A review on analytical models for functionally graded structures

Published Jul 31, 2025
https://doi.org/10.46223/HCMCOUJS.acs.en.15.2.74.2025
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Received: 2025-04-22
Accepted: 2025-06-02
Published: 2025-07-31
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Imene Ait Sidhoum - aitsidhoum.imene@cuniv-naama.dz
Artificial Intelligence Laboratory for Mechanical and Civil Structures and Soil University, Algeria; Material and Hydrology Laboratory University of Sidi Bel Abbes, Algeria
Fouad Bourada
Material and Hydrology Laboratory University of Sidi Bel Abbes, Algeria
Abdelouahed Tounsi
Material and Hydrology Laboratory University of Sidi Bel Abbes, Algeria; King Fahd University of Petroleum & Minerals, Saudi Arabia

Abstract

The Functionally Graded Materials (FGMs) are considered an advanced composite, as initially proposed in Japan. The most interesting advantage of this type of material is the continuity in a desired direction of the structural component (Shell, plate, or beam). Functionally Graded (FG) plates find applications in various engineering sectors and industries, including mechanical, biomedical, nuclear, and aerospace engineering. This increase in technical applications of FGMs has attracted the attention of many scientists. In this paper, the various analytical theories named Equivalent Single Layer (ESL) and three-dimensional (3D) elasticity are presented and explained. Also, bibliographical research of the behaviors of the laminated composite, FGM in one- and two-dimensional structures, such as (beam, plate/shell) under various loading types, axial, transversal, thermal, and thermomechanical.

Keywords

3D elasticity, ESL, functionally graded materials, laminated composite, thermomechanical

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This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

How to Cite
Ait sidhoum, I., Bourada, F., & Tounsi, A. (2025). A review on analytical models for functionally graded structures. HCMCOU Journal of Science - Advances in Computational Structures, 15(2), 41–54. https://doi.org/10.46223/HCMCOUJS.acs.en.15.2.74.2025
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References

  1. Abrate, S. (2008). Functionally graded plates behave like homogeneous plates. Composite Part B-Engineering, 39(1), 151-158. DOI: https://doi.org/10.1016/j.compositesb.2007.02.026
  2. Akbarzadeh, A. H., Zad, S. H., Eslami, M. R., & Sadighi, M. (2011). Mechanical behaviour of functionally graded plates under static and dynamic loading. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 225(2), 326-333. DOI: https://doi.org/10.1243/09544062JMES2111
  3. Akbaş, Ş. D. (2018). Forced vibration analysis of functionally graded porous deep beams. Composite Structures, 186, 293-302. https://doi.org/10.1016/j.compstruct.2017.12.013 DOI: https://doi.org/10.1016/j.compstruct.2017.12.013
  4. Alibeigloo, A. (2010). Exact solution for thermo-elastic response of functionally graded rectangular plates. Composite Structure, 92(1),113-21. DOI: https://doi.org/10.1016/j.compstruct.2009.07.003
  5. Alizadeh, M., & Fattahi, A. M. (2019). Non-classical plate model for FGMs. Engineering with Computers, 35(1), 215-228. https://doi.org/10.1007/s00366-018-0594-6 DOI: https://doi.org/10.1007/s00366-018-0594-6
  6. Amini, M. H., Soleimani, M., & Rastgoo, A. (2009). Three-dimensional free vibration analysis of functionally graded material plates resting on an elastic foundation. Smart Materials and Structures, 18(8), Article 085015. DOI: https://doi.org/10.1088/0964-1726/18/8/085015
  7. Avcar, M. (2019). Free vibration of imperfect sigmoid and power law functionally graded beams. Steel and Composite Structures, 30(6), 603-615. https://doi.org/10.12989/scs.2019.30.6.603
  8. Aydogdu, M. (2007). Three-dimensional vibration analysis of functionally graded plates under various boundary conditions. Composites of Reinforced Plastics and Composites, 26(18), 1847-63.
  9. Baferani, A. H., Saidi, A. R., & Ehteshami, H. (2011). Accurate solution for free vibration analysis of functionally graded thick rectangular plates resting on elastic foundation. Composite Structures, 93(7), 1842-1853. DOI: https://doi.org/10.1016/j.compstruct.2011.01.020
  10. Batra, R. C., & Vidoli, S. (2002). Higher-order piezoelectric plate theory derived from a three-dimensional variational principle. AIAA Journal, 40, Article 2. DOI: https://doi.org/10.2514/3.15002
  11. Bodaghi, M., & Saidi, A. (2010). Levy-type solution for buckling analysis of thick functionally graded rectangular plates based on the higher-order shear deformation plate theory. Applied Mathematical Modelling, 34(11), 3659-3673. DOI: https://doi.org/10.1016/j.apm.2010.03.016
  12. Bourada, F., Amara, K., & Tounsi, A. (2016). Buckling analysis of isotropic and orthotropic plates using a novel four variable refined plate theory. Steel and Composite Structures, 21(6), 1287-1306. DOI: https://doi.org/10.12989/scs.2016.21.6.1287
  13. Chakraverty, S., & Pradhan, K. K. (2014). Free vibration of functionally graded thin rectangular plates resting on Winkler elastic foundation with general boundary conditions using Rayleigh-Ritz method. International Journal of Applied Mechanics , 6(4), Article 1450043. DOI: https://doi.org/10.1142/S1758825114500434
  14. Chen, C. S., Hsu, C. Y., & Tzou, G. J. (2009). Vibration and stability of functionally graded plates based on a higher-order deformation theory. Journal of Reinforced Plastics and Composites, 28(10), 1215-1234. DOI: https://doi.org/10.1177/0731684408088884
  15. Cheng, Z. Q., & Batra, R. C. (2000). Deflection relationship between the homogenous Kirchhoff plate theory and different functionally graded plates theories. Archive of Applied Mechanics, 52(1), 143-158.
