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Displacement-based and stress-based analytical approaches for nonlinear bending analysis of functionally graded porous plates resting on elastic substrate

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Abstract

For the first time, displacement-based and stress-based approaches are used simultaneously to analyze the nonlinear bending response of functionally graded porous plates within the framework of first-order shear deformation theory. Three configurations of porosity distributions including uniform, nonuniform symmetric, and nonuniform asymmetric are selected. Utilizing the Galerkin method, the neutral surface position concept, and the solutions in terms of Fourier series, the nonlinear algebraic governing equations are derived and then solved by the Newton–Raphson method. Very good agreement is found in the comparison between results obtained by using two proposed approaches, and in comparisons with those of existing ones in the literature. Parametric studies are conducted to evaluate the influence of the porosity coefficient, porosity distribution configurations, geometrical parameters, Pasternak elastic foundation, and boundary conditions on the load–deflection and load-bending moment curves.

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References

  1. Smith, B., et al.: Steel foam for structures: A review of applications, manufacturing and material properties. J. Constr. Steel Res. 71, 1–10 (2012)

    Article  Google Scholar 

  2. Ashby, M.F., et al.: Metal Foams: A Design Guide. Elsevier, Amsterdam (2000)

    Google Scholar 

  3. Banhart, J.: Manufacture, characterisation and application of cellular metals and metal foams. Prog. Mater Sci. 46(6), 559–632 (2001)

    Article  Google Scholar 

  4. 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 

  5. 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 

  6. 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 

  7. Wang, Y., Wu, D.: Free vibration of functionally graded porous cylindrical shell using a sinusoidal shear deformation theory. Aerosp. Sci. Technol. 66, 83–91 (2017)

    Article  Google Scholar 

  8. 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 

  9. Zine, A., et al.: Bending analysis of functionally graded porous plates via a refined shear deformation theory. Comput. Concr. 26(1), 63–74 (2020)

    MathSciNet  Google Scholar 

  10. 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. (2020). https://doi.org/10.1007/s40997-020-00413-1

    Article  Google Scholar 

  11. Masjedi, P.K., Maheri, A., Weaver, P.M.: Large deflection of functionally graded porous beams based on a geometrically exact theory with a fully intrinsic formulation. Appl. Math. Model. 76, 938–957 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  12. Praveen, G., Reddy, J.: Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates. Int. J. Solids Struct. 35(33), 4457–4476 (1998)

    Article  MATH  Google Scholar 

  13. Aliaga, J., Reddy, J.: Nonlinear thermoelastic analysis of functionally graded plates using the third-order shear deformation theory. Int. J. Comput. Eng. Sci. 5(04), 753–779 (2004)

    Google Scholar 

  14. Zhao, X., Liew, K.M.: Geometrically nonlinear analysis of functionally graded plates using the element-free kp-Ritz method. Comput. Methods Appl. Mech. Eng. 198(33), 2796–2811 (2009)

    Article  MATH  Google Scholar 

  15. Barbosa, J., Ferreira, A.: Geometrically nonlinear analysis of functionally graded plates and shells. Mech. Adv. Mater. Struct. 17(1), 40–48 (2009)

    Article  Google Scholar 

  16. Behjat, B., Khoshravan, M.: Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates. Compos. Struct. 94(3), 874–882 (2012)

    Article  Google Scholar 

  17. Singh, J., Shukla, K.: Nonlinear flexural analysis of functionally graded plates under different loadings using RBF based meshless method. Eng. Anal. Bound. Elem. 36(12), 1819–1827 (2012)

    Article  MATH  Google Scholar 

  18. Zhu, P., Zhang, L., Liew, K.: Geometrically nonlinear thermomechanical analysis of moderately thick functionally graded plates using a local Petrov-Galerkin approach with moving Kriging interpolation. Compos. Struct. 107, 298–314 (2014)

    Article  Google Scholar 

  19. Yin, S., et al.: Geometrically nonlinear analysis of functionally graded plates using isogeometric analysis. Eng. Comput. (2015). https://doi.org/10.1108/EC-09-2013-0220

    Article  Google Scholar 

  20. Heydari, M.M., et al.: Nonlinear bending analysis of functionally graded CNT-reinforced composite Mindlin polymeric temperature-dependent plate resting on orthotropic elastomeric medium using GDQM. Nonlinear Dyn. 79(2), 1425–1441 (2015)

    Article  Google Scholar 

  21. Phung-Van, P., et al.: Geometrically nonlinear analysis of functionally graded plates using a cell-based smoothed three-node plate element (CS-MIN3) based on the C0-HSDT. Comput. Methods Appl. Mech. Eng. 270, 15–36 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Nourmohammadi, H., Behjat, B.: Geometrically nonlinear analysis of functionally graded piezoelectric plate using mesh-free RPIM. Eng. Anal. Bound. Elem. 99, 131–141 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  23. TonThat, H.L., Nguyen-Van, H., Chau-Dinh, T.: Nonlinear bending analysis of functionally graded plates using SQ4T elements based on twice interpolation strategy. J. Appl. Comput. Mech. 6(1), 125–136 (2020)

    Google Scholar 

  24. Shen, H.-S.: Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments. Int. J. Mech. Sci. 44(3), 561–584 (2002)

    Article  MATH  Google Scholar 

  25. Yang, J., Shen, H.-S.: Nonlinear bending analysis of shear deformable functionally graded plates subjected to thermo-mechanical loads under various boundary conditions. Compos. B Eng. 34(2), 103–115 (2003)

    Article  Google Scholar 

  26. Shen, H.-S., Wang, Z.-X.: Nonlinear bending of FGM plates subjected to combined loading and resting on elastic foundations. Compos. Struct. 92(10), 2517–2524 (2010)

    Article  Google Scholar 

  27. Van Tung, H., Duc, N.D.: Nonlinear analysis of stability for functionally graded plates under mechanical and thermal loads. Compos. Struct. 92(5), 1184–1191 (2010)

    Article  Google Scholar 

  28. Thang, P.-T., Nguyen-Thoi, T., Lee, J.: Closed-form expression for nonlinear analysis of imperfect sigmoid-FGM plates with variable thickness resting on elastic medium. Compos. Struct. 143, 143–150 (2016)

    Article  Google Scholar 

  29. Duc, N.D., Bich, D.H., Cong, P.H.: Nonlinear thermal dynamic response of shear deformable FGM plates on elastic foundations. J. Therm. Stresses 39(3), 278–297 (2016)

    Article  Google Scholar 

  30. Woo, J., Meguid, S., Ong, L.: Nonlinear free vibration behavior of functionally graded plates. J. Sound Vib. 289(3), 595–611 (2006)

    Article  Google Scholar 

  31. Wu, T.-L., Shukla, K., Huang, J.H.: Post-buckling analysis of functionally graded rectangular plates. Compos. Struct. 81(1), 1–10 (2007)

    Article  Google Scholar 

  32. Alinia, M., Ghannadpour, S.: Nonlinear analysis of pressure loaded FGM plates. Compos. Struct. 88(3), 354–359 (2009)

    Article  Google Scholar 

  33. Woo, J., Meguid, S.: Nonlinear analysis of functionally graded plates and shallow shells. Int. J. Solids Struct. 38(42–43), 7409–7421 (2001)

    Article  MATH  Google Scholar 

  34. Khabbaz, R.S., Manshadi, B.D., Abedian, A.: Nonlinear analysis of FGM plates under pressure loads using the higher-order shear deformation theories. Compos. Struct. 89(3), 333–344 (2009)

    Article  Google Scholar 

  35. Benatta, M.A., et al.: Nonlinear bending analysis of functionally graded plates under pressure loads using a four variable refined plate theory. Int. J. Comput. Methods 11(04), 1350062 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Dinh Duc, N., et al.: Nonlinear dynamic response of functionally graded porous plates on elastic foundation subjected to thermal and mechanical loads. J. Appl. Comput. Mech. 4(4), 245–259 (2018)

    Google Scholar 

  37. Tu, T.M., et al.: Nonlinear buckling and post-buckling analysis of imperfect porous plates under mechanical loads. J. Sandw. Struct. Mater. (2018). https://doi.org/10.1177/1099636218789612

    Article  Google Scholar 

  38. Cong, P.H., et al.: Nonlinear thermomechanical buckling and post-buckling response of porous FGM plates using Reddy’s HSDT. Aerosp. Sci. Technol. 77, 419–428 (2018)

    Article  Google Scholar 

  39. Phung-Van, P., et al.: Isogeometric nonlinear transient analysis of porous FGM plates subjected to hygro-thermo-mechanical loads. Thin-Walled Struct. 148, 106497 (2020)

