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Analytical solution for the micropolar cylindrical shell: Carrera unified formulation (CUF) approach

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Abstract

Here, an analytical form Navier solution for higher-order micropolar theory of cylindrical shell was developed based on the CUF approach. The cases of complete linear expansion and model based on Timoshenko-Mindlin shear deformation hypothesis are considered in detail. The two-dimensional system of differential equations obtained here using the principle of virtual displacements for the theory of micropolar elastic cylindrical shell of a higher order is solved here for the case of freely supported cylindrical shell using the Navier variable separation method. Some numerical examples were performed and the influence of the rotational field, as well as the micropolar couple stress on the stress–strain state, was analyzed. The equations presented here can be used for the stress–strain calculation and for thin-walled structures modeling in macro-, micro-, and nanoscale, with considering effects of micropolar couple stress and rotation.

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Acknowledgements

This work was supported by the visiting professor grants provided by Politecnico di Torino Research Excellence 2021 and the Committee of Science and Technology of Mexico (Ciencia Basica, Ref. No 256458), which are gratefully acknowledged.

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Authors and Affiliations

  1. Department of Aeronautics and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129, Torino, Italy

    E. Carrera & V. V. Zozulya

  2. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Nesterov Str. 3, Kyiv, 252680, Ukraine

    V. V. Zozulya

  3. Department of Mechanical Engineering, College of Engineering, Prince Mohammad Bin Fahd University, Al Khobar, Kingdom of Saudi Arabia

    E. Carrera

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  1. E. Carrera
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Correspondence to V. V. Zozulya.

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Communicated by Andreas Öchsner.

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Appendices

Appendix 1. Coefficients of the fundamental nucleus matrixes \({\mathbf{K}}_{\tau ,s}^{n,m,loc} \)

