Geometrically nonlinear rapid surface heating in FGM hermetic capsule

Abstract

The present study examines the geometrically non-linear dynamic response of hermetic capsule construction made of functionally graded materials subjected to thermal shock. The Voigt and Toloukian models are used to derive the material properties, where dependence of the properties to position and temperature is included. The one-dimensional transient heat transfer equation is established. The Crank-Nicholson approximation and Picard’s iterative method are used to solve this nonlinear equation using the GDQ numerical method in accordance with the temperature dependence of the material properties. After obtaining the temperature distribution along the thickness, it is possible to determine the thermal force and moment. Using the first-order shear theory to calculate the displacement field and the von Kármán form of geometry non-linearity, the equations of motion are determined. The Newton-Raphson iterative approach and the $\beta$-Newmark time estimate approach are used to solve the non-linear coupled equations of motion. The influencing factors on the reaction of the structure, such as the radius of the sphere and the length of the cylinder, the power law index, and the shell thickness are determined after the equations, techniques, and findings are validated.

Appendix

The governing equations of motion in the cylindrical shell using the GDQ method may be expressed as

$$\begin{aligned}&A_{11}\left( \sum _{j=1}^{N_{x}}{\overline{B}}_{ij}u_{j}+\left( \sum _{j=1}^{N_{x}}{\overline{A}}_{ij}w_{j}\right) \left( \sum _{k=1}^{N_{x}}{\overline{B}}_{ik}w_{k}\right) \right) +B_{11}\left( \sum _{j=1}^{N_{x}}{\overline{B}}_{ij}\psi _{j}\right) \nonumber \\&\quad +A_{12}\left( \dfrac{1}{R}\sum _{j=1}^{N_{x}}{\overline{A}}_{ij}w_{j} \right) =I_{1}\ddot{u}_{0,i}+I_{2}\ddot{\psi _{x,i}} \end{aligned}$$

(A1)

$$\begin{aligned}&A_{11}\left( -\dfrac{1}{R^{2}}w_{i}+\left( \sum _{j=1}^{N_{x}}{\overline{B}}_{ij}u_{j}\right) \left( \sum _{k=1}^{N_{x}}{\overline{A}}_{ik}w_{k}\right) +\left( \sum _{j=1}^{N_{x}}{\overline{A}}_{ij}u_{j}\right) \left( \sum _{k=1}^{N_{x}}{\overline{B}}_{ik}w_{k}\right) \right. \nonumber \\&\quad \left. +\dfrac{3}{2} \left( \sum _{j=1}^{N_{x}}{\overline{A}}_{ij}w_{j}\right) \left( \sum _{k=1}^{N_{x}}{\overline{A}}_{ik}w_{k}\right) \left( \sum _{l=1}^{N_{x}}{\overline{B}}_{il}w_{l}\right) \right) \nonumber \\&\quad +B_{11}\left( \left( \sum _{j=1}^{N_{x}}{\overline{B}}_{ij}\psi _{j}\right) \left( \sum _{k=1}^{N_{x}}{\overline{A}}_{ik}w_{k}\right) +\left( \sum _{j=1}^{N_{x}}{\overline{A}}_{ij}\psi _{j}\right) \left( \sum _{k=1}^{N_{x}}{\overline{B}}_{ik}w_{k}\right) \right) \nonumber \\&\quad +\dfrac{A_{12}}{R}\left( -\sum _{j=1}^{N_{x}}{\overline{A}}_{ij}u_{j}+w_{i}\left( \sum _{j=1}^{N_{x}}{\overline{B}}_{ij}w_{j}\right) + \dfrac{1}{2}\left( \sum _{j=1}^{N_{x}}{\overline{A}}_{ij}w_{j}\right) \left( \sum _{k=1}^{N_{x}}{\overline{A}}_{ik}w_{k}\right) \right) \nonumber \\&\quad +\dfrac{B_{12}}{R}\left( -\sum _{j=1}^{N_{x}}{\overline{A}}_{ij}\psi _{j}\right) +A_{55}\left( \sum _{j=1}^{N_{x}}{\overline{A}}_{ij}\psi _{j}+\sum _{j=1}^{N_{x}}{\overline{B}}_{ij}w_{j}\right) -N^{T}\left( \sum _{j=1}^{N_{x}}{\overline{B}}_{ij}w_{j}-\dfrac{1}{R}\right) =I_{1}\ddot{w}_{0,i} \end{aligned}$$

