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Vibration characteristics of microplates with GNPs-reinforced epoxy core bonded to piezoelectric-reinforced CNTs patches

  • Forsat, Masoud (Department of Mechanical and Industrial Engineering Qatar University) ;
  • Musharavati, Farayi (Department of Mechanical and Industrial Engineering Qatar University) ;
  • Eltai, Elsadig (Department of Mechanical and Industrial Engineering Qatar University) ;
  • Zain, Azlan Mohd (UTM Big Data Centre, Universiti Teknologi Malaysia) ;
  • Mobayen, Saleh (Future Technology Research Center, National Yunlin University of Science and Technology) ;
  • Mohamed, Abdeliazim Mustafa (Department of Civil Engineering, College of Engineering, Prince Sattam bin Abdulaziz University)
  • Received : 2020.09.11
  • Accepted : 2020.12.03
  • Published : 2021.08.25

Abstract

In the current study, vibration characteristics of a three-layered rectangular microplate with Graphene nanoplatelets (GNPs)-reinforced Epoxy core which is fully bonded to piezoelectric-reinforced single-walled Carbon nanotubes (SWCNTs) patches are provided. The face sheets are subjected to the electric field and the microplate is assumed to be in a thermal environment and also, is located on the visco-Pasternak model of the elastic substrate. The GNPs and SWCNTs are dispersed through the core's and face's thickness according to the given functions. To account the shear deformation effect, tangential shear deformation theory (TGSDT) as a higher-order theory is employed and the modified strain gradient theory (MSGT) with tree independent length-scale parameters is selected to capture the size effect. Using the extended form of Hamilton's principle and variational formulation, the governing motion equations are derived and solved mathematically via Navier's scheme for simply supported edges microplate. By ensuring the validity of the results after comparing them in a simpler state with previously published ones, the effects of the most prominent parameters on the results are investigated. It is seen GNPs and CNTs dispersion patterns play an important role in the microplate vibrational behavior, as well as temperature variations. Since the under consideration microstructure can be accounted as smart structures, therefore, the outcomes of this study may help to design and create more efficient engineering structures, such as sensors and actuators and also micro/nano electromechanical systems.

Keywords

References

  1. Abazid, M.A. (2019), "The Nonlocal Strain Gradient Theory for Hygrothermo-Electromagnetic Effects on Buckling, Vibration and Wave Propagation in Piezoelectromagnetic Nanoplates", Int. J. Appl. Mech., 11(7), 1950067. https://doi.org/10.1142/S1758825119500674
  2. Akgoz, B. and Civalek, O. (2018), "Vibrational characteristics of embedded microbeams lying on a two-parameter elastic foundation in thermal environment", Compos. Part B: Eng., 150, 68-77. https://doi.org/10.1016/j.compositesb.2018.05.049
  3. Allahkarami, F. and Nikkhah-Bahrami, M. (2018), "The effects of agglomerated CNTs as reinforcement on the size-dependent vibration of embedded curved microbeams based on modified couple stress theory", Mech. Adv. Mater. Struct., 25(12), 995-1008. https://doi.org/10.1080/15376494.2017.1323144
