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Buckling of bi-coated functionally graded porous nanoplates via a nonlocal strain gradient quasi-3D theory

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Abstract

With the increasing use of coated functionally graded materials (FGMs) in various industrial engineering, their accurate modeling is very theoretically challenging and therefore has attracted the attention of many scholars, especially for multilayer coated FGM nanostructures. To address this challenge, a new nanoplate model is proposed herein to characterize the buckling behavior of bilayer FG porous plates, which is capable of both geometrically thickness-stretching and physically microstructure-dependent effects. The materials are graded continuously through 2-directional by using a power law function. Two types of coated FG plates are investigated, Hardcore and Softcore FG plates. Based on the generalized field of displacement, a Quasi-3D higher-order shear deformation plate theory is proposed in this work by reducing the number of variables from six to five variables. The equilibrium equations are performed based on the virtual work principle and solved using the Galerkin method to cover various boundary conditions. The accuracy of the proposed solution was validated and it is in good agreement with the counterparts available in the open literature. The effects of microstructure-dependent length parameters, geometric parameters, and material property changes on the critical buckling load are studied in detail.

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References

  1. Garg, A., Chalak, H.D., Belarbi, M.O., Zenkour, A.M.: A parametric analysis of free vibration and bending behavior of sandwich beam containing an open-cell metal foam core. Arch. Civ. Mech. Eng. 22(1), 56 (2022). https://doi.org/10.1007/s43452-021-00368-3

    Article  Google Scholar 

  2. Garg, A., Belarbi, M.O., Chalak, H.D., Li, L., Sharma, A., Avcar, M., Sharma, N., Paruthi, S., Gulia, R.: Buckling and free vibration analysis of bio-inspired laminated sandwich plates with helicoidal/Bouligand face sheets containing softcore. Ocean Eng. 270, 113684 (2023). https://doi.org/10.1016/j.oceaneng.2023.113684

    Article  Google Scholar 

  3. Bensaid, I., Daikh, A.A., Drai, A.: Size-dependent free vibration and buckling analysis of sigmoid and power law functionally graded sandwich nanobeams with microstructural defects. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 234(18), 3667–3688 (2020)

    Google Scholar 

  4. Belarbi, M.-O., Khechai, A., Bessaim, A., Houari, M.-S.-A., Garg, A., Hirane, H., Chalak, H.: Finite element bending analysis of symmetric and non-symmetric functionally graded sandwich beams using a novel parabolic shear deformation theory. Proc. Inst. Mech. Eng. Part L J. Mater. Des. Appl. 235(11), 2482–2504 (2021)

    Google Scholar 

  5. Garg, A., Belarbi, M.-O., Chalak, H.D., Chakrabarti, A.: A review of the analysis of sandwich FGM structures. Compos. Struct. 258, 113427 (2021). https://doi.org/10.1016/j.compstruct.2020.113427

    Article  Google Scholar 

  6. Garg, A., Chalak, H.D., Li, L., Belarbi, M.O., Sahoo, R., Mukhopadhyay, T.: Vibration and buckling analyses of sandwich plates containing functionally graded metal foam core. Acta Mech. Solida Sin. 35(4), 1–16 (2022). https://doi.org/10.1007/s10338-021-00295-z

    Article  Google Scholar 

  7. Daikh, A.-A., Belarbi, M.-O., Ahmed, D., Houari, M.S.A., Avcar, M., Tounsi, A., Eltaher, M.A.: Static analysis of functionally graded plate structures resting on variable elastic foundation under various boundary conditions. Acta Mech. 234(2), 775–806 (2023). https://doi.org/10.1007/s00707-022-03405-1

    Article  MathSciNet  Google Scholar 

  8. Daikh, A.A., Houari, M.S.A., Belarbi, M.O., Mohamed, S.A., Eltaher, M.A.: Static and dynamic stability responses of multilayer functionally graded carbon nanotubes reinforced composite nanoplates via quasi 3D nonlocal strain gradient theory. Def. Technol. (2021). https://doi.org/10.1016/j.dt.2021.09.011

    Article  Google Scholar 

  9. Aifantis, E.C.: Strain Gradient Interpretation of Size Effects, in Fracture Scaling, pp. 299–314. Springer, Berlin (1999)

    Google Scholar 

  10. Gao, H., Huang, Y., Nix, W., Hutchinson, J.: Mechanism-based strain gradient plasticity—I. Theory. J. Mech. Phys. Solids 47(6), 1239–1263 (1999)

