gradient theory", Thin Wall. Struct., 122, 378-386. http://doi.org/10.1016/j.tws.2017.10.037
  103. Zenkour, A.M. (2014a), "Torsional analysis of heterogeneous magnetic circular cylinder", Steel Compos. Struct., 17(4), 535-548. http://doi.org/10.12989/scs.2014.17.4.535
  104. Zenkour, A.M. (2014b), "Exact solution of thermal stress problem of an inhomogeneous hygrothermal piezoelectric hollow cylinder", Appl. Math. Modell., 38(24), 6133-6143. https://doi.org/10.1016/j.apm.2014.05.028
  105. Zerrouki, R., Karas, A., Zidour, M., Bousahla, A.A., Tounsi, A., Bourada, F., Tounsi, A., Benrahou, K.H. and Mahmoud, S.R. (2021), "Effect of nonlinear FG-CNT distribution on mechanical properties of functionally graded nano-composite beam", Struct. Eng. Mech., 78(2), 117-124. https://doi.org/10.12989/sem.2021.78.2.117
  106. Zhao, X., Q., Zhang, D., Chen and P., Lu (2010), "Enhanced mechanical properties of graphene based poly (vinyl alcohol) composites", Macromolecules, 43, 2357-2363. https://doi.org/10.1021/ma902862u
  107. Zhao, L.C., Chen, S.S., Xu, Y.P., Tahouneh, V. (2021), "Vibration analysis of damaged core laminated curved panels with functionally graded sheets and finite length", Steel Compos. Struct., 38, 477-496. https://doi.org/10.12989/scs.2021.38.5.477
  108. Zienkiewicz, O.C., Taylor, R.L. and Zhu, J.Z. (2005), The Finite Element Method: Its Basis and Fundamentals, Elsevier.