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Static buckling, vibration analysis and optimization of nanocomposite multilayer perovskite solar cell

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Abstract

In this study, analytical solutions for the static buckling and vibration analysis of nanocomposite multilayer perovskite solar cell (NMPSC) on elastic foundations are presented. The NMPSC consists of six isotropic layers which are including Au, PEDOT:PSS, perovskite, PCBM, ITO and glass. The basic equations are derived based on the Reddy’s higher order shear deformation plate theory with the consideration of Pasternak-type elastic foundations interaction, von-Kármán strain terms, initial imperfection and damping. The NMPSC is subjected to the uniformly distributed external pressure, axial compressive load and exposed to uniform temperature rise. The relationship between deflection amplitude and time, axial compressive loading and deflection amplitude, the frequency ratio and amplitude as well as the expressions of the natural frequency and critical buckling load are obtained by using Galerkin and Runge–Kutta methods. Bees Algorithm is used to maximize the natural frequency and critical buckling load of the NMPSC depending on ten geometrical and material parameters. The effects of temperature increment, elastic foundations, initial imperfection and geometrical parameters on the static buckling and vibration characteristics of the NMPSC are discussed through the parametric studies.

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References

  1. Kurukavak, K.Ç., Yılmaz, T., Büyükbekar, A., Kuş, M.: Effect of different terminal groups of phenyl boronic acid self-assembled monolayers on the photovoltaic performance of organic solar cells. Opt. Mater. (Amst.) 112, 110783 (2021). https://doi.org/10.1016/j.optmat.2020.110783

    Article  Google Scholar 

  2. Rana, D., Materny, A.: Effect of static external electric field on bulk and interfaces in organic solar cell systems: a density-functional-theory-based study. Spectrochim. Acta Part A Mol. Biomol. Spectrosc. 253, 119565 (2021). https://doi.org/10.1016/j.saa.2021.119565

    Article  Google Scholar 

  3. Huang, T., Bai, Y., Wang, J., Wang, F., Dai, M., Han, F., Du, S.: Optimizing binding energy and electron-hole pair binding distance for efficient organic solar cells with low voltage loss. Sol. Energy 230, 628–634 (2021). https://doi.org/10.1016/j.solener.2021.10.002

    Article  Google Scholar 

  4. Li, Q., Wang, Q., Wu, D., Chen, X., Yu, Y., Gao, W.: Geometrically nonlinear dynamic analysis of organic solar cell resting on Winkler–Pasternak elastic foundation under thermal environment. Compos. Part B Eng. 163, 121–129 (2019). https://doi.org/10.1016/j.compositesb.2018.11.022

    Article  Google Scholar 

  5. Li, Q., Wu, D., Gao, W., Tin-Loi, F.: Size-dependent instability of organic solar cell resting on Winkler–Pasternak elastic foundation based on the modified strain gradient theory. Int. J. Mech. Sci. 177, 105306 (2020). https://doi.org/10.1016/j.ijmecsci.2019.105306

    Article  Google Scholar 

  6. Keskin, A.V., Gençten, M., Bozar, S., Arvas, M.B., Güneş, S., Sahin, Y.: Preparation of anatase form of TiO2 thin film at room temperature by electrochemical method as an alternative electron transport layer for inverted type organic solar cells. Thin Solid Films 706, 138093 (2020). https://doi.org/10.1016/j.tsf.2020.138093

    Article  Google Scholar 

  7. Chaudhary, D.K., Dhawan, P.K., Patel, S.P., Bhasker, H.P.: Large area semitransparent inverted organic solar cells with enhanced operational stability using TiO2 electron transport layer for building integrated photovoltaic devices. Mater. Lett. 283, 128725 (2021). https://doi.org/10.1016/j.matlet.2020.128725

    Article  Google Scholar 

  8. Kaçuş, H., Baltakesmez, A., Çaldıran, Z., Aydoğan, Ş, Yılmaz, M., Sevim, M.: Optical and electrical characterization of organic solar cells obtained using gold and silver metal nanoparticles. Mater. Today Proc. 46, 6986–6990 (2021). https://doi.org/10.1016/j.matpr.2021.03.276