    MathSciNet  MATH  Google Scholar 

  11. Gurtin, M.E., Ian Murdoch, A.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57(4), 291–323 (1975)

    MathSciNet  MATH  Google Scholar 

  12. Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14(6), 431–440 (1978)

    MATH  Google Scholar 

  13. Eringen, A.C.: Theory of micropolar plates. Z. Angew. Math. Phys. 18(1), 12–30 (1967)

    Google Scholar 

  14. Yang, F., Chong, A., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39(10), 2731–2743 (2002)

    MATH  Google Scholar 

  15. Askes, H., Aifantis, E.C.: Gradient elasticity and flexural wave dispersion in carbon nanotubes. Phys. Rev. B 80(19), 195412 (2009)

    Google Scholar 

  16. Eringen, A.C.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10(1), 1–16 (1972)

    MathSciNet  MATH  Google Scholar 

  17. Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54(9), 4703–4710 (1983)

    Google Scholar 

  18. Belarbi, M.-O., Daikh, A.A., Garg, A., Merzouki, T., Chalak, H., Hirane, H.: Nonlocal finite element model for the bending and buckling analysis of functionally graded nanobeams using a novel shear deformation theory. Compos. Struct. 264, 113712 (2021)

    Google Scholar 

  19. Esen, I., Daikh, A.A., Eltaher, M.A.: Dynamic response of nonlocal strain gradient FG nanobeam reinforced by carbon nanotubes under moving point load. Eur. Phys. J. Plus 136(4), 1–22 (2021)

    Google Scholar 

  20. Daikh, A.A., Houari, M.S.A., Eltaher, M.A.: A novel nonlocal strain gradient Quasi-3D bending analysis of sigmoid functionally graded sandwich nanoplates. Compos. Struct. 262, 113347 (2021). https://doi.org/10.1016/j.compstruct.2020.113347

    Article  Google Scholar 

  21. Benzair, A., Tounsi, A., Besseghier, A., Heireche, H., Moulay, N., Boumia, L.: The thermal effect on vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory. J. Phys. D Appl. Phys. 41(22), 225404 (2008)

    Google Scholar 

  22. Berrabah, H., Tounsi, A., Semmah, A., Adda Bedia, E.: Comparison of various refined nonlocal beam theories for bending, vibration and buckling analysis of nanobeams. Struct. Eng. Mech. Int. J. 48(3), 351–365 (2013)

    Google Scholar 

  23. Van Vinh, P., Tounsi, A., Belarbi, M.-O.: On the nonlocal free vibration analysis of functionally graded porous doubly curved shallow nanoshells with variable nonlocal parameters. Eng. Comput. (2022). https://doi.org/10.1007/s00366-022-01687-6

    Article  Google Scholar 

  24. Vinh, P.V., Belarbi, M.-O., Tounsi, A.: Wave propagation analysis of functionally graded nanoplates using nonlocal higher-order shear deformation theory with spatial variation of the nonlocal parameters. Waves Random Complex Media (2022). https://doi.org/10.1080/17455030.2022.2036387

    Article  Google Scholar 

  25. Ghayesh, M.H., Farajpour, A.: A review on the mechanics of functionally graded nanoscale and microscale structures. Int. J. Eng. Sci. 137, 8–36 (2019)

    MathSciNet  MATH  Google Scholar 

  26. Garg, A., Chalak, H., Zenkour, A., Belarbi, M.-O., Houari, M.-S.-A.: A review of available theories and methodologies for the analysis of nano isotropic, nano functionally graded, and CNT reinforced nanocomposite structures. Arch. Comput. Methods Eng. (2021). https://doi.org/10.1007/s11831-021-09652-0

    Article  Google Scholar 

  27. Reddy, J., Pang, S.: Nonlocal continuum theories of beams for the analysis of carbon nanotubes. J. Appl. Phys. 103(2), 023511 (2008)

    Google Scholar 

  28. Reddy, J.: Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. Int. J. Eng. Sci. 48(11), 1507–1518 (2010)

    MathSciNet  MATH  Google Scholar 

  29. Peddieson, J., Buchanan, G.R., McNitt, R.P.: Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41(3–5), 305–312 (2003)

    Google Scholar 

  30. Rahmani, A., Babaei, A., Faroughi, S.: Vibration characteristics of functionally graded micro-beam carrying an attached mass. Mech. Adv. Compos. Struct. 7(1), 49–58 (2020)

