Smart Structures and Systems

  • 제29권3호
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  • Pages.395-420
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  • 2022
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  • 1738-1584(pISSN)
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  • 1738-1991(eISSN)
테크노프레스 (Techno-Press)
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DOI QR코드

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A new hybrid HSDT for bending, free vibration, and buckling analysis of FGM plates (2D & quasi-3D)

  • Belkhodja, Y. (Laboratory of Science and Technology Environment and Valorization, Faculty of Sciences and Technology, Abdelhamid Ibn Badis University) ;
  • Ouinas, D. (Laboratory of Science and Technology Environment and Valorization, Faculty of Sciences and Technology, Abdelhamid Ibn Badis University) ;
  • Fekirini, H. (Mechanics and physics of materials Laboratory, Mechanical Engineering Department, Faculty of Technology, Djillali Liabes University) ;
  • Olay, J.A. Vina (Materials Science and Metallurgical Engineering Department, University of Oviedo, Viesques Campus) ;
  • Achour, B. (Civil Engineering Department, University of Ha'il, KSA) ;
  • Touahmia, M. (Civil Engineering Department, University of Ha'il, KSA) ;
  • Boukendakdji, M. (Civil Engineering Department, University of Ha'il, KSA)
  • 투고 : 2020.04.29
  • 심사 : 2021.11.24
  • 발행 : 2022.03.25
⟨ 이전 논문 다음 논문 ⟩

초록

A new hybrid quasi-3D and 2D high-order shear deformation theory is studied in this mathematical formulation, for an investigation of the bending, free vibrations and buckling influences on a functionally graded material plate. The theoretical formulation has been begun by a displacement field of five unknowns, governing the transverse displacement across the thickness of the plate by bending, shearing and stretching. The transverse shear deformation effect has been taken into consideration, satisfying the stress-free boundary conditions, especially on plate free surfaces as parabolic variation through its thickness. Thus, the mechanical properties of the functionally graded plate vary across the plate thickness, following three distributions forms: the power law, exponential form and the Mori-Tanaka scheme. The mechanical properties are used to develop the equations of motion, obtained from the Hamilton principle, and solved by applying the Navier-type solution for simply supported boundary conditions. The results obtained are compared with other solutions of 2D, 3D and quasi-3D plate theories have been found in the literature.

키워드

과제정보

The research reported herein was funded by the Deanship of Scientific Research at the University of Hail, Saudi Arabia, through the project number RG- 20098. The authors would like to express their deepest gratitude to the Deanship of Scientific Research and to the College of Engineering at the University of Hail for providing necessary support to conducting this research.

참고문헌

  1. Abualnour, M., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2018), "A novel quasi-3D trigonometric plate theory for free vibration analysis of advanced composite plates", Compos. Struct., 184, 688-697. https://doi.org/10.1016/j.compstruct.2017.10.047
  2. Atmane, H.A., Tounsi, A. and Mechab, I. (2010), "Free vibration analysis offunctionally graded plates resting on Winkler-Pasternak elastic foundations using a new shear deformation theory", Int. J. Mech. Mater. Des., 6(2), 113-121. https://doi.org/10.1007/s10999-010-9110-x
  3. Ameur, M., Tounsi, A., Mechab, I. and El Bedia, A.A. (2011), "A new trigonometric shear deformation theory for bending analysis of functionally graded plates resting on elastic foundations", KSCE J. Civil Eng., 15(8), 1405-1414. https://doi.org/10.1007/s12205-011-1361-z
  4. Amir, S., Arshid, E., Rasti-Alhosseini, S.A. and Loghman, A. (2020), "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
  5. Arani, A.G., Cheraghbak, A. and Kolahchi, R. (2016), "Dynamic buckling of FGM viscoelastic nano-plates resting on orthotropic elastic medium based on sinusoidal shear deformation theory", Struct. Eng. Mech., Int. J., 60(3), 489-505. https://doi.org/10.12989/sem.2016.60.3.489