  16. Croce, L. D., & Venini, P. (2004). Finite elements for functionally graded Reissner-Mindlinplates. Computer Methods in Applied Mechanics and Engineering, 193(9), 705-725. DOI: https://doi.org/10.1016/j.cma.2003.09.014
  17. Ding, X. H., Liu, H. B., She, G. L., & Fei, W. (2023). Wave propagation of FG-CNTRC plates in thermal environment using the high-order shear deformation plate theory. Computers and Concrete, 32(2), 207-215. https://doi.org/10.12989/cac.2023.32.2.207
  18. Ebrahimi, F., & Dashti, S. (2015). Free vibration analysis of a rotating non-uniform functionally graded beam. Steel and Composite Structures, 19(5), 1279-1298. https://doi.org/10.12989/scs.2015.19.5.1279 DOI: https://doi.org/10.12989/scs.2015.19.5.1279
  19. Eltaher, M. A., Khater, M. E., Park, S., Abdel-Rahman, E., & Yavuz, M. (2016). On the static stability of nonlocal nanobeams using higher-order beam theories. Advances in Nano Research, 4(1), 51-64. DOI: https://doi.org/10.12989/anr.2016.4.1.051
  20. Eringen, A. C. (1972). Nonlocal polar elastic continua. International Journal of Engineering Science, 10(1), 1-16.  https://doi.org/10.1016/0020-7225(72)90070-5   DOI: https://doi.org/10.1016/0020-7225(72)90070-5
  21. Eringen, A. C. (1983). On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54(9), 4703-4710. https://doi.org/10.1063/1.332803 DOI: https://doi.org/10.1063/1.332803
  22. Fattahi, A. M., Sahmani, S., & Ahmed, N. A. (2019). Nonlocal strain gradient beam model for nonlinear secondary resonance analysis of functionally graded porous micro/nanobeams under periodic hard excitations. Mechanics Based Design of Structures and Machines, 48(4), 403-432. https://doi.org/10.1080/15397734.2019.1624176 DOI: https://doi.org/10.1080/15397734.2019.1624176
  23. Fazeli, H., Adamian, A., & Hosseini-Sianaki, A. (2022). Vibration analysis of multi-directional functionally graded nanoplates with initial geometric imperfection using Quasi-3d theory based on an Isogeometric approach. Journal of Solid and Fluid Mechanics, 11(6), 111-126. https://doi.org/10.22044/JSFM.2022.10631.3355
  24. Feldman, E., & Aboudi, J. (1997). Buckling analysis of functionally graded plates subjected to uniaxial loading. Composite Structures, 38(1), 29-36. DOI: https://doi.org/10.1016/S0263-8223(97)00038-X
  25. Foroughi, H., & Azhari, M. (2014). Mechanical buckling and free vibration of thick functionally graded plates resting on elastic foundation using the higher order B-spline finite strip method. Meccanica An International Journal of Theoretical and Applied Mechanics AIMETA, 49(4), 981-993. DOI: https://doi.org/10.1007/s11012-013-9844-2
  26. Ganapathi, M., Prakash, T., & Sundararajan, N. (2006). Influence of functionally graded material on buckling of skew plates under mechanical loads. Journal of Engineering Mechanics, 132(8), 902-905. DOI: https://doi.org/10.1061/(ASCE)0733-9399(2006)132:8(902)
  27. Ghannadpour, S. A. M, & Alinia, M. M. (2006). Large deflection behavior of functionally graded plates under pressure loads. Composite Structures, 75(1/4), 67-71. DOI: https://doi.org/10.1016/j.compstruct.2006.04.004
  28. Gilhooley, D. F., Batra, R. C., Xiao, J. R., McCarthy, M. A., & Gillespie, Jr. J. W. (2007). Analysis of thick functionally graded plates by using higher-order shear and normal deformable plate theory and MLPG method with radial basis functions. Composite Structures, 75(1/4), 67-71. DOI: https://doi.org/10.1016/j.compstruct.2006.07.007
  29. Gulshan, T. M. N. A., Chakrabarti, A., & Sheikh, A. H.  (2013). Analysis of functionally graded plates using higher order shear deformation theory. Applied Mathematical Modelling, 37(18/19), 8484-8494. DOI: https://doi.org/10.1016/j.apm.2013.03.058
  30. Hadji, L., Madan, R., Atmane, H. A., Bernard, F., Zouatnia, N., & Safa, A. (2024). Thermal buckling Analysis of functionally graded plates using trigonometric shear deformation theory for temperature-dependent material properties. Structural Engineering and Mechanics, 91(6), 539-549. https://doi.org/10.12989/SEM.2024.91.6.539
  31. Haj mohammad, M. H., Azizkhani, M. B., & Kolahchi, R. (2018). Multiphase nanocomposite viscoelastic laminated conical shells subjected to magneto-hygrothermal loads: Dynamic buckling analysis. International Journal of Mechanical Sciences, 137, 205-213. DOI: https://doi.org/10.1016/j.ijmecsci.2018.01.026