    Google Scholar 

  31. Ansari, R., Sahmani, S., Arash, B.: Nonlocal plate model for free vibrations of single-layered graphene sheets. Phys. Lett. A 375(1), 53–62 (2010)

    Google Scholar 

  32. Reddy, J.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45(2–8), 288–307 (2007)

    MATH  Google Scholar 

  33. Arefi, M., Kiani, M., Zamani, M.: Nonlocal strain gradient theory for the magneto-electro-elastic vibration response of a porous FG-core sandwich nanoplate with piezomagnetic face sheets resting on an elastic foundation. J. Sandw. Struct. Mater. 22(7), 2157–2185 (2020)

    Google Scholar 

  34. Hadji, L., Avcar, M., Zouatnia, N.: Natural frequency analysis of imperfect FG sandwich plates resting on Winkler–Pasternak foundation. Mater. Today Proc. 53, 153–160 (2022)

    Google Scholar 

  35. Hadji, L., Avcar, M.: Free vibration analysis of FG porous sandwich plates under various boundary conditions. J. Appl. Comput. Mech. 7(2), 505–519 (2021)

    Google Scholar 

  36. Sobhani, E., Avcar, M.: Natural frequency analysis of imperfect GNPRN conical shell, cylindrical shell, and annular plate structures resting on Winkler–Pasternak Foundations under arbitrary boundary conditions. Eng. Anal. Bound. Elem. 144, 145–164 (2022)

    MathSciNet  Google Scholar 

  37. Civalek, Ö., Avcar, M.: Free vibration and buckling analyses of CNT reinforced laminated non-rectangular plates by discrete singular convolution method. Eng. Comput. 38(Suppl 1), 489–521 (2022)

    Google Scholar 

  38. Nian, Y., Wan, S., Wang, X., Zhou, P., Avcar, M., Li, M.: Study on crashworthiness of nature-inspired functionally graded lattice metamaterials for bridge pier protection against ship collision. Eng. Struct. 277, 115404 (2023)

    Google Scholar 

  39. Avcar, M., Hadji, L., Akan, R.: The influence of Winkler–Pasternak elastic foundations on the natural frequencies of imperfect functionally graded sandwich beams. Geomech. Eng. 31(1), 99–112 (2022)

    Google Scholar 

  40. Benachour, A., Tahar, H.D., Atmane, H.A., Tounsi, A., Ahmed, M.S.: A four variable refined plate theory for free vibrations of functionally graded plates with arbitrary gradient. Compos. B Eng. 42(6), 1386–1394 (2011)

    Google Scholar 

  41. Ramteke, P.M.: Effect of grading pattern and porosity on the Eigen characteristics of porous functionally graded structure. Steel Compos. Struct. Int. J. 33(6), 865–875 (2019)

    Google Scholar 

  42. Ramteke, P.M., Panda, S.K., Patel, B.: Nonlinear eigenfrequency characteristics of multi-directional functionally graded porous panels. Compos. Struct. 279, 114707 (2022)

    Google Scholar 

  43. Ramteke, P.M., Panda, S.K.: Free vibrational behaviour of multi-directional porous functionally graded structures. Arab. J. Sci. Eng. 46, 7741–7756 (2021)

    Google Scholar 

  44. Ramteke, P.M., Mahapatra, B.P., Panda, S.K., Sharma, N.: Static deflection simulation study of 2D Functionally graded porous structure. Mater. Today Proc. 33, 5544–5547 (2020)

    Google Scholar 

  45. Ramteke, P.M., Patel, B., Panda, S.K.: Time-dependent deflection responses of porous FGM structure including pattern and porosity. Int. J. Appl. Mech. 12(09), 2050102 (2020)

    Google Scholar 

  46. Hissaria, P., Ramteke, P.M., Hirwani, C.K., Mahmoud, S., Kumar, E.K., Panda, S.K.: Numerical investigation of eigenvalue characteristics (vibration and buckling) of damaged porous bidirectional FG panels. J. Vib. Eng. Technol. (2022). https://doi.org/10.1007/s42417-022-00677-8

    Article  Google Scholar 

  47. Ramteke, P.M., Mehar, K., Sharma, N., Panda, S.: Numerical prediction of deflection and stress responses of functionally graded structure for grading patterns (power-law, sigmoid, and exponential) and variable porosity (even/uneven). Sci. Iran. 28(2), 811–829 (2021)