    Article  Google Scholar 

  9. Dat, N.D., Anh, V.M., Quan, T.Q., Pham, T.D., Duc, N.D.: Nonlinear stability and optimization of thin nanocomposite multilayer organic solar cell using Bees Algorithm. Thin-Walled Struct. 149, 106520 (2021). https://doi.org/10.1016/j.tws.2019.106520

    Article  Google Scholar 

  10. Shin, D.H., Jang, C.W., Ko, J.S., Choi, S.H.: Enhancement of efficiency and stability in organic solar cells by employing MoS2 transport layer, graphene electrode, and graphene quantum dots-added active layer. Appl. Surf. Sci. 538, 148155 (2021). https://doi.org/10.1016/j.apsusc.2020.148155

    Article  Google Scholar 

  11. Raman, R.K., Gurusamy Thangavelu, S.A., Venkataraj, S., Krishnamoorthy, A.: Materials, methods and strategies for encapsulation of perovskite solar cells: from past to present. Renew. Sustain. Energy Rev. 151, 111608 (2021). https://doi.org/10.1016/j.rser.2021.111608

    Article  Google Scholar 

  12. Girish, K.H.: Advances in surface passivation of perovskites using organic halide salts for efficient and stable solar cells. Surf. Interfaces 26, 101420 (2021). https://doi.org/10.1016/j.surfin.2021.101420

    Article  Google Scholar 

  13. Guo, Z., Lin, B.: Machine learning stability and band gap of lead-free halide double perovskite materials for perovskite solar cells. Sol. Energy 228, 689–699 (2021). https://doi.org/10.1016/j.solener.2021.09.030

    Article  Google Scholar 

  14. Wang, S., Cao, F., Wu, Y., Zhang, X., Zou, J., Lan, Z., et al.: Multifunctional 2D perovskite capping layer using cyclohexylmethylammonium bromide for highly efficient and stable perovskite solar cells. Mater. Today Phys. 21, 100543 (2021). https://doi.org/10.1016/j.mtphys.2021.100543

    Article  Google Scholar 

  15. Zhang, Y., Xu, L., Wu, Y., Zhou, Q., Shi, Z., Zhuang, X., et al.: Double-layer synergistic optimization by functional black phosphorus quantum dots for high-efficiency and stable planar perovskite solar cells. Nano Energy 90, 106610 (2021). https://doi.org/10.1016/j.nanoen.2021.106610

    Article  Google Scholar 

  16. Huang, Z.L., Chen, C.M., Lin, Z.K., Yang, S.H.: Efficiency enhancement of regular-type perovskite solar cells based on Al-doped ZnO nanorods as electron transporting layers. Superlattices Microstruct. 102, 94–102 (2017). https://doi.org/10.1016/j.spmi.2016.12.012

    Article  Google Scholar 

  17. Li, Q., Tian, Y., Wu, D., Gao, W., Yu, Y., Chen, X., Yang, C.: The nonlinear dynamic buckling behaviour of imperfect solar cells subjected to impact load. Thin-Walled Struct. 169, 108317 (2021). https://doi.org/10.1016/j.tws.2021.108317

    Article  Google Scholar 

  18. Cho, S., Lee, H., Seo, Y., Na, S.: Multifunctional passivation agents for improving efficiency and stability of perovskite solar cells: synergy of methyl and carbonyl groups. Appl. Surf. Sci. 575, 151740 (2022). https://doi.org/10.1016/j.apsusc.2021.151740

    Article  Google Scholar 

  19. Varghese, A.S., Panda, S.: Stability analysis on the flow of thin second-grade fluid over a heated inclined plate with variable fluid properties. Int. J. Non Linear Mech. 133, 103711 (2021). https://doi.org/10.1016/j.ijnonlinmec.2021.103711

    Article  Google Scholar 

  20. Noroozi, A.R., Malekzadeh, P., Dimitri, R., Tornabene, F.: Meshfree radial point interpolation method for the vibration and buckling analysis of FG-GPLRC perforated plates under an in-plane loading. Eng. Struct. 221, 111000 (2020). https://doi.org/10.1016/j.engstruct.2020.111000