  6. Arefi, M. and Allam, M.N.M. (2015), "Nonlinear responses of an arbitrary FGP circular plate resting on the Winkler-Pasternak foundation", Smart Struct., Syst., Int. J., 16(1), 81-100. https://doi.org/10.12989/sss.2015.16.1.081
  7. Aydogdu, M. (2009), "A new shear deformation theory for laminated composite plates", Compos. Struct., 89(1), 94-101. https://doi.org/10.1016/j.compstruct.2008.07.008
  8. Baferani, A.H., Saidi, A.R. and Ehteshami, H. (2011), "Accurate solution for free vibration analysis of functionally graded thick rectangular plates resting on elastic foundation", Compos. Struct., 93, 1842-1853. https://doi.org/10.1016/j.compstruct.2011.01.020
  9. Barati, M.R. and Shahverdi, H. (2016), "A four-variable plate theory for thermal vibration of embedded FG nanoplates under non-uniform temperature distributions with different boundary conditions", Struct. Eng. Mech., Int. J., 60(4), 707-727. https://doi.org/10.12989/sem.2016.60.4.707
  10. Belabed, Z., Houari, M.S.A., Tounsi, A., Mahmoud, S.R. and Beg, O.A. (2014), "An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates", Compos. Part B Eng., 60, 274-283. https://doi.org/10.1016/j.compositesb.2013.12.057
  11. Beldjelili, Y., Tounsi, A. and Mahmoud, S.R. (2016), "Hygrothermo-mechanical bending of S-FGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory", Smart Struct., Syst., Int. J., 18(4), 755-786. https://doi.org/10.12989/sss.2016.18.4.755
  12. Belkhodja, Y., Ouinas, D., Zaoui, F.Z. and Fekirini, H. (2019), "An exponential-trigonometric higher order shear deformation theory (HSDT) for bending, free vibration, and buckling analysis of functionally graded materials (FGMs) plates", Adv. Compos. Lett., 28, 1-19. https://doi.org/10.1177/0963693519875739
  13. Benyoucef, S., Mechab, I., Tounsi, A., Fekrar, A. and Atmane, H.A. (2010), "Bending of thick functionally graded plates resting on Winkler-Pasternak elastic foundations", Mech. Compos. Mater., 46, 425-434. https://doi.org/10.1007/s11029-010-9159-5
  14. Bouderba, B., Houari, M.S.A. and Tounsi, A. (2013), "Thermomechanical bending response of FGM thick plates resting on Winkler-Pasternak elastic foundations", Steel Compos. Struct., Int. J., 14(1), 85-104. https://doi.org/10.12989/scs.2013.14.1.085
  15. Bourada, M., Tounsi, A., Houari, M.S.A. and Adda, B.E.A. (2012), "A new four-variable refined plate theory for thermal buckling analysis of functionally graded sandwich plates", J. Sandw. Struct. Mater., 14(1), 5-33. https://doi.org/10.1177%2F1099636211426386 https://doi.org/10.1177%2F1099636211426386
  16. Carrera, E., Brischetto, S. and Robaldo, A. (2008), "Variable kinematic model for the analysis of functionally graded material plates", AIAA J., 46(1), 194-203. https://doi.org/10.2514/1.32490
  17. Carrera, E., Brischetto, S., Cinefra, M. and Soave, M. (2011), "Effects of thickness stretching in functionally graded plates and shells", Compos. Part B, 42(2), 123-133. https://doi.org/10.1016/j.compositesb.2010.10.005
  18. Celebi, K., Yarimpabuc, D. and Keles, I. (2016), "A unified method for stresses in FGM sphere with exponentially-varying properties", Struct. Eng. Mech., Int. J., 57(5), 823-835. https://doi.org/10.12989/sem.2016.57.5.823
  19. Chen, Y.Z. (2018), "Transfer matrix method for solution of FGMs thick-walled cylinder with arbitrary inhomogeneous elastic response", Smart Struct. Syst., Int. J., 21(4), 469-477. https://doi.org/10.12989/sss.2018.21.4.469
  20. Chikh, A., Tounsi, A., Hebali, H. and Mahmoud, S.R. (2017), "Thermal buckling analysis of cross-ply laminated plates using a simplified HSDT", Smart Struct. Syst., Int. J., 19(3), 289-297. https://doi.org/10.12989/sss.2017.19.3.289
  21. Daikh, A.A. and Zenkour, A.M. (2019), "Effect of porosity on the bending analysis of various functionally graded sandwich plates", Mater. Res. Express, 6(6), 065703. https://doi.org/10.1088/2053-1591/ab0971
  22. Darabi, A. and Vosoughi, A.R. (2016), "Hybrid inverse method for small scale parameter estimation of FG nanobeams", Steel Compos. Struct., Int. J., 20(5), 1119-1131. https://doi.org/10.12989/scs.2016.20.5.1119
