## 3 The Actuating Equation of an MFC Laminated Plate Structure
\[\xi_{1}=\frac{2z-R-r}{R-r},\xi_{2}=\frac{z}{r}, \tag{7}\]  
According to deformation compatibility and stress balance, the shear stress between the upper surface of the MFC and the lower surface of the plate is 0, and the displacement of the raised layer between the MFC and the plate structure should remain continuous and their shear stresses should be equal. The relevant parameters of the local displacement distribution function can be obtained from the above-mentioned boundary conditions:  
\[ \begin{array}{l}a_{1}=\frac{(r-R)\left(\left(3G_{p}-4G_{s}\right)r\cos\left( \frac{mr}{R+r}\right)+\left(3G_{p}r+G_{s}(R-r)\right)\cos\left(\frac{\pi R}{R+ r}\right)\right)}{12G_{p}r+1G_{s}(R-r)}\\ a_{2}=\frac{r\left(\left(-7G_{p}r+3G_{p}R+6G_{s}(r-R)\right)\cos\left(\frac{\pi R }{R+r}\right)+G_{p}(R-r)\cos\left(\frac{\pi R}{R+r}\right)\right)}{12G_{p}r+ 12G_{s}(R-r)}\end{array} . \tag{8}\]  
To establish the relationship between shearing strain and displacement, the MFC is used as the object to obtain its shearing strain in \(xz\) plane with Equation (5):  
\[\gamma_{zx}(x,y,z)=\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x }=\frac{1}{\Lambda_{px}}\left(u_{R}-u_{r}+\int_{r}^{R}\frac{\partial w(x,y)} {\partial x}dz\right)\frac{df_{x}(z)}{dz}, \tag{9}\]  
\[\Lambda_{px}=\Lambda_{py}=\frac{R+r}{\pi}\left(\sin\left(\frac{\pi R}{R+r} \right)-\sin\left(\frac{\pi r}{R+r}\right)\right)+2a_{1}, \tag{10}\]  
where \(u_{R}\) and \(u_{r}\) represent the displacements at the positions of \(z=R\) and \(z=r\) along the direction of the \(x\) axis. The stress balance equation of the upper surface of the MFC is obtained according to the stress balance of the MFC:  
\[\frac{d\sigma_{px,R}}{dx}+\frac{G_{p}\Omega_{px,R}}{\Lambda_{px}}\left(u_{R} -u_{r}+\int_{r}^{R}\frac{\partial w(x,y)}{\partial x}dz\right)=0, \tag{11}\]  
\[\Omega_{px,R}=\Omega_{py,R}=\frac{8b_{1}}{\left(R-r\right)^{2}}-\frac{\pi}{R +r}\sin\left(\frac{\pi R}{R+r}\right), \tag{12}\]  
where \(\sigma_{px,R}\) is the normal stress of the upper surface of the MFC along the direction of the \(x\) axis. Similarly, the shearing strain of the plate structure in the \(xz\) plane is:  
\[\gamma_{zx}(x,y,z)=\frac{1}{\Lambda_{xx}}\left(u_{r}-u_{-r}+\int_{-r}^{r} \frac{\partial w(x,y)}{\partial x}dz\right)\frac{df_{x}(z)}{dz}, \tag{13}\]  
\[\Lambda_{xx}=\Lambda_{sy}=2a_{2}+\frac{R+r}{\pi}\sin\left(\frac{\pi r}{R+r} \right), \tag{14}\]  
where \(u_{-r}\) represents the displacement at the position of \(z=-r\) along the direction of the \(x\) axis. Similarly, the stress balance equation of the upper surface of the plate can be obtained with its upper surface as the object:  
\[\frac{d\sigma_{xx,r}}{dx}+\frac{G_{s}\Omega_{sx,r}}{\Lambda_{sx}}\left(u_{r} -u_{-r}+\int_{-r}^{r}\frac{\partial w(x,y)}{\partial x}dz\right)=0, \tag{15}\]  
\[\Omega_{sx,r}=\Omega_{sy,r}=-\frac{2b_{2}}{r^{2}}+\frac{\pi}{R+r}\sin\left( \frac{\pi r}{R+r}\right), \tag{16}\]  
where \(\sigma_{sx,R}\) is the normal stress of the upper surface of the plate along the direction of the \(x\) axis. According to the equal shear stress on the raised interface, Equations (5), (9) and (13) are combined:  
\[ \begin{array}{l}G_{p}\bigg{(}\psi_{x}(x,y)+\frac{\partial w(x,y)}{\partial x} \bigg{)}\frac{d\varepsilon_{x}(z)}{dz}\bigg{|}_{z=r+}=G_{s}\bigg{(}\psi_{x}(x,y )+\frac{\partial w(x,y)}{\partial x}\bigg{)}\frac{d\varepsilon_{x}(z)}{dz} \bigg{|}_{z=r-}\\ \frac{G_{p}}{\Lambda_{px}}\bigg{(}u_{R}-u_{r}+\int_{r}^{R}\frac{\partial w(x, y)}{\partial x}dz\bigg{)}\frac{d\varepsilon_{x}(z)}{dz}=\frac{G_{s}}{\Lambda_{xx}} \bigg{(}u_{r}-u_{-r}+\int_{-r}^{r}\frac{\partial w(x,y)}{\partial x}dz\bigg{)} \frac{d\varepsilon_{x}(z)}{dz}\ \,\end{array}  \tag{17}\]  
where \(z=r+\) represents the lower surface of the MFC; \(z=r-\) represents the upper surface of the plate structure.