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Stability analysis of a sandwich composite magnetostrictive nanoplate coupled with FG porous facesheets

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Abstract

The main aim of this study is to examine the buckling behavior of a composite material that has both magnetostrictive capabilities and functionally graded facesheets. The effective material parameters of the functionally graded layer are determined using the power-law model. Eringen’s nonlocal theory has been used for the quantification of the small-scale parameter. In contrast, the suggested system is based on the theoretical framework established by Winkler and Pasternak, which incorporates the analysis of an elastic medium. The use of higher-order sinusoidal shear deformation theory has been employed to derive the governing equation. This governing equation is then solved analytically using the Galerkin solution method, considering various boundary conditions. In order to assess the precision and effectiveness of the ongoing inquiry, the findings are juxtaposed with the existing literature articles. Furthermore, this study examines the impact of many factors, including aspect ratio, velocity feedback gain, and foundation, on the critical buckling load. The findings of the present research indicate that there is a positive correlation between the porosity volume parameter and the buckling load of the structure. The current study aims to provide engineers and designers with a better understanding and predictive capability about buckling response. This knowledge may be advantageous in the design of nanoscale systems, including highly sought-after technologies like as sensors and actuators.

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References

  1. Liu, G.R., Chen, X.L., Reddy, J.N.: Buckling of symmetrically laminated composite plates using the element-free Galerkin method. Int. J. Struct. Stab. Dyn. 02(03), 281–294 (2002). https://doi.org/10.1142/s0219455402000634

    Article  Google Scholar 

  2. Liew, K.M., Peng, L.X., Kitipornchai, S.: Buckling analysis of corrugated plates using a mesh-free Galerkin method based on the first-order shear deformation theory. Comput. Mech. 38(1), 61–75 (2006). https://doi.org/10.1007/s00466-005-0721-2

    Article  Google Scholar 

  3. Kim, S.-E., Thai, H.-T., Lee, J.: Buckling analysis of plates using the two variable refined plate theory. Thin-Walled Struct. 47(4), 455–462 (2009). https://doi.org/10.1016/j.tws.2008.08.002

    Article  Google Scholar 

  4. Lei, Z.X., Liew, K.M., Yu, J.L.: Buckling analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free kp-Ritz method. Compos. Struct. 98, 160–168 (2013). https://doi.org/10.1016/j.compstruct.2012.11.006

    Article  Google Scholar 

  5. Karimi, M., Shahidi, A.R.: Buckling analysis of skew magneto-electro-thermo-elastic nanoplates considering surface energy layers and utilizing the Galerkin method. Appl. Phys. A 124(10), 681 (2018). https://doi.org/10.1007/s00339-018-2088-1

    Article  Google Scholar 

  6. Karami, B., Janghorban, M., Tounsi, A.: Galerkin’s approach for buckling analysis of functionally graded anisotropic nanoplates/different boundary conditions. Eng. Comput. 35(4), 1297–1316 (2019). https://doi.org/10.1007/s00366-018-0664-9

    Article  Google Scholar 

  7. Zenkour, A.M., Radwan, A.F.: Hygrothermo-mechanical buckling of FGM plates resting on elastic foundations using a quasi-3D model. Int. J. Comput. Methods Eng. Sci. Mech. 20(2), 85–98 (2019). https://doi.org/10.1080/15502287.2019.1568618

    Article  MathSciNet  Google Scholar 

  8. Zenkour, A.M., Radwan, A.F.: Bending and buckling analysis of FGM plates resting on elastic foundations in hygrothermal environment. Arch. Civ. Mech. Eng. 20(4), 112 (2020). https://doi.org/10.1007/s43452-020-00116-z

    Article  Google Scholar 

  9. Rouabhia, A., Chikh, A., Bousahla, A.A., Bourada, F., Heireche, H., Tounsi, A., Benrahou, K.H., Tounsi, A., Al-Zahrani, M.M.: Physical stability response of a SLGS resting on viscoelastic medium using nonlocal integral first-order theory. Steel Compos. Struct. 37(6), 695–709 (2020)

    Google Scholar 

  10. Bourada, F., Bousahla, A.A., Tounsi, A., Tounsi, A., Tahir, S.I., Al-Osta, M.A., Do-Van, T.: An integral quasi-3D computational model for the hygro-thermal wave propagation of imperfect FGM sandwich plates. Comput. Concr. 32(1), 61–74 (2023)

    Google Scholar 

  11. Mudhaffar, I.M., Chikh, A., Tounsi, A., Al-Osta, M.A., Al-Zahrani, M.M., Al-Dulaijan, S.U.: Impact of viscoelastic foundation on bending behavior of FG plate subjected to hygro-thermo-mechanical loads. Struct. Eng. Mech. 86(2), 167 (2023)