  23. 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. https://doi.org/10.12989/sss.2017.20.3.351
  24. Ferreira, A.J.M., Batra, R.C., Roque, C.M.C., Qian, L.F. and Martins, P.A.L.S. (2005), "Static analysis of functionally graded plates using third-order shear deformation theory", Compos. Struct., 69, 449-457. https://doi.org/10.1016/j.compstruct.2004.08.003
  25. Fukui, Y. (1991), "Fundamental investigation of functionally gradient material manufacturing system using centrifugal force", JSME Int. J. Ser. 3, Vib., Control Eng., Eng., Ind., 34(1), 144-148. https://doi.org/10.1299/jsmec1988.34.144
  26. Fukushima, T., Kuroda, S. and Kitahara, S. (1990), "Gradient coatings formed by plasma twin torches and those properties", Proceedings of the First International Symposium on Functionally Gradient Materials, Tokyo, Japan, pp. 145-150.
  27. Ghugal, Y.M. and Sayyad, A.S. (2010), "A static flexure of thick isotropic plates using trigonometric shear deformation theory", J. Solid Mech., 2(1), 79-90.
  28. Guerroudj, H.Z., Yeghnem, R., Kaci, A., Zaoui, F.Z., Benyoucef, S. and Tounsi, A. (2018), "Eigenfrequencies of advanced composite plates using an efficient hybrid quasi-3D shear deformation theory", Smart Struct. Syst., Int. J., 22(1), 121-132. https://doi.org/10.12989/sss.2018.22.1.121
  29. Hanifi, H.A.L., Kaci, A. and Tounsi, A. (2017), "On the size-dependent behavior of functionally graded micro-beams with porosities", Struct. Eng. Mech., Int. J., 64(5), 527-541. https://doi.org/10.12989/scs.2017.64.5.527
  30. Hebali, H., Tounsi, A., Houari, M.S.A., Bessaim, A. and Bedia, E.A.A. (2014), "New quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates", (ASCE). J. Eng. Mech., 140(2), 374-383. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000665
  31. Hebbar, N., Hebbar, I., Ouinas D. and Bourada, M. (2020), "Numerical modeling of bending, buckling, and vibration of functionally graded beams by using a higher-order shear deformation theory", Frattura ed Integrita Strutturale, 14(52), 230-246. https://doi.org/10.3221/IGF-ESIS.52.18
  32. Hosseini-Hashemi, S., Fadaee, M. and Atashipour, S.R. (2011a), "Study on the free vibration of thick functionally graded rectangular plates according to a new exact closed form procedure", Compos. Struct., 93(2), 722-735. https://doi.org/10.1016/j.compstruct.2010.08.007
  33. Hosseini-Hashemi, S., Fadaee, M. and Atashipour, S.R. (2011b), "A new exact analytical approach for free vibration of Reissner-Mindlin functionally graded rectangular plates", Int. J. Mech. Sci., 53(1), 11-22. https://doi.org/10.1016/j.ijmecsci.2010.10.002
  34. Hosseini-Hashemi, S., Salehipour, H. and Atashipour, S.R. (2012), "Exact three-dimensional free vibration analysis of thick homogeneous plates coated by a functionally graded layer", Acta Mech., 223, 2153-2166. https://doi.org/10.1007/s00707-012-0683-3
  35. Iurlaro, L., Gherlone, M. and Di Sciuva, M. (2014), "Bending and free vibration analysis of functionally graded sandwich plates using the Refined Zigzag theory", J Sandw. Struct. Mater., 16(6), 669-699. https://doi.org/10.1177/1099636214548618
  36. Jha, D.K., Tarun, K. and Singh, R.K. (2012), "Higher order shear and normal deformation theory for natural frequency of functionally graded rectangular plates", Nucl. Eng. Des., 250, 8-13. https://doi.org/10.1016/j.nucengdes.2012.05.001
  37. Kar, V.R. and Panda, S.K. (2016a), "Post-buckling behaviour of shear deformable functionally graded curved shell panel under edge compression", Int. J. Mech. Sci., 115-116, 318-324. https://doi.org/10.1016/j.ijmecsci.2016.07.014
  38. Kar, V.R. and Panda, S.K. (2016b), "Geometrical nonlinear free vibration analysis of FGM spherical panel under nonlinear thermal loading with TD and TID properties", J. Therm. Stress., 39(8), 942-959. https://doi.org/10.1080/01495739.2016.1188623
  39. Kawasaki, A. and Watanabe, R. (1988), "Powder metallurgical fabrication of the thermal-stress relief type of functionally gradient materials", Sintering'87, London, UK, Volume 2, pp. 1197-1202.