    Article  Google Scholar 

  40. Nguyen, N.V., et al.: Geometrically nonlinear polygonal finite element analysis of functionally graded porous plates. Adv. Eng. Softw. 126, 110–126 (2018)

    Article  Google Scholar 

  41. Ansari, R., et al.: Nonlinear bending analysis of arbitrary-shaped porous nanocomposite plates using a novel numerical approach. Int. J. Non-Linear Mech. 126, 103556 (2020)

    Article  Google Scholar 

  42. Nematollahi, M.S., et al.: Nonlinear vibration of functionally graded graphene nanoplatelets polymer nanocomposite sandwich beams. Appl. Sci. 10(16), 5669 (2020)

    Article  Google Scholar 

  43. Selvamani, R., et al.: Nonlinear magneto-thermo-elastic vibration of mass sensor armchair carbon nanotube resting on an elastic substrate. Curved Layer. Struct. 7(1), 153–165 (2020)

    Article  Google Scholar 

  44. Zhang, D.-G., Zhou, Y.-H.: A theoretical analysis of FGM thin plates based on physical neutral surface. Comput. Mater. Sci. 44(2), 716–720 (2008)

    Article  Google Scholar 

  45. Zhang, D.-G.: Modeling and analysis of FGM rectangular plates based on physical neutral surface and high order shear deformation theory. Int. J. Mech. Sci. 68, 92–104 (2013)

    Article  Google Scholar 

  46. 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 

  47. 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 

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

    MATH  Google Scholar 

  49. Larbi, L.O., et al.: An efficient shear deformation beam theory based on neutral surface position for bending and free vibration of functionally graded beams#. Mech. Based Des. Struct. Mach. 41(4), 421–433 (2013)

    Article  Google Scholar 

  50. Reddy, J.N.: Theory and Analysis of Elastic Plates and Shells. CRC Press, Boca Raton (2006)

    Book  Google Scholar 

  51. Reddy, J.N.: Energy Principles and Variational Methods in Applied Mechanics. Wiley, Hoboken (2017)

    Google Scholar 

  52. Sobhy, M.: Buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions. Compos. Struct. 99, 76–87 (2013)

    Article  Google Scholar 

  53. Meziane, M.A.A., Abdelaziz, H.H., Tounsi, A.: An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions. J. Sandw. Struct. Mater. 16(3), 293–318 (2014)

    Article  Google Scholar 

  54. Brush, D.O., Almroth, B.O.: Buckling of Bars, Plates, and Shells, vol. 6. McGraw-Hill, New York (1975)

    MATH  Google Scholar 

  55. Librescu, L., Stein, M.: A geometrically nonlinear theory of transversely isotropic laminated composite plates and its use in the post-buckling analysis. Thin-Walled Struct. 11(1–2), 177–201 (1991)

    Article  Google Scholar 

  56. Shen, H.-S.: Thermal postbuckling behavior of shear deformable FGM plates with temperature-dependent properties. Int. J. Mech. Sci. 49(4), 466–478 (2007)

    Article  Google Scholar 

  57. 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 

  58. 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 

  59. Lei, X.-Y., Huang, M.-K., Wang, X.: Geometrically nonlinear analysis of a Reissner type plate by the boundary element method. Comput. Struct. 37(6), 911–916 (1990)

    Article  Google Scholar 

  60. Azizian, Z., Dawe, D.: Geometrically nonlinear analysis of rectangular mindlin plates using the finite strip method. Comput. Struct. 21(3), 423–436 (1985)

    Article  MATH  Google Scholar 

  61. Talha, M., Singh, B.: Nonlinear mechanical bending of functionally graded material plates under transverse loads with various boundary conditions. Int. J. Model. Simul. Sci. Comput. 2(02), 237–258 (2011)

    Article  Google Scholar 

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Acknowledgements

This research is funded by Hanoi University of Civil Engineering (HUCE) under Grant Number: 42-2021/KHXD-TĐ

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Authors and Affiliations

  1. Faculty of Industrial and Civil Engineering, Hanoi University of Civil Engineering, Hanoi, Vietnam

    Nguyen Van Long, Tran Minh Tu & Huong Quy Truong

  2. Faculty of Civil Engineering, Vinh University, Vinh, Vietnam

    Le Thanh Hai

  3. School of Mechanical Engineering, Vietnam Maritime University, Haiphong, Vietnam

    Vu Thi Thu Trang

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  1. Nguyen Van Long
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  2. Tran Minh Tu
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Appendices

Appendix A: The functions in Eq. (17)

$$ \tilde{l}_{mn}^{(11)} (x,y) = \tilde{A}_{11} U_{1m}^{\prime \prime } U_{2n} + \tilde{A}_{66} U_{1m} U_{2n}^{\prime \prime } ;\;\tilde{l}_{mn}^{(12)} (x,y) = \left( {\tilde{A}_{12} + \tilde{A}_{66} } \right)V_{1m}^{^{\prime}} V_{2n}^{^{\prime}} ; $$
$$ \tilde{h}_{mnpq}^{(13)} (x,y) = \tilde{A}_{11} X_{m}^{^{\prime}} Y_{n} X_{p}^{^{\prime\prime}} Y_{q} + \tilde{A}_{12} X_{m} Y_{n}^{^{\prime}} X_{p}^{^{\prime}} Y_{q}^{^{\prime}} + \tilde{A}_{66} \left( {X_{m}^{^{\prime}} Y_{n} X_{p} Y_{q}^{^{\prime\prime}} + X_{m} Y_{n}^{^{\prime}} X_{p}^{^{\prime}} Y_{q}^{^{\prime}} } \right); $$
$$ \tilde{l}_{mn}^{(21)} (x,y) = \left( {\tilde{A}_{12} + \tilde{A}_{66} } \right)U_{1m}^{^{\prime}} U_{2n}^{^{\prime}} ;\;\tilde{l}_{mn}^{(22)} (x,y) = \tilde{A}_{11} V_{1m} V_{2n}^{{^{\prime\prime}}} + \tilde{A}_{66} V_{1m}^{{^{\prime\prime}}} V_{2n} ; $$
$$ \tilde{h}_{mnpq}^{(23)} (x,y) = \tilde{A}_{22} X_{m} Y_{n}^{^{\prime}} X_{p} Y_{q}^{^{\prime\prime}} + \tilde{A}_{12} X_{m}^{^{\prime}} Y_{n} X_{p}^{^{\prime}} Y_{q}^{^{\prime}} + \tilde{A}_{66} \left( {X_{m}^{^{\prime}} Y_{n} X_{p}^{^{\prime}} Y_{q}^{^{\prime}} + X_{m} Y_{n}^{^{\prime}} X_{p}^{^{\prime\prime}} Y_{q} } \right); $$
$$ \tilde{l}_{mn}^{(33)} (x,y) = \tilde{A}_{44}^{s} X_{m} Y_{n}^{^{\prime\prime}} + \tilde{A}_{55}^{s} X_{m}^{^{\prime\prime}} Y_{n} - K_{w} X_{m} Y_{n} + K_{sx} X_{m}^{^{\prime\prime}} Y_{n} + K_{sy} X_{m} Y_{n}^{^{\prime\prime}} ; $$
$$ \tilde{l}_{mn}^{(34)} (x,y) = \tilde{A}_{55}^{s} X_{m}^{^{\prime\prime}} Y_{n} ;\;\tilde{l}_{mn}^{(35)} (x,y) = \tilde{A}_{44}^{s} X_{m} Y_{n}^{^{\prime\prime}} ; $$
$$ \tilde{h}_{mnpq}^{(31)} (x,y) = \tilde{A}_{11} U_{1m}^{^{\prime}} U_{2n} X_{p}^{^{\prime\prime}} Y_{q} + \tilde{A}_{12} U_{1m}^{^{\prime}} U_{2n} X_{p} Y_{q}^{^{\prime\prime}} + 2\tilde{A}_{66} U_{1m} U_{2n}^{^{\prime}} X_{p}^{^{\prime}} Y_{q}^{^{\prime}} ; $$
$$ \tilde{h}_{mnpq}^{(32)} (x,y) = \tilde{A}_{12} V_{1m} V_{2n}^{^{\prime}} X_{p}^{^{\prime\prime}} Y_{q} + \tilde{A}_{11} V_{1m} V_{2n}^{^{\prime}} X_{p} Y_{q}^{^{\prime\prime}} + 2\tilde{A}_{66} V_{1m}^{^{\prime}} V_{2n} X_{p}^{^{\prime}} Y_{q}^{^{\prime}} ; $$
$$ \begin{gathered} \tilde{p}_{mnpqrs}^{(33)} (x,y) = \frac{1}{2}\tilde{A}_{11} X_{m}^{^{\prime}} Y_{n} X_{p}^{^{\prime}} Y_{q} X_{r}^{^{\prime\prime}} Y_{s} + \frac{1}{2}\tilde{A}_{12} X_{m} Y_{n}^{^{\prime}} X_{p} Y_{q}^{^{\prime}} X_{r}^{^{\prime\prime}} Y_{s} + \frac{1}{2}\tilde{A}_{12} X_{m}^{^{\prime}} Y_{n} X_{p}^{^{\prime}} Y_{q} X_{r} Y_{s}^{^{\prime\prime}} \\ + \frac{1}{2}\tilde{A}_{11} X_{m} Y_{n}^{^{\prime}} X_{p} Y_{q}^{^{\prime}} X_{r} Y_{s}^{^{\prime\prime}} + 2\tilde{A}_{66} X_{m}^{^{\prime}} Y_{n} X_{p} Y_{q}^{^{\prime}} X_{r}^{^{\prime}} Y_{s}^{^{\prime}} ; \\ \end{gathered} $$
$$ \tilde{l}_{mn}^{(43)} (x,y) = - \tilde{A}_{55}^{s} X_{m}^{^{\prime}} Y_{n} ;\;\tilde{l}_{mn}^{(44)} (x,y) = \tilde{C}_{11} X_{m}^{\prime \prime \prime } Y_{n} - \tilde{A}_{55} X_{m}^{^{\prime}} Y_{n} + \tilde{C}_{66} X_{m}^{^{\prime}} Y_{n}^{\prime \prime } ; $$
$$ \tilde{l}_{mn}^{(45)} (x,y) = \left( {\tilde{C}_{12} + \tilde{C}_{66} } \right)X_{m}^{^{\prime}} Y_{n}^{^{\prime\prime}} ;\;\tilde{l}_{mn}^{(53)} (x,y) = - \tilde{A}_{44}^{s} X_{m} Y_{n}^{^{\prime}} ;\;\tilde{l}_{mn}^{(54)} (x,y) = \left( {\tilde{C}_{12} + \tilde{C}_{66} } \right)X_{m}^{^{\prime\prime}} Y_{n}^{^{\prime}} ; $$
$$ \tilde{l}_{mn}^{(55)} (x,y) = \tilde{C}_{66} X_{m}^{\prime \prime } Y_{n}^{^{\prime}} + \tilde{C}_{22} X_{m} Y_{n}^{\prime \prime \prime } - \tilde{A}_{44}^{s} X_{m} Y_{n}^{^{\prime}} $$

