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Static analysis of functionally graded saturated porous plate rested on pasternak elastic foundation by using a new quasi-3D higher-order shear deformation theory

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Abstract

A new quasi-3D higher-order shear deformation theory is introduced to investigate the static behaviour of functionally graded saturated porous (FGSP) plate resting on Pasternak’s elastic foundation for the first time. The governing equations are derived from eleven-unknowns higher-order shear deformation theory and using Biot’s poroelasticity theory taking into account transverse shear stress-free boundary conditions on the top and bottom surface of the plate. Three porosity distribution patterns of FGSP materials namely uniform, non-uniform symmetric and non-uniform asymmetric are considered. Navier’s technique is employed to obtain an analytical solution. The present results are compared with 3D and higher-order solutions available in the existing literature to validate the proposed model. Parametric studies show efficiency of proposed quasi-3D plate theory in analyzing FGSP thick plates, and exploring the effects of material, geometrical and elastic foundation parameters, as well as fluid compressibility, stretching effect on transverse displacement and stress field.

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References

  1. Magnucki, K., Stasiewicz, P.: Elastic buckling of a porous beam. J. Theor. Appl. Mech. 42(4), 859–868 (2004)

    MATH  Google Scholar 

  2. Magnucki, K., Malinowski, M., Kasprzak, J.: Bending and buckling of a rectangular porous plate. Steel Compos. Struct. 6(4), 319–333 (2006)

    Article  Google Scholar 

  3. Chen, D., Yang, J., Kitipornchai, S.: Elastic buckling and static bending of shear deformable functionally graded porous beam. Compos. Struct. 133, 54–61 (2015)

    Article  Google Scholar 

  4. Chen, D., Yang, J., Kitipornchai, S.: Free and forced vibrations of shear deformable functionally graded porous beams. Int. J. Mech. Sci. 108, 14–22 (2016)

    Article  Google Scholar 

  5. Ebrahimi, F., Dabbagh, A., Rastgoo, A.: Vibration analysis of porous metal foam shells rested on an elastic substrate. J. Strain Anal. Eng. Des. 54(3), 199–208 (2019)

    Article  Google Scholar 

  6. Li, H., et al.: Vibration analysis of functionally graded porous cylindrical shell with arbitrary boundary restraints by using a semi analytical method. Compos. B Eng. 164, 249–264 (2019)

    Article  Google Scholar 

  7. Binh, C.T., et al.: Nonlinear vibration of functionally graded porous variable thickness toroidal shell segments surrounded by elastic medium including the thermal effect. Compos. Struct. 255, 112891 (2021)

    Article  Google Scholar 

  8. Tu, T.M., et al.: Nonlinear buckling and post-buckling analysis of imperfect porous plates under mechanical loads. J. Sandwich Struct. Mater. 22(6), 1910–1930 (2020)

    Article  Google Scholar 

  9. Dang, X.-H., et al.: Free vibration characteristics of rotating functionally graded porous circular cylindrical shells with different boundary conditions. Iran. J. Sci. Technol. Trans. Mech. Eng. 1–17 (2020)

  10. Khatounabadi, M., Jafari, M., Asemi, K.: Low-velocity impact analysis of functionally graded porous circular plate reinforced with graphene platelets. Waves Random Complex Med. 1–27 (2022)

  11. Biot, M.A.: Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26(2), 182–185 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kiarasi, F., et al.: A review on functionally graded porous structures reinforced by graphene platelets. J. Comput. Appl. Mech. 52(4), 731–750 (2021)

    MathSciNet  Google Scholar 

  13. Babaei, M., et al.: Functionally graded saturated porous structures: a review. J. Comput. Appl. Mech. 53(2), 297–308 (2022)

    Google Scholar 

  14. Theodorakopoulos, D., Beskos, D.: Flexural vibrations of poroelastic plates. Acta Mech. 103(1), 191–203 (1994)

    Article  MATH  Google Scholar 

  15. Etchessahar, M., Sahraoui, S., Brouard, B.: Bending vibrations of a rectangular poroelastic plate. Comptes Rendus Acad. Sci. Ser. IIB-Mech. 329(8), 615–620 (2001)