    Article  Google Scholar 

  21. Moradi-Dastjerdi, R., Behdinan, K.: Stability analysis of multifunctional smart sandwich plates with graphene nanocomposite and porous layers. Int. J. Mech. Sci. 167, 105283 (2020). https://doi.org/10.1016/j.ijmecsci.2019.105283

    Article  Google Scholar 

  22. Shen, H.S., Xiang, Y., Fan, Y.: A novel technique for nonlinear dynamic instability analysis of FG-GRC laminated plates. Thin-Walled Struct. 139, 389–397 (2019). https://doi.org/10.1016/j.tws.2019.03.010

    Article  Google Scholar 

  23. Stojanović, V., Petković, M.D., Deng, J.: Stability of parametric vibrations of an isolated symmetric cross-ply laminated plate. Compos. Part B Eng. 167, 631–642 (2019). https://doi.org/10.1016/j.compositesb.2019.02.041

    Article  Google Scholar 

  24. Kolahchi, R., Kolahdouzan, F.: A numerical method for magneto-hygro-thermal dynamic stability analysis of defective quadrilateral graphene sheets using higher order nonlocal strain gradient theory with different movable boundary conditions. Appl. Math. Model. 91, 458–475 (2021). https://doi.org/10.1016/j.apm.2020.09.060

    Article  MathSciNet  MATH  Google Scholar 

  25. Tubaldi, E., Alijani, F., Amabili, M.: Non-linear vibrations and stability of a periodically supported rectangular plate in axial flow. Int. J. Non Linear Mech. 66, 54–65 (2014). https://doi.org/10.1016/j.ijnonlinmec.2013.12.004

    Article  Google Scholar 

  26. Duc, N.D., Lam, P.T., Quan, T.Q., Quang, P.M., Quyen, N.V.: Nonlinear post-buckling and vibration of 2D penta-graphene composite plates. Acta Mech. 231, 539–559 (2020). https://doi.org/10.1007/s00707-019-02546-0

    Article  MathSciNet  MATH  Google Scholar 

  27. Sofiyev, A.H.: On the solution of the dynamic stability of heterogeneous orthotropic visco-elastic cylindrical shells. Compos. Struct. 206, 124–130 (2018). https://doi.org/10.1016/j.compstruct.2018.08.027

    Article  Google Scholar 

  28. Namvar, A.R., Vosoughi, A.R.: Design optimization of moderately thick hexagonal honeycomb sandwich plate with modified multi-objective particle swarm optimization by genetic algorithm (MOPSOGA). Compos. Struct. 252, 112626 (2020). https://doi.org/10.1016/j.compstruct.2020.112626

    Article  Google Scholar 

  29. Innami, M., Honda, S., Sasaki, K., Narita, Y.: Analysis and optimization for vibration of laminated rectangular plates with blended layers. Compos. Struct. 274, 114400 (2021). https://doi.org/10.1016/j.compstruct.2021.114400

    Article  Google Scholar 

  30. Tu, T.M., Anh, P.H., Loi, N.V., Tuan, T.A.: Optimization of stiffeners for maximum fundamental frequency of cross-ply laminated cylindrical panels using social group optimization and smeared stiffener method. Thin-Walled Struct. 120, 172–179 (2017). https://doi.org/10.1016/j.tws.2017.08.033

    Article  Google Scholar 

  31. Anh, V.M., Quan, T.Q., Tran, P.: Nonlinear vibration and geometric optimization of nanocomposite multilayer organic solar cell under wind loading. Thin-Walled Struct. 158, 107199 (2021). https://doi.org/10.1016/j.tws.2020.107199

    Article  Google Scholar 

  32. Jing, Z., Sun, Q., Zhang, Y., Liang, K.: Buckling optimization of composite rectangular plates by sequential permutation search with bending-twisting correction. Appl. Math. Model. 100, 751–779 (2021). https://doi.org/10.1016/j.apm.2021.07.031

    Article  MathSciNet  MATH  Google Scholar 

  33. Pham, T.D., Castellani, M.: The Bees Algorithm: modelling foraging behavior to solve continuous optimization problems. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 223, 2919–2938 (2009). https://doi.org/10.1243/09544062JMES1494