    Google Scholar 

  12. Tounsi, A., Bousahla, A.A., Tahir, S.I., Mostefa, A.H., Bourada, F., Al-Osta, M.A., Tounsi, A.: Influences of different boundary conditions and hygro-thermal environment on the free vibration responses of FGM sandwich plates resting on viscoelastic foundation. Int. J. Struct. Stab. Dyn. (2023). https://doi.org/10.1142/s0219455424501177

    Article  Google Scholar 

  13. Tounsi, A., Mostefa, A.H., Attia, A., Bousahla, A.A., Bourada, F., Tounsi, A., Al-Osta, M.A.: Free vibration investigation of functionally graded plates with temperaturedependent properties resting on a viscoelastic foundation. Struct. Eng. Mech. 86(1), 1 (2023)

    Google Scholar 

  14. Ebrahimi, F., Barati, M.R.: Hygrothermal buckling analysis of magnetically actuated embedded higher order functionally graded nanoscale beams considering the neutral surface position. J. Therm. Stress. 39(10), 1210–1229 (2016). https://doi.org/10.1080/01495739.2016.1215726

    Article  Google Scholar 

  15. Shafiei, H., Setoodeh, A.: Nonlinear free vibration and post-buckling of FG-CNTRC beams on nonlinear foundation. Steel Compos. Struct. 24(1), 65–77 (2017)

    Article  Google Scholar 

  16. Malikan, M., Dastjerdi, S.: Analytical buckling of FG nanobeams on the basis of a new one variable first-order shear deformation beam theory. Int. J. Eng. Appl. Sci. 10(1), 21–34 (2018)

    Google Scholar 

  17. Taati, E.: On buckling and post-buckling behavior of functionally graded micro-beams in thermal environment. Int. J. Eng. Sci. 128, 63–78 (2018). https://doi.org/10.1016/j.ijengsci.2018.03.010

    Article  MathSciNet  Google Scholar 

  18. Ahmed, R.A., Fenjan, R.M., Faleh, N.M.: Analyzing post-buckling behavior of continuously graded FG nanobeams with geometrical imperfections. Geomech. Eng. 17(2), 175–180 (2019)

    Google Scholar 

  19. Khosravi, S., Arvin, H., Kiani, Y.: Interactive thermal and inertial buckling of rotating temperature-dependent FG-CNT reinforced composite beams. Compos. B Eng. 175, 107178 (2019). https://doi.org/10.1016/j.compositesb.2019.107178

    Article  Google Scholar 

  20. Abo-bakr, R.M., Abo-bakr, H.M., Mohamed, S.A., Eltaher, M.A.: Optimal weight for buckling of FG beam under variable axial load using Pareto optimality. Compos. Struct. 258, 113193 (2021). https://doi.org/10.1016/j.compstruct.2020.113193

    Article  Google Scholar 

  21. Carrera, E., Demirbas, M.D.: Evaluation of bending and post-buckling behavior of thin-walled FG beams in geometrical nonlinear regime with CUF. Compos. Struct. 275, 114408 (2021). https://doi.org/10.1016/j.compstruct.2021.114408

    Article  Google Scholar 

  22. Bellifa, H., Chikh, A., Bousahla, A.A., Bourada, F., Tounsi, A., Benrahou, K.H., Al-Zahrani, M., Tounsi, A.: Influence of porosity on thermal buckling behavior of functionally graded beams. Smart Struct. Syst. 27(4), 719–728 (2021)

    Google Scholar 

  23. Huang, Y., Karami, B., Shahsavari, D., Tounsi, A.: Static stability analysis of carbon nanotube reinforced polymeric composite doubly curved micro-shell panels. Arch. Civ. Mech. Eng. 21(4), 139 (2021). https://doi.org/10.1007/s43452-021-00291-7

    Article  Google Scholar 

  24. Kumar, Y., Gupta, A., Tounsi, A.: Size-dependent vibration response of porous graded nanostructure with FEM and nonlocal continuum model. Adv. Nano Res. 11(1), 001 (2021)

    Google Scholar 

  25. Cuong-Le, T., Nguyen, K.D., Le-Minh, H., Phan-Vu, P., Nguyen-Trong, P., Tounsi, A.: Nonlinear bending analysis of porous sigmoid FGM nanoplate via IGA and nonlocal strain gradient theory. Adv. Nano Res. 12(5), 441 (2022)

    Google Scholar 

  26. Garg, A., Belarbi, M.-O., Tounsi, A., Li, L., Singh, A., Mukhopadhyay, T.: Predicting elemental stiffness matrix of FG nanoplates using Gaussian process regression based surrogate model in framework of layerwise model. Eng. Anal. Bound. Elem. 143, 779–795 (2022). https://doi.org/10.1016/j.enganabound.2022.08.001

    Article  MathSciNet  Google Scholar 

  27. Liu, G., Wu, S., Shahsavari, D., Karami, B., Tounsi, A.: Dynamics of imperfect inhomogeneous nanoplate with exponentially-varying properties resting on viscoelastic foundation. Eur. J. Mech. A. Solids 95, 104649 (2022). https://doi.org/10.1016/j.euromechsol.2022.104649