$$\begin{aligned} K^{n,m,\tau ,s}_{u_{x} ,u_{x} }= & {} C_{55} J_{\tau _{r} ,s_{r} }^{u_{x} ,u_{x} } +C_{11} J_{\tau ,s}^{u_{x} ,u_{x} } \frac{\pi ^{2}n^{2}}{L^{2} }+C_{44} J_{\tau ,s}^{u_{x} ,u_{x} } \frac{\pi ^{2}m^{2}}{R^{2}\varphi _{0}^{2} },\\ K^{n,m,\tau ,s}_{u_{x} ,u_{\varphi } }= & {} \left( {C_{12} +C_{44}^{T} } \right) \frac{\pi ^{2}nm}{LR\varphi _{0} }J_{\tau ,s}^{u_{x} ,u_{\varphi } }, \\ K^{n,m,\tau ,s}_{u_{x} ,u_{r} }= & {} \left( {C_{55}^{T} J_{\tau _{r} ,s}^{u_{x} ,u_{r} } -J_{\tau ,s}^{u_{x} ,u_{r} } C_{12} /R-C_{13} J_{\tau ,s_{r} }^{u_{x} ,u_{r} } } \right) \frac{\pi n}{L}, K^{n,m,\tau ,s}_{u_{x} ,\omega _{x} } =0,\\ K^{n,m,\tau ,s}_{u_{x} ,\omega _{y} }= & {} \left( {C_{55}^{T} -C_{55} } \right) J_{\tau _{z} ,s}^{u_{x} ,\omega _{y} } , K^{n,m,\tau ,s}_{u_{x} ,\omega _{z} } =\left( {C_{44} -C_{44}^{T} } \right) J_{\tau ,s}^{u_{x} ,\omega _{z} } \frac{\pi m}{R\varphi _{0} }, \\ K^{n,m,\tau ,s}_{u_{\varphi } ,u_{x} }= & {} \left( {C_{12} +C_{44}^{T} } \right) J_{\tau ,s}^{u_{\varphi } ,u_{x} } \frac{\pi ^{2}nm}{LR\varphi _{0} },\\ K^{n,m,\tau ,s}_{u_{\varphi } ,u_{\varphi } }= & {} \frac{J_{\tau _{r} ,s_{r} }^{u_{\varphi } ,u_{\varphi } } C_{66} -\left( {J_{\tau ,s_{r} }^{u_{\varphi } ,u_{\varphi } } +J_{\tau _{r} ,s}^{u_{\varphi } ,u_{\varphi } } } \right) C_{66}^{T} }{R^{2}}+C_{44} J_{\tau ,s}^{u_{\varphi } ,u_{\varphi } } \frac{\pi ^{2}n^{2}}{L^{2} }+C_{22} J_{\tau ,s}^{u_{\varphi } ,u_{\varphi } } \frac{\pi ^{2}m^{2}}{R^{2}\varphi _{0}^{2} },\\ K^{n,m,\tau ,s}_{u_{\varphi } ,u_{r} }= & {} -\frac{\pi m\left( {J_{\tau ,s}^{u_{\varphi } ,u_{r} } C_{22} +J_{\tau ,s_{r} }^{u_{\varphi } ,u_{r} } RC_{23} +J_{\tau _{r} ,s_{r} }^{u_{\varphi } ,u_{r} } C_{66} -J_{\tau _{r} ,s}^{u_{\varphi } ,u_{r} } C_{66}^{T} } \right) }{R^{2}\varphi _{0} }, K^{n,m,\tau ,s}_{u_{\varphi } ,\omega _{\varphi } } =0,\\ K^{n,m,\tau ,s}_{u_{\varphi } ,\omega _{x} }= & {} \left( {C_{66} -C_{66}^{T} } \right) \left( {J_{\tau _{r} ,s}^{u_{\varphi } ,\omega _{x} } +J_{\tau ,s}^{u_{\varphi } ,\omega _{x} } /R} \right) , K^{n,m,\tau ,s}_{u_{\varphi } ,\omega _{r} } =\left( {C_{44}^{T} -C_{44} } \right) J_{\tau ,s}^{u_{\varphi } ,\omega _{r} } \frac{\pi n}{L}, \\ K^{n,m,\tau ,s}_{u_{r} ,u_{x} }= & {} \left( {C_{55}^{T} J_{\tau ,s_{r} }^{u_{r} ,u_{x} } -C_{13} J_{\tau _{r} ,s}^{u_{r} ,u_{x} } +C_{12} J_{\tau ,s}^{u_{r} ,u_{x} } /R} \right) \frac{\pi n}{L},\\ K^{n,m,\tau ,s}_{u_{r} ,u_{\varphi } }= & {} \left( {C_{66}^{T} J_{\tau ,s_{r} }^{u_{r} ,u_{\varphi } } -C_{23} J_{\tau _{r} ,s}^{u_{r} ,u_{\varphi } } -C_{22} J_{\tau ,s}^{u_{r} ,u_{\varphi } } /R} \right) \frac{\pi m}{R\varphi _{0} },\\ K^{n,m,\tau ,s}_{u_{r} ,u_{r} }= & {} \frac{C_{22} J_{\tau ,s}^{u_{r} ,u_{r} } }{R^{2}}+\frac{\left( {J_{\tau _{r} ,s}^{u_{r} ,u_{r} } +J_{\tau ,s_{r} }^{u_{r} ,u_{r} } } \right) C_{23} }{R}+C_{33} J_{\tau _{r} ,s_{r} }^{u_{r} ,u_{r} } +C_{55} J_{\tau ,s}^{u_{r} ,u_{r} } \frac{\pi ^{2}n^{2}}{L^{2} }+C_{66} J_{\tau ,s}^{u_{r} ,u_{r} }\\ \frac{\pi ^{2}m^{2}}{R^{2}\varphi _{0}^{2} }, K^{n,m,\tau ,s}_{u_{r} ,\omega _{r} } =0, \\ K^{n,m,\tau ,s}_{u_{r} ,\omega _{x} }= & {} \left( {C_{66} -C_{66}^{T} } \right) J_{\tau ,s}^{u_{r} ,\omega _{x} } \frac{\pi m}{R\varphi _{0} }, K^{n,m,\tau ,s}_{u_{r} ,\omega _{\varphi } } =\left( {C_{55} -C_{55}^{T} } \right) J_{\tau ,s}^{u_{r} ,\omega _{\varphi } } \frac{\pi n}{L}, \\ K^{n,m,\tau ,s}_{\omega _{x} ,u_{x} } = 0, K^{n,m,\tau ,s}_{\omega _{x} ,u_{\varphi } }= & {} \left( {C_{66} -C_{66}^{T} } \right) \left( {J_{\tau ,s_{r} }^{\omega _{x} ,u_{\varphi } } +J_{\tau ,s}^{\omega _{x} ,u_{\varphi } } /R} \right) \quad L^{n,m,\tau ,s}_{\omega _{x} ,u_{r} } =\left( {C_{66}^{T} -C_{66} } \right) J_{\tau ,s}^{\omega _{x} ,u_{z} } \frac{\pi m}{R\varphi _{0} }, \\ K^{n,\tau ,s}_{\omega _{x} ,\omega _{x} }= & {} A_{55} J_{\tau _{r} ,s_{r} }^{\omega _{x} ,\omega _{x} } +J_{\tau ,s}^{\omega _{x} ,\omega _{x} } \left( {2\left( {C_{66} -C_{66}^{T} } \right) +\pi ^{2}\left( {\frac{n^{2}A_{11} }{L^{2} }+\frac{m^{2}A_{44} }{R^{2}\varphi _{0}^{2} }} \right) } \right) ,\\ K^{n,m,\tau ,s}_{\omega _{x} ,\omega _{\varphi } }= & {} \left( {A_{11} +A_{44}^{T} } \right) J_{\tau ,s}^{\omega _{x} ,\omega _{\varphi } } \frac{\pi ^{2}nm}{LR\varphi _{0} },\\ K^{n,m,\tau ,s}_{\omega _{x} ,\omega _{r} }= & {} \left( {A_{12} J_{\tau _{x} ,s}^{\omega _{x} ,\omega _{r} } /R+A_{13} J_{\tau _{x} ,s_{r} }^{\omega _{x} ,\omega _{r} } -C_{55}^{T} J_{\tau _{r} ,s_{x} }^{\omega _{x} ,\omega _{r} } } \right) \frac{\pi n}{L}, \\ K^{n,m,\tau ,s}_{\omega _{\varphi } ,u_{x} }= & {} \left( {C_{55}^{T} -C_{55} } \right) J_{\tau ,s_{r} }^{\omega _{\varphi } ,u_{x} } , K^{n,m,\tau ,s}_{\omega _{\varphi } ,u_{\varphi } } =0, \quad K^{n,m,\tau ,s}_{\omega _{\varphi } ,u_{r} } =\left( {C_{55}^{T} -C_{55} } \right) J_{\tau ,s}^{\omega _{\varphi } ,u_{r} } \frac{\pi n}{L},\\ K^{n,m,\tau ,s}_{\omega _{\varphi } ,\omega _{x} }= & {} \left( {A_{12} +A_{44}^{T} } \right) J_{\tau ,s}^{\omega _{\varphi } ,\omega _{x} } \frac{\pi ^{2}nm}{LR\varphi _{0} },\\ K^{n,m,\tau ,s}_{\omega _{\varphi } ,\omega _{r} }= & {} \left( {A_{23} J_{\tau ,s_{r} }^{\omega _{\varphi } ,\omega _{r} } +A_{66} J_{\tau ,s}^{\omega _{\varphi } ,\omega _{r} } /R-A_{66}^{T} J_{\tau _{r} ,s}^{\omega _{\kappa } ,\omega _{r} } } \right) \frac{\pi m}{R\varphi _{0} }, \\ K^{n,m,\tau ,s}_{\omega _{\varphi } ,\omega _{\varphi } }= & {} \left( {\frac{J_{\tau ,s}^{\omega _{\varphi } ,\omega _{\varphi } } }{R^{2}}+J_{\tau _{r} ,s_{r} }^{\omega _{\varphi } ,\omega _{\varphi } } } \right) A_{66} -\frac{\left( {J_{\tau ,s_{r} }^{\omega _{\varphi } ,\omega _{\varphi } } +J_{\tau _{r} ,s}^{\omega _{\varphi } ,\omega _{\varphi } } } \right) A_{66}^{T} }{R}\\&\quad +\, J_{\tau ,s}^{\omega _{\varphi } ,\omega _{\varphi } } \left( {2C_{55} -2C_{55}^{T} +\pi ^{2}\left( {\frac{n^{2}A_{44} }{L^{2} }+\frac{m^{2}A_{22} }{R^{2}\varphi _{0}^{2} }} \right) } \right) , \\ K^{n,m,\tau ,s}_{\omega _{r} ,u_{x} }= & {} \left( {C_{44}^{T} -C_{44} } \right) J_{\tau ,s}^{\omega _{r} ,u_{x} } \frac{\pi m}{R\varphi _{0} }, K^{n,m,\tau ,s}_{\omega _{r} ,u_{\varphi } } =\left( {C_{44} -C_{44}^{T} } \right) J_{\tau ,s}^{\omega _{r} ,u_{\varphi } } \frac{\pi n}{L}, K^{n,m,\tau ,s}_{\omega _{r} ,u_{r} } =0,\\ K^{n,m,\tau ,s}_{\omega _{r} ,\omega _{x} }= & {} \left( {A_{12} J_{\tau ,s}^{\omega _{r} ,\omega _{x} } /R+A_{13} J_{\tau _{r} ,s}^{\omega _{r} ,\omega _{x} } +A_{55}^{T} J_{\tau ,s_{r} }^{\omega _{r} ,\omega _{x} } } \right) \frac{\pi n}{L},\\ K^{n,m,\tau ,s}_{\omega _{r} ,\omega _{\varphi } }= & {} \left( {A_{22} J_{\tau ,s}^{\omega _{r} ,\omega _{\varphi } } /R+A_{23} J_{\tau _{r} ,s}^{\omega _{r} ,\omega _{\varphi } } -A_{66}^{T} J_{\tau ,s_{r} }^{\omega _{r} ,\omega _{\varphi } } } \right) \frac{\pi m}{\varphi _{0} }, \\ K^{n,m,\tau ,s}_{\omega _{r} ,\omega _{r} }= & {} \frac{A_{22} J_{\tau ,s}^{\omega _{r} ,\omega _{r} } }{R^{2}}+\frac{\left( {J_{\tau _{r} ,s}^{\omega _{r} ,\omega _{r} } +J_{\tau ,s_{r} }^{\omega _{r} ,\omega _{r} } } \right) A_{23} }{R}+A_{33} J_{\tau _{r} ,s_{r} }^{\omega _{r} ,\omega _{r} } +2\left( {C_{44} -C_{44}^{T} } \right) J_{\tau ,s}^{\omega _{r} ,\omega _{r} } \\&\quad +A_{55} J_{\tau ,s}^{\omega _{r} ,\omega _{r} } \frac{\pi ^{2}n^{2}}{L^{2} }+A_{66} J_{\tau ,s}^{\omega _{r} ,\omega _{r} } \frac{\pi ^{2}m^{2}}{R^{2}\varphi _{0}^{2} }. \end{aligned}$$