    Article  Google Scholar 

  34. Brush, D.D., Almroth, B.O.: Buckling of Bars, Plates and Shells. Mc. Graw-Hill, New York (1975)

    Book  MATH  Google Scholar 

  35. Reddy, J.N.: Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. CRC Press, Boca Raton (2004)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mechanical Engineering and Mechatronics, Phenikaa University, Hanoi, 12116, Vietnam

    Tran Quoc Quan

  2. Phenikaa Institute for Advanced Study (PIAS), Phenikaa University, Hanoi, 12116, Vietnam

    Tran Quoc Quan

  3. Faculty of Civil Engineering – VNU Hanoi, University of Engineering and Technology, 144 – Xuan Thuy, Cau Giay, Hanoi, Vietnam

    Ngo Dinh Dat & Nguyen Dinh Duc

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  1. Tran Quoc Quan
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Appendices

Appendix A

$$\begin{aligned} E_{11}& = R_{44} - 6c_{1} Y_{44} + 9c_{1}^{2} X_{44} ,\,\,E_{12} = R_{55} - 6c_{1} Y_{55} + 9c_{1}^{2} X_{55} ,\,\, \\ E_{13}& = - c_{1}^{2} (Z_{11} H_{15}^{{}} + Z_{12} H_{25}^{{}} + O_{11} ),\,\, \\ E_{14} & = - c_{1}^{2} (4Z_{66} H_{33}^{{}} + 4O_{66} + Z_{11} H_{16} + Z_{12} H_{26} + 2O_{12} + Z_{12} H_{15} + Z_{22} H_{25} ),\,\, \\ E_{15} & = - c_{1}^{2} (Z_{12} H_{16} + Z_{22} H_{26} + O_{22} ),\,\, \\ E_{16} & = c_{1} (Z_{11} H_{13} - c_{1} Z_{11} H_{15} + X_{11} - c_{1} O_{11} + Z_{12} H_{23} - c_{1} Z_{12} H_{25} ), \\ E_{17} & = c_{1} (2Z_{66} H_{32} - 2c_{1} Z_{66} H_{33} + 2X_{66} - 2c_{1} O_{66} + c_{1} Z_{12} H_{13} \\ & - c_{1} Z_{12} H_{15} + X_{12} - c_{1} O_{12} + Z_{22} H_{23} - c_{1} Z_{22} H_{25} ), \\ E_{18} & = c_{1} \left( {Z_{12} H_{14} - c_{1} Z_{12} H_{16} + Z_{22} H_{24} - c_{1} Z_{22} H_{26} + X_{22} - c_{1} O_{22} } \right),\,\, \\ E_{19} & = c_{1} (2Z_{66} H_{32} - 2c_{1} Z_{66} H_{33} + 2X_{66} \\ & - 2c_{1} O_{66} + Z_{11} H_{14} - c_{1} Z_{11} H_{16} + Z_{12} H_{24} - c_{1} Z_{12} H_{26} + X_{12} - c_{1} O_{12} ),\,\, \\ E_{110} & = - c_{1} (Z_{11} H_{12} - Z_{12} H_{21} ),\,E_{111} = - c_{1} (2Z_{66} H_{31} - Z_{11} H_{11} + 2Z_{12} H_{12} - Z_{22} H_{21} ),\,\, \\ E_{112} & = c_{1} (Z_{12} H_{11} - Z_{22} H_{12} ),E_{21} = - R_{44} + 6c_{1} Y_{44} - 9c_{1}^{2} X_{44} ,\,\, \\ E_{22} & = - c_{1} (T_{11} H_{15} + X_{11} + T_{12} H_{25} - c_{1} Z_{11} H_{15} - c_{1} O_{11} - c_{1} Z_{12} H_{25} ), \\ E_{23} & = - c_{1} (T{}_{11}H_{16} + T_{12} H_{26} + X_{12} + 2T_{66} H_{33} + 2X_{66} - 2c_{1} Z_{66} H_{33} \\ & - 2c_{1} O_{66} - c_{1} Z_{11} H_{16} - c_{1} Z_{12} H_{26} - c_{1} O_{12} ), \\ E_{24} & = T_{11} H_{13} - c_{1} T_{11} H_{15} + Y_{11} - c_{1} X_{11} + T_{12} H_{23} - c_{1} T_{12} H_{25} \\ & - c_{1} Z_{11} H_{13} + c_{1}^{2} Z_{11} H_{15} - c_{1} X_{11} + c_{1}^{2} O_{11} - c_{1} Z_{12} H_{23} + c_{1}^{2} Z_{12} H_{25} ,\,\, \\ E_{25} & = T_{66} H_{32} - c_{1} T_{66} H_{33} + Y_{66} - c_{1} X_{66} - c_{1} Z_{66} H_{32} + c_{1}^{2} Z_{66} H_{33} - c_{1} X_{66} + c_{1}^{2} O_{66} , \\ E_{26} & = T_{11} H_{14} - c_{1} T_{11} H_{16} + T_{12} H_{24} - c_{1} T_{12} H_{26} + Y_{12} - c_{1} X_{12} + T_{66} H_{32} - c_{1} T_{66} H_{33} \\ & + Y_{66} - c_{1} X_{66} - c_{1} Z_{66} H_{32} + c_{1}^{2} Z_{66} H_{33} - c_{1} X_{66} + c_{1}^{2} O_{66} - c_{1} Z_{11} H_{14} \\ & + c_{1}^{2} Z_{11} H_{16} - c_{1} Z_{12} H_{24} + c_{1}^{2} Z_{12} H_{26} - c_{1} X_{12} + c_{1}^{2} O_{12} , \\ \end{aligned}$$
$$\begin{aligned} E_{27} & = - T_{11} H_{12} + T_{12} H_{21} + c_{1} Z_{11} H_{12} - c_{1} Z_{12} H_{21} ,\,\, \\ E_{28} & = T_{11} H_{11} - T_{12} H_{12} - T_{66} H_{31} - c_{1} Z_{11} H_{11} + c_{1} Z_{12} H_{12} \\ & + c_{1} Z_{66} H_{31} ,\,\,E_{31} = - R_{55} + 6c_{1} Y_{55} - 9c_{1}^{2} X_{55} ,\,\, \\ E_{32} & = - c_{1} (2T_{66} H_{33} + 2X_{66} + T_{12} H_{15} + X_{12} + T_{22} H_{25} \\ & - 2c_{1} Z_{66} H_{33} - 2c_{1} O_{66} - c_{1} Z_{12} H_{15} - c_{1} O_{12} - c_{1} X_{22} H_{25} ),\,\, \\ E_{33} & = - c_{1} (T_{12} H_{16} + T_{22} H_{26} + X_{22} - c_{1} Z_{12} H_{16} \\ & - c_{1} Z_{22} H_{26} - c_{1} O_{22} ),\,\,E_{34} = T_{66} H_{32} - c_{1} T_{66} H_{33} + Y_{66} - c_{1} X_{66} + T_{12} H_{13} \\ & - c_{1} T_{12} H_{15} + Y_{12} - c_{1} X_{12} + T_{22} H_{23} - c_{1} T_{22} H_{25} - c_{1} Z_{66} H_{32} + c_{1}^{2} Z_{66} H_{33} \\ & - c_{1} X_{66} + c_{1}^{2} O_{66} - c_{1} Z_{12} H_{13} + c_{1}^{2} Z_{12} H_{15} - c_{1} X_{12} + c_{1}^{2} O_{12} - c_{1} Z_{22} H_{23} + c_{1}^{2} Z_{22} H_{25} ,\,\, \\ E_{35} & = T_{66} H_{32} - c_{1} T_{66} H_{33} + Y_{66} - c_{1} X_{66} - c_{1} Z_{66} H_{32} + c_{1}^{2} Z_{66} H_{33} - c_{1} X_{66} + c_{1}^{2} O_{66} , \\ E_{36} & = T_{12} H_{14} - c_{1} T_{12} H_{16} + T_{22} H_{24} - c_{1} T_{22} H_{26} + Y_{22} - c_{1} F_{22} - c_{1} Z_{12} H_{14} + c_{1}^{2} Z_{12} H_{16} - c_{1} Z_{22} H_{24} \\ & + c_{1}^{2} Z_{22} H_{26} - c_{1} X_{22} + c_{1}^{2} H_{22} ,\,\,E_{37} = - T_{66} I_{31} - T_{12} H_{12} + T_{22} H_{21} + c_{1} Z_{66} H_{31} + c_{1} Z_{12} H_{12} - c_{1} Z_{22} H_{21} , \\ E_{38} & = T_{12} H_{11} - T_{22} H_{12} - c_{1} Z_{12} H_{11} + c_{1} Z_{22} H_{12} . \\ \end{aligned}$$