Appendix B: The coefficients in Eq. (18)

$$ \begin{gathered} \left\{ {\tilde{L}_{mnij}^{(11)} ,\tilde{L}_{mnij}^{(12)} ,\tilde{H}_{mnpqij}^{(13)} } \right\} = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left\{ {\tilde{l}_{mn}^{(11)} ,\tilde{l}_{mn}^{(12)} ,\tilde{h}_{mnpq}^{(13)} } \right\}U_{1i} U_{2j} {\text{d}}x{\text{d}}y} } ; \hfill \\ \left\{ {\tilde{L}_{mnij}^{(21)} ,\tilde{L}_{mnij}^{(22)} ,\tilde{H}_{mnpqij}^{(23)} } \right\} = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left\{ {\tilde{l}_{mn}^{(21)} ,\tilde{l}_{mn}^{(22)} ,\tilde{h}_{mnpq}^{(23)} } \right\}V_{1i} V_{2j} {\text{d}}x{\text{d}}y} } ; \hfill \\ \left\{ {\tilde{L}_{mnij}^{(33)} ,\tilde{L}_{mnij}^{(34)} ,\tilde{L}_{mnij}^{(35)} ,\tilde{H}_{mnpqij}^{(31)} ,\tilde{H}_{mnpqij}^{(32)} ,\tilde{P}_{mnpqrsij}^{(33)} } \right\} = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left\{ {\tilde{l}_{mn}^{(33)} ,\tilde{l}_{mn}^{(34)} ,\tilde{l}_{mn}^{(35)} ,\tilde{h}_{mnpq}^{(31)} ,\tilde{h}_{mnpq}^{(32)} ,\tilde{p}_{mnpqrs}^{(33)} } \right\}X_{i} Y_{j} {\text{d}}x{\text{d}}y} } ; \hfill \\ \tilde{F}_{ij} = \int\limits_{0}^{a} {\int\limits_{0}^{b} {qX_{i} Y_{j} {\text{d}}x{\text{d}}y} } ; \hfill \\ \left\{ {\tilde{L}_{mnij}^{(43)} ,\tilde{L}_{mnij}^{(44)} ,\tilde{L}_{mnij}^{(45)} } \right\} = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left\{ {\tilde{l}_{mn}^{(11)} ,\tilde{l}_{mn}^{(12)} ,\tilde{h}_{mnpq}^{(13)} ,\tilde{l}_{mn}^{(43)} ,\tilde{l}_{mn}^{(44)} ,\tilde{l}_{mn}^{(45)} } \right\}X_{i}^{^{\prime}} Y_{j} {\text{d}}x{\text{d}}y} } ; \hfill \\ \left\{ {\tilde{L}_{mnij}^{(53)} ,\tilde{L}_{mnij}^{(54)} ,\tilde{L}_{mnij}^{(55)} } \right\} = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left\{ {\tilde{l}_{mn}^{(11)} ,\tilde{l}_{mn}^{(12)} ,\tilde{h}_{mnpq}^{(13)} ,\tilde{l}_{mn}^{(53)} ,\tilde{l}_{mn}^{(54)} ,\tilde{l}_{mn}^{(55)} } \right\}X_{i} Y_{j}^{^{\prime}} {\text{d}}x{\text{d}}y} } \hfill \\ \end{gathered} $$

Appendix C: The coefficients in Eq. (30) for SSSS-SB boundary condition

$$ \chi_{1} = \frac{{\frac{{\tilde{D}}}{4}\left( {\alpha_{p} \alpha_{r} \lambda_{q} \lambda_{s} - \alpha_{p}^{2} \lambda_{s}^{2} } \right)}}{{\left[ {\left( {\alpha_{p} - \alpha_{r} } \right)^{2} + \left( {\lambda_{q} - \lambda_{s} } \right)^{2} } \right]^{2} }};\;({\text{with}}\;p = r\;{\text{v}}\`a \;q = s:\;\chi_{1} = 0); $$
$$ \chi_{2} = \frac{{\frac{{\tilde{D}}}{4}\left( {\alpha_{p} \alpha_{r} \lambda_{q} \lambda_{s} - \alpha_{p}^{2} \lambda_{s}^{2} } \right)}}{{\left[ {\left( {\alpha_{p} + \alpha_{r} } \right)^{2} + \left( {\lambda_{q} + \lambda_{s} } \right)^{2} } \right]^{2} }};\;\chi_{3} = \frac{{\frac{{\tilde{D}}}{4}\left( {\alpha_{p} \alpha_{r} \lambda_{q} \lambda_{s} + \alpha_{p}^{2} \lambda_{s}^{2} } \right)}}{{\left[ {\left( {\alpha_{p} - \alpha_{r} } \right)^{2} + \left( {\lambda_{q} + \lambda_{s} } \right)^{2} } \right]^{2} }}; $$
$$ \chi_{4} = \frac{{\frac{{\tilde{D}}}{4}\left( {\alpha_{p} \alpha_{r} \lambda_{q} \lambda_{s} + \alpha_{p}^{2} \lambda_{s}^{2} } \right)}}{{\left[ {\left( {\alpha_{p} + \alpha_{r} } \right)^{2} + \left( {\lambda_{q} - \lambda_{s} } \right)^{2} } \right]^{2} }} $$