    Google Scholar 

  48. Ramteke, P.M., Sharma, N., Choudhary, J., Hissaria, P., Panda, S.K.: Multidirectional grading influence on static/dynamic deflection and stress responses of porous FG panel structure: a micromechanical approach. Eng. Comput. 38(Suppl 4), 3077–3097 (2022)

    Google Scholar 

  49. Ramteke, P.M., Kumar, V., Sharma, N., Panda, S.K.: Geometrical nonlinear numerical frequency prediction of porous functionally graded shell panel under thermal environment. Int. J. Non-Linear Mech. 143, 104041 (2022)

    Google Scholar 

  50. Ramteke, P.M., Patel, B., Panda, S.K.: Nonlinear eigenfrequency prediction of functionally graded porous structure with different grading patterns. Waves Random Complex Media (2021). https://doi.org/10.1080/17455030.2021.2005850

    Article  Google Scholar 

  51. Malhari Ramteke, P., Kumar Panda, S., Sharma, N.: Nonlinear vibration analysis of multidirectional porous functionally graded panel under thermal environment. AIAA J. 60(8), 4923–4933 (2022)

    Google Scholar 

  52. Sahoo, B., Sharma, N., Sahoo, B., Ramteke, P.M., Panda, S.K., Mahmoud, S.: Nonlinear vibration analysis of FGM sandwich structure under thermal loadings. Structures 44(3), 1392–1402 (2022)

    Google Scholar 

  53. Choudhary, J., Patle, B.K., Ramteke, P.M., Hirwani, C.K., Panda, S.K., Katariya, P.V.: Static and dynamic deflection characteristics of cracked porous FG panels. Int. J. Appl. Mech. 14(7), 2250076 (2022)

    Google Scholar 

  54. Ramteke, P.M., Panda, S.K.: Nonlinear static and dynamic response prediction of bidirectional doubly-curved porous FG panel and experimental validation. Compos. A Appl. Sci. Manuf. 166, 107388 (2023)

    Google Scholar 

  55. Daneshmehr, A., Rajabpoor, A.: Stability of size dependent functionally graded nanoplate based on nonlocal elasticity and higher order plate theories and different boundary conditions. Int. J. Eng. Sci. 82, 84–100 (2014)

    Google Scholar 

  56. Dastjerdi, S., Akgöz, B.: New static and dynamic analyses of macro and nano FGM plates using exact three-dimensional elasticity in thermal environment. Compos. Struct. 192, 626–641 (2018)

    Google Scholar 

  57. Daikh, A.A., Bachiri, A., Houari, M.S.A., Tounsi, A.: Size dependent free vibration and buckling of multilayered carbon nanotubes reinforced composite nanoplates in thermal environment. Mech. Based Des. Struct. Mach. 50(4), 1371–1399 (2022)

    Google Scholar 

  58. Daikh, A.A., Drai, A., Bensaid, I., Houari, M.S.A., Tounsi, A.: On vibration of functionally graded sandwich nanoplates in the thermal environment. J. Sandw. Struct. Mater. 23(6), 2217–2244 (2021)

    Google Scholar 

  59. Zhu, J., Lai, Z., Yin, Z., Jeon, J., Lee, S.: Fabrication of ZrO2–NiCr functionally graded material by powder metallurgy. Mater. Chem. Phys. 68(1–3), 130–135 (2001)

    Google Scholar 

  60. Mechab, I., Mechab, B., Benaissa, S., Serier, B., Bouiadjra, B.B.: Free vibration analysis of FGM nanoplate with porosities resting on Winkler Pasternak elastic foundations based on two-variable refined plate theories. J. Braz. Soc. Mech. Sci. Eng. 38(8), 2193–2211 (2016)

    MATH  Google Scholar 

  61. Daikh, A.A., Houari, M.S.A., Tounsi, A.: Buckling analysis of porous FGM sandwich nanoplates due to heat conduction via nonlocal strain gradient theory. Eng. Res. Express 1(1), 015022 (2019)

    Google Scholar 

  62. Melaibari, A., Abo-bakr, R.M., Mohamed, S., Eltaher, M.: Static stability of higher order functionally graded beam under variable axial load. Alex. Eng. J. 59(3), 1661–1675 (2020)

    Google Scholar 

  63. Melaibari, A., Khoshaim, A.B., Mohamed, S.A., Eltaher, M.A.: Static stability and of symmetric and sigmoid functionally graded beam under variable axial load. Steel Compos. Struct. 35(5), 671–685 (2020)

