## 3 The Actuating Equation of an MFC Laminated Plate Structure
Therefore, the normal strain at the border position on the upper surface of the MFC can be regarded as a constant value:  
\[\overline{\varepsilon}_{px,e}=\varepsilon_{x0}+\frac{Q_{12}}{Q_{11}}\Big{(} \varepsilon_{y0}-\varepsilon_{py,c}\Big{)},\overline{\varepsilon}_{py,e}= \varepsilon_{y0}+\frac{Q_{21}}{Q_{22}}\Big{(}\varepsilon_{x0}-\varepsilon_{ px,c}\Big{)}. \tag{30}\]  
Equations (25) and (30) are substituted into Equation (22) to obtain the normal strain of the upper surface of the MFC along the direction of the \(x\) axis:  
\[\varepsilon_{px,R}=\frac{\overline{\varepsilon}_{px,e}\cosh(\alpha x)}{\cosh (0.5\alpha l_{2})}+\varepsilon_{px,c}\bigg{(}1-\frac{\cosh(\alpha x)}{\cosh(0.5 \alpha l_{2})}\bigg{)}. \tag{31}\]  
Equations (9), (11) and (31) are combined to obtain the interfacial shear stress between the MFC and the plate structure along the direction of the \(x\) axis, namely, the actuating stress of the MFC along the direction of the \(x\) axis:  
\[\tau_{x}=\frac{\alpha f_{r}Q_{11}^{E}\cosh(\alpha x)}{\Omega_{px,R}\cosh(0.5 \alpha l_{2})}\Big{(}\varepsilon_{px,c}-\overline{\varepsilon}_{py,c}\Big{)}, \tag{32}\]  
where,  
\[f_{r}=1-\frac{4r^{2}}{\left(R+r\right)^{2}}+\frac{2a_{1}-4b_{1}}{h_{p}}, \tag{33}\]  
In a similar way, the interfacial shear stress along the direction of the \(y\) axis is obtained, namely, the actuating stress of the MFC along the direction of the \(y\) axis:  
\[\tau_{y}=\frac{\beta frQ_{22}^{E}\sinh(\beta y)}{\Omega_{py,R}\cosh(0.5\beta l _{1})}\Big{(}\varepsilon_{py,c}-\overline{\varepsilon}_{py,e}\Big{)}, \tag{34}\]  
where,  
\[\beta=\sqrt{\frac{G_{s}\Omega_{sy,r}}{Q_{s}\Lambda_{py}}-\frac{G_{p}\Omega_{ py,R}}{Q_{22}^{E}\Lambda_{py}}},\Omega_{sy,r}=\Omega_{xx,r},\Lambda_{sy}= \Lambda_{xx},\Omega_{py,R}=\Omega_{px,R},\Lambda_{py}=\Lambda_{px}. \tag{35}\]  
According to Equations (25), (26) and (30), \(\varepsilon_{px,c}\), \(\varepsilon_{py,c}\), \(\overline{\varepsilon}_{px,e}\) and \(\overline{\varepsilon}_{py,e}\) all include an influence of the piezoelectric effect in the directions of the \(x\) and \(y\) axes, and the magnitude of the interfacial shear stress \(\tau_{x}\) and \(\tau_{y}\) is related to the piezoelectric effect in the directions of the \(x\) and \(y\) axes. According  
Figure 4: The stress of the infinitesimal element at the border position on the upper surface of the MFC. to Equations (32) and (34), the total actuating forces \(T_{x}\) and \(T_{y}\) of the MFC at the center of the MFC laminated plate structure can be obtained:  
\[T_{x}=\frac{f_{l1}Q_{11}^{E}}{\Omega_{px,R}}\Big{(}\varepsilon_{px,c}-\overline{ \varepsilon}_{px,c}\Big{)},T_{y}=\frac{f_{l2}Q_{22}^{E}}{\Omega_{py,R}}\Big{(} \varepsilon_{py,c}-\overline{\varepsilon}_{py,c}\Big{)}. \tag{36}\]  
Equation (36) indicates that the MFC actuating force is related to the elasticity modulus and the thickness of the plate structure. To explore their relationship, we have analyzed the relationship between the elasticity modulus of the plate, the thickness of plate and the total actuating force of the M2814-P1 type and the M2814-P2 type of MFC along the direction of the \(x\) axis under a voltage of 1500 V, as shown in Figure 5. It shows that the total actuating force of the MFC increases with the elasticity modulus and the thickness of the beam structure at first, then its growth rate gradually slows down and tends to become stable. There is a non-linear relationship between the actuating force of the MFC, the structural elasticity modulus and the thickness. A comparison has been made between the distribution formula of the mean normal stress based on SSDT and that based on CPT to analyze the distribution regularities of the mean normal stress of the MFC in MFC laminated plates.