Appendix B

$$\begin{aligned} \aleph_{11} & = - k_{1} - k_{2} \left( {\lambda_{m}^{2} + \delta_{n}^{2} } \right) + E_{13} \lambda_{m}^{4} + E_{14} \lambda_{m}^{2} \delta_{n}^{2} + E_{15} \delta_{n}^{4} + E_{110} C_{1} \lambda_{m}^{4} + E_{111} C_{1} \lambda_{m}^{2} \delta_{n}^{2} + X_{112} C_{1} \delta_{n}^{4} , \\ \aleph_{12} & = - E_{11} \lambda_{m} + E_{16} \lambda_{m}^{3} + E_{17} \lambda_{m} \delta_{n}^{2} + E_{110} C_{2} \lambda_{m}^{4} + E_{111} C_{2} \lambda_{m}^{2} \delta_{n}^{2} + E_{112} C_{2} \delta_{n}^{4} , \\ \aleph_{13} & = - E_{12} \delta_{n} + E_{18} \delta_{n}^{3} + E_{19} \lambda_{m}^{2} \delta_{n}^{{}} + E_{110} C_{3} \lambda_{m}^{4} + E_{111} C_{3} \lambda_{m}^{2} \delta_{n}^{2} + E_{112} C_{3} \delta_{n}^{4} , \\ \aleph_{14} & = \frac{{32C_{2} \lambda_{m} \delta_{n} }}{3ab},\,\,l_{15} = \frac{{32C_{3} \lambda_{m} \delta_{n} }}{3ab},\,\,l_{1} = - E_{11} \lambda_{m}^{2} - E_{12} \delta_{n}^{2} ,\,\,l_{2} = \frac{{32C_{1} \lambda_{m} \delta_{n} }}{3ab}, \\ l_{3} & = - \frac{{8E_{110} \lambda_{m} \delta_{n} }}{{3abH_{21}^{{}} }} - \frac{{8E_{112} \lambda_{m} \delta_{n} }}{{3abH_{11}^{{}} }},\,\,l_{4} = - \frac{{\lambda_{m}^{4} }}{{16H_{11}^{{}} }} - \frac{{\delta_{n}^{4} }}{{16H_{21}^{{}} }},\,l_{5} = \frac{16}{{mn\pi^{2} }}, \\ \aleph_{21} & = - \lambda_{m}^{3} (E_{22} + C_{1} E_{27} ) - \lambda_{m} \delta_{n}^{2} (E_{23} + C_{1} E_{28} ),\,\, \\ \aleph_{22} & = E_{21} - E_{24} \lambda_{m}^{2} - E_{25} \delta_{n}^{2} - E_{27} C_{2} \lambda_{m}^{3} - e_{28} C_{2} \lambda_{m} \delta_{n}^{2} , \\ \aleph_{23} & = - E_{26} \lambda_{m} \delta_{n} - E_{27} C_{3} \lambda_{m}^{3} - E_{28} C_{3} \lambda_{m} \delta_{n}^{2} ,\,\,l_{6} = E_{21} \lambda_{m} ,\,\,l_{7} = \frac{{8E_{27} \delta_{n} }}{{3abH_{21}^{{}} }},\,\,\aleph_{31} = - \delta_{n}^{3} (E_{33} + C_{1} E_{38} ) \\ & - \lambda_{m}^{2} \delta_{n}^{{}} (E_{32} + C_{1} E_{37} ),\,\aleph_{32} = - E_{34} \lambda_{m} \delta_{n} - E_{38} C_{2} \delta_{n}^{3} - E_{37} C_{2} \lambda_{m}^{2} \delta_{n}^{{}} ,\,\,\aleph_{33} = E_{31} - E_{35} \lambda_{m}^{2} - E_{36} \delta_{n}^{2} \\ & - E_{38} C_{3} \delta_{n}^{3} - E_{37} C_{3} \lambda_{m}^{2} \delta_{n}^{{}} ,\,\,l_{8} = E_{31} \delta_{n} ,\,\,l_{9} = \frac{{8E_{38} \lambda_{m} }}{{3abH_{11}^{{}} }}. \\ \end{aligned}$$