  32. Haj mohammad, M. H., Farrokhian, A., & Kolahchi, R. (2018). Smart control and vibration of viscoelastic actuator-multiphase nanocomposite conical shells-sensor considering hygrothermal load based on layerwise theory. Aerospace Science and Technology, 78, 260-270. DOI: https://doi.org/10.1016/j.ast.2018.04.030
  33. Haj mohammad, M. H., Kolahchi, R., Zarei, M. S., & Maleki, M. (2018). Earthquake induced dynamic deflection of submerged viscoelastic cylindrical shell reinforced by agglomerated CNTs considering thermal and moisture effects. Composite Structures, 187, 498-508. DOI: https://doi.org/10.1016/j.compstruct.2017.12.004
  34. Haj mohammad, M. H., Maleki, M., & Kolahchi, R. (2018). Seismic response of underwater concrete pipes conveying fluid covered with nano-fiber reinforced polymer layer. Soil Dynamics and Earthquake Engineering, 110, 18-27. DOI: https://doi.org/10.1016/j.soildyn.2018.04.002
  35. Haj mohammad, M. H., Zarei, M. S., Nouri, A., & Kolahchi, R. (2017). Dynamic buckling of sensor/functionally graded-carbon nanotube-reinforced laminated plates/actuator based on sinusoidal-visco-piezoelasticity theories. Journal of Sandwich Structures & Materials. https://doi.org/10.1177/1099636217720373 DOI: https://doi.org/10.1177/1099636217720373
  36. Hosseini-Hashemi, S., Fadaee, M., & Atashipour, S. R. (2011). A new exact analytical approach for free vibration of Reissner-Mindlin functionally graded rectangular plates. International Journal of Mechanical Sciences, 53(1), 11-22. DOI: https://doi.org/10.1016/j.ijmecsci.2010.10.002
  37. Hosseini-Hashemi, S., Rokni Damavandi Taher, H., Akhavan, H., & Omidi, M. (2010). Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory. Applied Mathematical Modelling, 34(5), 1276-1291. DOI: https://doi.org/10.1016/j.apm.2009.08.008
  38. Hu, Y., & Zhang, X. (2011). Parametric vibrations and stability of a functionally graded plate. Mechanics Based Design of Structures and Machines, 39(3),367-377. DOI: https://doi.org/10.1080/15397734.2011.557970
  39. Jari, H., Atri, H. R., & Shojaee, S. (2015). Nonlinear thermal analysis of functionally graded material plates using a NURBS based Isogeometric approach. Composite Structures, 119, 333-345. DOI: https://doi.org/10.1016/j.compstruct.2014.09.006
  40. Javaheri, R., & Eslami, M. (2002). Buckling of functionally graded plates under in-plane compressive loading. Journal of Applied Mathematics and Mechanics, 82(4), 277-283.  https://doi.org/10.1002/1521-4001(200204)82:4%3C277::AID-ZAMM277%3E3.0.CO;2-Y DOI: https://doi.org/10.1002/1521-4001(200204)82:4<277::AID-ZAMM277>3.0.CO;2-Y
  41. Jha, D. K., Kant, T., & Singh, R. K. (2013). Free vibration response of functionally graded thick plates with shear and normal deformations effects. Composite Structures, 96, 799-823. https://doi.org/10.1016/j.compstruct.2012.09.034 DOI: https://doi.org/10.1016/j.compstruct.2012.09.034
  42. Kar, V. R., & Panda, S. K. (2015). Nonlinear flexural vibration of shear deformable functionally graded spherical shell panel. Steel and Composite Structures, 18(3), 693-709. https://doi.org/10.12989/scs.2015.18.3.693 DOI: https://doi.org/10.12989/scs.2015.18.3.693
  43. Karamanli, A., & Vo, T. P. (2021). A Quasi-3D theory for functionally graded porous microbeams based on the modified strain gradient theory. Composite Structures, 257, Article 113066.  https://doi.org/10.1016/j.compstruct.2020.113066 DOI: https://doi.org/10.1016/j.compstruct.2020.113066
  44. Karamanli, A., Vo, T. P., & Eltaher, M. A. (2024). Comprehensive analysis of bio-inspired laminated composites plates using a Quasi-3D theory and higher order FE models. Thin-Walled Structures, Article 111735.  https://doi.org/10.1016/j.tws.2024.111735 DOI: https://doi.org/10.1016/j.tws.2024.111735
  45. Karami, B., & Janghorban, M. (2016). Effect of magnetic field on the wave propagation in nanoplates based on strain gradient theory with one parameter and two-variable refined plate theory. Modern Physics Letters B, 30(36), Article 1650421. https://doi.org/10.1142/S0217984916504212 DOI: https://doi.org/10.1142/S0217984916504212
