## 2 Formulations
\begin{array}{l}\varepsilon_{x}{=}\frac{\partial u_{0}}{\partial x}+z\frac{ \partial\varphi_{x}}{\partial x}-c_{1}z^{3}(\frac{\partial\varphi_{x}}{ \partial x}+\frac{\partial^{2}w}{\partial x^{2}})+\frac{1}{2}(\frac{\partial w }{\partial x})^{2},\\ \varepsilon_{\theta}{=}\frac{1}{R}\frac{\partial v_{0}}{\partial x}+z\frac{ \partial\varphi_{y}}{\partial\theta}-c_{1}z^{3}(\frac{\partial\varphi_{x}}{ \partial\theta}+\frac{1}{R}\frac{\partial^{2}w}{\partial\theta^{2}})+\frac{1 }{2}\frac{1}{R^{2}}(\frac{\partial w}{\partial\theta})^{2},\\ \varepsilon_{zz}=0,\\ \varepsilon_{x\theta}{=}\frac{1}{R}[\frac{\partial u_{0}}{\partial\theta}+z \frac{\partial\varphi_{x}}{\partial\theta}-c_{1}z^{3}(\frac{\partial\varphi_ {x}}{\partial\theta}+\frac{\partial^{2}w}{\partial x\theta\theta})]\\ +\frac{\partial v_{0}}{\partial x}+2\frac{\partial\varphi_{x}}{\partial x}-c_ {1}z^{3}(\frac{\partial\varphi_{x}}{\partial\theta}+\frac{1}{R}\frac{ \partial^{2}w}{\partial x\theta\theta})+(\frac{\partial w}{\partial x})(\frac {1}{R}\frac{\partial w}{\partial\theta}),\\ \varepsilon_{\theta z}{=}\frac{-v_{0}}{R}+\varphi_{y}-3c_{1}z^{2}(\varphi_{y} +\frac{\partial w}{\partial x})+\frac{\partial w}{\partial x}.\end{array}  \tag{7}\]  
After substituting (3) and (7) into (6), dynamic partial differential equilibrium equations in TSDT for magneto-stritive FGM cylindrical shells can be used. The \(\overline{Q}_{i^{\prime}\bar{i}}\) and \(\overline{Q}_{i^{\prime}\bar{i}}\), stiffness integrals are used in the following parameter forms:  
\[ \begin{array}{l}h^{*}\\ (A_{i^{\prime}\bar{i}^{\prime}},B_{i^{\prime}\bar{i}^{\prime}},D_{i^{ \prime}\bar{i}^{\prime}},E_{i^{\prime}\bar{i}^{\prime}},F_{i^{\prime}\bar{i}^ {\prime}},H_{i^{\prime}\bar{i}^{\prime}})=\int\frac{\overline{2} _{h^{*}}}{\overline{2}_{h^{*}}}\ \overline{Q}_{i^{\prime}\bar{i}^{\prime}}(1,z,z^{2},z^{3},z^{4},z^{6})dz, \text{ with }\\ \text{ subscripts }(i^{*},\vec{j^{*}}=1,2,6),\\ h^{*}\\ (A_{i^{*}\bar{i}^{\prime}},B_{i^{*}\bar{i}^{\prime}},D_{i^{\prime}\bar{i}^{ \prime}},E_{i^{\prime}\bar{i}^{\prime}},F_{i^{\prime}\bar{i}^{\prime}},H_{i^{ \prime}\bar{i}^{\prime}})=\int\frac{\overline{2}_{h^{*}}}{\overline{2}_{h^{*} }}\ k_{\alpha}\overline{Q}_{i^{\prime}\bar{i}^{\prime}}(1,z,z^{2},z^{3},z^{4}, z^{5})dz,\text{ with }\\ \text{ subscripts }(i^{*},\vec{j^{*}}=4,5),\end{array}  \tag{8}\]  
where \(k_{\alpha}\) denotes the advanced nonlinear shear correction, which is in function of \(T\), \(h_{3}\), \(h^{*}\), \(c_{1}\), and \(R_{n}\). Stiffness \(\overline{Q}_{i^{\prime}\bar{i}^{\prime}}\) and \(\overline{Q}_{i^{\prime}\bar{i}^{\prime}}\) in the magnetostrictive FGM thick-thickness shell with non-neglected term \(z/R\) can be used in simpler expressions. Thus, simpler \(\overline{Q}_{i^{\prime}\bar{i}^{\prime}}\), \(\overline{Q}_{i^{\prime}\bar{i}^{\prime}}\), expressions are then used to compute stress, stiffness integrals parameters \(A_{i^{\prime}\bar{i}^{\prime}}\), \(B_{i^{\prime}\bar{i}^{\prime}}\), \(D_{i^{\prime}\bar{i}^{\prime}}\), \(E_{i^{\prime}\bar{i}^{\prime}}\), \(F_{i^{\prime}\bar{i}^{\prime}}\) and \(A_{i^{\prime}\bar{i}^{\prime}}\), \(B_{i^{\prime}\bar{i}^{\prime}}\), \(D_{i^{\prime}\bar{i}^{\prime}}\), \(E_{i^{\prime}\bar{i}^{\prime}}\), \(F_{i^{\prime}\bar{i}^{\prime}}\), \(H_{i^{\prime}\bar{i}^{\prime}}\), \(e\).g., the \(A_{11}\), \(E_{11}\), \(F_{11}\), \(H_{11}\), \(H_{44}\), \(A_{12}\), \(B_{12}\), \(D_{12}\), \(E_{12}\)\(F_{12}\), \(H_{12}\), \(A_{55}\), \(B_{55}\), \(D_{55}\), \(E_{55}\), \(F_{55}\), and \(H_{55}\) can be obtained. The GDQ method was presented to solve dynamic partial differential equations into dynamic discrete equations in matrix form as follows for a grid point \((i,j)\) of magnetostrictive FGM cylindrical shell [20; 23; 24]:  
\[[A]\{W^{*}\}=\{B\}, \tag{9}\]  
in which \([A]\) is the matrix in \(N^{**}\)-th by \(N^{**}\)-th with dimension of \(N^{**}\) = \(5(N-2)(M-2)\), which contains stiffness integrals parameters, density integrals parameters, \(c_{1}\) term, and GDQ weighting parameters.