(A2)

$$\begin{aligned}&B_{11}\left( \sum _{j=1}^{N_{x}}{\overline{B}}_{ij}u_{j}+ \left( \sum _{j=1}^{N_{x}}{\overline{B}}_{ij}w_{j}\right) \left( \sum _{k=1}^{N_{x}}{\overline{A}}_{ik}w_{k}\right) \right) +D_{11}\left( \sum _{j=1}^{N_{x}}{\overline{B}}_{ij}\psi _{j}\right) \nonumber \\&\quad +B_{12}\left( \dfrac{1}{R}\sum _{j=1}^{N_{x}}{\overline{A}}_{ij}w_{j}\right) -A_{55}\left( \psi _{i}+\sum _{j=1}^{N_{x}}{\overline{A}}_{ij}w_{j}\right) =I_{2}\ddot{u}_{0,i}+I_{3}\ddot{\psi }_{x,i} \end{aligned}$$

(A3)

Similarly, the governing equations of motion in the spherical shell using the GDQ method may be expressed as

$$\begin{aligned}&\dfrac{1}{R^{2}}\left\{ A_{11}\sum _{j=1}^{N_{\phi }}{\overline{B}}_{ij}u_{j}+A_{11}\cot \left( \phi \right) \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}u_{j}-\dfrac{A_{12}}{\sin ^{2}\left( \phi \right) }u_{i}+\left( A_{12}-A_{11}\right) \cot ^{2}\left( \phi \right) u_{i}-A_{55}u_{i}+N^{T}u_{i}\right. \nonumber \\&\quad \left. +\dfrac{ A_{11}-3 A_{12}}{2R}\cot \left( \phi \right) u_{i}u_{i}-\dfrac{ A_{11}+ A_{12}}{R}u_{i}w_{i}-\dfrac{ A_{11}-2 A_{12}}{R}\cot \left( \phi \right) u_{i}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j} \right) \right. \nonumber \\&\quad \left. -\dfrac{ A_{11}}{R}u_{i}\left( \sum _{j=1}^{N_{\phi }}{\overline{B}}_{ij}w_{j} \right) -\dfrac{ A_{11}}{2R^{2}}\left[ u_{i}^3 -3u_{i}^2 \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j} \right) \right. \right. \nonumber \\&\quad + \left. \left. 3u_{i}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j} \right) \left( \sum _{k=1}^{N_{\phi }}{\overline{A}}_{ik}w_{k} \right) -\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j} \right) \left( \sum _{k=1}^{N_{\phi }}{\overline{A}}_{ik}w_{k} \right) \left( \sum _{l=1}^{N_{\phi }}{\overline{A}}_{il}w_{l} \right) \right] -\dfrac{B_{11}}{R}u_{i}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}\psi _{j} \right) \right. \nonumber \\&\quad - \left. \dfrac{B_{12}}{R}\cot \left( \phi \right) u_{i}\psi _{i} +B_{11}\sum _{j=1}^{N_{\phi }}{\overline{B}}_{ij}\psi _{j} +B_{11}\cot \left( \phi \right) \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}\psi _{j}-\dfrac{B_{12}}{\sin ^{2}\left( \phi \right) }\psi _{i} +\left( B_{12}-B_{11}\right) \cot ^{2}\left( \phi \right) \psi _{i} \right. \nonumber \\&\quad \left. + A_{55}R\psi _{i}+\dfrac{B_{11}}{R}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}\psi _{j} \right) \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j} \right) +\dfrac{B_{12}}{R}\cot \left( \phi \right) \psi _{i} \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j} \right) \right. \nonumber \\&\quad \left. +\left( A_{11}+A_{12}+A_{55}-N^{T}\right) \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j} \right) +\dfrac{A_{11}}{R} \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j} \right) \left( \sum _{j=1}^{N_{\phi }}{\overline{B}}_{ij}w_{j} \right) \right. \nonumber \\&\quad \left. +\dfrac{\left( A_{11}-A{12}\right) }{2R}\cot \left( \phi \right) \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j} \right) \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j} \right) +\dfrac{\left( A_{11}+A_{12}\right) }{R}w_{i} \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j} \right) \right\} =I_{1}\ddot{ u}_{0,i}+I_{1}\ddot{ \psi }_i \end{aligned}$$