    Google Scholar 

  16. Jabbari, M., et al.: Thermal buckling analysis of functionally graded thin circular plate made of saturated porous materials. J. Therm. Stress. 37(2), 202–220 (2014)

    Article  Google Scholar 

  17. Feyzi, M., Khorshidvand, A.: Axisymmetric post-buckling behavior of saturated porous circular plates. Thin-Walled Struct. 112, 149–158 (2017)

    Article  Google Scholar 

  18. Rezaei, A., Saidi, A.: On the effect of coupled solid-fluid deformation on natural frequencies of fluid saturated porous plates. Eur. J. Mech. A/Solids 63, 99–109 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jabbari, M., Joubaneh, E.F., Mojahedin, A.: Thermal buckling analysis of porous circular plate with piezoelectric actuators based on first order shear deformation theory. Int. J. Mech. Sci. 83, 57–64 (2014)

    Article  MATH  Google Scholar 

  20. Jabbari, M., Rezaei, M., Mojahedin, A.: Mechanical buckling of FG saturated porous rectangular plate with piezoelectric actuators. Iran. J. Mech. Eng. Trans. ISME 17(2), 46–66 (2016)

    Google Scholar 

  21. Rad, E.S., et al.: Shear deformation theories for elastic buckling of fluid-infiltrated porous plates: an analytical approach. Compos. Struct. 254, 112829 (2020)

    Article  Google Scholar 

  22. Mojahedin, A., et al.: Buckling analysis of functionally graded circular plates made of saturated porous materials based on higher order shear deformation theory. Thin-Walled Struct. 99, 83–90 (2016)

    Article  Google Scholar 

  23. Soleimani-Javid, Z., et al.: On the higher-order thermal vibrations of FG saturated porous cylindrical micro-shells integrated with nanocomposite skins in viscoelastic medium. Def. Technol. (2021)

  24. Sharifan, M.H., Jabbari, M.: Mechanical buckling analysis of saturated porous functionally graded elliptical plates subjected to in-plane force resting on two parameters elastic foundation based on HSDT. J. Pressure Vessel Technol. 142(4), 041302 (2020)

    Article  Google Scholar 

  25. Ebrahimi, F., Habibi, S.: Deflection and vibration analysis of higher-order shear deformable compositionally graded porous plate. Steel Compos. Struct. 20(1), 205–225 (2016)

    Article  Google Scholar 

  26. Akbari, H., Azadi, M., Fahham, H.: Free vibration analysis of thick sandwich cylindrical panels with saturated FG-porous core. Mech. Based Des. Struct. Mach. 50(4), 1268–1286 (2022)

    Article  Google Scholar 

  27. Babaei, M., Asemi, K., Kiarasi, F.: Dynamic analysis of functionally graded rotating thick truncated cone made of saturated porous materials. Thin-Walled Struct. 164, 107852 (2021)

    Article  Google Scholar 

  28. Babaei, M., Hajmohammad, M.H., Asemi, K.: Natural frequency and dynamic analyses of functionally graded saturated porous annular sector plate and cylindrical panel based on 3D elasticity. Aerosp. Sci. Technol. 96, 105524 (2020)

    Article  Google Scholar 

  29. Babaei, M., Asemi, K., Kiarasi, F.: Static response and free-vibration analysis of a functionally graded annular elliptical sector plate made of saturated porous material based on 3D finite element method. Mech. Based Des. Struct. Mach. 1–25 (2020)

  30. Kiarasi, F., et al.: Three-dimensional buckling analysis of functionally graded saturated porous rectangular plates under combined loading conditions. Appl. Sci. 11(21), 10434 (2021)