  40. Kiran, M.C. and Kattimani, S.C. (2018), "Free vibration and static analysis of functionally graded skew magneto-electro-elastic plate", Smart Struct. Syst., Int. J., 21(4), 493-519. https://doi.org/10.12989/sss.2018.21.4.493
  41. Koizumi, M. (1997), "FGM activities in Japan", Compos. Part B, 28(1-2), 1-4. https://doi.org/10.1016/S1359-8368(96)00016-9
  42. Kolahchi, R., Safari, M. and Esmailpour, M. (2016), "Dynamic stability analysis of temperature-dependent functionally graded CNT-reinforced visco-plates resting on orthotropic elastomeric medium", Compos. Struct., 150, 255-265. https://doi.org/10.1016/j.compstruct.2016.05.023
  43. Kolahchi, R., Zarei, M.S., Hajmohammad, M.H. and Oskouei, A.N. (2017), "Visco-nonlocal-refined Zigzag theories for dynamic buckling of laminated nanoplates using differential cubature-Bolotin methods", Thin-Wall. Struct., 113, 162-169. https://doi.org/10.1016/j.tws.2017.01.016
  44. Lal, R. and Saini, R. (2020), "Vibration analysis of FGM circular plates under non-linear temperature variation using generalized differential quadrature rule", Appl. Acoustics, 158, 107027. https://doi.org/10.1016/j.apacoust.2019.107027
  45. Le, C.I., Pham, V.N. and Nguyen, D.K. (2020), "Free vibration of FGSW plates partially supported by Pasternak foundation based on refined shear deformation theories", Math. Problems Eng., 13 p. https://doi.org/10.1155/2020/7180453
  46. Madani, H., Hosseini, H. and Shokravi, M. (2016), "Differential cubature method for vibration analysis of embedded FG-CNT-reinforced piezoelectric cylindrical shells subjected to uniform and non-uniform temperature distributions", Steel Compos. Struct., Int. J., 22(4), 889-913. https://doi.org/10.12989/scs.2016.22.4.889
  47. Mahesh, V., Kattimani, S., Harursampath, D. and Trung, N.T. (2019), "Coupled evaluation of the free vibration characteristics of magneto-electro-elastic skew plates in hygrothermal environment", Smart Struct. Syst., Int. J., 24(2), 267-292. https://doi.org/10.12989/sss.2019.24.2.267
  48. Mantari, J.L. and Soares, C.G. (2012a), "Bending analysis of thick exponentially graded plates using a new trigonometric higher order shear deformation theory", Compos. Struct., 94(6), 1991-2000. https://doi.org/10.1016/j.compstruct.2012.01.005
  49. Mantari, J.L. and Soares, C.G. (2012b), "Generalized hybrid quasi-3D shear deformation theory for the static analysis of advanced composite plates", Compos. Struct., 94(8), 2561-2575. https://doi.org/10.1016/j.compstruct.2012.02.019
  50. Mantari, J.L. and Soares, C.G. (2013), "A novel higher-order shear deformation theory with stretching effect for functionally graded plates", Compos. Part B, 45(1), 268-281. https://doi.org/10.1016/j.compositesb.2012.05.036
  51. Mantari, J.L. and Soares, C.G. (2014), "Optimized sinusoidal higher order shear deformation theory for the analysis of functionally graded plates and shell", Compos. Part B, 56, 126-136. https://doi.org/10.1016/j.compositesb.2013.07.027
  52. Mantari, J.L., Oktem, A.S. and Soares, C.G. (2012), "Bending response of functionally graded plates by using a new higher order shear deformation theory", Compos. Struct., 94(2), 714-723. https://doi.org/10.1016/j.compstruct.2011.09.007
  53. Matsunaga, H. (2008), "Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory", Compos. Struct., 82(4), 499-512. https://doi.org/10.1016/j.compstruct.2007.01.030
  54. Meftah, A., Bakora, A., Zaoui, F.Z., Tounsi, A. and Adda Bedia, E.A. (2017), "A non-polynomial four variable refined plate theory for free vibration of functionally graded thick rectangular plates on elastic foundation", Steel Compos. Struct., Int. J., 23(3), 317-330. https://doi.org/10.12989/scs.2017.23.3.317
  55. Miyamoto, Y., Nakanishi, H., Tanaka, I., Okamoto, T. and Yamada, O. (1990), "Gas pressure combustion sintering of TiC-Ni FGM", Proceedings of the First International Symposium, FGM, Tokyo, Japan, pp. 257-262.