  4. Amir, S., Khorasani, M. and BabaAkbar-Zarei, H. (2018), "Buckling analysis of nanocomposite sandwich plates with piezoelectric face sheets based on flexoelectricity and first-order shear deformation theory", J. Sandw. Struct. Mater., 109963621879538. https://doi.org/10.1177/1099636218795385
  5. Amir, S., Arshid, E. and Ghorbanpour Arani, M.R. (2019a), "Sizedependent magneto-electro-elastic vibration analysis of FG saturated porous annular/ circular micro sandwich plates embedded with nano-composite face sheets subjected to multiphysical pre loads", Smart Struct. Syst., 23(5), 429-447. https://doi.org/10.12989/sss.2019.23.5.429
  6. Amir, S., Soleimani-Javid, Z. and Arshid, E. (2019b), "Sizedependent free vibration of sandwich micro beam with porous core subjected to thermal load based on SSDBT", ZAMM Zeitschrift Fur Angewandte Mathematik Und Mechanik, 99(9), 1-21. https://doi.org/10.1002/zamm.201800334
  7. Amir, S., Bidgoli, E.M.R. and Arshid, E. (2020a), "Size-dependent vibration analysis of a three-layered porous rectangular nano plate with piezo-electromagnetic face sheets subjected to pre loads based on SSDT", Mech. Adv. Mater. Struct., 27(8), 605-619. https://doi.org/10.1080/15376494.2018.1487612
  8. Amir, S., Vossough, A.R., Vossough, H. and Arshid, E. (2020b), "Nonlinear magneto-nonlocal vibration analysis of coupled piezoelectric micro-plates reinforced with agglomerated CNTs", Mech. Adv. Compos. Struct., 7(1), 109-119. https://doi.org/10.22075/macs.2019.16632.1185
  9. Amir, S., Arshid, E. and Khoddami Maraghi, Z. (2020c), "Free vibration analysis of magneto-rheological smart annular threelayered plates subjected to magnetic field in viscoelastic medium", Smart Struct. Syst., 25(5), 581-592. https://doi.org/10.12989/sss.2020.25.5.581
  10. Amir, S., Arshid, E., Khoddami Maraghi, Z., Loghman, A. and Ghorbanpour Arani, A. (2020d), "Vibration analysis of magnetorheological fluid circular sandwich plates with magnetostrictive facesheets exposed to monotonic magnetic field located on visco-Pasternak substrate", JVC/J. Vib. Control, 26(17-18), 1523-1537. https://doi.org/10.1177/1077546319899203
  11. Amir, S., Arshid, E., Rasti-Alhosseini, S.M.A. and Loghman, A. (2020e), "Quasi-3D tangential shear deformation theory for size-dependent free vibration analysis of three-layered FG porous micro rectangular plate integrated by nano-composite faces in hygrothermal environment", J. Thermal Stress., 43(2), 133-156. https://doi.org/10.1080/01495739.2019.1660601
  12. Arefi, M., Mohammad-Rezaei Bidgoli, E. and Rabczuk, T. (2019), "Thermo-mechanical buckling behavior of FG GNP reinforced micro plate based on MSGT", Thin-Wall. Struct., 142, 444-459. https://doi.org/10.1016/j.tws.2019.04.054
  13. Arshid, E. and Khorshidvand, A.R. (2018), "Free vibration analysis of saturated porous FG circular plates integrated with piezoelectric actuators via differential quadrature method", Thin-Wall. Struct., 125, 220-233. https://doi.org/10.1016/j.tws.2018.01.007
  14. Arshid, E., Kiani, A. and Amir, S. (2019a), "Magneto-electroelastic vibration of moderately thick FG annular plates subjected to multi physical loads in thermal environment using GDQ method by considering neutral surface", Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, 233(10), 2140-2159. https://doi.org/10.1177/1464420719832626
  15. Arshid, E., Khorshidvand, A.R. and Khorsandijou, S.M. (2019b), "The effect of porosity on free vibration of SPFG circular plates resting on visco-Pasternak elastic foundation based on CPT, FSDT and TSDT", Struct. Eng. Mech., Int. J., 70(1), 97-112. https://doi.org/10.12989/sem.2019.70.1.097