(A4)

$$\begin{aligned}&\dfrac{1}{R^{2}}\left\{ \left( -\left( A_{11}+A_{12}+ A_{55}\right) +N^{T}\right) \cot \left( \phi \right) u_{i}+\left( -\left( A_{11}+A_{12}+ A_{55}\right) +N^{T}\right) \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}u_{j}\right. \nonumber \\&\quad \left. -2\left( A_{11}+A_{12}\right) w_{i}+\left( A_{55}-N^{T}\right) \cot \left( \phi \right) \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}+\left( A_{55}-N^{T}\right) \sum _{j=1}^{N_{\phi }}{\overline{B}}_{ij} w_{j}\right. \nonumber \\ {}&\qquad \left. \left( R A_{55}-\left( B_{11}+ B_{12}\right) \right) \cot \left( \phi \right) \psi _{i} +\left( RA_{55} -\left( B_{11}+ B_{12}\right) \right) \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij} \psi _{j} \right. \nonumber \\ {}&\quad \left. -\dfrac{\left( A_{11}-A_{12}\right) }{2R}u_{i}^2-\left( A_{11}+2A_{12} \right) \cot \left( \phi \right) \dfrac{1}{R}u_{i}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{kj}u_{j} \right) -\dfrac{A_{11}}{R}u_{i}\left( \sum _{j=1}^{N_{\phi }}{\overline{B}}_{kj}u_{j} \right) \right. \nonumber \\ {}&\quad \left. -\dfrac{\left( A_{11}+A_{12} \right) }{R}\cot \left( \phi \right) u_{i}w_{i} - \dfrac{A_{12}}{R}u_{i}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{kj}w_{j} \right) +\dfrac{A_{12}}{R}\cot \left( \phi \right) u_{i}\left( \sum _{j=1}^{N_{\phi }}{\overline{B}}_{kj}w_{j} \right) \right. \nonumber \\ {}&\quad \left. + \left( -B_{12}\cot ^{2}\left( \phi \right) +\dfrac{B_{12}}{\sin \left( \phi \right) } \right) \dfrac{1}{R}u_{i}\psi _{i}-\left( B_{11}+B_{12} \right) \cot \left( \phi \right) \dfrac{1}{R}u_{i}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{kj}\psi _{j} \right) \right. \nonumber \\ {}&\quad \left. -B_{11}\dfrac{1}{R}u_{i}\left( \sum _{j=1}^{N_{\phi }}{\overline{B}}_{kj}\psi _{j} \right) -\dfrac{A_{11} }{R}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}u_{j}\right) \left( \sum _{k=1}^{N_{\phi }}{\overline{A}}_{ij}u_{k}\right) -\dfrac{\left( A_{11}+A_{12} \right) }{R}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}u_{j}\right) w_{i}\right. \nonumber \\&\quad \left. + \dfrac{\left( A_{11}+A_{12} \right) }{R}\cot \left( \phi \right) \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}u_{j}\right) \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) +\dfrac{A_{11}}{R}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}u_{j}\right) \left( \sum _{j=1}^{N_{\phi }}{\overline{B}}_{ij}w_{j}\right) \right. \nonumber \\&\quad \left. -\dfrac{B_{12}}{R}\cot \left( \phi \right) \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}u_{j}\right) \psi _{i}-\dfrac{B_{11}}{R}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}u_{j}\right) \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}\psi _{j}\right) +\dfrac{A_{11}}{R}\left( \sum _{j=1}^{N_{\phi }}{\overline{B}}_{ij}u_{j}\right) \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) \right. \nonumber \\&\quad \left. +\dfrac{\left( A_{11}+A_{12} \right) }{R}\cot \left( \phi \right) w_{i} \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) +\dfrac{\left( A_{11}+A_{12} \right) }{R}w_{i}\left( \sum _{j=1}^{N_{\phi }}{\overline{B}}_{ij}w_{j}\right) \right. \nonumber \\&\quad \left. +\dfrac{\left( A_{11} +A_{12}\right) }{2R} \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) - \dfrac{B_{12}}{R}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) \psi _{i}\right. \nonumber \\&\quad \left. +\left( B_{11}+B_{12} \right) \cot \left( \phi \right) \dfrac{1}{R}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}\psi _{j}\right) +B_{11}\dfrac{1}{R}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) \left( \sum _{j=1}^{N_{\phi }}{\overline{B}}_{ij}\psi _{j}\right) \right. \nonumber \\&\quad \left. +\dfrac{A_{11}}{2R^{2}}\left[ -\cot \left( \phi \right) \left( u_{i}^3-3u_{i}^2 \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) +3u_{i}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) \left( \sum _{k=1}^{N_{\phi }}{\overline{A}}_{ik}w_{k}\right) \right. \right. \right. \nonumber \\&\quad \left. \left. \left. -\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) \left( \sum _{k=1}^{N_{\phi }}{\overline{A}}_{ik}w_{k}\right) \left( \sum _{l=1}^{N_{\phi }}{\overline{A}}_{il}w_{l}\right) \right) -6u_{i}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) \left( \sum _{j=1}^{N_{\phi }}{\overline{B}}_{ij}w_{j}\right) \right. \right. \nonumber \\&\quad \left. \left. +6u_{i}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}u_{j}\right) \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) +3u_{i}^2 \left( \sum _{j=1}^{N_{\phi }}{\overline{B}}_{ij}w_{j}\right) -3u_{i}^2\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}u_{j}\right) \right. \right. \nonumber \\&\quad \left. \left. -3\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}u_{j}\right) \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) \left( \sum _{k=1}^{N_{\phi }}{\overline{A}}_{ik}w_{k}\right) +3\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) \left( \sum _{j=1}^{N_{\phi }}{\overline{B}}_{ij}w_{j}\right) \right] \right. \nonumber \\&\quad \left. +B_{12}\cot \left( \phi \right) \dfrac{1}{R}\left( \sum _{j=1}^{N_{\phi }}{\overline{B}}_{ij}w_{j}\right) \psi _{i}+B_{11}\dfrac{1}{R}\left( \sum _{j=1}^{N_{\phi }}{\overline{B}}_{ij}w_{j}\right) \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}\psi _{j}\right) +2RN^{T}\right\} =I_{1}\ddot{w}_{0,i} \end{aligned}$$

