## 3 Governing Equations of the Problem
\begin{array}{l}\frac{h_{b}}{2}\left(\frac{N_{r}^{t}-N_{d}^{t}}{r}+\frac{\partial N _{r}^{t}}{\partial r}\right)+\frac{M_{r}^{t}-M_{d}^{t}}{r}+\frac{\partial M_{r} ^{t}}{\partial r}-Q_{rz}^{t}\ =\ \left(I_{0}^{t}\frac{h_{b}}{2}+I_{1}^{t}\right)\ddot{u}_{0}\\ +\frac{h_{b}}{2}\left(\frac{h_{b}}{2}I_{0}^{t}+I_{1}^{t}\right)\ddot{\theta}_{ r}^{b}+\left[I_{2}^{t}+h_{t}I_{1}^{t}+\left(\frac{h_{b}}{2}\right)^{2}I_{0}^{t} \right]\ddot{\theta}_{r}^{t},\\ \\ \frac{h_{c}}{2}\left(\frac{N_{r}^{t}-N_{d}^{t}}{r}+\frac{\partial N_{r}^{t}}{ \partial r}\right)-\frac{h_{c}}{2}\left(\frac{N_{r}^{t}-N_{b}^{t}}{r}+\frac{ \partial N_{r}^{t}}{\partial r}\right)+\frac{M_{r}^{t}-M_{b}^{t}}{r}+\frac{ \partial M_{r}^{t}}{\partial r}-Q_{rz}^{c}=\\ \left(\frac{h_{c}}{2}I_{0}^{t}+I_{1}^{c}-\frac{h_{b}}{2}I_{0}^{b}\right)\ddot{u }_{0}+\frac{h_{c}}{2}\left(I_{1}^{t}+\frac{h_{b}}{2}I_{0}^{t}\right)\ddot{ \theta}_{r}^{t}+\left[\left(\frac{h_{c}}{2}\right)^{2}\left(I_{0}^{t}+I_{0}^{ b}\right)+I_{2}^{t}\right]\ddot{\theta}_{r}^{c}\\ -\frac{h_{c}}{2}\left(I_{1}^{t}-\frac{h_{b}}{2}I_{0}^{b}\right)\ddot{\theta}_{ r}^{b},\\ \\ -\frac{h_{c}}{2}\left(\frac{N_{r}^{t}-N_{b}^{t}}{r}+\frac{\partial N_{r}^{t}}{ \partial r}\right)+\frac{M_{r}^{t}-M_{b}^{t}}{r}+\frac{\partial M_{r}^{t}}{ \partial r}-Q_{rz}^{b}\ =\ \left(I_{1}^{b}-\frac{h_{b}}{2}I_{0}^{b}\right)\ddot{u}_{0}\\ +\left[I_{2}^{b}-h_{b}I_{1}^{b}+\left(\frac{h_{b}}{2}\right)^{2}I_{0}^{b} \right]\ddot{\theta}_{r}^{b}+\frac{h_{c}}{2}\left(\frac{h_{b}}{2}I_{0}^{b}-I_{ 1}^{b}\right)\ddot{\theta}_{r}^{c},\\ \\ \frac{\partial Q_{r}^{b}}{\partial r}+\frac{Q_{r}^{b}}{r}+\frac{\partial Q_{r}^{ c}}{\partial r}+\frac{Q_{r}^{c}}{r}+\frac{Q_{r}^{c}}{\partial r}+\frac{Q_{r}^{t} }{r}-(N_{t}+N_{m}+N_{e}+N_{c})\left(\frac{\partial^{2}w}{\partial r^{2}}+\frac {1}{r}\frac{\partial w}{\partial r}\right)\\ +k_{w}w-k_{g}\left(\frac{\partial^{2}w}{\partial r^{2}}+\frac{\partial w}{ \partial r}\right)\ =\ \left(I_{0}^{t}+I_{0}^{b}+I_{0}^{c}\right)\ddot{w}_{0},\\ \\ \int_{-\frac{h_{b}}{2}}^{\frac{h_{b}}{2}}\left[\cos\left(\frac{\pi\xi_{b}^{b}}{ h_{b}}\right)\frac{\partial D_{r}}{\partial\tau}+\frac{\pi}{h_{b}}\sin\left(\frac{\pi \xi_{b}^{b}}{h_{b}}\right)D_{z}\right]\,d\xi^{b}\ =\ 0,\\ \\ \int_{-\frac{h_{b}}{2}}^{\frac{h_{b}}{2}}\left[\cos\left(\frac{\pi\xi_{b}^{t}}{ h_{t}}\right)\frac{\partial D_{r}}{\partial\tau}-\frac{\pi}{h_{t}}\sin\left(\frac{\pi \xi_{t}^{t}}{h_{t}}\right)D_{z}\right]\,d\xi^{t}\ =\ 0,\\ \\ \int_{-\frac{h_{b}}{2}}^{\frac{h_{b}}{2}}\left[\cos\left(\frac{\pi\xi_{b}^{b}}{ h_{b}}\right)\frac{\partial B_{r}}{\partial\tau}+\frac{\pi}{h_{b}}\sin\left(\frac{\pi \xi_{b}^{b}}{h_{b}}\right)B_{z}\right]\,d\xi^{b}\ =\ 0,\\ \\ \int_{-\frac{h_{b}}{2}}^{\frac{h_{b}}{2}}\left[\cos\left(\frac{\pi\xi_{b}^{t}}{ h_{t}}\right)\frac{\partial B_{r}}{\partial\tau}-\frac{\pi}{h_{t}}\sin\left(\frac{\pi \xi_{t}^{t}}{h_{t}}\right)B_{z}\right]\,d\xi^{t}\ =\ 0,\\ \\ \int_{-\frac{h_{b}}{2}}^{\frac{h_{b}}{2}}\left[\cos\left(\frac{\pi\xi_{b}^{t}}{ h_{t}}\right)\frac{\partial B_{r}}{\partial\tau}-\frac{\pi}{h_{t}}\sin\left(\frac{\pi \xi_{t}^{t}}{h_{t}}\right)B_{z}\right]\,d\xi^{t}\ =\ 0,\\ \\ \end{array}  \tag{40}\]  
The stress resultants \(M\), \(N\) and \(Q\) can be described as follows:  
\[\int_{-\frac{h_{b}}{2}}^{\frac{h_{b}}{2}}\left[\cos\left(\frac{\pi\xi_{t}^{t}}{ h_{t}}\right)\frac{\partial B_{r}}{\partial\tau}-\frac{\pi}{h_{t}}\sin\left(\frac{\pi \xi_{t}^{t}}{h_{t}}\right)B_{z}\right]\,d\xi^{t}\ =\ 0,\\ \\ \ Q_{r}^{t}\ =\ \int_{-\frac{h_{r}}{2}}^{\frac{h_{r}}{2}}\tau_{rz}^{i}\,d\xi^{i},\ \ i\ =\ t,c,b\ \ j\ =\ r,\theta\\ \\ \end{array} \tag{41}\]  
and  
\[I_{f}^{i}\ =\ \int_{-\frac{h_{r}}{2}}^{\frac{h_{r}}{2}}\rho^{i}\zeta^{\bar{\xi} \bar{\xi}}d\xi^{i},\ \ f\ =\ 0,1,2 \tag{42}\]  
More details about these terms can be found in Appendix A. Various boundary conditions at both edges of the sandwich plate are, thus, assumed as follows [18]:  
\[\left\{