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Higher-order buckling analysis of FG porous cylindrical micro-shells integrated with GPLs-RC patches in hygrothermal environment immersed on Kerr foundation

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Abstract

This study investigates the buckling behavior of functionally graded (FG) porous micro-shell covered with nanocomposite facesheets; graphene nanoplatelets (GPLs) are hired to reinforce and strengthen the faces. This micro-shell is under a hygrothermal environment and rested on Kerr foundations which consist of two rows of springs and one shear layer. Due to considering the size effect for this microstructure, modified couple stress theory is implemented in the strain energy. Reddy’s theory in the Cartesian coordinate system is applied to analyze shear stress distribution through z-direction. In the end, the Navier’s solution method is employed to solve equations analytically for simply supported edges. Moreover, the validity of the outcomes is confirmed by comparing them with a previous published study. After validation, the key findings of the research are examined; it highlights the influences of GPLs of facesheets and porous distribution on the stiffness of the structure, and the impact of Kerr foundation on the result is the most significant outcomes. The main implications show that the symmetric porosity distribution and FG V-A GPLs distribution make this engineering structure become stiffer which consequently leads to an increase in critical buckling load.

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Acknowledgements

This research has been jointly supported by The China’s Liaoning Province “**ng Liao Talents Plan” Youth Top-notch Talent Funding Project under Grant No. XLYC2007146 and Science and Technology Research Plan of China Railway 19th Bureau Co. Ltd under Grant No.2021-B03.

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No funding was received for conducting this study.

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Authors and Affiliations

  1. School of Architecture and Art, Zhejiang Business Technology Institute, Ningbo, 315012, Zhejiang, People’s Republic of China

    Yi-Fan Ying

  2. The Third Engineering Co., Ltd., China Railway 19th Bureau Group, Shenyang, 110136, Liaoning, People’s Republic of China

    Li-Cai Zhao

  3. Department of Civil and Construction Engineering, National Taiwan University of Science and Technology, No.43, Sec.4, Keelung Road, Taipei City, Taiwan

    Yi-Fan Ying & Li-Cai Zhao

  4. Department of Nuclear and Renewable Energy, Ural Federal University Named After the First President of Russia, Boris Yeltsin, 19 Mira Street, Yekaterinburg, 620002, Russia

    Abhinav Kumar

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  1. Yi-Fan Ying
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  3. Abhinav Kumar
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Correspondence to Li-Cai Zhao.

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Appendices

Appendix A

Κijk that are presented in Eq. (22) are defined as follows:

$${\kappa }_{\mathrm{xxx}}=\frac{{\partial }^{2}}{\partial {x}^{2}}u,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\kappa }_{xx\alpha }=\frac{{\partial }^{2}}{\partial {x}^{2}}v,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\kappa }_{zz\alpha }=\frac{{\partial }^{2}}{\partial {z}^{2}}v,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\kappa }_{\mathrm{xxz}}=\frac{{\partial }^{2}}{\partial {x}^{2}}w,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\kappa }_{xz\alpha }={\kappa }_{zx\alpha }=\frac{{\partial }^{2}}{\partial z\partial x}v,$$
$${\kappa }_{\mathrm{zxx}}={\kappa }_{\mathrm{xzx}}=\frac{{\partial }^{2}}{\partial x\partial z}u,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\kappa }_{\mathrm{zzz}}=\frac{{\partial }^{2}}{\partial {z}^{2}}w\text{\hspace{0.17em}},\text{\hspace{0.17em}\hspace{0.17em}}{\kappa }_{\mathrm{zzx}}=\frac{{\partial }^{2}}{\partial {z}^{2}}u,\text{\hspace{0.17em}\hspace{0.17em}}{\kappa }_{\mathrm{xzz}}={\kappa }_{\mathrm{zxz}}=\frac{{\partial }^{2}}{\partial z\partial x}w,$$
$${\kappa }_{x\theta \theta }={\kappa }_{\alpha x\alpha }=\left(\frac{{\partial }^{2}}{\partial x\partial \alpha }v+\frac{\partial }{\partial x}w\right){\left(R\left(1+\frac{z}{R}\right)\right)}^{-1},$$
$${\kappa }_{\alpha xx}={\kappa }_{x\alpha x}=\left(\frac{{\partial }^{2}}{\partial x\partial \alpha }u\right){\left(R\left(1+\frac{z}{R}\right)\right)}^{-1},$$
$${\kappa }_{\alpha \alpha x}=\left(\frac{{\partial }^{2}}{\partial {\alpha }^{2}}u{\left(R\left(1+\frac{z}{R}\right)\right)}^{-1}+\frac{\partial }{\partial z}u\right){\left(R\left(1+\frac{z}{R}\right)\right)}^{-1},$$
$${\kappa }_{x\alpha z}={\kappa }_{\alpha xz}=\left(\frac{{\partial }^{2}}{\partial x\partial \alpha }w-\frac{\partial }{\partial x}v\right){\left(R\left(1+\frac{z}{R}\right)\right)}^{-1},$$
$${\kappa }_{\alpha \alpha z}=\left(\frac{{\partial }^{2}}{\partial {\alpha }^{2}}w-2\frac{\partial }{\partial \alpha }v+R\left(1+\frac{z}{R}\right)\frac{\partial }{\partial z}w-w\right){\left(R\left(1+\frac{z}{R}\right)\right)}^{-2},$$
$${\kappa }_{z\alpha \alpha }={\kappa }_{\alpha z\alpha }=\left(\frac{{\partial }^{2}}{\partial z\partial \alpha }v-\frac{\partial }{\partial \alpha }v{\left(R\left(1+\frac{z}{R}\right)\right)}^{-1}+\frac{\partial }{\partial z}w-w{\left(R\left(1+\frac{z}{R}\right)\right)}^{-1}\right){\left(R\left(1+\frac{z}{R}\right)\right)}^{-1},$$
$${\kappa }_{z\alpha z}={\kappa }_{\alpha zz}=\left(\frac{{\partial }^{2}}{\partial z\partial \alpha }w-\frac{\partial }{\partial \alpha }w{\left(R\left(1+\frac{z}{R}\right)\right)}^{-1}-\frac{\partial }{\partial z}v+v{\left(R\left(1+\frac{z}{R}\right)\right)}^{-1}\right){\left(R\left(1+\frac{z}{R}\right)\right)}^{-1},$$
$${\kappa }_{\alpha zx}=\left(\frac{{\partial }^{2}}{\partial z\partial \alpha }u-\frac{\partial }{\partial \alpha }u{\left(R\left(1+\frac{z}{R}\right)\right)}^{-1}\right){\left(R\left(1+\frac{z}{R}\right)\right)}^{-1},$$
$${\kappa }_{z\alpha x}=\left(\frac{{\partial }^{2}}{\partial z\partial \alpha }u-\frac{\partial }{\partial \alpha }u{\left(R\left(1+\frac{z}{R}\right)\right)}^{-1}\right){\left(R\left(1+\frac{z}{R}\right)\right)}^{-1},$$
$${\kappa }_{\alpha \alpha \alpha }=\left(\frac{{\partial }^{2}}{\partial {\alpha }^{2}}v+2\frac{\partial }{\partial \alpha }w+R\left(1+\frac{z}{R}\right)\frac{\partial }{\partial z}v-v\right){\left(R\left(1+\frac{z}{R}\right)\right)}^{-2},$$

