## 2 Formulations
\begin{array}{l}\frac{\partial N_{xx}}{\partial x}+\frac{1}{R}\frac{ \partial N_{\theta\theta}}{\partial\theta^{2}}=I_{0}\frac{\partial^{2}u_{0}}{ \partial t^{2}}+I_{1}\frac{\partial^{2}\rho_{\xi}}{\partial t^{2}}-c_{1}I_{3} \frac{\partial^{2}}{\partial t^{2}}(\frac{\partial uv}{\partial x})\\ \frac{\partial N_{xx}}{\partial x}+\frac{1}{R}\frac{\partial N_{\theta\theta}} {\partial\theta}=I_{0}\frac{\partial^{2}v_{0}}{\partial x^{2}}+I_{1}\frac{ \partial^{2}\rho_{\xi}}{\partial x^{2}}-c_{1}I_{3}\frac{\partial^{2}}{\partial x ^{2}}(\frac{1}{R}\frac{\partial uv}{\partial\theta})\\ \frac{\partial Q_{x}}{\partial x}+\frac{1}{R}\frac{\partial Q_{\theta}}{ \partial\theta}+c_{1}(\frac{\partial^{2}P_{xx}}{\partial x^{2}}+\frac{2}{R} \frac{P_{\theta\theta}}{\partial\theta^{2}}+\frac{1}{R^{2}}\frac{\partial^{2} P_{\theta\theta}}{\partial x^{2}})+q=I_{0}\frac{\partial^{2}w}{\partial t^{2}}-c_{1}^{2}I_{6} \frac{\partial^{2}}{\partial t^{2}}(\frac{\partial^{2}w}{\partial x^{2}}+ \frac{1}{R^{2}}\frac{\partial^{2}w}{\partial\theta^{2}})\\ +c_{1}[I_{3}\frac{\partial^{2}}{\partial t^{2}}(\frac{\partial u_{0}}{ \partial x}+\frac{1}{R}\frac{\partial u_{0}}{\partial\theta})+J_{4}\frac{ \partial^{2}}{\partial t^{2}}(\frac{\partial v}{\partial x}+\frac{1}{R}\frac{ \partial u_{0}}{\partial\theta})],\\ \frac{\partial M_{xx}}{\partial x}+\frac{1}{R}\frac{\partial M_{\theta\theta}} {\partial\theta}-\overline{Q}_{x}=\frac{\partial^{2}t}{\partial t^{2}}(I_{1} u_{0}+K_{2}\rho_{\theta}-c_{1}I_{4}\frac{\partial uv}{\partial x})\\ \frac{\partial M_{\theta\theta}}{\partial x}+\frac{1}{R}\frac{\partial M_{\theta \theta}}{\partial\theta}-\overline{Q}_{\theta}=\frac{\partial^{2}t}{\partial t ^{2}}(I_{1}v_{0}+K_{2}\rho_{\theta}-c_{1}I_{4}\frac{\partial uv}{\partial x}) \end{array}  \tag{6}\]  
where \(\overline{M}_{\alpha\beta}=M_{\alpha\theta}-c_{1}P_{\alpha\beta}\), \(\overline{Q}_{\alpha}=Q_{\alpha}-3c_{1}R_{\alpha}\), \((\alpha,\beta=x,\theta)\); the stress resultants \(N_{xx}\), \(N_{\theta\theta}\), \(N_{x\theta}\), \(M_{xx}\), \(M_{\theta\theta}\), \(M_{xy}\), \(P_{xx}\), \(P_{\theta\theta}\), \(R_{\theta}\), \(R_{x}\), \(Q_{\theta}\), and \(Q_{x}\) are due to three effects of the stiffness with strains, the stiffness coupled with temperature and thermal expansion coefficients, and the actuated magnetostrictive loads, which can be listed as follows:  
\[\left\{ \begin{array}{l}N_{xx}\\ N_{\theta\theta}\\ N_{x\theta}\end{array} \right\}=\int_{-\frac{\theta^{*}}{2}}^{\frac{b^{*}}{2}} \left\{ \begin{array}{l}\sigma_{x}\\ \sigma_{\theta}\\ \sigma_{x\theta}\end{array} \right\}dz,\ \left\{ \begin{array}{l}M_{xx}\\ M_{\theta\theta}\\ M_{x\theta}\end{array} \right\}=\int_{-\frac{\theta^{*}}{2}}^{\frac{b^{*}}{2}} \left\{ \begin{array}{l}\sigma_{x}\\ \sigma_{\theta}\\ \sigma_{x\theta}\end{array} \right\}dzdz,\ \left\{ \begin{array}{l}P_{xx}\\ P_{\theta\theta}\\ P_{x\theta}\end{array} \right\}=\int_{-\frac{b^{*}}{2}}^{\frac{b^{*}}{2}} \left\{ \begin{array}{l}\sigma_{x}\\ \sigma_{\theta}\\ \sigma_{x\theta}\end{array} \right\}z^{3}dz,\\ \left\{ \begin{array}{l}\sigma_{x\theta}\\ \sigma_{x\theta}\end{array} \right\}z^{3}dz,\\ \end{array}\right.\]  
The integral density parameters \(I_{i}=\sum\limits_{k=1}^{N^{*}}\int_{k}^{k+1}\rho^{(k)}z^{i}dz\), \((i=0,1,2,\cdots,6)\), where \(N^{*}\) denotes the layers total number, and \(\rho^{(k)}\) denotes the density in (_k_)-th ply. \(J_{i}=I_{i}-c_{1}I_{i+2}\), with subscript \(i=1\). \(K_{2}=I_{2}-2c_{1}I_{4}+c_{1}^{2}I_{6}\). \(q\) is the applied external pressure load. Strain-displacement expressions with \(\frac{\partial v_{0}}{\partial z}=\frac{-v_{0}}{R}\), \(\frac{\partial u_{0}}{\partial z}=\frac{-u_{0}}{R}\) and \(\frac{\partial v}{\partial z}=\frac{\partial\varphi_{x}}{\partial z}=\frac{ \partial\varphi_{y}}{\partial z}=0\) can be expressed in the following:  
\[