  46. Karami, B., Janghorban, M., & Tounsi, A. (2018). Variational approach for wave dispersion in anisotropic doubly-curved nanoshells based on a new nonlocal strain gradient higher order shell theory. Thin-Walled Structures, 129, 251-264. https://doi.org/10.1016/j.tws.2018.02.025  DOI: https://doi.org/10.1016/j.tws.2018.02.025
  47. Karami, B., Shahsavari, D., & Li, L. (2018). Hygrothermal wave propagation in viscoelastic graphene under in-plane magnetic field based on nonlocal strain gradient theory. Physica E: Low-dimensional Systems and Nanostructures, 97, 317-327. https://doi.org/10.1016/j.physe.2017.11.020 DOI: https://doi.org/10.1016/j.physe.2017.11.020
  48. Karami, B., Shahsavari, D., Janghorban, M., & Li, L. (2018). Wave dispersion of mounted graphene with initial stress. Thin-Walled Structures, 122, 102-111. DOI: https://doi.org/10.1016/j.tws.2017.10.004
  49. Kiani, Y., Bagherizadeh, E., & Eslami, M. R. (2011). Thermal buckling of clamped thin rectangular FGM plates resting on Pasternak elastic foundation (Three approximate analytical solutions), ZAMM. Journal of Applied Mathematics and Mechanics, 91(7), 581-93. https://doi.org/10.1002/zamm.201000184 DOI: https://doi.org/10.1002/zamm.201000184
  50. Kiani, Y., Shakeri, M., & Eslami, M. R. (2012). Thermoelastic free vibration and dynamic behaviour of an FGM doubly curved panel via the analytical hybrid Laplace-Fourier transformation. Acta Mechanica, 223, 1199-1218. DOI: https://doi.org/10.1007/s00707-012-0629-9
  51. Kim, K. D., Lomboy, G. R., & Han, S. C. (2008). Geometrically nonlinear analysis of Functionally Graded Material (FGM) plates and shells using a four-node quasi-conforming shell element. Journal of Composite Materials, 42(5), 485-511. https://doi.org/10.1177/0021998307086211 DOI: https://doi.org/10.1177/0021998307086211
  52. Kim, S. E., Thai, H. T., & Lee, J. (2009). Buckling analysis of plates using the two variable refined plate theory. Thin-Walled Structures, 47, 455-462. DOI: https://doi.org/10.1016/j.tws.2008.08.002
  53. Kirchhoff, G. R. (1850). Uber das gleichgewicht und die bewegung einer elastischen scheibe [On the equilibrium and motion of an elastic disc]. Journal für die reine und angewandte Mathematik (Crelle), 40, 51-88. DOI: https://doi.org/10.1515/crll.1850.40.51
  54. Kolahchi, R. (2017). A comparative study on the bending, vibration and buckling of viscoelastic sandwich nanoplates based on different nonlocal theories using DC, HDQ and DQ methods. Aerospace Science and Technology, 66, 235-248. https://doi.org/10.1016/j.ast.2017.03.016 DOI: https://doi.org/10.1016/j.ast.2017.03.016
  55. Kolahchi, R., Keshtegar, B., & Fakhar, M. H. (2017). Optimization of dynamic buckling for sandwich nanocomposite plates with sensor and actuator layer based on sinusoidal-visco-piezoelasticity theories using Grey Wolf algorithm. Journal of Sandwich Structures & Materials. https://doi.org/10.1177/109963621773107 DOI: https://doi.org/10.1177/1099636217731071
  56. Kolahchi, R., Zarei, M. S., Hajmohammad, M. H., & Nouri, A. (2017). Wave propagation of embedded viscoelastic FG-CNT-reinforced sandwich plates integrated with sensor and actuator based on refined zigzag theory. International Journal of Mechanical Sciences, 130, 534-545. https://doi.org/10.1016/j.ijmecsci.2017.06.039 DOI: https://doi.org/10.1016/j.ijmecsci.2017.06.039
  57. Kolahchi, R., Zarei, M. S., Hajmohammad, M. H., & Oskouei, A. N. (2017). Visco-nonlocal-refined Zigzag theories for dynamic buckling of laminated nanoplates using differential cubature-Bolotin methods. Thin-Walled Structures, 113, 162-169. https://doi.org/10.1016/j.tws.2017.01.016 DOI: https://doi.org/10.1016/j.tws.2017.01.016
  58. Li, M., Soares, C. G., & Yan, R. (2021). Free vibration analysis of FGM plates on Winkler/Pasternak/Kerr foundation by using a simple Quasi-3D HSDT. Composite Structures, 264, Article 113643.  https://doi.org/10.1016/j.compstruct.2021.113643 DOI: https://doi.org/10.1016/j.compstruct.2021.113643
  59. Li, Q., Iu V. P., & Kou K. P. (2008). Three-dimensional vibration analysis of functionally graded material sandwich plates. Journal of Sound and Vibration, 311(1/2), 498-51. https://doi.org/10.1016/j.jsv.2007.09.018 DOI: https://doi.org/10.1016/j.jsv.2007.09.018