Appendix B

$$\delta u:\frac{1}{4}\frac{{K}_{15}}{R}\frac{{\partial }^{2}}{\partial x\partial \alpha }{\varphi }_{\alpha }-\frac{1}{4}\frac{{K}_{1}}{R}\frac{{\partial }^{4}}{\partial {x}^{3}\partial \alpha }{\varphi }_{\alpha }-\frac{1}{2}\frac{{K}_{16}}{{R}^{2}}\frac{{\partial }^{3}}{\partial x\partial {\alpha }^{2}}w+\frac{1}{4}\frac{{K}_{1}}{{R}^{2}}\frac{{\partial }^{4}}{\partial {x}^{2}\partial {\alpha }^{2}}{\varphi }_{x}-\frac{1}{4}\frac{{K}_{0}}{R}\frac{{\partial }^{4}}{\partial {x}^{3}\partial \alpha }v+\frac{1}{4}\frac{{K}_{0}}{{R}^{2}}\frac{{\partial }^{4}}{\partial {x}^{2}\partial {\alpha }^{2}}u+\frac{1}{4}\frac{{K}_{15}}{{R}^{2}}\frac{{\partial }^{2}}{\partial {\alpha }^{2}}{\varphi }_{x}-\frac{1}{4}\frac{{K}_{1}}{{R}^{3}}\frac{{\partial }^{4}}{\partial x\partial {\alpha }^{3}}{\varphi }_{\alpha }+\frac{1}{4}\frac{{K}_{1}}{{R}^{4}}\frac{{\partial }^{4}}{\partial {\alpha }^{4}}{\varphi }_{x}-\frac{1}{4}\frac{{K}_{0}}{{R}^{3}}\frac{{\partial }^{4}}{\partial x\partial {\alpha }^{3}}v+\frac{1}{4}\frac{{K}_{0}}{{R}^{4}}\frac{{\partial }^{4}}{\partial {\alpha }^{4}}u+\frac{1}{4}\frac{{K}_{6}}{{R}^{3}}\frac{{\partial }^{2}}{\partial {\alpha }^{2}}{\varphi }_{x}-\frac{3}{2}\frac{{K}_{7}}{{R}^{3}}\frac{{\partial }^{3}}{\partial x\partial {\alpha }^{2}}w+\frac{3}{4}\frac{{K}_{1}}{{R}^{3}}\frac{{\partial }^{2}}{\partial x\partial \alpha }{\varphi }_{\alpha }-\frac{{K}_{0}}{{R}^{3}}\frac{{\partial }^{3}}{\partial x\partial {\alpha }^{2}}w+\frac{3}{4}\frac{{K}_{0}}{{R}^{3}}\frac{{\partial }^{2}}{\partial x\partial \alpha }v+\frac{5}{4}\frac{{K}_{6}}{{R}^{2}}\frac{{\partial }^{2}}{\partial x\partial \alpha }{\varphi }_{\alpha }-\frac{3}{4}\frac{{K}_{2}}{{R}^{4}}\frac{{\partial }^{3}}{\partial x\partial {\alpha }^{2}}w-{C}_{110}\frac{{\partial }^{2}}{\partial {x}^{2}}u-{C}_{111}\frac{{\partial }^{2}}{\partial {x}^{2}}{\varphi }_{x}+{C}_{112}\frac{{\partial }^{3}}{\partial {x}^{3}}w-\frac{{C}_{120}}{R}\frac{{\partial }^{2}}{\partial x\partial \alpha }v-\frac{{C}_{121}}{R}\frac{{\partial }^{2}}{\partial x\partial \alpha }{\varphi }_{\alpha }+\frac{{C}_{122}}{{R}^{2}}\frac{{\partial }^{3}}{\partial x\partial {\alpha }^{2}}w-\frac{{C}_{120}}{R}\frac{\partial }{\partial x}w-\frac{{C}_{660}}{R}\frac{{\partial }^{2}}{\partial x\partial \alpha }v-\frac{{C}_{661}}{R}\frac{{\partial }^{2}}{\partial x\partial \alpha }{\varphi }_{\alpha }+2\frac{{C}_{662}}{{R}^{2}}\frac{{\partial }^{3}}{\partial x\partial {\alpha }^{2}}w-\frac{{C}_{660}}{{R}^{2}}\frac{{\partial }^{2}}{\partial {\alpha }^{2}}u-\frac{{C}_{661}}{{R}^{2}}\frac{{\partial }^{2}}{\partial {\alpha }^{2}}{\varphi }_{x}=0$$
$$\delta v:-\frac{1}{4}\frac{{K}_{15}}{{R}^{2}}{\varphi }_{\alpha }+\frac{{K}_{16}}{{R}^{3}}\frac{\partial }{\partial \theta }w-\frac{{K}_{6}}{{R}^{3}}{\varphi }_{\alpha }+\frac{{K}_{7}}{{R}^{4}}\frac{\partial }{\partial \theta }w+\frac{{K}_{1}}{{R}^{4}}{\varphi }_{\alpha }-\frac{{K}_{2}}{{R}^{5}}\frac{\partial }{\partial \theta }w-\frac{{K}_{0}}{{R}^{4}}\frac{\partial }{\partial \theta }w+\frac{{K}_{0}v}{{R}^{4}}-\frac{{K}_{15}}{R}\frac{{\partial }^{2}}{\partial x\partial \theta }{\varphi }_{x}+\frac{{K}_{1}}{{R}^{2}}\frac{{\partial }^{4}}{\partial {x}^{2}\partial {\theta }^{2}}{\varphi }_{\alpha }+\frac{{K}_{0}}{{R}^{2}}\frac{{\partial }^{4}}{\partial {x}^{2}\partial {\theta }^{2}}v+2\frac{{K}_{7}}{{R}^{2}}\frac{{\partial }^{3}}{\partial {x}^{2}\partial \theta }w-\frac{1}{4}\frac{{K}_{1}}{{R}^{3}}\frac{{\partial }^{4}}{\partial x\partial {\theta }^{3}}{\varphi }_{x}-\frac{{K}_{6}}{R}\frac{{\partial }^{2}}{\partial {x}^{2}}{\varphi }_{\alpha }-\frac{7}{4}\frac{{K}_{1}}{{R}^{2}}\frac{{\partial }^{2}}{\partial {x}^{2}}{\varphi }_{\alpha }-\frac{7}{4}\frac{{K}_{0}}{{R}^{2}}\frac{{\partial }^{2}}{\partial {x}^{2}}{v}_{0}+2\frac{{K}_{0}}{{R}^{2}}\frac{{\partial }^{3}}{\partial {x}^{2}\partial \theta }w+\frac{{K}_{2}}{{R}^{3}}\frac{{\partial }^{3}}{\partial {x}^{2}\partial \theta }w-\frac{{K}_{6}}{{R}^{2}}\frac{{\partial }^{2}}{\partial x\partial \theta }{\varphi }_{x}+\frac{3}{4}\frac{{K}_{1}}{{R}^{3}}\frac{{\partial }^{2}}{\partial x\partial \theta }{\varphi }_{x}+\frac{3}{4}\frac{{K}_{0}}{{R}^{3}}\frac{{\partial }^{2}}{\partial x\partial \theta }u-\frac{1}{4}\frac{{K}_{6}}{{R}^{3}}\frac{{\partial }^{2}}{\partial {\theta }^{2}}{\varphi }_{\alpha }-\frac{{K}_{1}}{{R}^{4}}\frac{{\partial }^{2}}{\partial {\theta }^{2}}{\varphi }_{\alpha }-\frac{{K}_{0}}{{R}^{3}}\frac{{\partial }^{4}}{\partial x\partial {\theta }^{3}}u+\frac{{K}_{2}}{{R}^{5}}\frac{{\partial }^{3}}{\partial {\theta }^{3}}w+\frac{{K}_{7}}{{R}^{4}}\frac{{\partial }^{3}}{\partial {\theta }^{3}}w+\frac{{K}_{0}}{{R}^{4}}\frac{{\partial }^{3}}{\partial {\theta }^{3}}w-\frac{{K}_{0}}{{R}^{4}}\frac{{\partial }^{2}}{\partial {\theta }^{2}}v+\frac{{K}_{16}}{R}\frac{{\partial }^{3}}{\partial {x}^{2}\partial \theta }w-\frac{{C}_{120}}{R}\frac{{\partial }^{2}}{\partial x\partial \theta }u-\frac{{C}_{121}}{R}\frac{{\partial }^{2}}{\partial x\partial \theta }{\varphi }_{x}+\frac{{C}_{122}}{R}\frac{{\partial }^{3}}{\partial {x}^{2}\partial \theta }w-\frac{{C}_{220}}{{R}^{2}}\frac{{\partial }^{2}}{\partial {\theta }^{2}}v-\frac{{C}_{221}}{{R}^{2}}\frac{{\partial }^{2}}{\partial {\theta }^{2}}{\varphi }_{\alpha }+\frac{{C}_{222}}{{R}^{3}}\frac{{\partial }^{3}}{\partial {\theta }^{3}}w-\frac{{C}_{220}}{{R}^{2}}\frac{\partial }{\partial \theta }w-{C}_{660}\frac{{\partial }^{2}}{\partial {x}^{2}}v-{C}_{661}\frac{{\partial }^{2}}{\partial {x}^{2}}{\varphi }_{\alpha }+2\frac{{C}_{662}}{R}\frac{{\partial }^{3}}{\partial {x}^{2}\partial \theta }w-\frac{{C}_{660}}{R}\frac{{\partial }^{2}}{\partial x\partial \theta }u-\frac{{C}_{661}}{R}\frac{{\partial }^{2}}{\partial x\partial \theta }{\varphi }_{x}=0$$
$$\begin{aligned} & \delta w:\frac{1}{4}\Bigg(\frac{{K}_{12}}{{R}^{2}}\frac{\partial }{\partial x}{\varphi }_{x}+\left(\frac{{K}_{12}}{{R}^{3}}-\frac{{K}_{15}}{{R}^{2}}\right)\frac{\partial }{\partial \alpha }{\varphi }_{\alpha }+\left(\frac{{K}_{2}}{{R}^{5}}-\frac{{K}_{16}}{{R}^{3}}\right)\frac{\partial }{\partial \alpha }v-\frac{{K}_{6}}{{R}^{3}}\frac{\partial }{\partial \alpha }{\varphi }_{\alpha }+\left(\frac{{K}_{5}}{{R}^{5}}-\frac{{K}_{18}}{{R}^{3}}\right)\frac{\partial }{\partial \alpha }{\varphi }_{\alpha}\\ & \qquad +\left(-\frac{{K}_{19}}{{R}^{3}}+\frac{{K}_{1}}{{R}^{4}}\right)\frac{\partial }{\partial \alpha }{\varphi }_{\alpha }-\left(-\frac{{K}_{10}}{{R}^{4}}-\frac{{K}_{23}}{R}+\frac{{K}_{9}}{{R}^{4}}\right)\frac{\partial }{\partial \alpha }{\varphi }_{\alpha }-\frac{{K}_{7}}{{R}^{4}}\frac{\partial }{\partial \alpha }v+\frac{{K}_{15}}{R}\frac{\partial }{\partial x}{\varphi }_{x}+\frac{{K}_{0}}{{R}^{4}}\frac{\partial }{\partial \alpha }v\\ & \qquad +\left(\frac{{K}_{21}}{{R}^{2}}+\frac{{K}_{27}}{{R}^{2}}\right)\frac{\partial }{\partial \alpha }{\varphi }_{\alpha }+\left(\frac{{K}_{26}}{R}+\frac{{K}_{27}}{R}+\frac{{K}_{6}}{{R}^{2}}{K}_{23}\right)\frac{\partial }{\partial x}{\varphi }_{x}+\left({K}_{0}+{C}_{115}\right)\frac{{\partial }^{4}}{\partial {x}^{4}}w+\left(-{K}_{24}-2\frac{{K}_{7}}{{R}^{2}}\right)\frac{{\partial }^{2}}{\partial {x}^{2}}w\\ & \qquad +\left(-{K}_{6}+{K}_{12}\right)\frac{{\partial }^{3}}{\partial {x}^{3}}{\varphi }_{x}+\left(5\frac{{K}_{10}}{{R}^{3}}-3\frac{{K}_{12}}{{R}^{2}}-3\frac{{K}_{6}}{{R}^{2}}+4\frac{{K}_{1}}{{R}^{3}}\right)\frac{{\partial }^{3}}{\partial x\partial {\alpha }^{2}}{\varphi }_{x}+\left(-\frac{{K}_{24}}{{R}^{2}}-\frac{{K}_{14}}{{R}^{4}}\right)\frac{{\partial }^{2}}{\partial {\alpha }^{2}}w\\ & \qquad +\left(6\frac{{K}_{7}}{{R}^{3}}-5\frac{{K}_{2}}{{R}^{4}}\right)\frac{{\partial }^{3}}{\partial x\partial {\alpha }^{2}}+\left(-3\frac{{K}_{5}}{{R}^{3}}-2\frac{{K}_{18}}{R}-11\frac{{K}_{9}}{{R}^{2}}-\frac{{C}_{124}}{R}-2\frac{{C}_{664}}{R}-3\frac{{K}_{6}}{R}\right)\frac{{\partial }^{3}}{\partial {x}^{2}\partial \alpha }{\varphi }_{\alpha }\\ & \qquad -\left(5\frac{{K}_{5}}{{R}^{4}}+5\frac{{K}_{12}}{R}+8\frac{{K}_{18}}{{R}^{2}}+6\frac{{K}_{9}}{{R}^{3}}\right)\frac{{\partial }^{3}}{\partial x\partial {\alpha }^{2}}{\varphi }_{x}+\left(2\frac{{K}_{16}}{{R}^{3}}+2\frac{{K}_{20}}{{R}^{4}}+2\frac{{K}_{7}}{{R}^{4}}\right)\frac{{\partial }^{2}}{\partial {\alpha }^{2}}w+14\frac{{K}_{7}}{{R}^{2}}\frac{{\partial }^{4}}{\partial {x}^{2}\partial {\alpha }^{2}}w\\ & \qquad +\left(-\frac{{K}_{14}}{{R}^{2}}-2\frac{{K}_{16}}{R}-\frac{{K}_{0}}{{R}^{2}}-2\frac{{K}_{28}}{R}\right)\frac{{\partial }^{2}}{\partial {x}^{2}}w+\left(2\frac{{K}_{11}}{{R}^{5}}-2\frac{{K}_{28}}{{R}^{3}}\right)\frac{{\partial }^{2}}{\partial {\alpha }^{2}}w\\ & \qquad +\left(-7\frac{{K}_{0}}{{R}^{2}}-3\frac{{K}_{2}}{{R}^{3}}-2\frac{{K}_{16}}{R}-11\frac{{K}_{7}}{{R}^{2}}\right)\frac{{\partial }^{3}}{\partial {x}^{2}\partial \alpha }v+\left(-\frac{{K}_{5}}{{R}^{5}}-\frac{{K}_{10}}{{R}^{4}}-\frac{{K}_{6}}{{R}^{3}}-\frac{{K}_{1}}{{R}^{4}}-\frac{{K}_{9}}{{R}^{4}}-\frac{{K}_{12}}{{R}^{3}}\right)\frac{{\partial }^{3}}{\partial {\alpha }^{3}}{\varphi }_{\alpha }\\ & \qquad +\left(-\frac{{K}_{2}}{{R}^{5}}-\frac{{K}_{7}}{{R}^{4}}-\frac{{K}_{0}}{{R}^{4}}\right)\frac{{\partial }^{3}}{\partial {\alpha }^{3}}v+\left(2\frac{{K}_{16}}{{R}^{2}}+4\frac{{K}_{0}}{{R}^{3}}\right)\frac{{\partial }^{3}}{\partial x\partial {\alpha }^{2}}u+\left(2\frac{{K}_{11}}{{R}^{5}}+2\frac{{K}_{7}}{{R}^{4}}+\frac{{K}_{4}}{{R}^{6}}+\frac{{K}_{0}}{{R}^{4}}\right)\frac{{\partial }^{4}}{\partial {\alpha }^{4}}w\\ & \qquad +\left(-\frac{{K}_{0}}{{R}^{4}}-2\frac{{K}_{2}}{{R}^{5}}-\frac{{K}_{4}}{{R}^{6}}\right)\frac{{\partial }^{2}}{\partial {\alpha }^{2}}w+\left(\frac{{K}_{14}}{{R}^{4}}+2\frac{{K}_{2}}{{R}^{5}}+\frac{{C}_{225}}{{R}^{4}}\right)\frac{{\partial }^{4}}{\partial {\alpha }^{4}}w\\ & \qquad +\left(8\frac{{K}_{11}}{{R}^{3}}+6\frac{{K}_{2}}{{R}^{3}}+8\frac{{K}_{14}}{{R}^{2}}+6\frac{{K}_{0}}{{R}^{2}}+8\frac{{K}_{4}}{{R}^{4}}\right)\frac{{\partial }^{4}}{\partial {x}^{2}\partial {\alpha }^{2}}w\\ & \qquad +\left(-8\frac{{K}_{10}}{{R}^{2}}-7\frac{{K}_{1}}{{R}^{2}}\right)\frac{{\partial }^{3}}{\partial {x}^{2}\partial \alpha }{\varphi }_{\alpha }+\left(-{C}_{5511}+2{C}_{559}-{C}_{550}-2\frac{{C}_{122}}{R}-{H}_{2}+{N}_{x}^{HT}\right)\frac{{\partial }^{2}}{\partial {x}^{2}}w\\ & \qquad -{C}_{112}\frac{{\partial }^{3}}{\partial {x}^{3}}u-{C}_{114}\frac{{\partial }^{3}}{\partial {x}^{3}}{\psi }_{x}-\frac{{C}_{224}}{{R}^{3}}\frac{{\partial }^{3}}{\partial {\alpha }^{3}}{\varphi }_{\alpha }+\left(-\frac{{C}_{440}}{{R}^{2}}+2\frac{{C}_{449}}{{R}^{2}}-\frac{{C}_{4411}}{{R}^{2}}-2\frac{{C}_{222}}{{R}^{3}}\right)\frac{{\partial }^{2}}{\partial {\alpha }^{2}}w\\ & \qquad +\left(-2\frac{{C}_{664}}{{R}^{2}}-\frac{{C}_{124}}{{R}^{2}}\right)\frac{{\partial }^{3}}{\partial x\partial {\alpha }^{2}}{\varphi }_{x}+\left(-2\frac{{C}_{662}}{{R}^{2}}-\frac{{C}_{122}}{{R}^{2}}\right)\frac{{\partial }^{3}}{\partial x\partial {\alpha }^{2}}u+\left(-\frac{{C}_{122}}{R}-\frac{{C}_{222}}{{R}^{3}}-2\frac{{C}_{662}}{R}\right)\frac{{\partial }^{3}}{\partial {x}^{2}\partial \alpha }v\\ & \qquad +\left(2\frac{{C}_{125}}{{R}^{2}}+4\frac{{C}_{665}}{{R}^{2}}\right)\frac{{\partial }^{4}}{\partial {x}^{2}\partial {\alpha }^{2}}w+\left({C}_{5510}-{C}_{556}+\frac{{C}_{121}}{R}\right)\frac{\partial }{\partial x}{\varphi }_{x}+\left(\frac{{C}_{221}}{{R}^{2}}-\frac{{C}_{446}}{R}+\frac{{C}_{4410}}{R}\right)\frac{\partial }{\partial \alpha }{\varphi }_{\alpha }\\ & \qquad +\frac{{C}_{120}}{R}\frac{\partial }{\partial x}u+\frac{{C}_{220}}{{R}^{2}}\frac{\partial }{\partial \alpha }v-\frac{1}{{R}^{2}}({H}_{2}+{N}_{\alpha }^{HT})\frac{{\partial }^{2}w}{\partial {\alpha }^{2}}+\left({H}_{1}+\frac{{C}_{220}}{{R}^{2}}\right)w=0\end{aligned}$$
$$\delta {\varphi }_{x}:-2\frac{{K}_{3}}{{R}^{4}}\frac{{\partial }^{2}}{\partial {\alpha }^{2}}{\varphi }_{x}-2\frac{{K}_{1}}{{R}^{4}}\frac{{\partial }^{2}}{\partial {\alpha }^{2}}u-\frac{{K}_{1}}{{R}^{3}}\frac{{\partial }^{3}}{\partial x\partial {\alpha }^{2}}w-\frac{{K}_{6}}{{R}^{2}}\frac{{\partial }^{2}}{\partial x\partial \alpha }v-2\frac{{K}_{13}}{{R}^{2}}\frac{{\partial }^{2}}{\partial {\alpha }^{2}}{\varphi }_{x}+\left(\frac{3}{4}\frac{{K}_{3}}{{R}^{3}}-\frac{7}{4}\frac{{K}_{8}}{{R}^{2}}\right)\frac{{\partial }^{2}}{\partial x\partial \alpha }{\varphi }_{\alpha }-\frac{1}{2}\frac{{K}_{18}}{{R}^{2}}\frac{{\partial }^{3}}{\partial x\partial {\alpha }^{2}}w+\frac{9}{2}\frac{{K}_{8}}{{R}^{3}}\frac{{\partial }^{2}}{\partial {\alpha }^{2}}{\varphi }_{x}-\frac{1}{4}\frac{{K}_{3}}{{R}^{3}}\frac{{\partial }^{4}}{\partial x\partial {\alpha }^{3}}{\varphi }_{\alpha }-\frac{1}{4}\frac{{K}_{1}}{R}\frac{{\partial }^{4}}{\partial {x}^{3}\partial \alpha }v+\frac{1}{4}\frac{{K}_{3}}{{R}^{4}}\frac{{\partial }^{4}}{\partial {\alpha }^{4}}{\varphi }_{x}-\frac{1}{4}\frac{{K}_{1}}{{R}^{3}}\frac{{\partial }^{4}}{\partial x\partial {\alpha }^{3}}v+\frac{5}{4}\frac{{K}_{13}}{R}\frac{{\partial }^{2}}{\partial x\partial \alpha }{\varphi }_{\alpha }+\frac{3}{4}\frac{{K}_{12}}{{R}^{2}}\frac{{\partial }^{3}}{\partial x\partial {\alpha }^{2}}w-\frac{5}{4}\frac{{K}_{10}}{{R}^{3}}\frac{{\partial }^{3}}{\partial x\partial {\alpha }^{2}}w-\frac{3}{2}\frac{{K}_{9}}{{R}^{3}}\frac{{\partial }^{3}}{\partial x\partial {\alpha }^{2}}w+\frac{1}{4}\frac{{K}_{1}}{{R}^{2}}\frac{{\partial }^{4}}{\partial {x}^{2}\partial {\alpha }^{2}}u+\frac{3}{4}\frac{{K}_{1}}{{R}^{3}}\frac{{\partial }^{2}}{\partial x\partial \alpha }v-\frac{1}{4}\frac{{K}_{3}}{R}\frac{{\partial }^{4}}{\partial {x}^{3}\partial \alpha }{\varphi }_{\alpha }+\frac{9}{4}\frac{{K}_{6}}{{R}^{3}}\frac{{\partial }^{2}}{\partial {\alpha }^{2}}u+\frac{1}{4}\frac{{K}_{15}}{{R}^{2}}\frac{{\partial }^{2}}{\partial {\alpha }^{2}}u-\frac{1}{4}\frac{{K}_{15}}{R}\frac{{\partial }^{2}}{\partial x\partial \alpha }v+\frac{1}{4}\frac{{K}_{3}}{{R}^{2}}\frac{{\partial }^{4}}{\partial {x}^{2}\partial {\alpha }^{2}}{\varphi }_{x}+\frac{1}{2}\frac{{K}_{17}}{{R}^{2}}\frac{{\partial }^{2}}{\partial {\alpha }^{2}}{\varphi }_{x}+\frac{1}{4}\frac{{K}_{1}}{{R}^{4}}\frac{{\partial }^{4}}{\partial {\alpha }^{4}}u+\frac{3}{4}\frac{{K}_{6}}{{R}^{2}}\frac{{\partial }^{3}}{\partial x\partial {\alpha }^{2}}w+\frac{5}{4}\frac{{K}_{5}}{{R}^{4}}\frac{{\partial }^{3}}{\partial x\partial {\alpha }^{2}}w+\frac{1}{4}{K}_{6}\frac{{\partial }^{3}}{\partial {x}^{3}}w-\frac{1}{4}{K}_{13}\frac{{\partial }^{2}}{\partial {x}^{2}}{\varphi }_{x}+\frac{1}{4}{K}_{12}\frac{{\partial }^{3}}{\partial {x}^{3}}w+\frac{1}{4}({K}_{22}+\frac{{K}_{13}}{{R}^{2}}+2\frac{{K}_{25}}{R}){\varphi }_{x}-\frac{1}{4}\left(\frac{{K}_{27}}{R}+\frac{{K}_{12}}{{R}^{2}}+\frac{{K}_{6}}{{R}^{2}}+{K}_{23}+\frac{{K}_{15}}{R}+\frac{{K}_{26}}{R}\right)\frac{\partial }{\partial x}w+\left(-{C}_{111}-\frac{{C}_{661}}{{R}^{2}}\right)\frac{{\partial }^{2}}{\partial {x}^{2}}u-{C}_{113}\frac{{\partial }^{2}}{\partial {x}^{2}}{\varphi }_{x}+{C}_{114}\frac{{\partial }^{3}}{\partial {x}^{3}}w+\left(-\frac{{C}_{123}}{R}-\frac{{C}_{663}}{R}\right)\frac{{\partial }^{2}}{\partial x\partial \alpha }{\varphi }_{\alpha }+\left(\frac{{C}_{124}}{{R}^{2}}+2\frac{{C}_{664}}{{R}^{2}}\right)\frac{{\partial }^{3}}{\partial x\partial {\alpha }^{2}}w+\left(-\frac{{C}_{121}}{R}+{C}_{556}-{C}_{5510}\right)\frac{\partial }{\partial x}w+{C}_{558}{\varphi }_{x}+\left(-\frac{{C}_{661}}{R}-\frac{{C}_{121}}{R}\right)\frac{{\partial }^{2}}{\partial x\partial \alpha }v-\frac{{C}_{663}}{{R}^{2}}\frac{{\partial }^{2}}{\partial {\alpha }^{2}}{\varphi }_{x}=0$$
$$\delta {\varphi }_{\alpha }:\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$$
$$\frac{1}{4}\Bigg({K}_{22}{\varphi }_{\alpha }-\frac{{K}_{15}}{{R}^{2}}v-\frac{{K}_{27}}{{R}^{2}}\frac{\partial }{\partial \alpha }w-\frac{{K}_{6}}{{R}^{3}}v+\frac{{K}_{19}}{{R}^{3}}\frac{\partial }{\partial \alpha }w+\frac{{K}_{1}}{{R}^{4}}v+\frac{{K}_{17}}{R}{\varphi }_{\alpha }+\frac{{K}_{10}}{{R}^{4}}\frac{\partial }{\partial \alpha }w+\frac{{K}_{13}}{{R}^{2}}{\varphi }_{\alpha }-\frac{{K}_{21}}{{R}^{2}}\frac{\partial }{\partial \alpha }w-\frac{{K}_{8}}{{R}^{3}}{\varphi }_{\alpha }+\frac{{K}_{3}}{{R}^{4}}{\varphi }_{\alpha }-\frac{{K}_{1}}{{R}^{4}}\frac{\partial }{\partial \alpha }w-\frac{{K}_{12}}{{R}^{3}}\frac{\partial }{\partial \alpha }w+\frac{{K}_{9}}{{R}^{4}}\frac{\partial }{\partial \alpha }w+\frac{{K}_{6}}{{R}^{3}}\frac{\partial }{\partial \alpha }w+\frac{{K}_{18}}{{R}^{3}}\frac{\partial }{\partial \alpha }w-\frac{{K}_{5}}{{R}^{5}}\frac{\partial }{\partial \alpha }w \Bigg)-{K}_{15}\frac{{\partial }^{2}}{\partial {x}^{2}}v+\frac{1}{4}{K}_{1}\frac{{\partial }^{4}}{\partial {x}^{4}}v-\frac{1}{2}{K}_{17}\frac{{\partial }^{2}}{\partial {x}^{2}}{\varphi }_{\alpha }-2{K}_{13}\frac{{\partial }^{2}}{\partial {x}^{2}}{\varphi }_{\alpha }+\frac{1}{4}{K}_{3}\frac{{\partial }^{4}}{\partial {x}^{4}}{\varphi }_{\alpha }+\frac{1}{4}\frac{{K}_{15}}{{R}^{2}}\frac{\partial }{\partial \alpha }w-\frac{1}{2}\frac{{K}_{17}{\varphi }_{\alpha }}{{R}^{2}}-\frac{1}{4}\frac{{K}_{23}}{R}\frac{\partial }{\partial \alpha }w+\frac{1}{4}\frac{{K}_{25}{\varphi }_{\alpha }}{R}+\frac{1}{4}\frac{{K}_{3}}{{R}^{2}}\frac{{\partial }^{4}}{\partial {x}^{2}\partial {\alpha }^{2}}{\varphi }_{\alpha }+\frac{3}{4}\frac{{K}_{1}}{{R}^{3}}\frac{{\partial }^{2}}{\partial x\partial \alpha }u+\frac{5}{2}\frac{{K}_{8}}{R}\frac{{\partial }^{2}}{\partial {x}^{2}}{\varphi }_{\alpha }+\frac{3}{4}\frac{{K}_{12}}{R}\frac{{\partial }^{3}}{\partial {x}^{2}\partial \alpha }w-\frac{5}{4}\frac{{K}_{6}}{R}\frac{{\partial }^{2}}{\partial {x}^{2}}v-\frac{1}{4}\frac{{K}_{3}}{R}\frac{{\partial }^{4}}{\partial {x}^{3}\partial \alpha }{\varphi }_{x}+\frac{1}{4}\frac{{K}_{9}}{{R}^{4}}\frac{{\partial }^{3}}{\partial {\alpha }^{3}}w+\frac{9}{4}\frac{{K}_{9}}{{R}^{2}}\frac{{\partial }^{3}}{\partial {x}^{2}\partial \alpha }w-\frac{7}{4}\frac{{K}_{8}}{{R}^{2}}\frac{{\partial }^{2}}{\partial x\partial \alpha }{\varphi }_{x}+\frac{5}{4}\frac{{K}_{13}}{R}\frac{{\partial }^{2}}{\partial x\partial \alpha }{\varphi }_{x}+\frac{1}{4}\frac{{K}_{15}}{R}\frac{{\partial }^{2}}{\partial x\partial \alpha }u-\frac{1}{4}\frac{{K}_{1}}{{R}^{3}}\frac{{\partial }^{4}}{\partial x\partial {\alpha }^{3}}u-\frac{3}{4}\frac{{K}_{6}}{{R}^{2}}\frac{{\partial }^{2}}{\partial x\partial \alpha }u+\frac{1}{4}\frac{{K}_{6}}{{R}^{3}}\frac{{\partial }^{3}}{\partial {\alpha }^{3}}w+\frac{1}{2}\frac{{K}_{18}}{R}\frac{{\partial }^{3}}{\partial {x}^{2}\partial \alpha }w-\frac{3}{2}\frac{{K}_{3}}{{R}^{2}}\frac{{\partial }^{2}}{\partial {x}^{2}}{\varphi }_{\alpha }+\frac{7}{4}\frac{{K}_{1}}{{R}^{2}}\frac{{\partial }^{3}}{\partial {x}^{2}\partial \alpha }w+\frac{1}{4}\frac{{K}_{1}}{{R}^{4}}\frac{{\partial }^{3}}{\partial {\alpha }^{3}}w+\frac{1}{4}\frac{{K}_{5}}{{R}^{5}}\frac{{\partial }^{3}}{\partial {\alpha }^{3}}w+\frac{3}{4}\frac{{K}_{5}}{{R}^{3}}\frac{{\partial }^{3}}{\partial {x}^{2}\partial \alpha }w-\frac{1}{4}\frac{{K}_{6}}{{R}^{3}}\frac{{\partial }^{2}}{\partial {\alpha }^{2}}v-\frac{1}{4}\frac{{K}_{3}}{{R}^{4}}\frac{{\partial }^{2}}{\partial {\alpha }^{2}}{\varphi }_{\alpha }-\frac{1}{4}\frac{{K}_{1}}{{R}^{4}}\frac{{\partial }^{2}}{\partial {\alpha }^{2}}v+2\frac{{K}_{10}}{{R}^{2}}\frac{{\partial }^{3}}{\partial {x}^{2}\partial \alpha }w+\frac{1}{4}\frac{{K}_{12}}{{R}^{3}}\frac{{\partial }^{3}}{\partial {\alpha }^{3}}w-\frac{3}{2}\frac{{K}_{1}}{{R}^{2}}\frac{{\partial }^{2}}{\partial {x}^{2}}v+\frac{1}{4}\frac{{K}_{10}}{{R}^{4}}\frac{{\partial }^{3}}{\partial {\alpha }^{3}}w-\frac{1}{4}\frac{{K}_{3}}{{R}^{3}}\frac{{\partial }^{4}}{\partial x\partial {\alpha }^{3}}{\varphi }_{x}-\frac{1}{4}\frac{{K}_{1}}{R}\frac{{\partial }^{4}}{\partial {x}^{3}\partial \alpha }{u}_{0}-\frac{1}{4}\frac{{K}_{13}}{{R}^{2}}\frac{{\partial }^{2}}{\partial {\alpha }^{2}}{\varphi }_{\alpha }+\frac{1}{4}\frac{{K}_{1}}{{R}^{2}}\frac{{\partial }^{4}}{\partial {x}^{2}\partial {\alpha }^{2}}v+\frac{3}{4}\frac{{K}_{6}}{R}\frac{{\partial }^{3}}{\partial {x}^{2}\partial \alpha }w-\frac{1}{2}\frac{{K}_{8}}{{R}^{3}}\frac{{\partial }^{2}}{\partial {\alpha }^{2}}{\varphi }_{\alpha }+\frac{3}{4}\frac{{K}_{3}}{{R}^{3}}\frac{{\partial }^{2}}{\partial x\partial \alpha }{\varphi }_{x}-\frac{{C}_{121}}{R}\frac{{\partial }^{2}}{\partial x\partial \alpha }u-\frac{{C}_{123}}{R}\frac{{\partial }^{2}}{\partial x\partial \alpha }{\varphi }_{x}+\frac{{C}_{124}}{R}\frac{{\partial }^{3}}{\partial {x}^{2}\partial \alpha }w-\frac{{C}_{221}}{{R}^{2}}\frac{{\partial }^{2}}{\partial {\alpha }^{2}}v-\frac{{C}_{223}}{{R}^{2}}\frac{{\partial }^{2}}{\partial {\alpha }^{2}}{\varphi }_{\alpha }+\frac{{C}_{224}}{{R}^{3}}\frac{{\partial }^{3}}{\partial {\alpha }^{3}}w-\frac{{C}_{221}}{{R}^{2}}\frac{\partial }{\partial \alpha }w-{C}_{661}\frac{{\partial }^{2}}{\partial {x}^{2}}v-{C}_{663}\frac{{\partial }^{2}}{\partial {x}^{2}}{\varphi }_{\alpha }+2\frac{{C}_{664}}{R}\frac{{\partial }^{3}}{\partial {x}^{2}\partial \alpha }w-\frac{{C}_{661}}{R}\frac{{\partial }^{2}}{\partial x\partial \theta \alpha }u-\frac{{C}_{663}}{R}\frac{{\partial }^{2}}{\partial x\partial \alpha }{\varphi }_{x}+\frac{{C}_{446}}{R}\frac{\partial }{\partial \alpha }w+{C}_{448}{\varphi }_{\alpha }-\frac{{C}_{4410}}{R}\frac{\partial }{\partial \alpha }w=0$$