(A5)

$$\begin{aligned}&\dfrac{1}{R^{2}}\left\{ B_{11}\sum _{j=1}^{N_{\phi }}{\overline{B}}_{ij}u_{j}+B_{11}\cot \left( \phi \right) \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}u_{j}-\dfrac{B_{12}}{\sin ^{2}\left( \phi \right) ^{2}}u_{i}+\left( B_{12}-B_{11}\right) \cot ^{2}\left( \phi \right) u_{i}+A_{55}Ru_{i} \right. \nonumber \\ {}&\quad \left. +\dfrac{ B_{11}-B_{12}}{2R}\cot \left( \phi \right) u_{i}^2+\dfrac{B_{11}}{R}u_{i}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}u_{j}\right) - \dfrac{ B_{11}-B_{12}}{R}\cot \left( \phi \right) u_{i}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) \right. \nonumber \\&\quad \left. -\dfrac{ B_{11}}{R}\cot \left( \phi \right) u_{i}\left( \sum _{j=1}^{N_{\phi }}{\overline{B}}_{ij}w_{j}\right) -\dfrac{ B_{11}}{R}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}u_{j}\right) \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) \right. \nonumber \\ {}&\quad \left. +D_{11}\sum _{j=1}^{N_{\phi }}{\overline{B}}_{ij}\psi _{j}+D_{11}\cot \left( \phi \right) \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}\psi _{j} -\dfrac{D_{12}}{\sin ^{2}\left( \phi \right) }\psi _{i} +\left( D_{12}-D_{11}\right) \cot ^{2}\left( \phi \right) \psi _{i} - A_{55}R^{2}\psi _{i} \right. \nonumber \\&\quad \left. + \left( B_{11}+B_{12} \right) \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) +\dfrac{B_{11}}{R} \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) \left( \sum _{j=1}^{N_{\phi }}{\overline{B}}_{ij}w_{j}\right) \right. \nonumber \\&\quad \left. +\dfrac{\left( B_{11}-B_{12}\right) }{2R}\cot \left( \phi \right) \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) \left( \sum _{k=1}^{N_{\phi }}{\overline{A}}_{ik}w_{k}\right) -RA_{55}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{ij}w_{j}\right) \right\} =I_{2}\ddot{u}_{0,i}+I_{3}\ddot{\psi }_i \end{aligned}$$

(A6)

where in the above equations, $u_i$, $w_i$, and $\psi _i$ are the magnitudes of $u_0$, $w_0$, and $\psi _\xi$ at $\xi _i$. Also, ${\overline{A}}$ and ${\overline{B}}$ are the weighting coefficients of the GDQ method associated to first and second derivatives.

Also, the boundary conditions should be discretized using GDQ method as follow

$$\begin{aligned} N_{x,s}&=A_{11}\left( \sum _{j=1}^{N_{x}}{\overline{A}}_{sj}u_{j}+\frac{1}{2}\left( \sum _{j=1}^{N_{x}}{\overline{A}}_{sj}w_{j}\right) \left( \sum _{k=1}^{N_{x}}{\overline{A}}_{sk}w_{k}\right) \right) +A_{12}\dfrac{1}{R}w_{s} \nonumber \\&\quad +B_{11}\sum _{j=1}^{N_{x}}{\overline{A}}_{sj}\psi _{j} -N^{T}, \quad s=1,N_{x} \end{aligned}$$

(A7)

$$\begin{aligned} Q_{xz,s}&=A_{55}\left( \sum _{j=1}^{N_{x}}{\overline{A}}_{sj}w_{j}+\psi _{s}\right) +\left( \sum _{l=1}^{N_{x}}{\overline{A}}_{sl}w_{l}\right) \left\{ A_{11}\left( \sum _{j=1}^{N_{x}}{\overline{A}}_{sj}u_{j}+\frac{1}{2}\left( \sum _{j=1}^{N_{x}}{\overline{A}}_{sj}w_{j}\right) \left( \sum _{k=1}^{N_{x}}{\overline{A}}_{sk}w_{k}\right) \right) \right. \nonumber \\&\quad \left. +A_{12}\left( \dfrac{1}{R}w_{s} \right) + B_{11}\sum _{j=1}^{N_{x}}{\overline{A}}_{sj}\psi _{j} -N^{T}\right\} , \quad s=1,N_{x} \end{aligned}$$

(A8)

$$\begin{aligned} M_{x,s}&=B_{11}\left( \sum _{j=1}^{N_{x}}{\overline{A}}_{sj}u_{j}+\frac{1}{2}\left( \sum _{j=1}^{N_{x}}{\overline{A}}_{sj}w_{j}\right) \left( \sum _{k=1}^{N_{x}}{\overline{A}}_{sk}w_{k}\right) \right) +B_{12}\left( \dfrac{1}{R}w_{s} \right) \nonumber \\&\quad + D_{11}\sum _{j=1}^{N_{x}}{\overline{A}}_{sj}\psi _{j}-M^{T}, \quad s=1,N_{x} \end{aligned}$$