  60. Lo, K. H., Christensen, R. M., & Wu, E. M. (1977). A high-order theory of plate deformation part 1: Homogeneous plates. Journal of Applied Mechanics, 44(4),663-668. https://doi.org/10.1115/1.3424154 DOI: https://doi.org/10.1115/1.3424154
  61. Lü, C. F., Lim, C. W., & Chen, W. Q. (2009). Exact solutions for free vibrations of functionally graded thick plates on elastic foundations. Journal Mechanics of Advanced Materials and Structures, 16(8), 576-584. https://doi.org/10.1080/15376490903138888 DOI: https://doi.org/10.1080/15376490903138888
  62. Madani, H., Hosseini, H., & 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 and Composite Structures, 22(4), 889-913. https://doi.org/10.12989/scs.2016.22.4.889 DOI: https://doi.org/10.12989/scs.2016.22.4.889
  63. Mahdavian, M. (2009). Buckling analysis of simply-supported functionally graded rectangular plates under non-uniform in-plane compressive loading. Journal of Solid Mechanics, 1(3), 213-225.
  64. Malekzadeh, P. (2009). Three-dimensional free vibration analysis of thick functionally graded plates on elastic foundations. Composite Structures, 89(3), 367-373. https://doi.org/10.1016/j.compstruct.2008.08.007 DOI: https://doi.org/10.1016/j.compstruct.2008.08.007
  65. Mantari, J. L., & Guedes Soares, C. (2012). Bending analysis of thick exponentially graded plates using a new trigonometric higher order shear deformation theory. Composite Structures, 94(6), 1991-2000. https://doi.org/10.1016/j.compstruct.2012.01.005 DOI: https://doi.org/10.1016/j.compstruct.2012.01.005
  66. Mantari, J. L., & Guedes Soares, C. (2013). Finite element formulation of a generalized higher order shear deformation theory for advanced composite plates. Composite Structures, 96, 545-553. https://doi.org/10.1016/j.compstruct.2012.08.004 DOI: https://doi.org/10.1016/j.compstruct.2012.08.004
  67. Mantari, J. L., Bonilla, E. M., & Guedes Soares, C. (2014). A new tangential-exponential higher order shear deformation theory for advanced composite plates. Composites Part B: Engineering, 60, 319-328. DOI: https://doi.org/10.1016/j.compositesb.2013.12.001
  68. Mian, M. A., & Spencer, A. (1998). Exact solutions for functionally graded and laminated elastic materials. Journal of the Mechanics and Physics of Solids, 46(12), 2283-2295. DOI: https://doi.org/10.1016/S0022-5096(98)00048-9
  69. Mindlin, R. D. (1951). Influence of rotary inertia and shear on flexural motions of isotropic elastic plates. Journal of Applied Mechanics, Transaction of American Society of Mechanical Engineers, 18(1), 31-38. DOI: https://doi.org/10.1115/1.4010217
  70. Mohammadi, M., Saidi, A. R., & Jomehzadeh, E. (2010). Levy solution for buckling analysis of functionally graded rectangular plates. Applied Composite Materials, 17(2), 81-93. DOI: https://doi.org/10.1007/s10443-009-9100-z
  71. Nguyen, K. T., Thai, T. H., & Vo, T. P. (2015). A refined higher-order shear deformation theory for bending, vibration and buckling analysis of functionally graded sandwich plates. Steel and Composite Structures, 18(1), 91-120. https://doi.org/10.12989/scs.2015.18.1.091 DOI: https://doi.org/10.12989/scs.2015.18.1.091
  72. Ootao, Y., & Tanigawa, Y. (1999). Three-dimensional transient thermal stresses of functionally graded rectangular plate due to partial heating. Journal of Thermal Stresses, 22(1), 35-55. DOI: https://doi.org/10.1080/014957399281048
  73. Pagano, N. J. (1970). Exact solutions for rectangular bidirectional composites and sandwich plates. Journal of Composite Materials, 4, 20-34. https://doi.org/10.1177/002199837000400102 DOI: https://doi.org/10.1177/002199837000400102
  74. Plevris, V., Hadji, L., & Madan, R. (2024). Exploring porosity impact on the free vibration of FG plates using trigonometric shear deformation theory. Structural Engineering and Mechanics, 92(3), 267-275. https://doi.org/10.12989/SEM.2024.92.3.267
  75. Praveen, G. N., & Reddy, J. N. (1998). Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates. International Journal of Solids and Structures, 35(33), 4457-4471. DOI: https://doi.org/10.1016/S0020-7683(97)00253-9