    Article  Google Scholar 

  31. Alhaifi, K., Arshid, E., Khorshidvand, A.R.: Large deflection analysis of functionally graded saturated porous rectangular plates on nonlinear elastic foundation via GDQM. Steel Compos. Struct. Int. J. 39(6), 795–809 (2021)

    Google Scholar 

  32. Babaei, M., Asemi, K., Safarpour, P.: Buckling and static analyses of functionally graded saturated porous thick beam resting on elastic foundation based on higher order beam theory. Iran. J. Mech. Eng. Trans. ISME 20(1), 94–112 (2019)

    Google Scholar 

  33. Babaei, M., Asemi, K., Safarpour, P.: Natural frequency and dynamic analyses of functionally graded saturated porous beam resting on viscoelastic foundation based on higher order beam theory. J. Solid Mech. 11(3), 615–634 (2019)

    Google Scholar 

  34. Barati, M.R., Zenkour, A.M.: Investigating post-buckling of geometrically imperfect metal foam nanobeams with symmetric and asymmetric porosity distributions. Compos. Struct. 182, 91–98 (2017)

    Article  Google Scholar 

  35. Chen, D., Kitipornchai, S., Yang, J.: Nonlinear free vibration of shear deformable sandwich beam with a functionally graded porous core. Thin-Walled Struct. 107, 39–48 (2016)

    Article  Google Scholar 

  36. Jabbari, M., et al.: Buckling analysis of a functionally graded thin circular plate made of saturated porous materials. J. Eng. Mech. 140(2), 287–295 (2014)

    Google Scholar 

  37. Jha, D., Kant, T., Singh, R.: Higher order shear and normal deformation theory for natural frequency of functionally graded rectangular plates. Nucl. Eng. Des. 250, 8–13 (2012)

    Article  Google Scholar 

  38. Jha, D., Kant, T., Singh, R.: Stress analysis of transversely loaded functionally graded plates with a higher order shear and normal deformation theory. J. Eng. Mech. 139(12), 1663–1680 (2013)

    Google Scholar 

  39. Kim, J., Reddy, J.: Analytical solutions for bending, vibration, and buckling of FGM plates using a couple stress-based third-order theory. Compos. Struct. 103, 86–98 (2013)

    Article  Google Scholar 

  40. Kant, T.: A critical review and some results of recently developed refined theories of fiber-reinforced laminated composites and sandwiches. Compos. Struct. 23(4), 293–312 (1993)

    Article  Google Scholar 

  41. Reddy, J., Kim, J.: A nonlinear modified couple stress-based third-order theory of functionally graded plates. Compos. Struct. 94(3), 1128–1143 (2012)

    Article  Google Scholar 

  42. Christensen, R.: A high-order theory of plate deformation. J. Appl. Mech. 669 (1977)

  43. Soares, C.M., Soares, C.M., Correia, V.F.: Multiple eigenvalue optimization of composite structures using discrete third order displacement models. Compos. Struct. 38(1–4), 99–110 (1997)

    Article  Google Scholar 

  44. Reddy, J.N.: A simple higher-order theory for laminated composite plates (1984)

  45. Nayak, A., Moy, S., Shenoi, R.: Free vibration analysis of composite sandwich plates based on Reddy’s higher-order theory. Compos. B Eng. 33(7), 505–519 (2002)

    Article  Google Scholar 

  46. Thinh, T.I., et al.: Vibration and buckling analysis of functionally graded plates using new eight-unknown higher order shear deformation theory. Lat. Am. J. Solids Struct. 13, 456–477 (2016)

    Article  Google Scholar 

  47. Duc, N.D., Cong, P.H., Quang, V.D.: Thermal stability of eccentrically stiffened FGM plate on elastic foundation based on Reddy’s third-order shear deformation plate theory. J. Therm. Stress. 39(7), 772–794 (2016)

    Article  Google Scholar 

  48. Tu, T.M., Quoc, T.H., Long, N.V.: Bending analysis of functionally graded plates using new eight-unknown higher order shear deformation theory. Struct. Eng. Mech 62(3), 311–324 (2017)