  56. Mohammadimehr, M. and Alimirzaei, S. (2017), "Buckling and free vibration analysis of tapered FG-CNTRC micro Reddy beam under longitudinal magnetic field using FEM", Smart Struct. Syst., Int. J., 19(3), 309-322. https://doi.org/10.12989/sss.2017.19.3.309
  57. Neves, A.M.A., Ferreira, A.J.M., Carrera, E., Roque, C.M.C., Cinefra, M., Jorge, R.M.N. and Soares, C.M.M. (2011), "Bending of FGM plates by a sinusoidal plate formulation and collocation with radial basis functions", Mech. Res. Commun., 38(5), 368-371. https://doi.org/10.1016/j.mechrescom.2011.04.011
  58. Neves, A.M.A., Ferreira, A.J.M., Carrera, E., Roque, C.M.C., Cinefra, M., Jorge, R.M.N. and Soares, C.M.M. (2012a), "A quasi-3D sinusoidal shear deformation theory for the static and free vibration analysis of functionally graded plates", Compos. Part B Eng., 43(2), 711-725. https://doi.org/10.1016/j.compositesb.2011.08.009
  59. Neves, A.M.A., Ferreira, A.J.M., Carrera, E., Cinefra, M., Roque, C.M.C., Jorge, R.M.N. and Soares, C.M.M. (2012b), "A quasi3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates", Compos. Struct., 94(5), 1814-1825. https://doi.org/10.1016/j.compstruct.2011.12.005
  60. Neves, A.M.A., Ferreira, A.J.M., Carrera, E., Cinefra, M., Roque, C.M.C., Jorge, R.M.N. and Soares, C.M. (2013), "Static, free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higher-order shear deformation theory and a meshless technique", Compos. Part B, 44(1), 657-674. https://doi.org/10.1016/j.compositesb.2012.01.089
  61. Nguyen, T.K. (2014), "A higher-order hyperbolic shear deformation plate model for analysis of functionally graded materials", Int. J. Mech. Mater. Des., 11(2), 203-219. https://doi.org/10.1007/s10999-014-9260-3
  62. Niino, A. and Maeda, S. (1990), "Recent development status of functionally gradient materials", ISIJ Int., 30(9), 699-703. https://doi.org/10.2355/isijinternational.30.699
  63. Qian, L.F., Batra, R.C. and Chen, L.M. (2004), "Static and dynamic deformations of thick functionally graded elastic plates by using higher-order shear and normal deformable plate theory and meshless local Petrov-Galerkin method", Compos. Part B, 35(6-8), 685-697. https://doi.org/10.1016/j.compositesb.2004.02.004
  64. Raminnea, M., Biglari, H. and Tahami, F.V. (2016), "Nonlinear higher order Reddy theory for temperaturedependent vibration and instability of embedded functionally graded pipes conveying fluid-nanoparticle mixture", Struct. Eng. Mech., Int. J., 59(1), 153-186. https://doi.org/10.12989/sem.2016.59.1.153
  65. Reddy, J.N. (2000), "Analysis of functionally graded plates", Int. J. Numer. Methods Eng., 47(1-3), 663-684. https://doi.org/10.1002/(SICI)1097-0207(20000110/30)47:1/3<663::AID-NME787>3.0.CO; 2-8
  66. Sasaki, M., Wang, Y., Hirano, T. and Hirai, T. (1989), "Design of SiC/C functionally gradient material and its preparation by chemical vapor deposition", J. Ceram. Soc. Japan, 97(1125), 539-543. https://doi.org/10.2109/jcersj.97.539
  67. Sata, N., Sanada, N., Hirano, T. and Niino, M. (1990), "Fabrication of a functionally gradient material by using a self-propagating reaction process", Proceedings of the First Int. Symp. On Combustion and Plasma Synthesis of High-Temperature Materials, pp. 195-203.