Appendix D: The coefficients in Eq. (31) for SCSC-SB boundary condition

$$ \chi_{1} = \frac{{\frac{{\tilde{D}}}{4}\left( {\alpha_{p} \alpha_{r} \lambda_{q} \lambda_{s} - \alpha_{p}^{2} \lambda_{s}^{2} } \right)}}{{\left[ {\left( {\alpha_{p} - \alpha_{r} } \right)^{2} + 4\left( {\lambda_{q} - \lambda_{s} } \right)^{2} } \right]^{2} }}; ({\text{with}}\;p = r\;{\text{v}}\`a \;q = s:\chi_{1} = 0); $$
$$ \chi_{2} = \frac{{\frac{{\tilde{D}}}{4}\left[ {\alpha_{p} \alpha_{r} \lambda_{q} \lambda_{s} + \alpha_{p}^{2} \lambda_{s}^{2} } \right]}}{{\left[ {\left( {\alpha_{p} + \alpha_{r} } \right)^{2} + 4\left( {\lambda_{q} + \lambda_{s} } \right)^{2} } \right]^{2} }};\;\chi_{3} = \frac{{ - \frac{{\tilde{D}}}{4}\left( {\alpha_{p} \alpha_{r} \lambda_{q} \lambda_{s} + \alpha_{p}^{2} \lambda_{s}^{2} } \right)}}{{\left[ {\left( {\alpha_{p} - \alpha_{r} } \right)^{2} + 4\left( {\lambda_{q} + \lambda_{s} } \right)^{2} } \right]^{2} }}; $$
$$ \chi_{4} = \frac{{ - \frac{{\tilde{D}}}{4}\left( {\alpha_{p} \alpha_{r} \lambda_{q} \lambda_{s} - \alpha_{p}^{2} \lambda_{s}^{2} } \right)}}{{\left[ {\left( {\alpha_{p} + \alpha_{r} } \right)^{2} + 4\left( {\lambda_{q} - \lambda_{s} } \right)^{2} } \right]^{2} }};\;\chi_{5} = \frac{{\frac{{\tilde{D}}}{2}\alpha_{p}^{2} \lambda_{s}^{2} }}{{\left[ {\left( {\alpha_{p} - \alpha_{r} } \right)^{2} + 4\lambda_{s}^{2} } \right]^{2} }};\;\chi_{6} = \frac{{ - \frac{{\tilde{D}}}{2}\alpha_{p}^{2} \lambda_{s}^{2} }}{{\left[ {\left( {\alpha_{p} + \alpha_{r} } \right)^{2} + 4\lambda_{s}^{2} } \right]^{2} }} $$

Appendix E: The coefficients in Eq. (32) for CCCC-SB boundary condition

$$ \chi_{1} = \frac{{\frac{{\tilde{D}}}{16}\left( {\alpha_{p} \alpha_{r} \lambda_{q} \lambda_{s} - \alpha_{p}^{2} \lambda_{s}^{2} } \right)}}{{\left[ {\left( {\alpha_{p} - \alpha_{r} } \right)^{2} + \left( {\lambda_{q} - \lambda_{s} } \right)^{2} } \right]^{2} }};\; ({\text{with}}\;p = r\;{\text{v}}\`a \;q = s:\;\chi_{1} = 0); $$
$$ \chi_{2} = \frac{{\frac{{\tilde{D}}}{16}\left( {\alpha_{p} \alpha_{r} \lambda_{q} \lambda_{s} - \alpha_{p}^{2} \lambda_{s}^{2} } \right)}}{{\left[ {\left( {\alpha_{p} + \alpha_{r} } \right)^{2} + \left( {\lambda_{q} + \lambda_{s} } \right)^{2} } \right]^{2} }};\;\chi_{3} = \frac{{ - \frac{{\tilde{D}}}{16}\left( {\alpha_{p} \alpha_{r} \lambda_{q} \lambda_{s} + \alpha_{p}^{2} \lambda_{s}^{2} } \right)}}{{\left[ {\left( {\alpha_{p} - \alpha_{r} } \right)^{2} + \left( {\lambda_{q} + \lambda_{s} } \right)^{2} } \right]^{2} }}; $$
$$ \chi_{4} = \frac{{ - \frac{{\tilde{D}}}{16}\left( {\alpha_{p} \alpha_{r} \lambda_{q} \lambda_{s} + \alpha_{p}^{2} \lambda_{s}^{2} } \right)}}{{\left[ {\left( {\alpha_{p} + \alpha_{r} } \right)^{2} + \left( {\lambda_{q} - \lambda_{s} } \right)^{2} } \right]^{2} }};\;\chi_{5} = \frac{{ - \frac{{\tilde{D}}}{4}\alpha_{p}^{2} \lambda_{s}^{2} }}{{\left( {\alpha_{p}^{2} + \lambda_{s}^{2} } \right)^{2} }};\;\chi_{6} = \frac{{\frac{{\tilde{D}}}{8}\alpha_{p}^{2} \lambda_{s}^{2} }}{{\left[ {\alpha_{p}^{2} + \left( {\lambda_{q} - \lambda_{s} } \right)^{2} } \right]^{2} }}; $$
$$ \chi_{7} = \frac{{\frac{{\tilde{D}}}{8}\alpha_{p}^{2} \lambda_{s}^{2} }}{{\left[ {\alpha_{p}^{2} + \left( {\lambda_{q} + \lambda_{s} } \right)^{2} } \right]^{2} }};\;\chi_{8} = \frac{{\frac{{\tilde{D}}}{8}\alpha_{p}^{2} \lambda_{s}^{2} }}{{\left[ {\left( {\alpha_{p} - \alpha_{r} } \right)^{2} + \lambda_{s}^{2} } \right]^{2} }};\;\chi_{9} = \frac{{\frac{{\tilde{D}}}{8}\alpha_{p}^{2} \lambda_{s}^{2} }}{{\left[ {\left( {\alpha_{p} + \alpha_{r} } \right)^{2} + \lambda_{s}^{2} } \right]^{2} }}; $$

Appendix F: The coefficients \(\zeta_{pqrs}^{(1)} ,\zeta_{pqrs}^{(2)}\) in Eq. (33)

6.1 Boundary condition SSSS-SB:

$$ \zeta_{pqrs}^{(1)} = \frac{1}{ab}\int\limits_{0}^{b} {\int\limits_{0}^{a} {\left\{ \begin{gathered} \chi_{1} \frac{{\left( {q - s} \right)^{2} \pi^{2} }}{{b^{2} }}\cos \frac{{\left( {p - r} \right)\pi x}}{a}\cos \frac{{\left( {q - s} \right)\pi y}}{b} \hfill \\ \quad + \chi_{2} \frac{{\left( {q + s} \right)^{2} \pi^{2} }}{{b^{2} }}\cos \frac{{\left( {p + r} \right)\pi x}}{a}\cos \frac{{\left( {q + s} \right)\pi y}}{b} \hfill \\ \quad + \chi_{3} \frac{{\left( {q + s} \right)^{2} \pi^{2} }}{{b^{2} }}\cos \frac{{\left( {p - r} \right)\pi x}}{a}\cos \frac{{\left( {q + s} \right)\pi y}}{b} \hfill \\ \quad + \chi_{4} \frac{{\left( {q - s} \right)^{2} \pi^{2} }}{{b^{2} }}\cos \frac{{\left( {p + r} \right)\pi x}}{a}\cos \frac{{\left( {q - s} \right)\pi y}}{b} \hfill \\ \quad + \frac{{\tilde{A}_{11} }}{2}\frac{p\pi }{a}\frac{r\pi }{a}\cos \frac{p\pi x}{a}\cos \frac{r\pi x}{a}\sin \frac{q\pi y}{b}\sin \frac{s\pi y}{b} \hfill \\ \quad + \frac{{\tilde{A}_{12} }}{2}\frac{q\pi }{b}\frac{s\pi }{b}\sin \frac{p\pi x}{a}\sin \frac{r\pi x}{a}\cos \frac{q\pi y}{b}\cos \frac{s\pi y}{b} \hfill \\ \end{gathered} \right\}{\text{d}}x{\text{d}}y} } ; $$
$$ \zeta_{pqrs}^{(2)} = \frac{1}{ab}\int\limits_{0}^{b} {\int\limits_{0}^{a} {\left\{ \begin{gathered} \chi_{1} \frac{{\left( {p - r} \right)^{2} \pi^{2} }}{{a^{2} }}\cos \frac{{\left( {p - r} \right)\pi x}}{a}\cos \frac{{\left( {q - s} \right)\pi y}}{b} \hfill \\ \quad + \chi_{2} \frac{{\left( {p + r} \right)^{2} \pi^{2} }}{{a^{2} }}\cos \frac{{\left( {p + r} \right)\pi x}}{a}\cos \frac{{\left( {q + s} \right)\pi y}}{b} \hfill \\ \quad + \chi_{3} \frac{{\left( {p - r} \right)^{2} \pi^{2} }}{{a^{2} }}\cos \frac{{\left( {p - r} \right)\pi x}}{a}\cos \frac{{\left( {q + s} \right)\pi y}}{b} \hfill \\ \quad + \chi_{4} \frac{{\left( {p + r} \right)^{2} \pi^{2} }}{{a^{2} }}\cos \frac{{\left( {p + r} \right)\pi x}}{a}\cos \frac{{\left( {q - s} \right)\pi y}}{b} \hfill \\ \quad + \frac{{\tilde{A}_{12} }}{2}\frac{p\pi }{a}\frac{r\pi }{a}\cos \frac{p\pi x}{a}\cos \frac{r\pi x}{a}\sin \frac{q\pi y}{b}\sin \frac{s\pi y}{b} \hfill \\ \quad + \frac{{\tilde{A}_{11} }}{2}\frac{q\pi }{b}\frac{s\pi }{b}\sin \frac{p\pi x}{a}\sin \frac{r\pi x}{a}\cos \frac{q\pi y}{b}\cos \frac{s\pi y}{b} \hfill \\ \end{gathered} \right\}{\text{d}}x{\text{d}}y} } ; $$