    Google Scholar 

  64. Belarbi, M.-O., Li, L., Ahmed-Houari, M.S., Garg, A., Chalak, H.D., Dimitri, R., Tornabene, F.: Nonlocal vibration of functionally graded nanoplates using a layerwise theory. Math. Mech. Solids 27(12), 2634–2661 (2022)

    MathSciNet  Google Scholar 

  65. Alazwari, M.A., Daikh, A.A., Eltaher, M.A.: Novel quasi 3D theory for mechanical responses of FG-CNTs reinforced composite nanoplates. Adv. Nano Res. 12(2), 117–137 (2022)

    Google Scholar 

  66. Lim, C., Zhang, G., Reddy, J.: A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J. Mech. Phys. Solids 78, 298–313 (2015)

    MathSciNet  MATH  Google Scholar 

  67. Melaibari, A., Daikh, A.A., Basha, M., Wagih, A., Othman, R., Almitani, K.H., Hamed, M.A., Abdelrahman, A., Eltaher, M.A.: A dynamic analysis of randomly oriented functionally graded carbon nanotubes/fiber-reinforced composite laminated shells with different geometries. Mathematics 10, 408 (2022)

    Google Scholar 

  68. Abdelhaffez, G.S., Daikh, A.A., Saleem, H.A., Eltaher, M.A.: Buckling of coated functionally graded spherical nanoshells rested on orthotropic elastic medium. Mathematics 11, 409 (2023)

    Google Scholar 

  69. Daikh, A.A., Belarbi, M.O., Ahmed, D., Houari, M.S.A., Avcar, A., Tounsi, M., Eltaher, M.A.: Static analysis of functionally graded plate structures resting on variable elastic foundation under various boundary conditions. Acta Mech 234, 775–806 (2023)

    MathSciNet  Google Scholar 

  70. Khadir, A.I., Daikh, A.A., Eltaher, M.A.: Novel four-unknowns quasi 3D theory for bending, buckling and free vibration of functionally graded carbon nanotubes reinforced composite laminated nanoplates. Adv. Nano Res. 11(6), 621–640 (2021)

    Google Scholar 

  71. Sobhy, M., Zenkour, A.M.: Porosity and inhomogeneity effects on the buckling and vibration of double-FGM nanoplates via a quasi-3D refined theory. Compos. Struct. 220, 289–303 (2019)

    Google Scholar 

  72. Sobhy, M., Radwan, A.F.: A new quasi 3D nonlocal plate theory for vibration and buckling of FGM nanoplates. Int. J. Appl. Mech. 9(01), 1750008 (2017)

    Google Scholar 

  73. Zenkour, A.M., Aljadani, M.H.: Buckling analysis of actuated functionally graded piezoelectric plates via a quasi-3D refined theory. Mech. Mater. 151, 103632 (2020)

    Google Scholar 

  74. Thai, H.-T., Kim, S.-E.: Closed-form solution for buckling analysis of thick functionally graded plates on elastic foundation. Int. J. Mech. Sci. 75, 34–44 (2013)

    Google Scholar 

  75. Yaghoobi, H., Fereidoon, A.: Mechanical and thermal buckling analysis of functionally graded plates resting on elastic foundations: An assessment of a simple refined nth-order shear deformation theory. Compos. B Eng. 62, 54–64 (2014)

    Google Scholar 

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Authors and Affiliations

  1. Department of Technology, University Centre of Naama, 45000, Naama, Algeria

    Ahmed Amine Daikh

  2. Laboratoire d’Etude des Structures et de Mécanique des Matériaux, Département de Génie Civil, Faculté des Sciences et de la Technologie, Université Mustapha Stambouli, B.P. 305, 29000, Mascara, Algérie

    Ahmed Amine Daikh

  3. Laboratoire de Recherche en Génie Civil, LRGC, Université de Biskra, B.P. 145, 07000, Biskra, Algeria

    Mohamed-Ouejdi Belarbi & Abdelhak Khechai

  4. State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China

    Li Li

  5. Civil Engineering Department, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, Saudi Arabia

    Hani M Ahmed

  6. Mechanical Design and Production Department, Faculty of Engineering, Zagazig University, P.O. Box 44519, Zagazig, Egypt

    Mohamed A Eltaher

  7. Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, Saudi Arabia

    Mohamed A Eltaher

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  1. Ahmed Amine Daikh
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  2. Mohamed-Ouejdi Belarbi
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Appendix A