Appendix 2. Coefficients of the fundamental nucleus matrixes \({\mathbf{K}}_{\tau ,s}^{n,m,loc} \)for CLEC.

For \(\tau =1\) and \(s=1\)

$$\begin{aligned} K^{n,m,1,1}_{u_{x} ,u_{x} }= & {} 2h\pi ^{2}\left( {\frac{n^{2}C_{11} }{L^{2} }+\frac{m^{2}C_{44} }{R^{2}\varphi _{0}^{2} }} \right) , K^{n,m,1,1}_{u_{x} ,u_{\varphi } } = \frac{2hmn\pi ^{2}\left( {C_{12} +C_{44}^{T} } \right) }{LR\varphi _{0} }, K^{n,m,1,1}_{u_{x} ,u_{r} } =-\frac{2hn\pi C_{12} }{RL},\\ K^{n,m,1,1}_{u_{x} ,\omega _{x} }= & {} 0, K^{n,m,1,1}_{u_{x} ,\omega _{y} } =2h\left( {C_{55}^{T} -C_{55} } \right) , K^{n,m,1,1}_{u_{x} ,\omega _{z} } =\frac{2hm\pi \left( {C_{44} -C_{44}^{T} } \right) }{R\varphi _{0} },\\ K^{n,m,1,1}_{u_{\varphi } ,u_{x} }= & {} \frac{2hmn\pi ^{2}\left( {C_{12} +C_{44}^{T} } \right) }{LR\varphi _{0} }\left( {C_{12} +C_{44}^{T} } \right) , K^{n,m,1,1}_{u_{\varphi } ,u_{\varphi } } =\frac{2hC_{66} }{R^{2}}+2h\pi ^{2}\left( {\frac{n^{2}C_{44} }{L^{2} }+\frac{m^{2}C_{22} }{R^{2}\varphi _{0}^{2} }} \right) , \\ K^{n,m,1,1}_{u_{\varphi } ,\omega _{\varphi } }= & {} 0, K^{n,m,1,1}_{u_{\varphi } ,u_{r} } =-\frac{m\pi \left( {2hC_{22} +2hC_{66} } \right) }{R^{2}\varphi _{0} }, K^{n,m,1,1}_{u_{\varphi } ,\omega _{x} } =-\frac{2h\left( {-C_{66} +C_{66}^{T} } \right) }{R},\\ K^{n,m,1,1}_{u_{\varphi } ,\omega _{r} }= & {} \frac{2hn\pi \left( {C_{44}^{T} -C_{44} } \right) }{L}, K^{n,m,1,1}_{u_{r} ,u_{x} } =-\frac{2hn\pi C_{12} }{RL}, K^{n,m,1,1}_{u_{r} ,u_{\varphi } } =-\frac{2hm\pi \left( {C_{22} +C_{66} } \right) }{R^{2}\varphi _{0} },\\ K^{n,m,1,1}_{u_{r} ,u_{r} }= & {} \frac{2hC_{22} }{R^{2}}+\frac{2hn^{2}\pi ^{2}C_{55} }{L^{2} }+\frac{2hm^{2}\pi ^{2}C_{66} }{R^{2}\varphi _{0}^{2} },\\ K^{n,m,1,1}_{u_{r} ,\omega _{x} }= & {} \frac{2hm\pi \left( {C_{66} -C_{66}^{T} } \right) }{R\varphi _{0} }J_{\tau ,s}^{u_{r} ,\omega _{x} } , K^{n,m,1,1}_{u_{r} ,\omega _{\varphi } } {=}\frac{2hn\pi \left( {C_{55} -C_{55}^{T} } \right) }{L}, K^{n,m,1,1}_{u_{r} ,\omega _{r} } =0, \\ K^{n,m,1,1}_{\omega _{x} ,u_{x} }= & {} 0, K^{n,m,1,1}_{\omega _{x} ,u_{\varphi } } =\frac{2h\left( {C_{66} -C_{66}^{T} } \right) }{R}, \quad L^{n,m,1,1}_{\omega _{x} ,u_{r} } {=}\frac{2hm\pi \left( {C_{66}^{T} -C_{66} } \right) }{R\varphi _{0} }, \\ K^{n,m,1,1}_{\omega _{x} ,\omega _{x} }= & {} 2h\left( {2\left( {C_{66} -C_{66}^{T} } \right) +\pi ^{2}\left( {\frac{n^{2}A_{11} }{L^{2} }+\frac{m^{2}A_{44} }{R^{2}\varphi _{0}^{2} }} \right) } \right) , K^{n,m,1,1}_{\omega _{x} ,\omega _{\varphi } } =\frac{2hmn\pi ^{2}\left( {A_{12} +A_{44}^{T} } \right) }{LR\varphi _{0} },\\ K^{n,m,1,1}_{\omega _{x} ,\omega _{r} }= & {} \frac{2hn\pi A_{12} }{RL}, K^{n,m,1,1}_{\omega _{\varphi } ,u_{x} } =0,\\ K^{n,m,1,1}_{\omega _{\varphi } ,u_{\varphi } }= & {} 0, \quad K^{n,m,1,1}_{\omega _{\varphi } ,u_{r} } =\frac{2hn\pi \left( {C_{55}^{T} -C_{55} } \right) }{L}, K^{n,m,1,1}_{\omega _{\varphi } ,\omega _{x} } =\frac{2hmn\pi ^{2}\left( {A_{12} +A_{44}^{T} } \right) }{LR\varphi _{0} },\\ K^{n,m,1,1}_{\omega _{\varphi } ,\omega _{\varphi } }= & {} \frac{2hA_{66} }{R^{2}}+2h\left( {2C_{55} -2C_{55}^{T} +\pi ^{2}\left( {\frac{n^{2}A_{44} }{L^{2} }+\frac{m^{2}A_{22} }{R^{2}\varphi _{0}^{2} }} \right) } \right) , K^{n,m,1,1}_{\omega _{\varphi } ,\omega _{r} } =\frac{2hm\pi A_{66} }{R^{2}\varphi _{0} }, \\ K^{n,m,1,1}_{\omega _{r} ,u_{x} }= & {} \frac{2hm\pi \left( {C_{44}^{T} -C_{44} } \right) }{R\varphi _{0} }, K^{n,m,1,1}_{\omega _{r} ,u_{\varphi } } =\frac{2hn\pi \left( {C_{44} -C_{44}^{T} } \right) }{L}, K^{n,m,1,1}_{\omega _{r} ,u_{r} } =0, K^{n,m,1,1}_{\omega _{r} ,\omega _{x} } =\frac{2hn\pi A_{12} }{RL},\\ K^{n,m,1,1}_{\omega _{r} ,\omega _{\varphi } }= & {} \frac{2hn\pi A_{12} }{RL}, K^{n,m,1,1}_{\omega _{r} ,\omega _{r} } =\frac{2hA_{22} }{R^{2}}+4h\left( {C_{44} -C_{44}^{T} } \right) +\frac{2hn^{2}\pi ^{2}A_{55} }{L^{2} }+\frac{2hm^{2}\pi ^{2}A_{66} }{R^{2}\varphi _{0}^{2} }. \end{aligned}$$