6.2 Boundary condition SCSC-SB:

$$ \zeta_{pqrs}^{(1)} = \frac{1}{ab}\int\limits_{0}^{b} {\int\limits_{0}^{a} {\left\{ \begin{gathered} \chi_{1} \frac{{4\left( {q - s} \right)^{2} \pi^{2} }}{{b^{2} }}\cos \frac{{\left( {p - r} \right)\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b} \hfill \\ + \chi_{2} \frac{{4\left( {q + s} \right)^{2} \pi^{2} }}{{b^{2} }}\cos \frac{{\left( {p + r} \right)\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b} \hfill \\ + \chi_{3} \frac{{4\left( {q + s} \right)^{2} \pi^{2} }}{{b^{2} }}\cos \frac{{\left( {p - r} \right)\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b} \hfill \\ + \chi_{4} \frac{{4\left( {q - s} \right)^{2} \pi^{2} }}{{b^{2} }}\cos \frac{{\left( {p + r} \right)\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b} \hfill \\ + \chi_{5} \frac{{4s^{2} \pi^{2} }}{{b^{2} }}\cos \frac{{\left( {p - r} \right)\pi x}}{a}\cos \frac{2s\pi y}{b} \hfill \\ + \chi_{6} \frac{{4s^{2} \pi^{2} }}{{b^{2} }}\cos \frac{{\left( {p + r} \right)\pi x}}{a}\cos \frac{2s\pi y}{b} \hfill \\ + \frac{{\tilde{A}_{11} }}{2}\frac{p\pi }{a}\frac{r\pi }{a}\cos \frac{p\pi x}{a}\cos \frac{r\pi x}{a}\sin^{2} \frac{q\pi y}{b}\sin^{2} \frac{s\pi y}{b} \hfill \\ + \frac{{\tilde{A}_{12} }}{2}\frac{q\pi }{b}\frac{s\pi }{b}\sin \frac{p\pi x}{a}\sin \frac{r\pi x}{a}\sin \frac{2q\pi y}{b}\sin \frac{2s\pi y}{b} \hfill \\ \end{gathered} \right\}{\text{d}}x{\text{d}}y} } ; $$
$$ \zeta_{pqrs}^{(2)} = \frac{1}{ab}\int\limits_{0}^{b} {\int\limits_{0}^{a} {\left\{ \begin{gathered} \chi_{1} \frac{{\left( {p - r} \right)^{2} \pi^{2} }}{{a^{2} }}\cos \frac{{\left( {p - r} \right)\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b} \hfill \\ + \chi_{2} \frac{{\left( {p + r} \right)^{2} \pi^{2} }}{{a^{2} }}\cos \frac{{\left( {p + r} \right)\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b} \hfill \\ + \chi_{3} \frac{{\left( {p - r} \right)^{2} \pi^{2} }}{{a^{2} }}\cos \frac{{\left( {p - r} \right)\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b} \hfill \\ + \chi_{4} \frac{{\left( {p + r} \right)^{2} \pi^{2} }}{{a^{2} }}\cos \frac{{\left( {p + r} \right)\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b} \hfill \\ + \chi_{5} \frac{{\left( {p - r} \right)^{2} \pi^{2} }}{{a^{2} }}\cos \frac{{\left( {p - r} \right)\pi x}}{a}\cos \frac{2s\pi y}{b} \hfill \\ + \chi_{6} \frac{{\left( {p + r} \right)^{2} \pi^{2} }}{{a^{2} }}\cos \frac{{\left( {p + r} \right)\pi x}}{a}\cos \frac{2s\pi y}{b} \hfill \\ + \frac{{\tilde{A}_{12} }}{2}\frac{p\pi }{a}\frac{r\pi }{a}\cos \frac{p\pi x}{a}\cos \frac{r\pi x}{a}\sin^{2} \frac{q\pi y}{b}\sin^{2} \frac{s\pi y}{b} \hfill \\ + \frac{{\tilde{A}_{11} }}{2}\frac{q\pi }{b}\frac{s\pi }{b}\sin \frac{p\pi x}{a}\sin \frac{r\pi x}{a}\sin \frac{2q\pi y}{b}\sin \frac{2s\pi y}{b} \hfill \\ \end{gathered} \right\}{\text{d}}x{\text{d}}y} } ; $$

6.3 Boundary condition CCCC-SB:

$$ \zeta_{pqrs}^{(1)} = \frac{1}{ab}\int\limits_{0}^{b} {\int\limits_{0}^{a} {\left\{ \begin{gathered} \chi_{1} \frac{{4\left( {q - s} \right)^{2} \pi^{2} }}{{b^{2} }}\cos \frac{{2\left( {p - r} \right)\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b} \hfill \\ + \chi_{2} \frac{{4\left( {q + s} \right)^{2} \pi^{2} }}{{b^{2} }}\cos \frac{{2\left( {p + r} \right)\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b} \hfill \\ + \chi_{3} \frac{{4\left( {q + s} \right)^{2} \pi^{2} }}{{b^{2} }}\cos \frac{{2\left( {p - r} \right)\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b} \hfill \\ + \chi_{4} \frac{{4\left( {q - s} \right)^{2} \pi^{2} }}{{b^{2} }}\cos \frac{{2\left( {p + r} \right)\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b} \hfill \\ + \chi_{5} \frac{{4s^{2} \pi^{2} }}{{b^{2} }}\cos \frac{2p\pi x}{a}\cos \frac{2s\pi y}{b} \hfill \\ + \chi_{6} \frac{{4\left( {q - s} \right)^{2} \pi^{2} }}{{b^{2} }}\cos \frac{2p\pi x}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b} \hfill \\ + \chi_{7} \frac{{4\left( {q + s} \right)^{2} \pi^{2} }}{{b^{2} }}\cos \frac{2p\pi x}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b} \hfill \\ + \chi_{8} \frac{{4s^{2} \pi^{2} }}{{b^{2} }}\cos \frac{{2\left( {p - r} \right)\pi x}}{a}\cos \frac{2s\pi y}{b} \hfill \\ + \chi_{9} \frac{{4s^{2} \pi^{2} }}{{b^{2} }}\cos \frac{{2\left( {p + r} \right)\pi x}}{a}\cos \frac{2s\pi y}{b} \hfill \\ + \frac{{\tilde{A}_{11} }}{2}\frac{p\pi }{a}\frac{r\pi }{a}\sin \frac{2p\pi x}{a}\sin \frac{2r\pi x}{a}\sin^{2} \frac{q\pi y}{b}\sin^{2} \frac{s\pi y}{b} \hfill \\ + \frac{{\tilde{A}_{12} }}{2}\frac{q\pi }{b}\frac{s\pi }{b}\sin^{2} \frac{p\pi x}{a}\sin^{2} \frac{r\pi x}{a}\sin \frac{2q\pi y}{b}\sin \frac{2s\pi y}{b} \hfill \\ \end{gathered} \right\}{\text{d}}x{\text{d}}y} } ; $$
$$ \zeta_{pqrs}^{(2)} = \frac{1}{ab}\int\limits_{0}^{b} {\int\limits_{0}^{a} {\left\{ \begin{gathered} \chi_{1} \frac{{4\left( {p - r} \right)^{2} \pi^{2} }}{{a^{2} }}\cos \frac{{2\left( {p - r} \right)\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b} \hfill \\ + \chi_{2} \frac{{4\left( {p + r} \right)^{2} \pi^{2} }}{{a^{2} }}\cos \frac{{2\left( {p + r} \right)\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b} \hfill \\ + \chi_{3} \frac{{4\left( {p - r} \right)^{2} \pi^{2} }}{{a^{2} }}\cos \frac{{2\left( {p - r} \right)\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b} \hfill \\ + \chi_{4} \frac{{4\left( {p + r} \right)^{2} \pi^{2} }}{{a^{2} }}\cos \frac{{2\left( {p + r} \right)\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b} \hfill \\ + \chi_{5} \frac{{4p^{2} \pi^{2} }}{{a^{2} }}\cos \frac{2p\pi x}{a}\cos \frac{2s\pi y}{b} \hfill \\ + \chi_{6} \frac{{4p^{2} \pi^{2} }}{{a^{2} }}\cos \frac{2p\pi x}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b} \hfill \\ + \chi_{7} \frac{{4p^{2} \pi^{2} }}{{a^{2} }}\cos \frac{2p\pi x}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b} \hfill \\ + \chi_{8} \frac{{4\left( {p - r} \right)^{2} \pi^{2} }}{{a^{2} }}\cos \frac{{2\left( {p - r} \right)\pi x}}{a}\cos \frac{2s\pi y}{b} \hfill \\ + \chi_{9} \frac{{4\left( {p + r} \right)^{2} \pi^{2} }}{{a^{2} }}\cos \frac{{2\left( {p + r} \right)\pi x}}{a}\cos \frac{2s\pi y}{b} \hfill \\ + \frac{{\tilde{A}_{12} }}{2}\frac{p\pi }{a}\frac{r\pi }{a}\sin \frac{2p\pi x}{a}\sin \frac{2r\pi x}{a}\sin^{2} \frac{q\pi y}{b}\sin^{2} \frac{s\pi y}{b} \hfill \\ + \frac{{\tilde{A}_{11} }}{2}\frac{q\pi }{b}\frac{s\pi }{b}\sin^{2} \frac{p\pi x}{a}\sin^{2} \frac{r\pi x}{a}\sin \frac{2q\pi y}{b}\sin \frac{2s\pi y}{b} \hfill \\ \end{gathered} \right\}{\text{d}}x{\text{d}}y} } ; $$