    Article  MathSciNet  Google Scholar 

  28. Van Vinh, P., Tounsi, A.: The role of spatial variation of the nonlocal parameter on the free vibration of functionally graded sandwich nanoplates. Eng. Comput. 38(5), 4301–4319 (2022). https://doi.org/10.1007/s00366-021-01475-8

    Article  Google Scholar 

  29. Van Vinh, P., Tounsi, A.: Free vibration analysis of functionally graded doubly curved nanoshells using nonlocal first-order shear deformation theory with variable nonlocal parameters. Thin-Walled Struct. 174, 109084 (2022). https://doi.org/10.1016/j.tws.2022.109084

    Article  Google Scholar 

  30. Hajlaoui, A., Chebbi, E., Dammak, F.: Buckling analysis of carbon nanotube reinforced FG shells using an efficient solid-shell element based on a modified FSDT. Thin-Walled Struct. 144, 106254 (2019). https://doi.org/10.1016/j.tws.2019.106254

    Article  Google Scholar 

  31. Shahgholian-Ghahfarokhi, D., Rahimi, G., Khodadadi, A., Salehipour, H., Afrand, M.: Buckling analyses of FG porous nanocomposite cylindrical shells with graphene platelet reinforcement subjected to uniform external lateral pressure. Mech. Based Des. Struct. Mach. 49(7), 1059–1079 (2021). https://doi.org/10.1080/15397734.2019.1704777

    Article  Google Scholar 

  32. Sofiyev, A.H., Tornabene, F., Dimitri, R., Kuruoglu, N.: Buckling behavior of FG-CNT reinforced composite conical shells subjected to a combined loading. Nanomaterials 10(3), 419 (2020)

    Article  Google Scholar 

  33. Hieu, P.T., Van Tung, H.: Thermal and thermomechanical buckling of shear deformable FG-CNTRC cylindrical shells and toroidal shell segments with tangentially restrained edges. Arch. Appl. Mech. 90(7), 1529–1546 (2020). https://doi.org/10.1007/s00419-020-01682-7

    Article  Google Scholar 

  34. Sun, J., Zhu, S., Tong, Z., Zhou, Z., Xu, X.: Post-buckling analysis of functionally graded multilayer graphene platelet reinforced composite cylindrical shells under axial compression. Proc. Royal Soc. A Math. Phys. Eng. Sci. 476(2243), 20200506 (2020). https://doi.org/10.1098/rspa.2020.0506

    Article  MathSciNet  Google Scholar 

  35. Al-Osta, M.A., Saidi, H., Tounsi, A., Al-Dulaijan, S., Al-Zahrani, M., Sharif, A., Tounsi, A.: Influence of porosity on the hygro-thermo-mechanical bending response of an AFG ceramic-metal plates using an integral plate model. Smart Struct. Syst. Int. J. 28(4), 499–513 (2021)

    Google Scholar 

  36. Tahir, S.I., Chikh, A., Tounsi, A., Al-Osta, M.A., Al-Dulaijan, S.U., Al-Zahrani, M.M.: Wave propagation analysis of a ceramic-metal functionally graded sandwich plate with different porosity distributions in a hygro-thermal environment. Compos. Struct. 269, 114030 (2021). https://doi.org/10.1016/j.compstruct.2021.114030

    Article  Google Scholar 

  37. Arshid, E., Khorasani, M., Soleimani-Javid, Z., Amir, S., Tounsi, A.: Porosity-dependent vibration analysis of FG microplates embedded by polymeric nanocomposite patches considering hygrothermal effect via an innovative plate theory. Eng. Comput. 38(5), 4051–4072 (2022). https://doi.org/10.1007/s00366-021-01382-y

    Article  Google Scholar 

  38. Katiyar, V., Gupta, A., Tounsi, A.: Microstructural/geometric imperfection sensitivity on the vibration response of geometrically discontinuous bi-directional functionally graded plates (2D FGPs) with partial supports by using FEM. Steel Compos. Struct. Int. J. 45(5), 621–640 (2022)

    Google Scholar 

  39. Van Vinh, P., Van Chinh, N., Tounsi, A.: Static bending and buckling analysis of bi-directional functionally graded porous plates using an improved first-order shear deformation theory and FEM. Eur. J. Mech. A. Solids 96, 104743 (2022). https://doi.org/10.1016/j.euromechsol.2022.104743

    Article  MathSciNet  Google Scholar 

  40. Addou, F.Y., Bourada, F., Meradjah, M., Bousahla, A.A., Tounsi, A., Ghazwani, M.H., Alnujaie, A.: Impact of porosity distribution on static behavior of functionally graded plates using a simple quasi-3D HSDT. Comput. Concr. 32(1), 87–97 (2023)

    Google Scholar 

  41. Alsubaie, A.M., Alfaqih, I., Al-Osta, M.A., Tounsi, A., Chikh, A., Mudhaffar, I.M., Tahir, S.: Porosity-dependent vibration investigation of functionally graded carbon nanotube-reinforced composite beam. Comput. Concr. 32(1), 75–85 (2023)