    Article  Google Scholar 

  49. Detournay, E., Cheng, A.H.D.: 5 - Fundamentals of poroelasticity. In: Fairhurst, C. (ed.) Analysis and Design Methods, pp. 113–171. Pergamon, Oxford (1993)

    Chapter  Google Scholar 

  50. Arshid, E., Khorshidvand, A.R.: Free vibration analysis of saturated porous FG circular plates integrated with piezoelectric actuators via differential quadrature method. Thin-Walled Struct. 125, 220–233 (2018)

    Article  Google Scholar 

  51. Ebrahimi, F., Habibi, S.: Deflection and vibration analysis of higher-order shear deformable compositionally graded porous plate. Steel Compos. Struct. 20, 205–225 (2016)

    Article  Google Scholar 

  52. Liew, K., Teo, T., Han, J.-B.: Three-dimensional static solutions of rectangular plates by variant differential quadrature method. Int. J. Mech. Sci. 43(7), 1611–1628 (2001)

    Article  MATH  Google Scholar 

  53. Civalek, Ö.: Three-dimensional vibration, buckling and bending analyses of thick rectangular plates based on discrete singular convolution method. Int. J. Mech. Sci. 49(6), 752–765 (2007)

    Article  Google Scholar 

  54. Werner, H.: A three-dimensional solution for rectangular plate bending free of transversal normal stresses. Int. J. Numer. Methods Biomed. Eng. 15(4), 295–302 (1999)

    MATH  Google Scholar 

  55. Thai, H.-T., Choi, D.-H.: A refined plate theory for functionally graded plates resting on elastic foundation. Compos. Sci. Technol. 71(16), 1850–1858 (2011)

    Article  Google Scholar 

  56. Zenkour, A.M.: The refined sinusoidal theory for FGM plates on elastic foundations. Int. J. Mech. Sci. 51(11–12), 869–880 (2009)

    Article  Google Scholar 

  57. Thai, H.-T., Choi, D.-H.: A simple first-order shear deformation theory for the bending and free vibration analysis of functionally graded plates. Compos. Struct. 101, 332–340 (2013)

    Article  Google Scholar 

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Acknowledgements

This research is funded by Ha Noi University of Civil Engineering (HUCE) under grant number 23-2022/KHXD-TĐ.

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Appendix: Definition of coefficient in Eq. (24)

Appendix: Definition of coefficient in Eq. (24)