Appendix G: The function \(g_{mnpqrs}^{(3)}\) in Eq. (34)

7.1 Boundary condition SSSS-SB:

$$ \tilde{g}_{mnpqrs}^{(3)} = \tilde{g}_{mnpqrs}^{(3a)} + \tilde{g}_{mnpqrs}^{(3b)} + \tilde{g}_{mnpqrs}^{(3c)} ; $$
$$ \begin{aligned} \tilde{g}_{{mnpqrs}}^{{(3a)}} = & \tilde{H}_{{1{\text{a}}}} \cos \frac{{\left( {p - r} \right)\pi x}}{a}\sin \frac{{m\pi x}}{a}\cos \frac{{\left( {q - s} \right)\pi y}}{b}\sin \frac{{n\pi y}}{b} \\ & + \tilde{H}_{{2{\text{a}}}} \cos \frac{{\left( {p + r} \right)\pi x}}{a}\sin \frac{{m\pi x}}{a}\cos \frac{{\left( {q + s} \right)\pi y}}{b}\sin \frac{{n\pi y}}{b} \\ & + \tilde{H}_{{3{\text{a}}}} \cos \frac{{\left( {p - r} \right)\pi x}}{a}\sin \frac{{m\pi x}}{a}\cos \frac{{\left( {q + s} \right)\pi y}}{b}\sin \frac{{n\pi y}}{b} \\ & + \tilde{H}_{{4{\text{a}}}} \cos \frac{{\left( {p + r} \right)\pi x}}{a}\sin \frac{{m\pi x}}{a}\cos \frac{{\left( {q - s} \right)\pi y}}{b}\sin \frac{{n\pi y}}{b}; \\ \tilde{g}_{{mnpqrs}}^{{(3b)}} = & - 2\tilde{H}_{{1b}} \sin \frac{{\left( {p - r} \right)\pi x}}{a}\cos \frac{{m\pi x}}{a}\sin \frac{{\left( {q - s} \right)\pi y}}{b}\cos \frac{{n\pi y}}{b} \\ & - 2\tilde{H}_{{2b}} \sin \frac{{\left( {p + r} \right)\pi x}}{a}\cos \frac{{m\pi x}}{a}\sin \frac{{\left( {q + s} \right)\pi y}}{b}\cos \frac{{n\pi y}}{b} \\ & - 2\tilde{H}_{{3b}} \sin \frac{{\left( {p - r} \right)\pi x}}{a}\cos \frac{{m\pi x}}{a}\sin \frac{{\left( {q + s} \right)\pi y}}{b}\cos \frac{{n\pi y}}{b} \\ & - 2\tilde{H}_{{4b}} \sin \frac{{\left( {p + r} \right)\pi x}}{a}\cos \frac{{m\pi x}}{a}\sin \frac{{\left( {q - s} \right)\pi y}}{b}\cos \frac{{n\pi y}}{b}; \\ \end{aligned} $$
$$ \tilde{g}_{mnpqrs}^{(3c)} = - \left( {\zeta_{pqrs}^{(1)} \alpha_{m}^{2} + \zeta_{pqrs}^{(2)} \lambda_{n}^{2} } \right)\sin \frac{m\pi x}{a}\sin \frac{n\pi y}{b}; $$

in which:

$$ \tilde{H}_{{1{\text{a}}}} = \chi_{1} \left[ {\left( {\lambda_{q} - \lambda_{s} } \right)^{2} \alpha_{m}^{2} + \left( {\alpha_{p} - \alpha_{r} } \right)^{2} \lambda_{n}^{2} } \right];\;\tilde{H}_{{2{\text{a}}}} = \chi_{2} \left[ {\left( {\lambda_{q} + \lambda_{s} } \right)^{2} \alpha_{m}^{2} + \left( {\alpha_{p} + \alpha_{r} } \right)^{2} \lambda_{n}^{2} } \right]; $$
$$ \tilde{H}_{{3{\text{a}}}} = \chi_{3} \left[ {\left( {\lambda_{q} + \lambda_{s} } \right)^{2} \alpha_{m}^{2} + \left( {\alpha_{p} - \alpha_{r} } \right)^{2} \lambda_{n}^{2} } \right];\;\tilde{H}_{{4{\text{a}}}} = \chi_{4} \left[ {\left( {\lambda_{q} - \lambda_{s} } \right)^{2} \alpha_{m}^{2} + \left( {\alpha_{p} + \alpha_{r} } \right)^{2} \lambda_{n}^{2} } \right];$$
$$ \tilde{H}_{1b} = \chi_{1} \left( {\alpha_{p} - \alpha_{r} } \right)\left( {\lambda_{q} - \lambda_{s} } \right)\alpha_{m} \lambda_{n} ;\;\tilde{H}_{2b} = \chi_{2} \left( {\alpha_{p} + \alpha_{r} } \right)\left( {\lambda_{q} + \lambda_{s} } \right)\alpha_{m} \lambda_{n} ; $$
$$ \tilde{H}_{3b} = \chi_{3} \left( {\alpha_{p} - \alpha_{r} } \right)\left( {\lambda_{q} + \lambda_{s} } \right)\alpha_{m} \lambda_{n} ;\;\tilde{H}_{4b} = \chi_{4} \left( {\alpha_{p} + \alpha_{r} } \right)\left( {\lambda_{q} - \lambda_{s} } \right)\alpha_{m} \lambda_{n} ; $$