  16. Arshid, E., Kiani, A., Amir, S. and Zarghami Dehaghani, M. (2019c), "Asymmetric free vibration analysis of first-order shear deformable functionally graded magneto-electro-thermoelastic circular plates", Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 233(16), 5659-5675. https://doi.org/10.1177/0954406219850598
  17. Arshid, E., Amir, S. and Loghman, A. (2020a), "Static and dynamic analyses of FG-GNPs reinforced porous nanocomposite annular micro-plates based on MSGT", Int. J. Mech. Sci., 180, 105656. https://doi.org/10.1016/j.ijmecsci.2020.105656
  18. Arshid, E., Amir, S. and Loghman, A. (2020b), "Bending and buckling behaviors of heterogeneous temperature-dependent micro annular/circular porous sandwich plates integrated by FGPEM nano-Composite layers", J. Sandw. Struct. Mater., 109963622095502. https://doi.org/10.1177/1099636220955027
  19. Behdinan, K., Moradi-Dastjerdi, R., Safaei, B., Qin, Z., Chu, F. and Hui, D. (2020), "Graphene and CNT impact on heat transfer response of nanocomposite cylinders", Nanotechnol. Reviews, 9(1), 41-52. https://doi.org/10.1515/ntrev-2020-0004
  20. Brush, D.O., Almroth, B.O. and Hutchinson, J.W. (1975), "Buckling of bars, plates, and shells", J. Appl. Mech., 42, 911. https://doi.org/10.1115/1.3423755
  21. Chikr, S.C., Kaci, A., Bousahla, A.A., Bourada, F., Tounsi, A., Bedia, E.A. and Tounsi, A. (2020), "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., 21(5), 471-487. https://doi.org/10.12989/GAE.2020.21.5.471
  22. Cinefra, M., Valvano, S. and Carrera, E. (2015), "A layer-wise MITC9 finite element for the free-vibration analysis of plates with piezo-patches", Int. J. Smart Nano Mater., 6(2), 85-104. https://doi.org/10.1080/19475411.2015.1037377
  23. Duc, N.D. (2018), "Nonlinear thermo-electro-mechanical dynamic response of shear deformable piezoelectric sigmoid functionally graded sandwich circular cylindrical shells on elastic foundations", J. Sandw. Struct. Mater., 20(3), 351-378. https://doi.org/10.1177/1099636216653266
  24. 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
  25. Emdadi, M., Mohammadimehr, M. and Navi, B.R. (2019), "Free vibration of an annular sandwich plate with CNTRC facesheets and FG porous cores using Ritz method", Adv. Nano Res., 7(2), 109-123. https://doi.org/10.12989/anr.2019.7.2.109
  26. 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
  27. Eringen, A.C. (2002), Nonlocal Continuum Field Theories, Springer Science & Business Media. https://doi.org/10.1007/b97697
  28. Eyvazian, A., Shahsavari, D. and Karami, B. (2020), "On the dynamic of graphene reinforced nanocomposite cylindrical shells subjected to a moving harmonic load", Int. J. Eng. Sci., 154, 103339. https://doi.org/10.1016/j.ijengsci.2020.103339
  29. Fattahi, A.M., Safaei, B. and Moaddab, E. (2019a), "The application of nonlocal elasticity to determine vibrational behavior of FG nanoplates", Steel Compos. Struct., 32(2), 281-292. https://doi.org/10.12989/scs.2019.32.2.281
  30. Fattahi, A.M., Safaei, B. and Ahmed, N.A. (2019b), "A comparison for the non-classical plate model based on axial buckling of single-layered graphene sheets", Eur. Phys. J. Plus, 134(11), 1-13. https://doi.org/10.1140/epjp/i2019-12912-7
  31. Ferreira, A.J.M., Fasshauer, G.E., Batra, R.C. and Rodrigues, J.D. (2008), "Static deformations and vibration analysis of composite and sandwich plates using a layerwise theory and RBF-PS discretizations with optimal shape parameter", Compos. Struct., 86(4), 328-343. https://doi.org/10.1016/J.COMPSTRUCT.2008.07.025