(A9)

$$\begin{aligned} N_{\phi ,s}&=\dfrac{A_{11}}{R}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{sj}u_{j}+w_{s}+\frac{1}{2R}\left( \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{sj}w_{j}\right) \left( \sum _{k=1}^{N_{\phi }}{\overline{A}}_{sk}w_{k}\right) -2\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{sj}w_{j}\right) u_{s}+u_{s}^2\right) \right) \nonumber \\&\quad +\dfrac{A_{12}}{R}\left( \cot \left( \phi \right) u_{s} +w_{s} \right) +\dfrac{B_{11}}{R}\sum _{j=1}^{N_{\phi }}{\overline{A}}_{sj}\psi _{j} +\dfrac{B_{12}}{R}\cot \left( \phi \right) \psi _{s}-N^{T}, \quad s=1,N_{\phi } \end{aligned}$$

(A10)

$$\begin{aligned} Q_{\phi z,s}&=\dfrac{A_{55}}{R}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{sj}w_{j}+\psi _{s}\right) +\dfrac{A_{11}}{R^{2}}\left\{ \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{sj}u_{j}\right) \left( \sum _{l=1}^{N_{\phi }}{\overline{A}}_{sl}w_{l}\right) -\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{sj}u_{j}\right) u_{s}\right. \nonumber \\&\quad \left. + \left( \sum _{l=1}^{N_{\phi }}{\overline{A}}_{sl}w_{l}\right) w_{s}-u_{s}w_{s}+\dfrac{1}{2R}\left( \left( \sum _{l=1}^{N_{\phi }}{\overline{A}}_{sl}w_{l}\right) \left( \sum _{l=1}^{N_{\phi }}{\overline{A}}_{sl}w_{l}\right) \left( \sum _{l=1}^{N_{\phi }}{\overline{A}}_{sl}w_{l}\right) \right. \right. \nonumber \\&\quad \left. \left. -3 \left( \sum _{l=1}^{N_{\phi }}{\overline{A}}_{sl}w_{l}\right) \left( \sum _{l=1}^{N_{\phi }}{\overline{A}}_{sl}w_{l}\right) u_{s}+3 \left( \sum _{l=1}^{N_{\phi }}{\overline{A}}_{sl}w_{l}\right) u_{s}^2- u_{s}^3\right) \right\} \nonumber \\ {}&\quad +\dfrac{A_{12}}{R^{2}}\cot \left( \phi \right) \left( \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{sj}w_{j}\right) u_{s}-u_{s}^2+\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{sj}w_{j}\right) w_{s}-u_{s}w_{s}\right) \nonumber \\&\quad +\dfrac{B_{11}}{R^{2}} \left( \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{sj}\psi _{j}\right) \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{sj}w_{j}\right) -\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{sj}\psi _{j}\right) u_{s}\right) \nonumber \\&\quad +\dfrac{B_{12}}{R^{2}}\cot \left( \phi \right) \left( \psi _{s} \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{sj}w_{j}\right) - \psi _{s} u_{s}\right) -\dfrac{N^{T}}{R}, \quad s=1,N_{\phi } \end{aligned}$$

(A11)

$$\begin{aligned} M_{\phi ,s}&=\dfrac{B_{11}}{R}\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{sj}u_{j}+w_{s}+\frac{1}{2R}\left( \left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{sj}w_{j}\right) \left( \sum _{k=1}^{N_{\phi }}{\overline{A}}_{sk}w_{k}\right) -2\left( \sum _{j=1}^{N_{\phi }}{\overline{A}}_{sj}w_{j}\right) u_{s}+u_{s}^2\right) \right) \nonumber \\&\quad +\dfrac{B_{12}}{R}\left( \cot \left( \phi \right) u_{s} +w_{s} \right) +\dfrac{D_{11}}{R}\sum _{j=1}^{N_{\phi }}{\overline{A}}_{sj}\psi _{j} +\dfrac{D_{12}}{R}\cot \left( \phi \right) \psi _{s}-M^{T}, \quad s=1,N_{\phi } \end{aligned}$$

(A12)