7.2 Boundary condition CCCC-SB:

$$ \tilde{g}_{mnpqrs}^{(3)} = \tilde{g}_{mnpqrs}^{(3a)} + \tilde{g}_{mnpqrs}^{(3b)} + \tilde{g}_{mnpqrs}^{(3c)} + \tilde{g}_{mnpqrs}^{(3d)} ; $$
$$ \begin{aligned} \tilde{g}_{{mnpqrs}}^{{(3a)}} = & - 8\chi _{1} \frac{{m^{2} \pi ^{2} }}{{a^{2} }}\frac{{\left( {q - s} \right)^{2} \pi ^{2} }}{{b^{2} }}\cos \frac{{2\left( {p - r} \right)\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b}\cos \frac{{2m\pi x}}{a}\sin ^{2} \frac{{n\pi y}}{b} \\ & - 8\chi _{2} \frac{{m^{2} \pi ^{2} }}{{a^{2} }}\frac{{\left( {q + s} \right)^{2} \pi ^{2} }}{{b^{2} }}\cos \frac{{2\left( {p + r} \right)\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b}\cos \frac{{2m\pi x}}{a}\sin ^{2} \frac{{n\pi y}}{b} \\ & - 8\chi _{3} \frac{{m^{2} \pi ^{2} }}{{a^{2} }}\frac{{\left( {q + s} \right)^{2} \pi ^{2} }}{{b^{2} }}\cos \frac{{2\left( {p - r} \right)\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b}\cos \frac{{2m\pi x}}{a}\sin ^{2} \frac{{n\pi y}}{b} \\ & - 8\chi _{4} \frac{{m^{2} \pi ^{2} }}{{a^{2} }}\frac{{\left( {q - s} \right)^{2} \pi ^{2} }}{{b^{2} }}\cos \frac{{2\left( {p + r} \right)\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b}\cos \frac{{2m\pi x}}{a}\sin ^{2} \frac{{n\pi y}}{b} \\ & - 8\chi _{5} \frac{{m^{2} \pi ^{2} }}{{a^{2} }}\frac{{s^{2} \pi ^{2} }}{{b^{2} }}\cos \frac{{2p\pi x}}{a}\cos \frac{{2s\pi y}}{b}\cos \frac{{2m\pi x}}{a}\sin ^{2} \frac{{n\pi y}}{b} \\ & - 8\chi _{6} \frac{{m^{2} \pi ^{2} }}{{a^{2} }}\frac{{\left( {q - s} \right)^{2} \pi ^{2} }}{{b^{2} }}\cos \frac{{2p\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b}\cos \frac{{2m\pi x}}{a}\sin ^{2} \frac{{n\pi y}}{b} \\ & - 8\chi _{7} \frac{{m^{2} \pi ^{2} }}{{a^{2} }}\frac{{\left( {q + s} \right)^{2} \pi ^{2} }}{{b^{2} }}\cos \frac{{2p\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b}\cos \frac{{2m\pi x}}{a}\sin ^{2} \frac{{n\pi y}}{b} \\ & - 8\chi _{8} \frac{{m^{2} \pi ^{2} }}{{a^{2} }}\frac{{s^{2} \pi ^{2} }}{{b^{2} }}\cos \frac{{2\left( {p - r} \right)\pi x}}{a}\cos \frac{{2s\pi y}}{b}\cos \frac{{2m\pi x}}{a}\sin ^{2} \frac{{n\pi y}}{b} \\ & - 8\chi _{9} \frac{{m^{2} \pi ^{2} }}{{a^{2} }}\frac{{s^{2} \pi ^{2} }}{{b^{2} }}\cos \frac{{2\left( {p + r} \right)\pi x}}{a}\cos \frac{{2s\pi y}}{b}\cos \frac{{2m\pi x}}{a}\sin ^{2} \frac{{n\pi y}}{b}; \\ \end{aligned} $$
$$ \begin{aligned} \tilde{g}_{{mnpqrs}}^{{(3b)}} = & - 8\chi _{1} \frac{{m\pi }}{a}\frac{{n\pi }}{b}\frac{{\left( {p - r} \right)\pi }}{a}\frac{{\left( {q - s} \right)\pi }}{b}\cos \frac{{2\left( {p - r} \right)\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b}\sin \frac{{2m\pi x}}{a}\sin \frac{{2n\pi y}}{b} \\ & - 8\chi _{2} \frac{{m\pi }}{a}\frac{{n\pi }}{b}\frac{{\left( {p + r} \right)\pi }}{a}\frac{{\left( {q + s} \right)\pi }}{b}\cos \frac{{2\left( {p + r} \right)\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b}\sin \frac{{2m\pi x}}{a}\sin \frac{{2n\pi y}}{b} \\ & - 8\chi _{3} \frac{{m\pi }}{a}\frac{{n\pi }}{b}\frac{{\left( {p - r} \right)\pi }}{a}\frac{{\left( {q + s} \right)\pi }}{b}\cos \frac{{2\left( {p - r} \right)\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b}\sin \frac{{2m\pi x}}{a}\sin \frac{{2n\pi y}}{b} \\ & - 8\chi _{4} \frac{{m\pi }}{a}\frac{{n\pi }}{b}\frac{{\left( {p + r} \right)\pi }}{a}\frac{{\left( {q - s} \right)\pi }}{b}\cos \frac{{2\left( {p + r} \right)\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b}\sin \frac{{2m\pi x}}{a}\sin \frac{{2n\pi y}}{b} \\ & - 8\chi _{5} \frac{{m\pi }}{a}\frac{{n\pi }}{b}\frac{{p\pi }}{a}\frac{{s\pi }}{b}\cos \frac{{2p\pi x}}{a}\cos \frac{{2s\pi y}}{b}\sin \frac{{2m\pi x}}{a}\sin \frac{{2n\pi y}}{b} \\ & - 8\chi _{6} \frac{{m\pi }}{a}\frac{{n\pi }}{b}\frac{{p\pi }}{a}\frac{{\left( {q - s} \right)\pi }}{b}\cos \frac{{2p\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b}\sin \frac{{2m\pi x}}{a}\sin \frac{{2n\pi y}}{b} \\ & - 8\chi _{7} \frac{{m\pi }}{a}\frac{{n\pi }}{b}\frac{{p\pi }}{a}\frac{{\left( {q + s} \right)\pi }}{b}\cos \frac{{2p\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b}\sin \frac{{2m\pi x}}{a}\sin \frac{{2n\pi y}}{b} \\ & - 8\chi _{8} \frac{{m\pi }}{a}\frac{{n\pi }}{b}\frac{{\left( {p - r} \right)\pi }}{a}\frac{{s\pi }}{b}\cos \frac{{2\left( {p - r} \right)\pi x}}{a}\cos \frac{{2s\pi y}}{b}\sin \frac{{2m\pi x}}{a}\sin \frac{{2n\pi y}}{b} \\ & - 8\chi _{9} \frac{{m\pi }}{a}\frac{{n\pi }}{b}\frac{{\left( {p + r} \right)\pi }}{a}\frac{{s\pi }}{b}\cos \frac{{2\left( {p + r} \right)\pi x}}{a}\cos \frac{{2s\pi y}}{b}\sin \frac{{2m\pi x}}{a}\sin \frac{{2n\pi y}}{b}; \\ \end{aligned} $$
$$ \begin{aligned} \tilde{g}_{{mnpqrs}}^{{(3c)}} = & - 8\chi _{1} \frac{{n^{2} \pi ^{2} }}{{b^{2} }}\frac{{\left( {p - r} \right)^{2} \pi ^{2} }}{{a^{2} }}\cos \frac{{2\left( {p - r} \right)\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b}\sin ^{2} \frac{{m\pi x}}{a}\cos \frac{{2n\pi y}}{b} \\ & - 8\chi _{2} \frac{{n^{2} \pi ^{2} }}{{b^{2} }}\frac{{\left( {p + r} \right)^{2} \pi ^{2} }}{{a^{2} }}\cos \frac{{2\left( {p + r} \right)\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b}\sin ^{2} \frac{{m\pi x}}{a}\cos \frac{{2n\pi y}}{b} \\ & - 8\chi _{3} \frac{{n^{2} \pi ^{2} }}{{b^{2} }}\frac{{\left( {p - r} \right)^{2} \pi ^{2} }}{{a^{2} }}\cos \frac{{2\left( {p - r} \right)\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b}\sin ^{2} \frac{{m\pi x}}{a}\cos \frac{{2n\pi y}}{b} \\ & - 8\chi _{4} \frac{{n^{2} \pi ^{2} }}{{b^{2} }}\frac{{\left( {p + r} \right)^{2} \pi ^{2} }}{{a^{2} }}\cos \frac{{2\left( {p + r} \right)\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b}\sin ^{2} \frac{{m\pi x}}{a}\cos \frac{{2n\pi y}}{b} \\ & - 8\chi _{5} \frac{{n^{2} \pi ^{2} }}{{b^{2} }}\frac{{p^{2} \pi ^{2} }}{{a^{2} }}\cos \frac{{2p\pi x}}{a}\cos \frac{{2s\pi y}}{b}\sin ^{2} \frac{{m\pi x}}{a}\cos \frac{{2n\pi y}}{b} \\ & - 8\chi _{6} \frac{{n^{2} \pi ^{2} }}{{b^{2} }}\frac{{p^{2} \pi ^{2} }}{{a^{2} }}\cos \frac{{2p\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b}\sin ^{2} \frac{{m\pi x}}{a}\cos \frac{{2n\pi y}}{b} \\ & - 8\chi _{7} \frac{{n^{2} \pi ^{2} }}{{b^{2} }}\frac{{p^{2} \pi ^{2} }}{{a^{2} }}\cos \frac{{2p\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b}\sin ^{2} \frac{{m\pi x}}{a}\cos \frac{{2n\pi y}}{b} \\ & - 8\chi _{8} \frac{{n^{2} \pi ^{2} }}{{b^{2} }}\frac{{\left( {p - r} \right)^{2} \pi ^{2} }}{{a^{2} }}\cos \frac{{2\left( {p - r} \right)\pi x}}{a}\cos \frac{{2s\pi y}}{b}\sin ^{2} \frac{{m\pi x}}{a}\cos \frac{{2n\pi y}}{b} \\ & - 8\chi _{9} \frac{{n^{2} \pi ^{2} }}{{b^{2} }}\frac{{\left( {p + r} \right)^{2} \pi ^{2} }}{{a^{2} }}\cos \frac{{2\left( {p + r} \right)\pi x}}{a}\cos \frac{{2s\pi y}}{b}\sin ^{2} \frac{{m\pi x}}{a}\cos \frac{{2n\pi y}}{b}; \\ \end{aligned} $$
$$ \tilde{g}_{mnpqrs}^{(3d)} = 2\zeta_{pqrs}^{(1)} \alpha_{m}^{2} \cos \frac{2m\pi x}{a}\sin^{2} \frac{n\pi y}{b} + 2\zeta_{pqrs}^{(2)} \lambda_{n}^{2} \sin^{2} \frac{m\pi x}{a}\cos \frac{2n\pi y}{b}; $$

