## 3 The Actuating Equation of an MFC Laminated Plate Structure
Equation (17) is simplified as:  
\[u_{r}=\frac{\Lambda_{xx}}{\Lambda_{px}}\bigg{(}u_{R}-u_{r}+\int_{r}^{R}\frac{ \partial w(x,y)}{\partial x}dz\bigg{)}+u_{-r}-\int_{-r}^{r}\frac{\partial w( x,y)}{\partial x}dz. \tag{18}\]  
Equation (18) is substituted into Equation (15) to eliminate the displacement \(u_{-r}\) of the lower surface of the plate along the direction of \(x\) axis, obtaining:  
\[\frac{d\sigma_{sx,r}}{dx}+\frac{G_{s}\Omega_{sx,r}}{\Lambda_{px}}\bigg{(}u_{R }-u_{r}+\int_{r}^{R}\frac{\partial w(x,y)}{\partial x}dz\bigg{)}=0. \tag{19}\]  
According to the stress-strain relationship, Equations (11) and (19) are translated into the system of strain partial differential equations of the upper surface of the MFC and the upper surface of the plate:  
\[ \begin{array}{l}\frac{d\varepsilon_{sx,r}}{dx^{2}}+\frac{G_{s}\Omega_{sx,r} h_{p}}{Q_{s}\Lambda_{px}}\bigg{(}\frac{\varepsilon_{sx,R}-\varepsilon_{sx,r}}{h_{p} }+\frac{\partial^{2}w(x,y)}{\partial x^{2}}\bigg{)}=0\\ \frac{d^{2}\varepsilon_{sx,R}}{dx^{2}}+\frac{G_{p}\Omega_{px,R}h_{p}}{Q_{11}^{2 }\Lambda_{px}}\bigg{(}\frac{\varepsilon_{sx,R}-\varepsilon_{sx,r}}{h_{p}}+ \frac{\partial^{2}w(x,y)}{\partial x^{2}}\bigg{)}=0\end{array} , \tag{20}\]  
where \(Q_{s}=\frac{E_{s}}{1-v_{s}^{2}}\). According to the piezoceramics actuating force formula put forward by [PERSON] et al. [19], the mutation of MFC stress-strain along the direction of the \(x\) axis appears only at the border position of the piezoelectric material, while its stress-strain at other positions almost remains constant. A proportional relationship exists between the MFC curvature and the normal strain distribution of the cross-section, and the normal strain distribution except that at border position remains constant. Thus, supposing that the curvature of the MFC laminated plate structure along the direction of the \(x\) axis is constant, then it is obtained by Equation (20):  
\[\frac{d^{4}\varepsilon_{sx,r}}{dx^{4}}+\frac{G_{s}\Omega_{sx,r}}{Q_{s}\Lambda_ {px}}\bigg{(}\varepsilon_{px,R}^{*}-\varepsilon_{sx,r}^{*}\bigg{)}=0,\frac{d ^{4}\varepsilon_{px,R}}{dx^{4}}+\frac{G_{p}\Omega_{px,R}}{Q_{11}^{E}\Lambda_{ px}}\bigg{(}\varepsilon_{px,R}^{*}-\varepsilon_{sx,r}^{*}\bigg{)}=0. \tag{21}\]  
Because no normal stress at the border position of the upper surface of the MFC and the normal strain along the direction of \(x\) axis is symmetric around the \(y\) axis, the following is obtained:  
\[\varepsilon_{px,R}=X_{1}\cosh(ax)+Y_{1},\varepsilon_{sx,r}=X_{2}\cosh(ax)+Y_{ 2}, \tag{22}\]  
where  
\[\alpha=\sqrt{\frac{G_{s}\Omega_{sx,r}}{Q_{s}\Lambda_{px}}\ \frac{G_{p}\Omega_{px,R}}{Q_{11}^{E}\Lambda_{px}}}. \tag{23}\]  
Equation (22) is the general solution form of the strain distribution of the upper surface of the MFC. To calculate the strain distribution of the upper surface of the MFC further, the strain at the midline position and the border position of the upper surface of the MFC should be obtained. Since the raised position between the MFC and the plate is symmetrical and no shear deformation is at the midline position, \(\psi_{x}(0,y)\) in Equation (5) is 0 and the midline position is shown in Figure 3. With Equation (5), the strain \(\varepsilon_{x,c}\) and \(\varepsilon_{y,c}\) can be obtained at the midline position of the MFC laminated plate  
The strain at the border position on the upper surface of the MFC is:  
\[ \begin{split}\varepsilon_{px,e}&=\varepsilon_{x0}+\frac{Q _{12}^{E}}{Q_{11}^{E}}\Big{(}\varepsilon_{y0}-\varepsilon_{py,c}\Big{)}\bigg{(} 1-\frac{\cosh(\beta y)}{\cosh(0.5\beta l_{1})}\bigg{)}\\ \varepsilon_{py,e}&=\varepsilon_{y0}+\frac{Q_{21}^{ E}}{Q_{22}^{E}}\Big{(}\varepsilon_{x0}-\varepsilon_{px,c}\Big{)}\bigg{(}1-\frac{ \cosh(\alpha x)}{\cosh(0.5\alpha l_{2})}\bigg{)}\end{split} . \tag{29}\]  
According to Equation (29), mutations of \(\varepsilon_{px,e}\) and \(\varepsilon_{py,e}\) exist at the two corners of the border position of the MFC in the directions of the \(x\) and \(y\) axes, but normal strain at other border positions remains essentially constant.