  76. Qian, L. F., Batra, R. C., & 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 Petrove Galerkin method. Composites Part B: Engineering, 35(6/8), 685 - 697. https://doi.org/10.1016/j.compositesb.2004.02.004 DOI: https://doi.org/10.1016/j.compositesb.2004.02.004
  77. Reddy, J. N. (1984). A simple higher-order theory for laminated composite plates. Journal Applied Mechanics, 51(4), 745-752.   DOI: https://doi.org/10.1115/1.3167719
  78. Reddy, J. N. (2004). Mechanics of laminated composite plates and shells: Theory and analysis. CRC Press. DOI: https://doi.org/10.1201/b12409
  79. Reddy, J. N., & Chin, C. D.  (1998). Thermo-mechanical analysis of functionally graded cylinders and plates. Journal of Thermal Stresses, 21(6), 593-626. DOI: https://doi.org/10.1080/01495739808956165
  80. Reissner, E. (1945). The effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics The American Society of Mechanical Engineers, 12(2), 69-77. DOI: https://doi.org/10.1115/1.4009435
  81. Reissner, E. (1947). On bending of elastic plates. Quarterly of Applied Mathematics, 5, 55-68.
    https://doi.org/10.1090/qam/20440 DOI: https://doi.org/10.1090/qam/20440
  82. Ruan, M., & Wang, Z. M. (2014). Transverse vibrations of moving skew plates made of functionally graded material.  Journal of Vibration and Control.  http://doi.org/10.1177/ 1077546314563967
  83. Sadoune, M., Tounsi, A., Houari, M. S. A., & Adda Bedia, E. A. (2014). A novel first-order shear deformation theory for laminated composite plates. Steel and Composite Structures, 17(3), 321-338. https://doi.org/10.12989/scs.2014.17.3.321 DOI: https://doi.org/10.12989/scs.2014.17.3.321
  84. Safaei, B., Sahmani, S., & Tofighi Asl, H. (2021). Quasi-3D nonlinear flexural response of isogeometric functionally graded CNT-reinforced plates with various shapes with variable thicknesses. Mechanics Based Design of Structures and Machines, 51(5), 2957-2981. https://doi.org/10.1080/15397734.2021.1999264 DOI: https://doi.org/10.1080/15397734.2021.1999264
  85. Saidi, A. R., & Jomehzadeh, E. (2009). On the analytical approach for the bending/stretching of linearly elastic functionally graded rectangular plates with two opposite edges simply supported. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 223(9), 2009-2016. DOI: https://doi.org/10.1243/09544062JMES1431
  86. Saidi, H., Tounsi, A., & Bousahla, A. A. (2016). A simple hyperbolic shear deformation theory for vibration analysis of thick functionally graded rectangular plates resting on elastic foundations. Geomechanics and Engineering, 11(2), 289-307. DOI: https://doi.org/10.12989/gae.2016.11.2.289
  87. Samsam Shariat, B. A., & Eslami, M. R. (2007). Buckling of thick functionally graded plates under mechanical and thermal loads. Composite Structures, 78(3), 433-439. DOI: https://doi.org/10.1016/j.compstruct.2005.11.001
  88. Shahsavari, D., Karami, B., & Li, L. (2018). Damped vibration of a graphene sheet using a higher-order nonlocal strain-gradient Kirchhoff plate model. Comptes Rendus Mécanique, 346, 1216-1232. DOI: https://doi.org/10.1016/j.crme.2018.08.011
  89. Shahsavari, D., Karami, B., & Mansouri, B. (2018). Shear buckling of single layer graphene sheets in hygrothermal environment resting on elastic foundation based on different nonlocal strain gradient theories. European Journal of Mechanics - A/Solids, 67, 200-214. DOI: https://doi.org/10.1016/j.euromechsol.2017.09.004
  90. Shimpi, R. P. (2002). Refined plate theory and its variants. American Institute of Aeronautics and Astronautics Journal, 40(1), 137-146. DOI: https://doi.org/10.2514/2.1622
  91. Shimpi, R. P., & Patel, H. G. (2006). Free vibrations of plate using two variable refined plate theory. Vib Journal of Sound and Vibration, 296(4),979-999. DOI: https://doi.org/10.1016/j.jsv.2006.03.030
  92. Shyang-Ho, C., & Yen-Ling, C. (2006). Mechanical behavior of functionally graded material plates under transverse load - Part I: Analysis. International Journal of Solids and Structures, 43(13), 3657-3674. DOI: https://doi.org/10.1016/j.ijsolstr.2005.04.011
  93. Sobhy, M. (2013). Buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions. Composite Structures, 99, 76-87. DOI: https://doi.org/10.1016/j.compstruct.2012.11.018