  32. Garcia-Macias, E., Rodriguez-Tembleque, L. and Saez, A. (2018), "Bending and free vibration analysis of functionally graded graphene vs. carbon nanotube reinforced composite plates", Compos. Struct., 186, 123-138. https://doi.org/10.1016/j.compstruct.2017.11.076
  33. Habibi, M., Taghdir, A. and Safarpour, H. (2019), "Stability analysis of an electrically cylindrical nanoshell reinforced with graphene nanoplatelets", Compos. Part B: Eng., 175, 107125. https://doi.org/10.1016/j.compositesb.2019.107125
  34. Hajmohammad, M.H., Zarei, M.S., Farrokhian, A. and Kolahchi, R. (2018), "A layerwise theory for buckling analysis of truncated conical shells reinforced by CNTs and carbon fibers integrated with piezoelectric layers in hygrothermal environment", Adv. Nano Res., 6(4), 299-321. https://doi.org/10.12989/anr.2018.6.4.299
  35. Iijima, S. (1991), "Helical microtubules of graphitic carbon", Nature, 354(6348), 56. https://doi.org/10.1038/354056a0
  36. Karami, B. and Shahsavari, D. (2020), "On the forced resonant vibration analysis of functionally graded polymer composite doubly-curved nanoshells reinforced with graphene-nanoplatelets", Comput. Methods Appl. Mech. Eng., 359, 112767. https://doi.org/10.1016/j.cma.2019.112767
  37. Karami, B., Shahsavari, D. and Janghorban, M. (2018), "A comprehensive analytical study on functionally graded carbon nanotube-reinforced composite plates", Aerosp. Sci. Technol., 82-83, 499-512. https://doi.org/10.1016/j.ast.2018.10.001
  38. Karami, B., Janghorban, M. and Tounsi, A. (2019a), "Galerkin's approach for buckling analysis of functionally graded anisotropic nanoplates/different boundary conditions", Eng. Comput., 35(4), 1297-1316. https://doi.org/10.1007/s00366-018-0664-9
  39. Karami, B., Janghorban, M. and Tounsi, A. (2019b), "On exact wave propagation analysis of triclinic material using threedimensional bi-Helmholtz gradient plate model", Struct. Eng. Mech., Int. J., 69(5), 487-497. https://doi.org/10.12989/sem.2019.69.5.487
  40. Karami, B., Janghorban, M. and Tounsi, A. (2019c), "Wave propagation of functionally graded anisotropic nanoplates resting on Winkler-Pasternak foundation", Struct. Eng. Mech., 70(1), 55-66. https://doi.org/10.12989/sem.2019.70.1.055
  41. Karami, B., Shahsavari, D., Janghorban, M. and Li, L. (2019d), "Elastic guided waves in fully-clamped functionally graded carbon nanotube-reinforced composite plates", Mater. Res. Express, 6(9), 0950a9. https://doi.org/10.1088/2053-1591/ab3474
  42. Karami, B., Shahsavari, D., Janghorban, M. and Li, L. (2019e), "Influence of homogenization schemes on vibration of functionally graded curved microbeams", Compos. Struct., 216, 67-79. https://doi.org/10.1016/j.compstruct.2019.02.089
  43. Karami, B., Shahsavari, D., Janghorban, M. and Tounsi, A. (2019f), "Resonance behavior of functionally graded polymer composite nanoplates reinforced with graphene nanoplatelets", Int. J. Mech. Sci., 156, 94-105. https://doi.org/10.1016/j.ijmecsci.2019.03.036
  44. Karami, B., Gheisari, P., Nazemosadat, S.M.R., Akbari, P., Shahsavari, D., Naghizadeh, M. and Naghizadeh, M. (2020), "Elastic wave characteristics of graphene nanoplatelets reinforced composite nanoplates", Struct. Eng. Mech., 74(6), 809-819. https://doi.org/10.12989/sem.2020.74.6.809.