For \(\tau =1\) and \(s=2\)

$$\begin{aligned} K^{n,m,1,2}_{u_{x} ,u_{x} }= & {} 0, K^{n,m,1,2}_{u_{x} ,u_{\varphi } } =0, K^{n,m,1,2}_{u_{x} ,u_{r} } =-\frac{2hn\pi C_{13} }{L}, K^{n,m,1,2}_{u_{x} ,\omega _{x} } =0, K^{n,m,1,2}_{u_{x} ,\omega _{y} } =0, K^{n,m,1,2}_{u_{x} ,\omega _{z} } =0, \\ K^{n,m,1,2}_{u_{\varphi } ,u_{x} }= & {} 0, K^{n,m,1,2}_{u_{\varphi } ,u_{\varphi } } =-\frac{2hC_{66}^{T} }{R}, K^{n,m,1,2}_{u_{\varphi } ,u_{r} } =-\frac{2hm\pi C_{23} }{R\varphi _{0} }, K^{n,m,1,2}_{u_{\varphi } ,\omega _{x} } =0, K^{n,m,1,2}_{u_{\varphi } ,\omega _{\varphi } } =0, K^{n,m,1,2}_{u_{\varphi } ,\omega _{r} } =0, \\ K^{n,m,1,2}_{u_{r} ,u_{x} }= & {} \frac{2hn\pi C_{55}^{T} }{L}, K^{n,m,1,2}_{u_{r} ,u_{\varphi } } =\frac{2hm\pi C_{66}^{T} }{R\varphi _{0} }, K^{n,m,1,2}_{u_{r} ,u_{r} } =\frac{2hC_{23} }{R}, K^{n,m,1,2}_{u_{r} ,\omega _{r} } =0, K^{n,m,1,2}_{u_{r} ,\omega _{x} } =0, K^{n,m,1,2}_{u_{r} ,\omega _{\varphi } } =0, \\ K^{n,m,1,2}_{\omega _{x} ,u_{x} }= & {} 0, K^{n,m,1,2}_{\omega _{x} ,u_{\varphi } } =2h\left( {C_{66} -C_{66}^{T} } \right) , \quad L^{n,m,1,2}_{\omega _{x} ,u_{r} } =0, K^{n,1,2}_{\omega _{x} ,\omega _{x} } =0, K^{n,m,0,1}_{\omega _{x} ,\omega _{\varphi } } =0, K^{n,m,1,2}_{\omega _{x} ,\omega _{r} } =\frac{2hn\pi A_{13} }{L}, \\ K^{n,m,1,2}_{\omega _{\varphi } ,u_{x} }= & {} 2h\left( {C_{55}^{T} -C_{55} } \right) , K^{n,m,1,2}_{\omega _{\varphi } ,u_{\varphi } } =0, \quad K^{n,m,1,2}_{\omega _{\varphi } ,u_{r} } =0, K^{n,m,1,2}_{\omega _{\varphi } ,\omega _{x} } =0, K^{n,m,1,2}_{\omega _{\varphi } ,\omega _{\varphi } } =\frac{2hm\pi A_{23} }{R\varphi _{0} }, K^{n,m,1,2}_{\omega _{\varphi } ,\omega _{r} } =-\frac{2hA_{66}^{T} }{R}, \\ K^{n,m,1,2}_{\omega _{r} ,u_{x} }= & {} 0, K^{n,m,0,1}_{\omega _{r} ,u_{\varphi } } =0, K^{n,m,1,2}_{\omega _{r} ,u_{r} } =0, K^{n,m,1,2}_{\omega _{r} ,\omega _{x} } =-\frac{2hn\pi A_{55}^{T} }{L}, K^{n,m,1,2}_{\omega _{r} ,\omega _{\varphi } } =-\frac{2hm\pi A_{66}^{T} }{R\varphi _{0} }, K^{n,m,1,2}_{\omega _{r} ,\omega _{r} } =\frac{2hA_{23} }{R}. \end{aligned}$$

]For \(\tau =2\) and \(s=1\)

$$\begin{aligned} K^{n,m,2,1}_{u_{x} ,u_{x} }= & {} 0, K^{n,m,2,1}_{u_{x} ,u_{\varphi } } =0, K^{n,m,2,1}_{u_{x} ,u_{r} } =\frac{2hn\pi C_{55}^{T} }{L}, K^{n,m,2,1}_{u_{x} ,\omega _{x} } =0, K^{n,m,2,1}_{u_{x} ,\omega _{y} } =0, K^{n,m,2,1}_{u_{x} ,\omega _{z} } =0, \\ K^{n,m,2,1}_{u_{\varphi } ,u_{x} }= & {} 0, K^{n,m,2,1}_{u_{\varphi } ,u_{\varphi } } =-\frac{2hC_{66}^{T} }{R}, K^{n,m,2,1}_{u_{\varphi } ,u_{r} } =\frac{2hm\pi C_{66}^{T} }{R\varphi _{0} } K^{n,m,2,1}_{u_{\varphi } ,\omega _{x} } =2h\left( {C_{66} -C_{66}^{T} } \right) , K^{n,m,2,1}_{u_{\varphi } ,\omega _{\varphi } } =0, K^{n,m,2,1}_{u_{\varphi } ,\omega _{r} } =0, \\ K^{n,m,2,1}_{u_{r} ,u_{x} }= & {} -\frac{2hn\pi C_{13} }{L}, K^{n,m,2,1}_{u_{r} ,u_{\varphi } } =-\frac{2hm\pi C_{23} }{R\varphi _{0} }, K^{n,m,2,1}_{u_{r} ,u_{r} } =\frac{2hC_{23} }{R}, K^{n,m,2,1}_{u_{r} ,\omega _{r} } =0, K^{n,m,2,1}_{u_{r} ,\omega _{x} } =0, K^{n,m,2,1}_{u_{r} ,\omega _{\varphi } } =0, \\ K^{n,m,2,1}_{\omega _{x} ,u_{x} }= & {} 0, K^{n,m,2,1}_{\omega _{x} ,u_{\varphi } } =0, \quad L^{n,m,2,1}_{\omega _{x} ,u_{r} } =0, K^{n,m,2,1}_{\omega _{x} ,\omega _{x} } =0, K^{n,m,2,1}_{\omega _{x} ,\omega _{\varphi } } =0, K^{n,m,2,1}_{\omega _{x} ,\omega _{r} } =-\frac{2hn\pi A_{55}^{T} }{L}, \\ K^{n,m,2,1}_{\omega _{\varphi } ,u_{x} }= & {} 0, K^{n,m,2,1}_{\omega _{\varphi } ,u_{\varphi } } =0, \quad K^{n,m,2,1}_{\omega _{\varphi } ,u_{r} } =0, K^{n,m,2,1}_{\omega _{\varphi } ,\omega _{x} } =0, K^{n,m,2,1}_{\omega _{\varphi } ,\omega _{\varphi } } =-\frac{2hA_{66}^{T} }{R}, K^{n,m,2,1}_{\omega _{\varphi } ,\omega _{r} } =-\frac{2hm\pi A_{66}^{T} }{R\varphi _{0} }, \\ K^{n,m,2,1}_{\omega _{r} ,u_{x} }= & {} 0, K^{n,m,2,1}_{\omega _{r} ,u_{\varphi } } =0, K^{n,m,2,1}_{\omega _{r} ,u_{r} } =0, K^{n,m,2,1}_{\omega _{r} ,\omega _{x} } =\frac{2hn\pi A_{13} }{L}, K^{n,m,2,1}_{\omega _{r} ,\omega _{\varphi } } =\frac{2hm\pi A_{23} }{R\varphi _{0} }, K^{n,m,2,1}_{\omega _{r} ,\omega _{r} } =\frac{2hA_{23} }{R}. \end{aligned}$$