    Google Scholar 

  42. Khorasani, M., Lampani, L., Tounsi, A.: A refined vibrational analysis of the FGM porous type beams resting on the silica aerogel substrate. Steel Compos. Struct. 47(5), 633–644 (2023)

    Google Scholar 

  43. Mesbah, A., Belabed, Z., Amara, K., Tounsi, A., Bousahla, A.A., Bourada, F.: Formulation and evaluation a finite element model for free vibration and buckling behaviours of functionally graded porous (FGP) beams. Struct. Eng. Mech. 86(3), 291 (2023)

    Google Scholar 

  44. Xia, L., Wang, R., Chen, G., Asemi, K., Tounsi, A.: The finite element method for dynamics of FG porous truncated conical panels reinforced with graphene platelets based on the 3-D elasticity. Adv. Nano Res. 14(4), 375 (2023)

    Google Scholar 

  45. Ren, H., Li, Z., Shu, X.: The numerical simulation of magnetoelastic buckling based on magnetostrictive material Model. J. Taiyuan Univ. Technol. 36(5), 561 (2005)

    Google Scholar 

  46. Ghorbanpour Arani, A., Abdollahian, M., Rahmati, A.: Nonlocal piezomagnetoelasticity theory for buckling analysis of piezoelectric/magnetostrictive nanobeams including surface effects. J. Solid Mech. 9(4), 707–729 (2017)

    Google Scholar 

  47. Tabbakh, M., Nasihatgozar, M.: Buckling analysis of nanocomposite plates coated by magnetostrictive layer. Smart Struct. Syst. Int. J. 22(6), 743–751 (2018)

    Google Scholar 

  48. Yuan, Y., Zhao, X., Zhao, Y., Sahmani, S., Safaei, B.: Dynamic stability of nonlocal strain gradient FGM truncated conical microshells integrated with magnetostrictive facesheets resting on a nonlinear viscoelastic foundation. Thin-Walled Struct. 159, 107249 (2021). https://doi.org/10.1016/j.tws.2020.107249

    Article  Google Scholar 

  49. Fan, L., Sahmani, S., Safaei, B.: Couple stress-based dynamic stability analysis of functionally graded composite truncated conical microshells with magnetostrictive facesheets embedded within nonlinear viscoelastic foundations. Eng. Comput. 37(2), 1635–1655 (2021). https://doi.org/10.1007/s00366-020-01182-w

    Article  Google Scholar 

  50. Reddy, J.N., Wang, C.M., Kitipornchai, S.: Axisymmetric bending of functionally graded circular and annular plates. Eur. J. Mech. A. Solids 18(2), 185–199 (1999). https://doi.org/10.1016/S0997-7538(99)80011-4

    Article  Google Scholar 

  51. Ebrahimi, F., Barati, M.R.: Electro-magnetic effects on nonlocal dynamic behavior of embedded piezoelectric nanoscale beams. J. Intell. Mater. Syst. Struct. 28(15), 2007–2022 (2017). https://doi.org/10.1177/1045389x16682850

    Article  Google Scholar 

  52. Ebrahimi, F., Farazmandnia, N., Kokaba, M.R., Mahesh, V.: Vibration analysis of porous magneto-electro-elastically actuated carbon nanotube-reinforced composite sandwich plate based on a refined plate theory. Eng. Comput. 37(2), 921–936 (2021). https://doi.org/10.1007/s00366-019-00864-4

    Article  Google Scholar 

  53. Touratier, M.: An efficient standard plate theory. Int. J. Eng. Sci. 29(8), 901–916 (1991). https://doi.org/10.1016/0020-7225(91)90165-Y

    Article  Google Scholar 

  54. Touratier, M.: A generalization of shear deformation theories for axisymmetric multilayered shells. Int. J. Solids Struct. 29(11), 1379–1399 (1992). https://doi.org/10.1016/0020-7683(92)90085-8

    Article  Google Scholar 

  55. Touratier, M.: A refined theory of laminated shallow shells. Int. J. Solids Struct. 29(11), 1401–1415 (1992). https://doi.org/10.1016/0020-7683(92)90086-9

    Article  Google Scholar 

  56. Ebrahimi, F., Ahari, M.F.: Dynamic analysis of sandwich magnetostrictive nanoplates with a mass–spring–damper stimulator. Int. J. Struct. Stab. Dyn. (2023). https://doi.org/10.1142/s0219455424501360

    Article  Google Scholar 

  57. Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10(3), 233–248 (1972). https://doi.org/10.1016/0020-7225(72)90039-0

    Article  MathSciNet  Google Scholar 

  58. Ebrahimi, F., Dabbagh, A., Rabczuk, T.: On wave dispersion characteristics of magnetostrictive sandwich nanoplates in thermal environments. Eur. J. Mech. A. Solids 85, 104130 (2021). https://doi.org/10.1016/j.euromechsol.2020.104130