  45. Kiani, Y. (2016), "Shear buckling of FG-CNT reinforced composite plates using Chebyshev-Ritz method", Compos. Part B: Eng., 105, 176-187. https://doi.org/10.1016/J.COMPOSITESB.2016.09.001
  46. Kolahdouzan, F., Gorbanpour Arani, A. and Abdollahian, M. (2018), "Buckling and free vibration analysis of FG-CNTRCmicro sandwich plate", Steel Compos. Struct., 26(3), 273-287. http://dx.doi.org/10.12989/scs.2018.26.3.273
  47. Li, Q., Wu, D., Gao, W., Tin-Loi, F., Liu, Z. and Cheng, J. (2019), "Static bending and free vibration of organic solar cell resting on Winkler-Pasternak elastic foundation through the modified strain gradient theory", Eur. J. Mech., A/Solids, 78, 103852. https://doi.org/10.1016/j.euromechsol.2019.103852
  48. Li, Q., Wu, D., Gao, W. and Tin-Loi, F. (2020), "Size-dependent instability of organic solar cell resting on Winkler-Pasternak elastic foundation based on the modified strain gradient theory", Int. J. Mech. Sci., 177, 105306. https://doi.org/10.1016/j.ijmecsci.2019.105306
  49. Lin, H.G., Cao, D.Q. and Xu, Y.Q. (2018), "Vibration, buckling and aeroelastic analyses of functionally graded multilayer graphene-nanoplatelets-reinforced composite plates embedded in piezoelectric layers", Int. J. Appl. Mech., 10(3), 1850023. https://doi.org/10.1142/S1758825118500230
  50. Liu, H., Wu, H. and Lyu, Z. (2020), "Nonlinear resonance of FG multilayer beam-type nanocomposites: Effects of graphene nanoplatelet-reinforcement and geometric imperfection", Aerosp. Sci. Technol., 98, 105702. https://doi.org/10.1016/j.ast.2020.105702
  51. Loghman, A. and Cheraghbak, A. (2018), "Agglomeration effects on electro-magneto-thermo elastic behavior of nano-composite piezoelectric cylinder", Polym. Compos., 39(5), 1594-1603. https://doi.org/10.1002/pc.24104
  52. Mao, J.J. and Zhang, W. (2019), "Buckling and post-buckling analyses of functionally graded graphene reinforced piezoelectric plate subjected to electric potential and axial forces", Compos. Struct., 216, 392-405. https://doi.org/10.1016/j.compstruct.2019.02.095
  53. Matouk, H., Bousahla, A.A., Heireche, H., Bourada, F., Bedia, E.A.A., 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., 8(4), 293-305. https://doi.org/10.12989/anr.2020.8.4.293
  54. Mehar, K., Panda, S.K., Bui, T.Q. and Mahapatra, T.R. (2017), "Nonlinear thermoelastic frequency analysis of functionally graded CNT-reinforced single/doubly curved shallow shell panels by FEM", J. Thermal Stress., 40(7), 899-916. https://doi.org/10.1080/01495739.2017.1318689
  55. Mirjavadi, S.S., Forsat, M., Barati, M.R. and Hamouda, A.M.S. (2020a), "Investigating nonlinear forced vibration behavior of multi-phase nanocomposite annular sector plates using Jacobi elliptic functions", Steel Compos. Struct., 36(1), 87-101. https://doi.org/10.12989/scs.2020.36.1.087
  56. Mirjavadi, S.S., Forsat, M., Barati, M.R., Hamouda, A., Mirjavadi, S.S., Forsat, M. and Hamouda, A. (2020b), "Nonlinear forced vibrations of multi-scale epoxy/CNT/fiberglass truncated conical shells and annular plates via 3D Mori-Tanaka scheme", Steel Compos. Struct., 35(6), 765-677. https://doi.org/10.12989/scs.2020.35.6.765
  57. Mirjavadi, S.S., Nikookar, M., Mollaee, S., Forsat, M., Barati, M.R., Hamouda, A.M.S. and Hamouda, A.M.S. (2020c), "Analyzing exact nonlinear forced vibrations of two-phase magneto-electro-elastic nanobeams under an elliptic-type force", Adv. Nano Res., 9(1), 47-58. https://doi.org/10.12989/anr.2020.9.1.047