Appendix A

The elements \({K}_{ij}\)

$$\begin{aligned} K_{11} & = A_{11} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{3} X_{m} }}{{\partial x^{3} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}Y_{n} {\text{d}}x{\text{d}}y + A_{66} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}\frac{{\partial X_{m} }}{\partial x}Y_{n} {\text{d}}x{\text{d}}y \\ & \quad - {\varvec{\lambda}}\left[ {\left( {A_{11} + A_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{3} X_{m} }}{{\partial x^{3} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}\frac{{\partial X_{m} }}{\partial x}Y_{n} {\text{d}}x{\text{d}}y} \right. \\ & \quad \left. { + A_{11} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{5} X_{m} }}{{\partial x^{5} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}Y_{n} {\text{d}}x{\text{d}}y + A_{66} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}\frac{{\partial X_{m} }}{\partial x}Y_{n} {\text{d}}x{\text{d}}y} \right] \\ \end{aligned}$$
$$\begin{aligned} K_{12} & = \left( {A_{12} + A_{66} } \right)\left( {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}\frac{{\partial X_{m} }}{\partial x}Y_{n} {\text{d}}x{\text{d}}y} \right. \\ & \quad \left. { - {\varvec{\lambda}}\left[ {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{3} X_{m} }}{{\partial x^{3} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}\frac{{\partial X_{m} }}{\partial x}Y_{n} {\text{d}}x{\text{d}}y + \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}\frac{{\partial X_{m} }}{\partial x}Y_{n} {\text{d}}x{\text{d}}y} \right]} \right) \\ \end{aligned}$$
$$\begin{aligned} K_{13} & = - B_{11} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{3} X_{m} }}{{\partial x^{3} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}Y_{n} {\text{d}}x{\text{d}}y - \left( {B_{12} + 2B_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}\frac{{\partial X_{m} }}{\partial x}Y_{n} {\text{d}}x{\text{d}}y \\ & \quad - {\varvec{\lambda}}\left[ { - \left( {B_{12} + 2B_{66} + B_{11} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{3} X_{m} }}{{\partial x^{3} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}\frac{{\partial X_{m} }}{\partial x}Y_{n} {\text{d}}x{\text{d}}y} \right. \\ & \quad \left. { - B_{11} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{5} X_{m} }}{{\partial x^{5} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}Y_{n} {\text{d}}x{\text{d}}y} \right] \\ \end{aligned}$$
$$\begin{aligned} K_{14} & = - C_{11} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{3} X_{m} }}{{\partial x^{3} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}Y_{n} {\text{d}}x{\text{d}}y - \left( {C_{12} + 2C_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}\frac{{\partial X_{m} }}{\partial x}Y_{n} {\text{d}}x{\text{d}}y \\ & \quad - {\varvec{\lambda}}\left[ { - \left( {B_{12} + 2B_{66} + B_{11} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{3} X_{m} }}{{\partial x^{3} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}\frac{{\partial X_{m} }}{\partial x}Y_{n} {\text{d}}x{\text{d}}y} \right. \\ & \quad \left. { - B_{11} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{5} X_{m} }}{{\partial x^{5} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}Y_{n} {\text{d}}x{\text{d}}y} \right] \\ \end{aligned}$$
$$\begin{aligned} K_{15} & = G_{13} \left( {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial X_{m} }}{\partial x}Y_{n} \frac{{\partial X_{m} }}{\partial x}Y_{n} {\text{d}}x{\text{d}}y - {\varvec{\lambda}}\left[ {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{3} X_{m} }}{{\partial x^{3} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}Y_{n} {\text{d}}x{\text{d}}y} \right.} \right. \\ & \quad \left. {\left. { + \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}\frac{{\partial X_{m} }}{\partial x}Y_{n} {\text{d}}x{\text{d}}y} \right]} \right) \\ \end{aligned}$$
$$\begin{aligned} K_{21} & = \left( {A_{12} + A_{66} } \right)\left( {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{1} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y} \right. \\ & \quad \left. { - {\varvec{\lambda}}\left[ {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y + \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{3} Y_{n} }}{{\partial y^{3} }}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y} \right]} \right) \\ \end{aligned}$$
$$\begin{aligned} K_{22} & = A_{22} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{3} Y_{n} }}{{\partial y^{3} }}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y + A_{66} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y \\ & \quad - {\varvec{\lambda}}\left[ {\left( {A_{22} + A_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{3} Y_{n} }}{{\partial y^{3} }}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y} \right. \\ & \quad \left. { + A_{22} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{5} Y_{n} }}{{\partial y^{5} }}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y + A_{66} \mathop \int \limits_{0}^{a} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y} \right] \\ \end{aligned}$$
$$\begin{aligned} K_{23} & = - B_{22} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{3} Y_{n} }}{{\partial y^{3} }}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y - \left( {B_{12} + 2B_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y \\ & \quad - \lambda \left[ { - B_{22} \left( {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{3} Y_{n} }}{{\partial y^{3} }}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y + \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{5} Y_{n} }}{{\partial y^{5} }}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y} \right)} \right. \\ & \quad \left. { - \left( {B_{12} + 2B_{66} } \right)\left( {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{3} X_{m} }}{{\partial x^{3} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y + \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y} \right)} \right] \\ \end{aligned}$$