7.3 Boundary condition SCSC-SB:

$$ \tilde{g}_{mnpqrs}^{(3)} = \tilde{g}_{mnpqrs}^{(3a)} + \tilde{g}_{mnpqrs}^{(3b)} + \tilde{g}_{mnpqrs}^{(3c)} + \tilde{g}_{mnpqrs}^{(3d)} ; $$
$$ \begin{aligned} \tilde{g}_{{mnpqrs}}^{{(3a)}} = & \chi _{1} \frac{{4\left( {q - s} \right)^{2} \pi ^{2} }}{{b^{2} }}\frac{{m^{2} \pi ^{2} }}{{a^{2} }}\cos \frac{{\left( {p - r} \right)\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b}\sin \frac{{m\pi x}}{a}\sin ^{2} \frac{{n\pi y}}{b} \\ & + \chi _{2} \frac{{4\left( {q + s} \right)^{2} \pi ^{2} }}{{b^{2} }}\frac{{m^{2} \pi ^{2} }}{{a^{2} }}\cos \frac{{\left( {p + r} \right)\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b}\sin \frac{{m\pi x}}{a}\sin ^{2} \frac{{n\pi y}}{b} \\ & + \chi _{3} \frac{{4\left( {q + s} \right)^{2} \pi ^{2} }}{{b^{2} }}\frac{{m^{2} \pi ^{2} }}{{a^{2} }}\cos \frac{{\left( {p - r} \right)\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b}\sin \frac{{m\pi x}}{a}\sin ^{2} \frac{{n\pi y}}{b} \\ & + \chi _{4} \frac{{4\left( {q - s} \right)^{2} \pi ^{2} }}{{b^{2} }}\frac{{m^{2} \pi ^{2} }}{{a^{2} }}\cos \frac{{\left( {p + r} \right)\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b}\sin \frac{{m\pi x}}{a}\sin ^{2} \frac{{n\pi y}}{b} \\ & + \chi _{5} \frac{{4s^{2} \pi ^{2} }}{{b^{2} }}\frac{{m^{2} \pi ^{2} }}{{a^{2} }}\cos \frac{{\left( {p - r} \right)\pi x}}{a}\cos \frac{{2s\pi y}}{b}\sin \frac{{m\pi x}}{a}\sin ^{2} \frac{{n\pi y}}{b} \\ & + \chi _{6} \frac{{4s^{2} \pi ^{2} }}{{b^{2} }}\frac{{m^{2} \pi ^{2} }}{{a^{2} }}\cos \frac{{\left( {p + r} \right)\pi x}}{a}\cos \frac{{2s\pi y}}{b}\sin \frac{{m\pi x}}{a}\sin ^{2} \frac{{n\pi y}}{b}; \\ \end{aligned} $$
$$ \begin{aligned} \tilde{g}_{{mnpqrs}}^{{(3b)}} = & - \chi _{1} \frac{{4\left( {p - r} \right)\pi }}{a}\frac{{\left( {q - s} \right)\pi }}{b}\frac{{m\pi }}{a}\frac{{n\pi }}{b}\cos \frac{{\left( {p - r} \right)\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b}\cos \frac{{m\pi x}}{a}\sin \frac{{2n\pi y}}{b} \\ & - \chi _{2} \frac{{4\left( {p + r} \right)\pi }}{a}\frac{{\left( {q + s} \right)\pi }}{b}\frac{{m\pi }}{a}\frac{{n\pi }}{b}\cos \frac{{\left( {p + r} \right)\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b}\cos \frac{{m\pi x}}{a}\sin \frac{{2n\pi y}}{b} \\ & - \chi _{3} \frac{{4\left( {p - r} \right)\pi }}{a}\frac{{\left( {q + s} \right)\pi }}{b}\frac{{m\pi }}{a}\frac{{n\pi }}{b}\cos \frac{{\left( {p - r} \right)\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b}\cos \frac{{m\pi x}}{a}\sin \frac{{2n\pi y}}{b} \\ & - \chi _{4} \frac{{4\left( {p + r} \right)\pi }}{a}\frac{{\left( {q - s} \right)\pi }}{b}\frac{{m\pi }}{a}\frac{{n\pi }}{b}\cos \frac{{\left( {p + r} \right)\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b}\cos \frac{{m\pi x}}{a}\sin \frac{{2n\pi y}}{b} \\ & - \chi _{5} \frac{{4\left( {p - r} \right)\pi }}{a}\frac{{s\pi }}{b}\frac{{m\pi }}{a}\frac{{n\pi }}{b}\cos \frac{{\left( {p - r} \right)\pi x}}{a}\cos \frac{{2s\pi y}}{b}\cos \frac{{m\pi x}}{a}\sin \frac{{2n\pi y}}{b} \\ & - \chi _{6} \frac{{4\left( {p + r} \right)\pi }}{a}\frac{{s\pi }}{b}\frac{{m\pi }}{a}\frac{{n\pi }}{b}\cos \frac{{\left( {p + r} \right)\pi x}}{a}\cos \frac{{2s\pi y}}{b}\cos \frac{{m\pi x}}{a}\sin \frac{{2n\pi y}}{b}; \\ \end{aligned} $$
$$ \begin{aligned} \tilde{g}_{{mnpqrs}}^{{(3c)}} = & - 2\chi _{1} \frac{{\left( {p - r} \right)^{2} \pi ^{2} }}{{a^{2} }}\frac{{n^{2} \pi ^{2} }}{{b^{2} }}\cos \frac{{\left( {p - r} \right)\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b}\sin \frac{{m\pi x}}{a}\cos \frac{{2n\pi y}}{b} \\ & - 2\chi _{2} \frac{{\left( {p + r} \right)^{2} \pi ^{2} }}{{a^{2} }}\frac{{n^{2} \pi ^{2} }}{{b^{2} }}\cos \frac{{\left( {p + r} \right)\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b}\sin \frac{{m\pi x}}{a}\cos \frac{{2n\pi y}}{b} \\ & - 2\chi _{3} \frac{{\left( {p - r} \right)^{2} \pi ^{2} }}{{a^{2} }}\frac{{n^{2} \pi ^{2} }}{{b^{2} }}\cos \frac{{\left( {p - r} \right)\pi x}}{a}\cos \frac{{2\left( {q + s} \right)\pi y}}{b}\sin \frac{{m\pi x}}{a}\cos \frac{{2n\pi y}}{b} \\ & - 2\chi _{4} \frac{{\left( {p + r} \right)^{2} \pi ^{2} }}{{a^{2} }}\frac{{n^{2} \pi ^{2} }}{{b^{2} }}\cos \frac{{\left( {p + r} \right)\pi x}}{a}\cos \frac{{2\left( {q - s} \right)\pi y}}{b}\sin \frac{{m\pi x}}{a}\cos \frac{{2n\pi y}}{b} \\ & - 2\chi _{5} \frac{{\left( {p - r} \right)^{2} \pi ^{2} }}{{a^{2} }}\frac{{n^{2} \pi ^{2} }}{{b^{2} }}\cos \frac{{\left( {p - r} \right)\pi x}}{a}\cos \frac{{2s\pi y}}{b}\sin \frac{{m\pi x}}{a}\cos \frac{{2n\pi y}}{b} \\ & - 2\chi _{6} \frac{{\left( {p + r} \right)^{2} \pi ^{2} }}{{a^{2} }}\frac{{n^{2} \pi ^{2} }}{{b^{2} }}\cos \frac{{\left( {p + r} \right)\pi x}}{a}\cos \frac{{2s\pi y}}{b}\sin \frac{{m\pi x}}{a}\cos \frac{{2n\pi y}}{b}; \\ \tilde{g}_{{mnpqrs}}^{{(3d)}} = & - \zeta _{{pqrs}}^{{(1)}} \alpha _{m}^{2} \sin \frac{{m\pi x}}{a}\sin ^{2} \frac{{n\pi y}}{b} + 2\zeta _{{pqrs}}^{{(2)}} \lambda _{n}^{2} \sin \frac{{m\pi x}}{a}\cos \frac{{2n\pi y}}{b}. \\ \end{aligned} $$

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Long, N.V., Tu, T.M., Truong, H.Q. et al. Displacement-based and stress-based analytical approaches for nonlinear bending analysis of functionally graded porous plates resting on elastic substrate. Acta Mech 233, 1689–1714 (2022). https://doi.org/10.1007/s00707-022-03196-5

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