  58. Mirsalehi, M., Azhari, M. and Amoushahi, H. (2017), "Buckling and free vibration of the FGM thin micro-plate based on the modified strain gradient theory and the spline finite strip method", Eur. J. Mech., A/Solids, 61, 1-13. https://doi.org/10.1016/j.euromechsol.2016.08.008
  59. Mirzaei, M. and Kiani, Y. (2016), "Free vibration of functionally graded carbon nanotube reinforced composite cylindrical panels", Compos. Struct., 142, 45-56. https://doi.org/10.1016/J.COMPSTRUCT.2015.12.071
  60. Mohammadimehr, M., Arshid, E., Alhosseini, S.M.A.R., Amir, S. and Arani, M.R.G. (2019), "Free vibration analysis of thick cylindrical MEE composite shells reinforced CNTs with temperature-dependent properties resting on viscoelastic foundation", Struct. Eng. Mech., 70(6), 683-702. https://doi.org/10.12989/sem.2019.70.6.683
  61. Mohammadzadeh-Keleshteri, M., Asadi, H. and Aghdam, M.M. (2017), "Geometrical nonlinear free vibration responses of FGCNT reinforced composite annular sector plates integrated with piezoelectric layers", Compos. Struct., 171, 100-112. https://doi.org/10.1016/J.COMPSTRUCT.2017.01.048
  62. Novoselov, K.S., Geim, A.K., Morozov, S.V., Jiang, D., Zhang, Y., Dubonos, S.V. and Firsov, A.A. (2004), "Electric field in atomically thin carbon films", Science, 306(5696), 666-669. https://doi.org/10.1126/science.1102896
  63. Paul, R., Kumbhakar, P. and Mitra, A.K. (2013), "A facile chemical synthesis of a novel photo catalyst: SWCNT/titania nanocomposite", Adv. Nano Res., 1(2), 71-82. https://doi.org/10.12989/anr.2013.1.2.071
  64. Refrafi, S., Bousahla, A.A., Bouhadra, A., Menasria, A., Bourada, F., Tounsi, A. and Tounsi, A. (2020), "Effects of hygro-thermomechanical conditions on the buckling of FG sandwich plates resting on elastic foundations", Comput. Concrete, 25(4), 311-325. https://doi.org/10.12989/cac.2020.25.4.311
  65. Safaei, B. (2020), "The effect of embedding a porous core on the free vibration behavior of laminated composite plates", Steel Compos. Struct., 35(5), 659-670. http://dx.doi.org/10.12989/scs.2020.35.5.659
  66. Safaei, B., Khoda, F.H. and Fattahi, A.M. (2019), "Non-classical plate model for single-layered graphene sheet for axial buckling", Adv. Nano Res., 7(4), 265-275. https://doi.org/10.12989/anr.2019.7.4.265
  67. Safarpour, H., Esmailpoor Hajilak, Z. and Habibi, M. (2019), "A size-dependent exact theory for thermal buckling, free and forced vibration analysis of temperature dependent FG multilayer GPLRC composite nanostructures restring on elastic foundation", Int. J. Mech. Mater. Des., 15(3), 569-583. https://doi.org/10.1007/s10999-018-9431-8
  68. Sahmani, S. and Aghdam, M.M. (2017a), "Axial postbuckling analysis of multilayer functionally graded composite nanoplates reinforced with GPLs based on nonlocal strain gradient theory", Eur. Phys. J. Plus, 132(11), 1-17. https://doi.org/10.1140/epjp/i2017-11773-4
  69. Sahmani, S. and Aghdam, M.M. (2017b), "Imperfection sensitivity of the size-dependent postbuckling response of pressurized FGM nanoshells in thermal environments", Arch. Civil Mech. Eng., 17(3), 623-638. https://doi.org/10.1016/j.acme.2017.01.004
  70. Sahmani, S., Aghdam, M.M. and Bahrami, M. (2016), "Sizedependent axial buckling and postbuckling characteristics of cylindrical nanoshells in different temperatures", Int. J. Mech. Sci., 107, 170-179. https://doi.org/10.1016/j.ijmecsci.2016.01.014