$$\begin{aligned} K_{24} & = - C_{22} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{3} Y_{n} }}{{\partial y^{3} }}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y - \left( {C_{12} + 2C_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y \\ & \quad - \lambda \left[ { - C_{22} \left( {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{3} Y_{n} }}{{\partial y^{3} }}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y + \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{5} Y_{n} }}{{\partial y^{5} }}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y} \right)} \right. \\ & \quad - \left. {\left( {C_{12} + 2C_{66} } \right)\left( {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y + \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{3} Y_{n} }}{{\partial y^{3} }}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y} \right)} \right] \\ \end{aligned}$$
$$\begin{aligned} K_{25} & = G_{23} \left( {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y - {\varvec{\lambda}}} \right. \\ & \quad \left. {\left[ {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y + \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{3} Y_{n} }}{{\partial y^{3} }}X_{m} \frac{{\partial Y_{n} }}{\partial y}{\text{d}}x{\text{d}}y} \right]} \right) \\ \end{aligned}$$
$$\begin{aligned} K_{13} & = B_{11} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{n} }}{{\partial x^{4} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y + \left( {B_{12} + 2B_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y \\ & \quad - \lambda \left[ {B_{11} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{6} X_{n} }}{{\partial x^{6} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y + \left( {B_{11} + B_{12} + 2B_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right. \\ & \quad \left. { + \left( {B_{12} + 2B_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right] \\ \end{aligned}$$
$$\begin{aligned} K_{23} & = B_{22} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y + \left( {B_{12} + 2B_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y \\ & \quad - \lambda \left[ {\left( {B_{22} + B_{12} + 2B_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right. \\ & \quad \left. { + \left( {B_{12} + 2B_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y + B_{22} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{6} Y_{n} }}{{\partial y^{6} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right] \\ \end{aligned}$$
$$\begin{aligned} K_{33} & = - D_{11} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y - D_{22} \left( {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right) \\ & \quad - 2\left( {D_{12} + 2D_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y \\ & \quad - \lambda \left[ { - D_{11} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{6} X_{m} }}{{\partial x^{6} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y - D_{22} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{6} Y_{n} }}{{\partial y^{6} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right. \\ & \quad - \left( {D_{11} + 2D_{12} + 4D_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y \\ & \quad \left. { - \left( {D_{22} + 2D_{12} + 4D_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right] \\ & \quad + \overline{N}_{xx}^{0} \left( {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right. \\ & \quad \left. { - \mu \left[ {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y + \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right]} \right) \\ & \quad + \overline{N}_{yy}^{0} \left( {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right. \\ & \quad \left. { - \mu \left[ {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y + \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right]} \right) \\ \end{aligned}$$
$$\begin{aligned} K_{34} & = - E_{11} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y - E_{22} \left( {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right) \\ & \quad - 2\left( {E_{12} + 2E_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y \\ & \quad - \lambda \left[ { - E_{11} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{6} X_{m} }}{{\partial x^{6} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y - E_{22} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{6} Y_{n} }}{{\partial y^{6} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right. \\ & \quad - \left( {E_{11} + 2E_{12} + 4E_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y \\ & \quad \left. { - \left( {E_{22} + 2E_{12} + 4E_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right] \\ \end{aligned}$$
$$\begin{aligned} K_{35} & = H_{13} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y + H_{23} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y \\ & \quad - \lambda \left[ {H_{13} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y + H_{23} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right. \\ & \quad \left. { + \left( {H_{13} + H_{23} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right] \\ \end{aligned}$$
$$\begin{aligned} K_{41} & = C_{11} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y + \left( {C_{12} + 2C_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y \\ & \quad - \lambda \left[ {C_{11} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{6} X_{m} }}{{\partial x^{6} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y + \left( {C_{12} + 2C_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right. \\ & \quad \left. { + \left( {C_{11} + C_{12} + 2C_{66} } \right)\left( {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right)} \right] \\ \end{aligned}$$