    Article  MathSciNet  Google Scholar 

  59. Ebrahimi, F., Farajzadeh Ahari, M.: Mechanics of Magnetostrictive Materials and Structures. CRC Press, Boca Raton (2023)

    Book  Google Scholar 

  60. Ebrahimi, F., Mollazeinal, A., Ahari, M.F.: Active vibration control of truncated conical porous smart composite shells. Int. J. Struct. Stab. Dyn. (2023). https://doi.org/10.1142/s0219455424501323

    Article  Google Scholar 

  61. Ebrahimi, F., Mollazeinal, A., Ahari, M.F.: Nonlinear vibration analysis of smart truncated conical porous composite shells reinforced with terfenol-D particles. Acta Mech. (2023). https://doi.org/10.1007/s00707-023-03746-5

    Article  Google Scholar 

  62. Ghorbani, K., Rajabpour, A., Ghadiri, M., Keshtkar, Z.: Investigation of surface effects on the natural frequency of a functionally graded cylindrical nanoshell based on nonlocal strain gradient theory. Eur. Phys. J. Plus 135(9), 701 (2020). https://doi.org/10.1140/epjp/s13360-020-00712-1

    Article  Google Scholar 

  63. Ebrahimi, F., Dabbagh, A., Tornabene, F., Civalek, O.: Hygro-thermal effects on wave dispersion responses of magnetostrictive sandwich nanoplates. Adv. Nano Res. 7(3), 157 (2019)

    Google Scholar 

  64. Ebrahimi, F., Ahari, M.F.: Magnetostriction-assisted active control of the multi-layered nanoplates: effect of the porous functionally graded facesheets on the system’s behavior. Eng. Comput. 39(1), 269–283 (2023). https://doi.org/10.1007/s00366-021-01539-9

    Article  Google Scholar 

  65. Ebrahimi, F., Ahari, M.F.: Active vibration control of the multilayered smart nanobeams: velocity feedback gain effects on the system’s behavior. Acta Mech. (2023). https://doi.org/10.1007/s00707-023-03769-y

    Article  Google Scholar 

  66. Thai, H.-T., Kim, S.-E.: A simple quasi-3D sinusoidal shear deformation theory for functionally graded plates. Compos. Struct. 99, 172–180 (2013). https://doi.org/10.1016/j.compstruct.2012.11.030

    Article  Google Scholar 

  67. Mahinzare, M., Akhavan, H., Ghadiri, M.: A nonlocal strain gradient theory for rotating thermo-mechanical characteristics on magnetically actuated viscoelastic functionally graded nanoshell. J. Intell. Mater. Syst. Struct. 31(12), 1511–1523 (2020). https://doi.org/10.1177/1045389x20924828

    Article  Google Scholar 

  68. Hebali, H., Tounsi, A., Houari, M.S.A., Bessaim, A., Bedia, E.A.A.: New quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates. J. Eng. Mech. 140(2), 374–383 (2014)

    Article  Google Scholar 

  69. Ahari, M.F., Ghadiri, M.: Resonator vibration of a magneto-electro-elastic nano-plate integrated with FGM layer subjected to the nano mass-Spring-damper system and a moving load. Waves Random Complex Media (2022). https://doi.org/10.1080/17455030.2022.2053233

    Article  Google Scholar 

  70. Rao, S.S.: Vibration of Continuous Systems. Wiley, Hoboken (2019)

    Book  Google Scholar 

  71. Ebrahimi, F., Dabbagh, A.: Mechanics of Multiscale Hybrid Nanocomposites, 1st edn. Elsevier, Amsterdam (2022)

    Google Scholar 

  72. Ebrahimi, F., Shafiei, M.-S., Ahari, M.F.: Vibration analysis of single and multi-walled circular graphene sheets in thermal environment using GDQM. Waves Random Complex Media (2022). https://doi.org/10.1080/17455030.2022.2067370

    Article  Google Scholar 

  73. Ebrahimi, F., Shafiee, M.-S., Ahari, M.F.: Buckling analysis of single and double-layer annular graphene sheets in thermal environment. Eng. Comput. 39(1), 625–639 (2023). https://doi.org/10.1007/s00366-022-01634-5

    Article  Google Scholar 

  74. Ebrahimi, F., Barati, M.R.: Temperature distribution effects on buckling behavior of smart heterogeneous nanosize plates based on nonlocal four-variable refined plate theory. Int. J. Smart Nano Mater. 7(3), 119–143 (2016). https://doi.org/10.1080/19475411.2016.1223203

    Article  Google Scholar 

  75. Ebrahimi, F., Ahari, M.F.: Dynamic analysis of meta-material plates with magnetostrictive face sheets. Int. J. Struct. Stab. Dyn. (2023). https://doi.org/10.1142/s0219455424501748