  68. Shimoda, N., Kitaguchi, S., Saito, T., Takigawa, H. and Koga, M. (1990), "Production of functionally gradient materials by applying low pressure plasma spray", Proceedings of the First International Symposium on Funtionally Gradient Materials, Sendai, Tokyo, Japan, pp. 151-156.
  69. Shokravi, M. and Jalili, N. (2017), "Dynamic buckling response of temperature-dependent functionally graded-carbon nanotubes-reinforced sandwich microplates considering structural damping", Smart Struct. Syst., Int. J., 20(5), 583-593. https://doi.org/10.12989/sss.2017.20.5.583
  70. Shufrin, I. and Eisenberger, M. (2005), "Stability and vibration of shear deformable plates-first order and higher order analyses", Int. J. Solids Struct. 42(3-4), 1225-1251. https://doi.org/10.1016/j.ijsolstr.2004.06.067
  71. Singh, V.K. and Panda, S.K. (2015), "Large amplitude free vibration analysis of laminated composite spherical shells embedded with piezoelectric layers", Smart Struct. Syst., Int. J., 16(5), 853-872. https://doi.org/10.12989/sss.2015.16.5.853
  72. Talha, M. and Singh, B.N. (2010), "Static response and free vibration analysis of FGM plates using higher order shear deformation theory", Appl. Math. Modell., 34(12), 3991-4011. https://doi.org/10.1016/j.apm.2010.03.034
  73. Thai, H.T. and Kim, S.E. (2013), "A simple higher-order shear deformation theory for bending and free vibration analysis of functionally graded plates", Compos. Struct., 96, 165-173. https://doi.org/10.1016/j.compstruct.2012.08.025
  74. Thai, H.T. and Vo, T.P. (2013), "A new sinusoidal shear deformation theory for bending, buckling, and vibration of functionally graded plates", Applied Math. Modell., 37(5), 3269-3281. https://doi.org/10.1016/j.apm.2012.08.008
  75. Tornabene, F. (2009), "Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution", Comput. Meth. Appl. Mech. Eng., 198(37-40), 2911-2935. https://doi.org/10.1016/j.cma.2009.04.011
  76. Trinh, T.H., Nguyen, D.K., Gan, B.S. and Alexandrov, S. (2016), "Post-buckling responses of elastoplastic FGM beams on nonlinear elastic foundation", Struct. Eng. Mech., Int. J., 58(3), 515-532. https://doi.org/10.12989/sem.2016.58.3.515
  77. Vaghefi, R., Baradaran, G.H. and Koohkan, H. (2010), "Three-dimensional static analysis of thick functionally graded plates by using meshless local Petrov - Galerkin (MLPG) method", Eng. Anal. Bound. Elem., 34(6), 564-573. https://doi.org/10.1016/j.enganabound.2010.01.005
  78. Vel, S.S. and Batra, R.C. (2004), "Three-dimensional exact solution for the vibration of functionally graded rectangular plates", J. Sound Vib., 272(3-5), 703-730. https://doi.org/10.1016/S0022-460X(03)00412-7
  79. Wu, C.P. and Li, H.Y. (2010), "An RMVT-based third-order shear deformation theory of multilayered functionally graded material plates", Compos. Struct., 92(10), 2591-2605. https://doi.org/10.1016/j.compstruct.2010.01.022
  80. Wu, C.P., Chiu, K.H. and Wang, Y.M. (2011), "RMVT-based meshless collocation and element free Galerkin methods for the quasi-3D analysis of multilayered composite and FGM plates", Compos. Struct., 93(2), 923-943. https://doi.org/10.1016/j.compstruct.2010.11.015