$$\begin{aligned} K_{42} & = C_{22} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y \\ & \quad + \left( {C_{12} + 2C_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y \\ & \quad - \lambda \left[ {C_{22} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{6} Y_{n} }}{{\partial y^{6} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y + \left( {C_{12} + 2C_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right. \\ & \quad \left. {\left( {C_{22} + C_{12} + 2C_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right] \\ \end{aligned}$$
$$\begin{aligned} K_{43} & = - E_{11} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y - E_{22} \left( {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right) \\ & \quad - 2\left( {E_{12} + 2E_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y \\ & \quad - \lambda \left[ { - E_{11} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{6} X_{m} }}{{\partial x^{6} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y - E_{22} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{6} Y_{n} }}{{\partial y^{6} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right. \\ & \quad - \left( {E_{11} + 2E_{12} + 4E_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y \\ & \quad \left. { - \left( {E_{22} + 2E_{12} + 4E_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right] \\ \end{aligned}$$
$$\begin{aligned} K_{44} & = - F_{11} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y - F_{22} \left( {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right) \\ & \quad - 2\left( {F_{12} + 2F_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y + J_{44} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y \\ & \quad + J_{55} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y - \lambda \left[ { - F_{11} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{6} X_{m} }}{{\partial x^{6} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right. \\ & \quad - F_{22} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{6} Y_{n} }}{{\partial y^{6} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y - \left( {F_{11} + 2F_{12} + 4F_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y \\ & \quad - \left( {F_{22} + 2F_{12} + 4F_{66} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y + J_{44} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y \\ & \quad + J_{55} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y + \left( {J_{44} + J_{55} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y \\ \end{aligned}$$
$$\begin{aligned} K_{45} & = \left( {I_{13} - J_{55} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y + \left( {I_{23} - J_{44} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y \\ & \quad - \left( {\lambda \left[ {\left( {I_{13} - J_{55} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y + \left( {I_{23} - J_{44} } \right)} \right.} \right. \\ & \quad \left. {\left. {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y\left( {I_{13} + I_{23} - J_{55} - J_{44} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right]} \right) \\ \end{aligned}$$
$$\begin{aligned} K_{51} & = - G_{13} \left( {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y - {\varvec{\lambda}}\left[ {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right.} \right. \\ & \quad \left. {\left. { + \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right]} \right) \\ \end{aligned}$$
$$\begin{aligned} K_{52} & = - G_{23} \left( {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y - {\varvec{\lambda}}\left[ {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right.} \right. \\ & \quad \left. {\left. { + \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right]} \right) \\ \end{aligned}$$
$$\begin{aligned} K_{53} & = H_{13} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y + H_{23} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y \\ & \quad - \lambda \left[ {H_{13} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y + H_{23} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right. \\ & \quad + \left. {\left( {H_{13} + H_{23} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right] \\ \end{aligned}$$
$$\begin{aligned} K_{54} & = \left( {I_{13} - J_{55} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y + \left( {I_{23} - J_{44} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y \\ & \quad - \left( {\lambda \left[ {\left( {I_{13} - J_{55} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y + \left( {I_{23} - J_{44} } \right)} \right.} \right. \\ & \quad \left. {\left. {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y\left( {I_{13} + I_{23} - J_{55} - J_{44} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right]} \right) \\ \end{aligned}$$
$$\begin{aligned} K_{55} & = - K_{33} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y + J_{44} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y + J_{55} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y \\ & \quad - \lambda \left[ { - K_{33} \left( {\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y + \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right) + J_{44} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} X_{m} \frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right. \\ & \quad \left. { + J_{55} \mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}Y_{n} X_{m} Y_{n} {\text{d}}x{\text{d}}y + \left( {J_{44} + J_{55} } \right)\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} \frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}X_{m} Y_{n} {\text{d}}x{\text{d}}y} \right] \\ \end{aligned}$$

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Daikh, A.A., Belarbi, MO., Khechai, A. et al. Buckling of bi-coated functionally graded porous nanoplates via a nonlocal strain gradient quasi-3D theory. Acta Mech 234, 3397–3420 (2023). https://doi.org/10.1007/s00707-023-03548-9

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