    Article  Google Scholar 

  76. Mizuji, Z.K., Ghadiri, M., Rajabpour, A., Ahari, M.F., Zajkani, A., Yazdinia, S.: Numerical modeling of a body vessel for dynamic study of a nano cylindrical shell carrying fluid and a moving nanoparticle. Eng. Anal. Bound. Elem. 152, 362–382 (2023). https://doi.org/10.1016/j.enganabound.2023.04.005

    Article  MathSciNet  Google Scholar 

  77. Rahimi, Y., Ghadiri, M., Rajabpour, A., Farajzadeh Ahari, M.: Temperature-dependent vibrational behavior of bilayer doubly curved micro-nano liposome shell: Simulation of drug delivery mechanism. J. Therm. Stress. 46(11), 1199–1226 (2023). https://doi.org/10.1080/01495739.2023.2232413

    Article  Google Scholar 

  78. Abolfathi, M., Alavi Nia, A.: Optimization of energy absorption properties of thin-walled tubes with combined deformation of folding and circumferential expansion under axial load. Thin-Walled Struct. 130, 57–70 (2018). https://doi.org/10.1016/j.tws.2018.05.011

    Article  Google Scholar 

  79. Abolfathi, M., Alavi Nia, A., Akhavan Attar, A., Abbasi, M.: Experimental and numerical investigation of the effect of the combined mechanism of circumferential expansion and folding on energy absorption parameters. Arch. Civ. Mech. Eng. 18(4), 1464–1477 (2018). https://doi.org/10.1016/j.acme.2018.05.004

    Article  Google Scholar 

  80. Abbasi, M., Alavi Nia, A., Abolfathi, M.: Experimental study on the high-velocity impact behavior of sandwich structures with an emphasis on the layering effects of foam core. J. Sandwich Struct. Mater. 23(1), 3–22 (2021). https://doi.org/10.1177/1099636218813412

    Article  Google Scholar 

  81. Kakavand, E., Seifi, R., Abolfathi, M.: An investigation on the crack growth in aluminum alloy 7075–T6 under cyclic mechanical and thermal loads. Theoret. Appl. Fract. Mech. 122, 103585 (2022). https://doi.org/10.1016/j.tafmec.2022.103585

    Article  Google Scholar 

  82. Ebrahimi, F., Jafari, A., Barati, M.R.: Vibration analysis of magneto-electro-elastic heterogeneous porous material plates resting on elastic foundations. Thin-Walled Struct. 119, 33–46 (2017). https://doi.org/10.1016/j.tws.2017.04.002

    Article  Google Scholar 

  83. Barati, M.R.: A general nonlocal stress-strain gradient theory for forced vibration analysis of heterogeneous porous nanoplates. Eur. J. Mech. A. Solids 67, 215–230 (2018). https://doi.org/10.1016/j.euromechsol.2017.09.001

    Article  MathSciNet  Google Scholar 

  84. Mohammadi, M., Saidi, A.R., Jomehzadeh, E.: A novel analytical approach for the buckling analysis of moderately thick functionally graded rectangular plates with two simply-supported opposite edges. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 224(9), 1831–1841 (2010). https://doi.org/10.1243/09544062jmes1804

    Article  Google Scholar 

  85. Bodaghi, M., Saidi, A.R.: Levy-type solution for buckling analysis of thick functionally graded rectangular plates based on the higher-order shear deformation plate theory. Appl. Math. Model. 34(11), 3659–3673 (2010). https://doi.org/10.1016/j.apm.2010.03.016

    Article  MathSciNet  Google Scholar 

  86. Thai, H.-T., Choi, D.-H.: An efficient and simple refined theory for buckling analysis of functionally graded plates. Appl. Math. Model. 36(3), 1008–1022 (2012). https://doi.org/10.1016/j.apm.2011.07.062

    Article  MathSciNet  Google Scholar 

  87. Sobhy, M.: A comprehensive study on FGM nanoplates embedded in an elastic medium. Compos. Struct. 134, 966–980 (2015). https://doi.org/10.1016/j.compstruct.2015.08.102

    Article  Google Scholar 

  88. Barati, M.R., Zenkour, A.M., Shahverdi, H.: Thermo-mechanical buckling analysis of embedded nanosize FG plates in thermal environments via an inverse cotangential theory. Compos. Struct. 141, 203–212 (2016). https://doi.org/10.1016/j.compstruct.2016.01.056

    Article  Google Scholar 

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Acknowledgements

The authors would like to thank the reviewers for their comments and suggestions to improve this article’s clarity.