in which the used coefficients are:

$$j=\frac{{h}_{c}}{2}+{h}_{f},\frac{{h}_{c}}{2},-\frac{{h}_{c}}{2},\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}i=\frac{{h}_{c}}{2},-\frac{{h}_{c}}{2},-\frac{{h}_{c}}{2}-{h}_{f}$$
$${C}_{b}={\int }_{i}^{j}{Q}_{11}\left(z\right)(\text{\hspace{0.17em}}1,f(z),g(z),f{(z)}^{2},f(z)g(z),g{(z)}^{2}){\text{d}}z,\text{\hspace{0.17em}}\text{\hspace{0.17em}\hspace{0.17em}}b=110...115$$
$${C}_{b}={\int }_{i}^{j}{Q}_{12}\left(z\right)(\text{\hspace{0.17em}}1,f(z),g(z),f{(z)}^{2},f(z)g(z),g{(z)}^{2}){\text{d}}z,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}b=120...125$$
$${C}_{b}={\int }_{i}^{j}{Q}_{22}\left(z\right)(\text{\hspace{0.17em}}1,f(z),g(z),f{(z)}^{2},f(z)g(z),g{(z)}^{2}){\text{d}}z,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}b=220...225$$
$${C}_{b}={\int }_{i}^{j}{Q}_{44}\left(z\right)(\text{\hspace{0.17em}}1,f(z),g(z),f{(z)}^{2},f(z)g(z),g{(z)}^{2},{f}{\prime}(z),f(z){f}{\prime}(z),{{f}{\prime}}{\prime}(z),{g}{\prime}(z)\text{\hspace{0.17em}}){\text{d}}z,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}b=440...449$$
$${C}_{b}={\int }_{i}^{j}{Q}_{44}\left(z\right)({f}{\prime}(z){g}{\prime}(z),{g}{{\prime\prime} }(z),f(z){g}{\prime}(z),{f}{\prime}(z)g(z),g(z){g}{\prime}(z))\mathrm{d}z,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}b=4411...44415$$
$${C}_{550},{C}_{556},{C}_{558},{C}_{559},{C}_{5510},{C}_{5511}=\text{\hspace{0.17em}}{\int }_{i}^{j}{Q}_{55}\left(z\right)(\text{\hspace{0.17em}}1,{f}{\mathrm{{\prime}}}(z),{{f}{\mathrm{{\prime}}}}{\mathrm{{\prime}}}(z),{g}{\mathrm{{\prime}}}(z),{f}{\mathrm{{\prime}}}(z){g}{\mathrm{{\prime}}}(z),{g}{{\prime\prime} }(z)){\text{d}}z,$$
$${C}_{660},{C}_{661},{C}_{662},{C}_{663},{C}_{664},{C}_{665}={\int }_{i}^{j}{Q}_{66}\left(z\right)(\text{\hspace{0.17em}}1,f(z),g(z),f{(z)}^{2},f(z)g(z),g{(z)}^{2}){\text{d}}z,$$
$${K}_{b}={\int }_{i}^{j}{{l}_{m}}^{2}\mu (z)(\text{\hspace{0.17em}}1,f(z),g(z),f{(z)}^{2},g{(z)}^{2},f(z)g(z),{f}{\prime}(z),{g}{\prime}(z),f(z){f}{\prime}(z),$$
$${g}{\prime}\left(z\right)f\left(z\right),g\left(z\right){f}{\prime}\left(z\right),{g}{\prime}\left(z\right)g\left(z\right),{f}{\prime}\left(z\right){g}{\prime}\left(z\right),{f}{\prime}{(}^{z},{g}{\prime}{(}^{z},{f}{{\prime\prime} }\left(z\right),{g}{{\prime\prime} }\left(z\right),$$
$$f\left(z\right){f}{{\prime\prime} }\left(z\right),{g}{{\prime\prime} }\left(z\right)f\left(z\right),g\left(z\right){f}{{\prime\prime} }\left(z\right),{g}{{\prime\prime} }\left(z\right)g\left(z\right)\text{\hspace{0.17em}},{g}{{\prime\prime} }\left(z\right){f}{\prime}\left(z\right),{f}{{\prime\prime} }{(}^{z},{f}{{\prime\prime} }\left(z\right){g}{{\prime\prime} }\left(z\right),$$
$${g}{{\prime\prime} }{(}^{z},{f}{{\prime\prime} }(z){f}{\prime}(z),\text{\hspace{0.17em}}{g}{{\prime\prime} }(z){f}{\prime}(z),{g}{{\prime\prime} }(z){f}{\prime}(z),\text{\hspace{0.17em}}{g}{{\prime\prime} }(z){g}{\prime}(z)){\text{d}}z,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}b=0...28$$

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Ying, YF., Zhao, LC. & Kumar, A. Higher-order buckling analysis of FG porous cylindrical micro-shells integrated with GPLs-RC patches in hygrothermal environment immersed on Kerr foundation. Acta Mech 235, 1785–1802 (2024). https://doi.org/10.1007/s00707-023-03809-7

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