  71. Sahmani, S., Fattahi, A.M. and Ahmed, N.A. (2019), "Analytical mathematical solution for vibrational response of postbuckled laminated FG-GPLRC nonlocal strain gradient micro-/nanobeams", Eng. Comput., 35(4), 1173-1189. https://doi.org/10.1007/s00366-018-0657-8
  72. Selim, B.A., Zhang, L.W. and Liew, K.M. (2016), "Vibration analysis of CNT reinforced functionally graded composite plates in a thermal environment based on Reddy's higher-order shear deformation theory", Compos. Struct., 156, 276-290. https://doi.org/10.1016/j.compstruct.2015.10.026
  73. Shen, H.-S., Xiang, Y., Lin, F. and Hui, D. (2017), "Buckling and postbuckling of functionally graded graphene-reinforced composite laminated plates in thermal environments", Compos. Part B: Eng., 119, 67-78. https://doi.org/10.1016/J.COMPOSITESB.2017.03.020
  74. Shingare, K.B. and Kundalwal, S.I. (2019), "Static and dynamic response of graphene nanocomposite plates with flexoelectric effect", Mech. Mater., 134, 69-84. https://doi.org/10.1016/j.mechmat.2019.04.006
  75. Sobhy, M. (2018), "Magneto-electro-thermal bending of FG-graphene reinforced polymer doubly-curved shallow shells with piezoelectromagnetic faces", Compos. Struct., 203, 844-860. https://doi.org/10.1016/j.compstruct.2018.07.056
  76. Talebizadehsardari, P., Eyvazian, A., Asmael, M., Karami, B., Shahsavari, D., and Mahani, R.B. (2020), "Static bending analysis of functionally graded polymer composite curved beams reinforced with carbon nanotubes", Thin Wall. Struct., 157, 107139. https://doi.org/10.1016/j.tws.2020.107139
  77. Thai, C.H., Ferreira, A.J.M. and Phung-Van, P. (2019), "Size dependent free vibration analysis of multilayer functionally graded GPLRC microplates based on modified strain gradient theory", Compos. Part B: Eng., 169, 174-188. https://doi.org/10.1016/j.compositesb.2019.02.048
  78. Van Thu, P. and Duc, N.D. (2016), "Nonlinear stability analysis of imperfect three-phase sandwich laminated polymer nanocomposite panels resting on elastic foundations in thermal environments", VNU J. Sci.: Math. Phys., 32(1), 20-36. http://js.vnu.edu.vn/index.php/MaP/article/view/423
  79. Yang, J., Chen, D. and Kitipornchai, S. (2018), "Buckling and free vibration analyses of functionally graded graphene reinforced porous nanocomposite plates based on Chebyshev-Ritz method", Compos. Struct., 193, 281-294. https://doi.org/10.1016/J.COMPSTRUCT.2018.03.090
  80. Yu, T., Hu, H., Zhang, J. and Bui, T.Q. (2019), "Isogeometric analysis of size-dependent effects for functionally graded microbeams by a non-classical quasi-3D theory", Thin-Wall. Struct., 138, 1-14. https://doi.org/10.1016/j.tws.2018.12.006
  81. Zenkour, A.M. (2016), "Buckling of a single-layered graphene sheet embedded in visco-Pasternak's medium via nonlocal first-order theory", Adv. Nano Res., 4(4), 309-326. https://doi.org/10.12989/anr.2016.4.4.309
  82. Zenkour, A.M. and Hafed, Z.S. (2020), "Bending response of functionally graded piezoelectric plates using a two variable shear deformation theory", Adv. Aircr. Spacecr. Sci., 7(2), 115-134. https://doi.org/10.12989/aas.2020.7.2.115
  83. Zhong, R., Wang, Q., Tang, J., Shuai, C. and Qin, B. (2018), "Vibration analysis of functionally graded carbon nanotube reinforced composites (FG-CNTRC) circular, annular and sector plates", Compos. Struct., 194, 49-67. https://doi.org/10.1016/j.compstruct.2018.03.104
  84. Zhu, P., Lei, Z.X. and Liew, K.M. (2012), "Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with first order shear deformation plate theory", Compos. Struct., 94(4), 1450-1460. https://doi.org/10.1016/j.compstruct.2011.11.010