  81. Xiang, S. and Kang, G.W. (2013), "A nth-order shear deformation theory for the bending analysis on the functionally graded plates", Eur. J. Mech. A/Solids, 37, 336-343. https://doi.org/10.1016/j.euromechsol.2012.08.005
  82. Xiang, S., Kang, G.W., Yang, M.S. and Zhao, Y. (2013), "Natural frequencies of sandwich plate with functionally graded face and homogeneous core", Compos. Struct., 96, 226-231. https://doi.org/10.1016/j.compstruct.2012.09.003
  83. Younsi, A., Tounsi, A., Zaoui, F.Z., Bousahla, A.A. and Mahmoud S.R. (2018), "Novel quasi-3D and 2D shear deformation theories for bending and free vibration analysis of FGM plates", Geomech. Eng., Int. J., 14(6), 519-532. https://doi.org/10.12989/gae.2018.14.6.519
  84. Yuki, M., Murayama, T., Irisawa, T., Kawasaki, A. and Watanabe, R. (1990), "FGM'90", Proceedings of the 1st International Symposium on Functionally Gradient Materials, Sendai, FGM Forum, Tokyo, Japan, pp. 203-208.
  85. Zafarmand, H. and Kadkhodayan, M. (2015), "Three dimensional elasticity solution for static and dynamic analysis of multidirectional functionally graded thick sector plates with general boundary conditions", Compos. Part B, 69, 592-602. https://doi.org/10.1016/j.compositesb.2014.10.048
  86. Zaoui, F.Z., Tounsi, A. and Ouinas, D. (2017), "Free vibration of functionally graded plates resting on elastic foundations based on quasi-3D hybrid-type higher order shear deformation theory", Smart Struct. Syst., Int. J., 20(4), 509-524. https://doi.org/10.12989/sss.2017.20.4.509
  87. 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. Part B, 159, 231-247. https://doi.org/10.1016/j.compositesb.2018.09.051
  88. Zaoui, F.Z., Tounsi, A., Ouinas, D. and Olay, J.A.V. (2020), "A refined HSDT for bending and dynamic analysis of FGM plates", Struct. Eng. Mech., Int. J., 74(1), 105-119. https://doi.org/10.12989/sem.2020.74.1.105
  89. Zenkour, A.M. (2006), "Generalized shear deformation theory for bending analysis of functionally graded plates", Appl. Math. Model., 30(1), 67-84. https://doi.org/10.1016/j.apm.2005.03.009
  90. Zenkour, A.M. (2007), "Benchmark trigonometric and 3-D elasticity solutions for an exponentially graded thick rectangular plate", Arch. Appl. Mech., 77(4), 197-214. https://doi.org/10.1007/s00419-006-0084-y
  91. Zenkour, A.M. (2009), "The refined sinusoidal theory for FGM plates on elastic foundations", Int. J. Mech. Sci., 51(11-12), 869-880. https://doi.org/10.1016/j.ijmecsci.2009.09.026
  92. Zhang, H., Jiang, J.K. and Zhang, Z.C. (2014), "Threedimensional elasticity solutions for bending of generally supported thick functionally graded plates", Appl. Math. Mech., 35(11), 1467-1478. https://doi.org/10.1007/s10483-014-1871-7
  93. Zhou, D., Cheung, Y.K., Au, F.T.K. and Lo, S.H. (2002), "Threedimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method", Int. J. Solids Struct., 39, 6339-6353. https://doi.org/10.1016/S0020-7683(02)00460-2
  94. Zidi, M., Tounsi, A., Houari, M.S.A., Adda Bedia, E.A. and Anwar Beg, O. (2014), "Bending analysis of FGM plates under hygrothermo-mechanical loading using a four variable refined plate theory", Aerosp. Sci. Technol., 34, 24-34. https://doi.org/10.1016/j.ast.2014.02.001