  94. Swaminathan, K., & Naveenkumar, D. T. (2014). Higher order refined computational models for the stability analysis of FGM plates: Analytical solutions. European Journal of Mechanics A/Solids, 47, 349-361. https://doi.org/10.1016/j.euromechsol.2014.06.003 DOI: https://doi.org/10.1016/j.euromechsol.2014.06.003
  95. Swaminathan, K., Naveenkumar, D. T., Zenkour, A. M., & Carrera, E. (2015). Stress, vibration and buckling analyses of FGM plates-A state-of-the-art review. Composite Structures, 120, 10-31. https://doi.org/10.1016/j.compstruct.2014.09.070  DOI: https://doi.org/10.1016/j.compstruct.2014.09.070
  96. Thai, H. T., & Choi, D. H. (2014). Improved refined plate theory accounting for effect of thickness stretching in functionally graded plates. Composites Part B: Engineering, 56, 705-716. DOI: https://doi.org/10.1016/j.compositesb.2013.09.008
  97. Thai, H. T., & Kim, S. E. (2013). Closed-form solution for buckling analysis of thick functionally graded plates on elastic foundation. International Journal of Mechanical Sciences, 75, 34-44. DOI: https://doi.org/10.1016/j.ijmecsci.2013.06.007
  98. Thai, H. T., & Vo, T. P. (2013). A new sinusoidal shear deformation theory for bending buckling, and vibration of functionally graded plates. Applied Mathematical Modelling, 37(5), 3269-3281. DOI: https://doi.org/10.1016/j.apm.2012.08.008
  99. Timesli, A. (2020). Prediction of the critical buckling load of SWCNT reinforced concrete cylindrical shell embedded in an elastic foundation. Computers and Concrete, 26(1), 53-62. https://doi.org/10.12989/cac.2020.26.1.053
  100. Tounsi, A., Houari, M. S. A., & Bessaim, A. (2016). A new 3-unknowns non-polynomial plate theory for buckling and vibration of functionally graded sandwich plate. Structural Engineering and Mechanics, 60(4), 547-565. DOI: https://doi.org/10.12989/sem.2016.60.4.547
  101. Tounsi, A., Houari, M. S. A., Benyoucef, S., & Adda Bedia, E. A. (2013). A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates. Aerospace Science and Technology, 24(1), 209-220. DOI: https://doi.org/10.1016/j.ast.2011.11.009
  102. Tran, L. V., Thai, C. H., & Nguyen-Xuan, H. (2013). An isogeometric finite element formulation for thermal buckling analysis of functionally graded plates. Finite Elements in Analysis and Design, 73, 65-76. DOI: https://doi.org/10.1016/j.finel.2013.05.003
  103. Uymaz, B., & Aydogdu, M. (2007). Three-dimensional vibration analyses of functionally graded plates under various boundary conditions. Journal of Reinforced Plastics and Composites, 26(18), 1847-63. DOI: https://doi.org/10.1177/0731684407081351
  104. Vel, S. S., & Batra, R. C. (2002). Three-dimensional analysis of transient thermal stresses in functionally graded plates. International Journal of Solids and Structures, 40(25), 7181-7196. DOI: https://doi.org/10.1016/S0020-7683(03)00361-5
  105. Vel, S. S., & Batra, R. C. (2004). Three-dimensional exact solution for the vibration of functionally graded rectangular plates. Journal of Sound and Vibration, 272(3/5), 703-730. DOI: https://doi.org/10.1016/S0022-460X(03)00412-7
  106. Vu, T. V., & Nguyen, H. V. (2022). Buckling analysis of functionally graded sandwich plates resting on Pasternak foundation using a novel refined Quasi-3D third-order shear deformation theory. Journal of Science and Technology in Civil Engineering (STCE)-HUCE, 16(1), 68-79.  https://doi.org/10.31814/stce.huce(nuce)2022-16(1)-06 DOI: https://doi.org/10.31814/stce.huce(nuce)2022-16(1)-06
  107. Vu, V. T., & Cao, L. H. (2023). Deflection and natural frequency analysis of FG porous plates embedded in elastic foundations using four-variable hyperbolic Quasi-3D theory.  Arabian Journal for Science and Engineering, 48, 5407-5445.  https://doi.org/10.1007/s13369-022-07423-y DOI: https://doi.org/10.1007/s13369-022-07423-y
  108. Wang, C. M. G. T., Lim, J., Reddy, N., & Lee, K. H. (2001). Relationships between bending solutions of Reissner and Mindlin plate theories.  Engineering Structures, 23(7), 838-849. https://doi.org/10.1016/S0141-0296(00)00092-4 DOI: https://doi.org/10.1016/S0141-0296(00)00092-4
  109. Whitney, J. M. (1969). The effect of transverse shear deformation on the bending of laminated plates. Journal of Composite Materials, 3(3), 534-547. DOI: https://doi.org/10.1177/002199836900300316