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  1. Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University, Qazvin, Iran

    Farzad Ebrahimi & Mehrdad Farajzadeh Ahari

  2. School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran

    Ali Dabbagh

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  1. Farzad Ebrahimi
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  2. Mehrdad Farajzadeh Ahari
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Appendix 1

Appendix 1

$$K_{11} = A_{11} \left( {\mathop \int \limits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{3} X_{m} }}{{\partial x^{3} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}Y_{n} } \right){\text{d}}x\,{\text{d}}y} \right) + A_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right)$$
$$K_{12} = A_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) + A_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right)$$
$$\begin{aligned} K_{13} = & - B_{11} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{3} X_{m} }}{{\partial x^{3} }}Y_{n} X_{m} Y_{n} } \right){\text{d}}x\,{\text{d}}y} \right) - B_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}Y_{n} X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & - B_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}Y_{n} X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$
$$K_{14} = S_{11} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{3} X_{m} }}{{\partial x^{3} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}Y_{n} } \right){\text{d}}x\,{\text{d}}y} \right) + S_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right)$$
$$K_{15} = S_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) + S_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right)$$
$$K_{21} = A_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right) + A_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right)$$
$$K_{22} = A_{22} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {X_{m} \frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{3} Y_{n} }}{{\partial y^{3} }}} \right){\text{d}}x\,{\text{d}}y} \right) + A_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right)$$
$$\begin{aligned} K_{23} = & - B_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} X_{m} \frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right) - B_{22} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {X_{m} Y_{n} X_{m} \frac{{\partial^{3} Y_{n} }}{{\partial y^{3} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & - 2B_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} X_{m} \frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$
$$K_{24} = S_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right) + S_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right)$$
$$K_{25} = S_{22} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {X_{m} \frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{3} Y_{n} }}{{\partial y^{3} }}} \right){\text{d}}x\,{\text{d}}y} \right) + S_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right)$$
$$\begin{aligned} K_{31} = & B_{11} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}Y_{n} } \right){\text{d}}x\,{\text{d}}y} \right) + B_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & + 2B_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$
$$\begin{aligned} K_{32} = & B_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) + B_{22} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {X_{m} \frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & + 2B_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$
$$\begin{aligned} K_{33} = & - D_{11} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}Y_{n} X_{m} Y_{n} } \right){\text{d}}x\,{\text{d}}y} \right) - 2D_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & - D_{22} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {X_{m} Y_{n} X_{m} \frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}} \right){\text{d}}x\,{\text{d}}y} \right) - 4D_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$
$$\begin{aligned} K_{34} = & M_{11} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}Y_{n} } \right){\text{d}}x\,{\text{d}}y} \right) + M_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & + 2M_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$
$$\begin{aligned} K_{35} = & M_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) + M_{22} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {X_{m} \frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & + 2M_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$
$$K_{41} = S_{11} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{3} X_{m} }}{{\partial x^{3} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}Y_{n} } \right){\text{d}}x\,{\text{d}}y} \right) + S_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right)$$
$$K_{42} = S_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) + S_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right)$$
$$\begin{aligned} K_{43} = & - M_{11} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{3} X_{m} }}{{\partial x^{3} }}Y_{n} X_{m} Y_{n} } \right){\text{d}}x\,{\text{d}}y} \right) - M_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}Y_{n} X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & - 2M_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}Y_{n} X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$
$$\begin{aligned} K_{44} = & O_{11} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{3} X_{m} }}{{\partial x^{3} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}Y_{n} } \right){\text{d}}x\,{\text{d}}y} \right) + O_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & - L_{55} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}Y_{n} } \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$
$$K_{45} = O_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) + O_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right)$$
$$K_{51} = S_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right) + S_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right)$$
$$K_{52} = S_{22} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {X_{m} \frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{3} Y_{n} }}{{\partial y^{3} }}} \right){\text{d}}x\,{\text{d}}y} \right) + S_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right)$$
$$\begin{aligned} K_{53} = & - M_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} X_{m} \frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right) - M_{22} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {X_{m} Y_{n} X_{m} \frac{{\partial^{3} Y_{n} }}{{\partial y^{3} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & - 2M_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} X_{m} \frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$
$$K_{54} = O_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right) + O_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right)$$
$$\begin{aligned} K_{55} = & O_{22} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {X_{m} \frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{3} Y_{n} }}{{\partial y^{3} }}} \right){\text{d}}x\,{\text{d}}y} \right) + O_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & - L_{44} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {X_{m} \frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$
$$C_{11} = C_{12} = C_{14} = C_{15} = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,C_{13} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} e_{31} \left( \alpha \right)K_{c} C\left( t \right){\text{d}}z$$
$$C_{21} = C_{22} = C_{24} = C_{25} = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,C_{23} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} e_{32} \left( \beta \right)K_{c} C\left( t \right){\text{d}}z$$
$$C_{31} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} e_{31} \left( \alpha \right)K_{c} C\left( t \right)dz\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,C_{32} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} e_{31} \left( \beta \right)K_{c} C\left( t \right){\text{d}}z$$
$$C_{33} = c_{d} \left( {1 + \mu^{2} \alpha^{2} + \mu^{2} \beta^{2} } \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,C_{34} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} ze_{31} \left( \alpha \right)K_{c} C\left( t \right){\text{d}}z$$
$$C_{35} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} ze_{32} \left( \beta \right)K_{c} C\left( t \right){\text{d}}z$$
$$C_{41} = C_{42} = C_{44} = C_{45} = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,C_{43} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} ze_{31} \left( \alpha \right)K_{c} C\left( t \right){\text{d}}z$$