For \(\tau =2\) and \(s=2\)

$$\begin{aligned} K^{n,m,2,2}_{u_{x} ,u_{x} }= & {} 2hC_{55} +\frac{2}{3}h^{3}\pi ^{2}\left( {\frac{n^{2}C_{11} }{L^{2}}+\frac{m^{2}C_{44} }{R^{2}\varphi _{0}^{2} }} \right) , K^{n,m,2,2}_{u_{x} ,u_{\varphi } } =\frac{2h^{3}mn\pi ^{2}\left( {C_{12} +C_{44}^{T} } \right) }{3LR\varphi _{0} }, K^{n,m,2,2}_{u_{x} ,u_{r} } =-\frac{2h^{3}n\pi C_{12} }{3LR}, K^{n,m,2,2}_{u_{x} ,\omega _{x} } =0,\\ K^{n,m,2,2}_{u_{x} ,\omega _{y} }= & {} \frac{2}{3}h^{3}\left( {C_{55}^{T} -C_{55} } \right) , K^{n,m,2,2}_{u_{x} ,\omega _{z} } =\frac{2h^{3}m\pi \left( {C_{44} -C_{44}^{T} } \right) }{3R\varphi _{0} }, \\ K^{n,m,2,2}_{u_{\varphi } ,u_{x} }= & {} \frac{2h^{3}mn\pi ^{2}\left( {C_{12} +C_{44}^{T} } \right) }{3LR\varphi _{0} }, K^{n,m,2,2}_{u_{\varphi } ,u_{\varphi } } =\left( {2h+\frac{2h^{3}}{3R^{2}}} \right) C_{66} +\frac{2}{3}h^{3}\pi ^{2}\left( {\frac{n^{2}C_{44} }{L^{2}}+\frac{m^{2}C_{22} }{R^{2}\varphi _{0}^{2} }} \right) , K^{n,m,2,2}_{u_{\varphi } ,\omega _{\varphi } } =0,\\ K^{n,m,2,2}_{u_{\varphi } ,u_{r} }= & {} -\frac{2h^{3}m\pi \left( {C_{22} +C_{66} } \right) }{3R^{2}\varphi _{0} }, K^{n,m,2,2}_{u_{\varphi } ,\omega _{x} } =\frac{2h^{3}}{3R}\left( {C_{66} -C_{66}^{T} } \right) , K^{n,m,2,2}_{u_{\varphi } ,\omega _{r} } =\frac{2h^{3}n\pi }{3L}\left( {C_{44}^{T} -C_{44} } \right) , \\ K^{n,m,2,2}_{u_{r} ,u_{x} }= & {} -\frac{2h^{3}n\pi C_{12} }{3LR}, K^{n,m,2,2}_{u_{r} ,u_{\varphi } } =-\frac{2h^{3}m\pi \left( {C_{22} +C_{66} } \right) }{3R^{2}\varphi _{0} }, K^{n,m,2,2}_{u_{r} ,\omega _{x} } =\frac{2h^{3}\pi m}{3R\varphi _{0} }\left( {C_{66} -C_{66}^{T} } \right) ,\\ K^{n,m,2,2}_{u_{r} ,u_{r} }= & {} \frac{2h^{3}C_{22} }{3R^{2}}+2hC_{33} +\frac{2h^{3}n^{2}\pi ^{2}C_{55} }{3L^{2}}+\frac{2h^{3}m^{2}\pi ^{2}C_{66} }{3R^{2}\varphi _{0}^{2} }, K^{n,m,2,2}_{u_{r} ,\omega _{\varphi } } =\frac{2h^{3}\pi n}{3L}\left( {C_{55} -C_{55}^{T} } \right) , \quad K^{n,m,2,2}_{u_{r} ,\omega _{r} } =0, \\ K^{n,m,2,2}_{\omega _{x} ,u_{x} }= & {} 0, K^{n,m,2,2}_{\omega _{x} ,u_{\varphi } } =\frac{2h^{3}\left( {C_{66} -C_{66}^{T} } \right) }{3R}, \quad L^{n,m,2,2}_{\omega _{x} ,u_{r} } =\frac{2h^{3}\pi m\left( {C_{66}^{T} -C_{66} } \right) }{3R\varphi _{0} }, K^{n,m,2,2}_{\omega _{x} ,\omega _{\varphi } } =\frac{2h^{3}\pi ^{2}nm\left( {A_{12} +A_{44}^{T} } \right) }{3LR\varphi _{0} }, \\ K^{n,2,2}_{\omega _{x} ,\omega _{x} }= & {} 2hA_{55} +\frac{2}{3}h^{3}\left( {2C_{66} -2C_{66}^{T} +\pi ^{2}\left( {\frac{n^{2}A_{11} }{L^{2}}+\frac{m^{2}A_{44} }{R^{2}\varphi _{0}^{2} }} \right) } \right) , K^{n,m,2,2}_{\omega _{x} ,\omega _{r} }= \frac{2h^{3}\pi nA_{12} }{3LR}, \\ K^{n,m,2,2}_{\omega _{\varphi } ,u_{x} }= & {} 0, K^{n,m,2,2}_{\omega _{\varphi } ,u_{\varphi } } =0, \quad K^{n,m,2,2}_{\omega _{\varphi } ,u_{r} } =\frac{2h^{3}n\pi \left( {C_{55}^{T} -C_{55} } \right) }{3L}, K^{n,m,2,2}_{\omega _{\varphi } ,\omega _{x} } =\frac{2h^{3}\pi ^{2}nm\left( {A_{12} +A_{44}^{T} } \right) }{3LR\varphi _{0} },\\ K^{n,m,2,2}_{\omega _{\varphi } ,\omega _{\varphi } }= & {} \left( {2h+\frac{2h^{3}}{3R^{2}}} \right) A_{66} +\frac{2}{3}h^{3}\left( {2C_{55} -2C_{55}^{T} +\pi ^{2}\left( {\frac{n^{2}A_{44} }{L^{2}}+\frac{m^{2}A_{22} }{R^{2}\varphi _{0}^{2} }} \right) } \right) , K^{n,m,2,2}_{\omega _{\varphi } ,\omega _{r} } =\frac{2h^{3}\pi mA_{66} }{3R^{2}\varphi _{0} }, \\ K^{n,m,2,2}_{\omega _{r} ,u_{x} }= & {} \frac{2h^{3}\pi m\left( {C_{44}^{T} -C_{44} } \right) }{3R\varphi _{0} }, K^{n,m,2,2}_{\omega _{r} ,u_{\varphi } } =\frac{2h^{3}\pi n\left( {C_{44}^{T} -C_{44} } \right) }{3L}, K^{n,m,2,2}_{\omega _{r} ,u_{r} } =0, K^{n,m,2,2}_{\omega _{r} ,\omega _{x} } =\frac{2h^{3}\pi nA_{12} }{3LR},\\ K^{n,m,2,2}_{\omega _{r} ,\omega _{\varphi } }= & {} \frac{2h^{3}m\pi \left( {A_{22} +A_{66} } \right) }{3R^{2}\varphi _{0} }, K^{n,m,2,2}_{\omega _{r} ,\omega _{r} } =\frac{2h^{3}A_{22} }{3R^{2}}+2hA_{33} +\frac{2h^{3}\pi ^{2}n^{2}A_{55} }{3L^{2}}+\frac{4h^{3}}{3}\left( {C_{44} -C_{44}^{T} } \right) +\frac{2h^{3}\pi ^{2}m^{2}A_{66} }{3R^{2}\varphi _{0}^{2} }. \end{aligned}$$

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Carrera, E., Zozulya, V.V. Analytical solution for the micropolar cylindrical shell: Carrera unified formulation (CUF) approach. Continuum Mech. Thermodyn. (2022). https://doi.org/10.1007/s00161-022-01156-x

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