Appendix C

$$a_{1} = - \frac{{\left( {\aleph_{21} \aleph_{33} - \aleph_{23} \aleph_{31} } \right)}}{{\left( {\aleph_{22} \aleph_{33} - \aleph_{23} \aleph_{32} } \right)}},\,\,a_{2} = - \frac{{\left( {l_{6} \aleph_{33} - l_{8} \aleph_{23} } \right)}}{{\left( {\aleph_{22} \aleph_{33} - \aleph_{23} \aleph_{32} } \right)}},\,$$
$$\,a_{3} = - \frac{{\left( {l_{7} \aleph_{33} - l_{9} \aleph_{23} } \right)}}{{\left( {\aleph_{22} \aleph_{33} - \aleph_{23} \aleph_{32} } \right)}},\,\,a_{4} = \frac{{\left( { - \lambda_{m} \overline{\overline{{A_{5} }}} \aleph_{33} + \delta_{n} \overline{\overline{{A_{5}^{*} }}} \aleph_{23} } \right)}}{{\left( {\aleph_{22} \aleph_{33} - \aleph_{23} \aleph_{32} } \right)}},$$
$$\begin{aligned} a_{5} & = \frac{{\left( {\aleph_{21} \aleph_{32} - \aleph_{22} \aleph_{31} } \right)}}{{\left( {\aleph_{22} \aleph_{33} - \aleph_{23} \aleph_{32} } \right)}},\,\,a_{6} = \frac{{\left( {l_{6} \aleph_{32} - l_{8} \aleph_{22} } \right)}}{{\left( {\aleph_{22} \aleph_{33} - \aleph_{23} \aleph_{32} } \right)}},\, \\ a_{7} & = \frac{{\left( {l_{7} \aleph_{32} - l_{9} \aleph_{22} } \right)}}{{\left( {\aleph_{22} \aleph_{33} - \aleph_{23} \aleph_{32} } \right)}},\,\,a_{8} = - \frac{{\left( { - \lambda_{m} \overline{\overline{{A_{5} }}} \aleph_{32} + \delta_{n} \overline{\overline{{A_{5}^{*} }}} \aleph_{22} } \right)}}{{\left( {\aleph_{22} \aleph_{33} - \aleph_{23} \aleph_{32} } \right)}}, \\ \end{aligned}$$
$$\begin{aligned} r_{11} & = \left( {\aleph_{11} + \aleph_{12} a_{1} + \aleph_{13} a_{5} } \right),\,\,r_{12} = \left( {\aleph_{1}^{1} + \aleph_{12} a_{2} + \aleph_{13} a_{6} } \right), \\ r_{13} & = \left( {l_{2}^{1} + \aleph_{14}^{1} a_{1} + \aleph_{15}^{1} a_{5} } \right),\,\,r_{14} = \left( {l_{3} + \aleph_{12} a_{3} + \aleph_{13} a_{7} } \right), \\ r_{15} & = \left( {\aleph_{14}^{1} a_{2} + \aleph_{15}^{1} a_{6} } \right),\,\,r_{16} = \left( {l_{4}^{1} + \aleph_{14}^{1} a_{3} + \aleph_{15}^{1} a_{7} } \right), \\ \overline{{A_{0} }} & = A_{0} - \left( {\aleph_{12} a_{4} + \aleph_{13} a_{8} } \right),\,\,A_{0}^{*} = \left( {\aleph_{14}^{1} a_{4} + \aleph_{15}^{1} a_{8} } \right). \\ \end{aligned}$$

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Quan, T.Q., Dat, N.D. & Duc, N.D. Static buckling, vibration analysis and optimization of nanocomposite multilayer perovskite solar cell. Acta Mech 234, 3893–3915 (2023). https://doi.org/10.1007/s00707-023-03588-1

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