$$C_{51} = C_{52} = C_{54} = C_{55} = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,C_{53} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} ze_{32} \left( \beta \right)K_{c} C\left( t \right){\text{d}}z$$
$$A_{11} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{11}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} C_{11}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{11}^{f} {\text{d}}z$$
$$A_{12} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{12}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} C_{12}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{12}^{f} {\text{d}}z$$
$$A_{22} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{22}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} C_{22}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{22}^{f} {\text{d}}z$$
$$A_{44} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{44}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} C_{44}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{44}^{f} {\text{d}}z$$
$$A_{55} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{55}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} C_{55}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{55}^{f} {\text{d}}z$$
$$A_{66} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{66}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} C_{66}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{66}^{f} {\text{d}}z$$
$$B_{11} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{11}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} zC_{11}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{11}^{f} {\text{d}}z$$
$$B_{12} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{12}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} zC_{12}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{12}^{f} {\text{d}}z$$
$$B_{22} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{22}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} zC_{22}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{22}^{f} {\text{d}}z$$
$$B_{44} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{44}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} zC_{44}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{44}^{f} {\text{d}}z$$
$$B_{55} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{55}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} zC_{22}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{55}^{f} {\text{d}}z$$
$$B_{66} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{66}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} zC_{66}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{66}^{f} {\text{d}}z$$
$$D_{11} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{11}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} z^{2} C_{11}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{11}^{f} {\text{d}}z$$
$$D_{12} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{12}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} z^{2} C_{12}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{12}^{f} {\text{d}}z$$
$$D_{22} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{22}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} z^{2} C_{22}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{22}^{f} {\text{d}}z$$
$$D_{44} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{44}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} z^{2} C_{44}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{44}^{f} {\text{d}}z$$
$$D_{55} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{55}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} z^{2} C_{55}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{55}^{f} {\text{d}}z$$
$$D_{66} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{66}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} z^{2} C_{66}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{66}^{f} {\text{d}}z$$
$$S_{11} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)C_{11}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)C_{11}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)C_{11}^{f} {\text{d}}z$$
$$S_{12} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)C_{12}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)C_{12}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)C_{12}^{f} {\text{d}}z$$
$$S_{22} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)C_{22}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)C_{22}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)C_{22}^{f} {\text{d}}z$$
$$S_{44} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)C_{44}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)C_{44}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)C_{44}^{f} {\text{d}}z$$
$$S_{55} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)C_{55}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)C_{55}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)C_{55}^{f} {\text{d}}z$$
$$S_{66} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)C_{66}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)C_{66}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)C_{66}^{f} {\text{d}}z$$
$$M_{11} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)zC_{11}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)zC_{11}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)zC_{11}^{f} {\text{d}}z$$
$$M_{12} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)zC_{12}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)zC_{12}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)zC_{12}^{f} {\text{d}}z$$
$$M_{22} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)zC_{22}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)zC_{22}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)zC_{22}^{f} {\text{d}}z$$
$$M_{44} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)zC_{44}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)zC_{44}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)zC_{44}^{f} {\text{d}}z$$
$$M_{55} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)zC_{55}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)zC_{22}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)zC_{55}^{f} {\text{d}}z$$
$$M_{66} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)zC_{66}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)zC_{66}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)zC_{66}^{f} {\text{d}}z$$
$$O_{11} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f(z)^{2} C_{11}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)^{2} C_{11}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)^{2} C_{11}^{f} {\text{d}}z$$
$$O_{12} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)^{2} C_{12}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f(z)^{2} C_{12}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)^{2} C_{12}^{f} {\text{d}}z$$
$$O_{22} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)^{2} C_{22}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)^{2} C_{22}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)^{2} C_{22}^{f} {\text{d}}z$$
$$O_{44} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)^{2} C_{44}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)^{2} C_{44}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f(z)^{2} C_{44}^{f} {\text{d}}z$$
$$O_{55} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)^{2} C_{55}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f(z)^{2} C_{55}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f(z)^{2} C_{55}^{f} {\text{d}}z$$
$$O_{66} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f(z)^{2} C_{66}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)^{2} C_{66}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)^{2} C_{66}^{f} {\text{d}}z$$
$$L_{44} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)^{\prime 2} C_{44}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)^{\prime 2} C_{44}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)^{\prime 2} C_{44}^{f} {\text{d}}z$$
$$L_{55} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)^{\prime 2} C_{55}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)^{\prime 2} C_{55}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)^{\prime 2} C_{55}^{f} {\text{d}}z$$

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Ebrahimi, F., Ahari, M.F. & Dabbagh, A. Stability analysis of a sandwich composite magnetostrictive nanoplate coupled with FG porous facesheets. Acta Mech 235, 2575–2597 (2024). https://doi.org/10.1007/s00707-023-03837-3

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