  110. Woo, J., Meguid, S. A., & Ong, L. S. (2006). Nonlinear free vibration behavior of functionally graded plates. Journal of Sound and Vibration, 289(3), 595-611. DOI: https://doi.org/10.1016/j.jsv.2005.02.031
  111. Xiang, S., Jin, Y. X., Bi, Z. Y., Jiang, S. X., & Yang, M. S. (2011). A n-order shear deformation theory for free vibration of functionally graded and composite sandwich plates. Composite Structures, 93(11), 2826-2832. DOI: https://doi.org/10.1016/j.compstruct.2011.05.022
  112. Xie, K., Wang, Y., Niu, H., & Chen, H. (2022). In-plane free vibrations of functionally graded sandwich arches using shear and Quasi-3D deformation theories. Journal of Sandwich Structures & Materials, 24(1), 536-564. https://doi.org/10.1177/10996362211021901 DOI: https://doi.org/10.1177/10996362211021901
  113. Yaghoobi, H., & Fereidoon, A. (2014). Mechanical and thermal buckling analysis of functionally graded plates resting on elastic foundations: An assessment of a simple refined nth-order shear deformation theory. Composites Part B: Engineering, 62, 54-64. DOI: https://doi.org/10.1016/j.compositesb.2014.02.014
  114. Yang, J., & Shen, H. S. (2002). Vibration characteristics and transient response of shear deformable functionally graded plates in thermal environments. Journal of Sound and Vibration, 255(3), 579-602. DOI: https://doi.org/10.1006/jsvi.2001.4161
  115. Yang, J., Liew, K. M., & Kitipornchai, S. (2005). Second-order statistics of the elastic buckling of functionally graded rectangular plates. Composites Science and Technology, 65(7/8), 1165-1175. DOI: https://doi.org/10.1016/j.compscitech.2004.11.012
  116. Yaylaci, M., Adiyaman, G., Oner, E., & Birinci, A. (2021). Investigation of continuous and discontinuous contact cases in the contact mechanics of graded materials using analytical method and FEM. Computer Concrete, 27(3), 199-210. https://doi.org/10.12989/cac.2021.27.3.199
  117. Yaylaci, M., Sabano, B. S., Ozdemir, M. E., & Birinci, A. (2022). Solving the contact problem of functionally graded layers resting on a HP and pressed with a uniformly distributed load by analytical and numerical methods. Structural Engineering and Mechanics, 82(3), 401-416. https://doi.org/10.12989/sem.2022.82.3.401
  118. Yaylaci, M., Yaylaci, E. U., Ozdemir, M. E., Ozturk, S., & Sesli, H. (2023). Vibration and buckling analyses of FGM beam with edge crack: Finite element and multilayer perceptron methods. Steel and Composite Structures, 46(4), 565-575. https://doi.org/10.12989/scs.2023.46.4.565
  119. Yaylaci, M., Yayli, M., Yaylaci, E. U., Olmez, H., & Birinci, A. (2021). Analyzing the contact problem of a functionally graded layer resting on an elastic half plane with theory of elasticity, finite element method and multilayer perceptron. Structural Engineering and Mechanics, 78(5), 585-597.  https://doi.org/10.12989/sem.2021.78.5.585
  120. Yukun, C., Guoyong, J., & Mingfei, C. (2021). A Quasi-3D dynamic model for free vibration analysis of rotating pre-twisted functionally graded blades. Journal of Sound and Vibration, 499, Article 115990.  https://doi.org/10.1016/j.jsv.2021.115990  DOI: https://doi.org/10.1016/j.jsv.2021.115990
  121. Yukun, C., Tiangui, Y., Guoyong, J., Heow, P. L., & Xianglong, M. (2022). A unified quasi-three-dimensional solution for vibration analysis of rotating pre-twisted laminated composite shell panels. Composite Structures, 282, Article 115072. https://doi.org/10.1016/j.compstruct.2021.115072. DOI: https://doi.org/10.1016/j.compstruct.2021.115072
  122. Zenkour, A. M. (2007). Benchmark trigonometric and 3-D elasticity solutions for an exponentially graded thick rectangular plate. Archive of Applied Mechanics, 77(4),197-214. DOI: https://doi.org/10.1007/s00419-006-0084-y
  123. Zhang, D. G., & Zhou, Y. H. (2008). A theoretical analysis of FGM thin plates based on physical neutral surface. Computational Materials Science, 44(2), 716-720. DOI: https://doi.org/10.1016/j.commatsci.2008.05.016
  124. Zhang, Y. W., Ding, H. X., She, G. L., & and Tounsi, A. (2023). Wave propagation of CNTRC beams resting on elastic foundation based on various higher-order beam theories. Geomechanics and Engineering, 33(4), 381-391.  https://doi.org/10.12989/gae.2023.33.4.381 

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