$$\begin{aligned} s_{{11}} & = A_{{11}} \alpha ^{2} + A_{{44}} \beta ^{2} ;\,s_{{12}} = s_{{21}} = \left( {A_{{12}} + A_{{44}} } \right)\alpha \beta ; \\ s_{{13}} & = s_{{31}} = - \frac{{D_{{11}} c_{2} }}{3}\alpha ^{3} - \left( {\frac{{D_{{12}} c_{2} }}{3} + \frac{{2D_{{44}} c_{2} }}{3}} \right)\alpha \beta ^{2} ; \\ s_{{14}} & = s_{{41}} = \left( {B_{{11}} - \frac{{D_{{11}} c_{2} }}{3}} \right)\alpha ^{2} + \left( {B_{{44}} - \frac{{D_{{44}} c_{2} }}{3}} \right)\beta ^{2} ; \\ s_{{15}} & = s_{{51}} = \left( {B_{{12}} + B_{{44}} - \frac{{D_{{12}} c_{2} }}{3} - \frac{{D_{{44}} c_{2} }}{3}} \right)\alpha \beta ; \\ s_{{16}} & = s_{{61}} = - \frac{{C_{{11}} }}{2}\alpha ^{3} - A_{{13}} \alpha - \left( {\frac{{C_{{12}} }}{2} + C_{{44}} } \right)\alpha \beta ^{2} ; \\ s_{{17}} & = s_{{71}} = - \frac{{D_{{11}} }}{3}\alpha ^{3} - 2B_{{13}} \alpha - \left( {\frac{{D_{{12}} }}{3} + \frac{{2D_{{44}} }}{3}} \right)\alpha \beta ^{2} ; \\ s_{{22}} & = A_{{44}} \alpha ^{2} + A_{{22}} \beta ^{2} ;\,s_{{24}} = s_{{42}} = \left( {B_{{21}} + B_{{44}} - \frac{{D_{{21}} c_{2} }}{3} - \frac{{D_{{44}} c_{2} }}{3}} \right)\alpha \beta ; \\ s_{{25}} & = s_{{52}} = \left( {B_{{44}} - \frac{{D_{{44}} c_{2} }}{3}} \right)\alpha ^{2} + \left( {B_{{22}} - \frac{{D_{{22}} c_{2} }}{3}} \right)\beta ^{2} ; \\ s_{{26}} & = s_{{62}} = - \left( {\frac{{C_{{21}} }}{2} + C_{{44}} } \right)\alpha ^{2} \beta - A_{{23}} \beta + \frac{{C_{{22}} }}{2}\beta ^{3} ; \\ s_{{27}} & = s_{{72}} = - \left( {\frac{{D_{{21}} }}{3} + \frac{{2D_{{44}} }}{3}} \right)\alpha ^{2} \beta - 2B_{{23}} \beta - \frac{{D_{{22}} }}{3}\beta ^{3} ; \\ s_{{33}} & = \frac{{G_{{11}} c_{2}^{2} }}{9}\alpha ^{4} + \left( {A_{{55}} + E_{{55}} c_{2}^{2} - 2C_{{55}} c_{2} } \right)\alpha ^{2} + \frac{{\left( {G_{{12}} + G_{{21}} + 4G_{{44}} } \right)c_{2}^{2} }}{9}\alpha ^{2} \beta ^{2} \\ & \quad + \left( {A_{{66}} + E_{{66}} c_{2}^{2} - 2C_{{66}} c_{2} } \right)\beta ^{2} + \frac{{G_{{22}} c_{2}^{2} }}{9}\beta ^{4} + k_{w} + k_{{sx}} \alpha ^{2} + k_{{sy}} \beta ^{2} ; \\ s_{{34}} & = s_{{43}} = \left( {\frac{{G_{{11}} c_{2}^{2} }}{9} - \frac{{E_{{11}} c_{2} }}{3}} \right)\alpha ^{3} + \left( {A_{{55}} - 2C_{{55}} c_{2} + E_{{55}} c_{2}^{2} } \right)\alpha \\ & \quad - \left( {\frac{{E_{{12}} c_{2} }}{3} + \frac{{2E_{{44}} c_{2} }}{3} - \frac{{G_{{12}} c_{2}^{2} }}{9} - \frac{{2G_{{44}} c_{2}^{2} }}{9}} \right)\alpha \beta ^{2} ; \\ s_{{35}} & = s_{{53}} = - \left( {\frac{{E_{{12}} c_{2} }}{3} + \frac{{2E_{{44}} c_{2} }}{3} - \frac{{G_{{12}} c_{2}^{2} }}{9} - \frac{{2G_{{44}} c_{2}^{2} }}{9}} \right)\alpha ^{2} \beta \\ & \quad + \left( {A_{{66}} - 2C_{{66}} c_{2} + E_{{66}} c_{2}^{2} } \right)\beta + \left( {\frac{{G_{{22}} c_{2}^{2} }}{9} - \frac{{E_{{22}} c_{2} }}{3}} \right)\beta ^{3} ; \\ s_{{36}} & = s_{{63}} = \frac{{F_{{11}} c_{2} }}{6}\alpha ^{4} + \frac{{D_{{13}} c_{2} }}{3}\alpha ^{2} + \left( {\frac{{F_{{12}} c_{2} }}{6} + \frac{{F_{{21}} c_{2} }}{6} - \frac{{2F_{{44}} c_{2} }}{3}} \right)\alpha ^{2} \beta ^{2} \\ & \quad + \frac{{D_{{23}} c_{2} }}{3}\beta ^{2} + \frac{{F_{{22}} c_{2} }}{6}\beta ^{4} - \frac{h}{2}\left( {k_{w} + k_{{sx}} \alpha ^{2} + k_{{sy}} \beta ^{2} } \right); \\ s_{{37}} & = s_{{73}} = \frac{{G_{{11}} c_{2} }}{9}\alpha ^{4} + \frac{{2E_{{13}} c_{2} }}{3}\alpha ^{2} + \left( {\frac{{G_{{12}} c_{2} }}{9} + \frac{{G_{{21}} c_{2} }}{9} + \frac{{4G_{{44}} c_{2} }}{9}} \right)\alpha ^{2} \beta ^{2} \\ & \quad + \frac{{2E_{{23}} c_{2} }}{3}\beta ^{2} + \frac{{G_{{22}} c_{2} }}{9}\beta ^{4} + \frac{{h^{2} }}{4}\left( {K_{w} + K_{{sx}} \alpha ^{2} + K_{{sy}} \beta ^{2} } \right); \\ s_{{44}} & = \left( {\frac{{G_{{11}} c_{2}^{2} }}{9} - \frac{{2E_{{11}} c_{2} }}{3} + C_{{11}} } \right)\alpha ^{2} + \left( {\frac{{G_{{44}} c_{2}^{2} }}{9} - \frac{{2E_{{44}} c_{2} }}{3} + C_{{44}} } \right)\beta ^{2} + E_{{55}} c_{2}^{2} - 2C_{{55}} c_{2} + A_{{55}} ; \\ s_{{45}} & = s_{{54}} = \left( {C_{{12}} + C_{{44}} + \frac{{G_{{12}} c_{2}^{2} }}{9} + \frac{{G_{{44}} c_{2}^{2} }}{9} - \frac{{2E_{{12}} c_{2} }}{3} - \frac{{2E_{{44}} c_{2} }}{3}} \right)\alpha \beta ; \\ s_{{45}} & = s_{{54}} = \left( {C_{{12}} + C_{{44}} + \frac{{G_{{12}} c_{2}^{2} }}{9} + \frac{{G_{{44}} c_{2}^{2} }}{9} - \frac{{2E_{{12}} c_{2} }}{3} - \frac{{2E_{{44}} c_{2} }}{3}} \right)\alpha \beta ; \\ s_{{46}} & = s_{{64}} = - \left( {\frac{{D_{{11}} }}{2} - \frac{{F_{{11}} c_{2} }}{6}} \right)\alpha ^{3} + \left( {\frac{{D_{{13}} c_{2} }}{3} - B_{{13}} } \right)\alpha - \left( {\frac{{D_{{12}} }}{2} + D_{{44}} - \frac{{F_{{12}} c_{2} }}{6} - \frac{{F_{{44}} c_{2} }}{3}} \right)\alpha \beta ^{2} ; \\ s_{{47}} & = s_{{74}} = - \left( {\frac{{E_{{11}} }}{3} - \frac{{G_{{11}} c_{2} }}{9}} \right)\alpha ^{3} + \left( {\frac{{2E_{{13}} c_{2} }}{3} - 2C_{{13}} } \right)\alpha \\ & \quad - \left( {\frac{{E_{{12}} }}{3} + \frac{{2E_{{44}} }}{3} - \frac{{G_{{12}} c_{2} }}{9} - \frac{{2G_{{44}} c_{2} }}{9}} \right)\alpha \beta ^{2} ; \\ s_{{55}} & = \left( {\frac{{G_{{44}} c_{2}^{2} }}{9} - \frac{{2E_{{44}} c_{2} }}{3} + C_{{44}} } \right)\alpha ^{2} + \left( {\frac{{G_{{22}} c_{2}^{2} }}{9} - \frac{{2E_{{22}} c_{2} }}{3} + C_{{22}} } \right)\beta ^{2} + E_{{66}} c_{2}^{2} - 2C_{{66}} c_{2} + A_{{66}} ; \\ s_{{56}} & = s_{{65}} = - \left( {\frac{{D_{{12}} }}{2} + D_{{44}} - \frac{{F_{{12}} c_{2} }}{6} - \frac{{F_{{44}} c_{2} }}{3}} \right)\alpha ^{2} \beta + \left( {\frac{{D_{{32}} c_{2} }}{3} - B_{{32}} } \right)\beta - \left( {\frac{{D_{{22}} }}{2} - \frac{{F_{{22}} c_{2} }}{6}} \right)\beta ^{3} ; \\ s_{{57}} & = s_{{75}} = - \left( {\frac{{E_{{21}} }}{3} + \frac{{2E_{{44}} }}{3} - \frac{{G_{{21}} c_{2} }}{9} - \frac{{2G_{{44}} c_{2} }}{9}} \right)\alpha ^{2} \beta \\ & \quad + \left( {\frac{{2E_{{23}} c_{2} }}{3} - 2C_{{23}} } \right)\beta - \left( {\frac{{E_{{22}} }}{3} - \frac{{G_{{22}} c_{2} }}{9}} \right)\beta ^{3} ; \\ s_{{66}} & = \frac{{E_{{11}} }}{4}\alpha ^{4} + \left( {\frac{{C_{{13}} }}{2} + \frac{{C_{{31}} }}{2}} \right)\alpha ^{2} + \left( {\frac{{E_{{12}} }}{4} + \frac{{E_{{21}} }}{4} + E_{{44}} } \right)\alpha ^{2} \beta ^{2} \\ & \quad + \left( {\frac{{C_{{23}} }}{2} + \frac{{C_{{32}} }}{2}} \right)\beta ^{2} + \frac{{E_{{22}} }}{4}\beta ^{4} + A_{{33}} + \frac{{h^{2} }}{4}\left( {k_{w} + k_{{sx}} \alpha ^{2} + k_{{sy}} \beta ^{2} } \right); \\ s_{{67}} & = s_{{76}} = \frac{{F_{{11}} }}{6}\alpha ^{4} + \left( {D_{{13}} + D_{{31}} } \right)\alpha ^{2} + \left( {\frac{{F_{{12}} }}{6} + \frac{{F_{{21}} }}{6} + \frac{{2F_{{44}} }}{3}} \right)\alpha ^{2} \beta ^{2} \\ & \quad + \left( {D_{{23}} + D_{{32}} } \right)\beta ^{2} + \frac{{F_{{22}} }}{6}\beta ^{4} + 2B_{{33}} - \frac{{h^{3} }}{8}\left( {k_{w} + k_{{sx}} \alpha ^{2} + k_{{sy}} \beta ^{2} } \right); \\ s_{{77}} & = \frac{{G_{{11}} }}{9}\alpha ^{4} + \left( {\frac{{2E_{{13}} }}{3} + \frac{{2E_{{31}} }}{3}} \right)\alpha ^{2} + \left( {\frac{{G_{{12}} }}{9} + \frac{{G_{{21}} }}{9} + \frac{{4G_{{44}} }}{9}} \right)\alpha ^{2} \beta ^{2} \\ & \quad + \left( {\frac{{2E_{{23}} }}{3} + \frac{{2E_{{32}} }}{3}} \right)\beta ^{2} + \frac{{G_{{22}} }}{9}\beta ^{4} + 4C_{{33}} + \frac{{h^{4} }}{{16}}\left( {k_{w} + k_{{sx}} \alpha ^{2} + k_{{sy}} \beta ^{2} } \right). \\ \end{aligned}$$

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Tru, V.N., Long, N.V., Tu, T.M. et al. Static analysis of functionally graded saturated porous plate rested on pasternak elastic foundation by using a new quasi-3D higher-order shear deformation theory. Arch Appl Mech 93, 2565–2583 (2023). https://doi.org/10.1007/s00419-023-02397-1

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