## 3 The Actuating Equation of an MFC Laminated Plate Structure  
The laminated plate is made by single-face stickup piece of MFC, whose size is shown in Figure 2, with the original point of coordinate axes in the central position of the plate. \(l_{1}\) and \(l_{2}\) are the MFC's effective width and length; \(R\) is the distance between the plate midline and the upper surface of the MFC; \(h_{p}\) is the thickness of the MFC; the thickness of the plate is \(2r\), the elasticity modulus is \(E_{s}\), the shear elasticity is \(G_{s}\), Poisson's ratio is \(\
u_{s}\). Figure 1: The structure chart of the macro fiber composite (MFC). (**a**) The P1 type of MFC; (**b**) The P2 type of MFC. The action of voltage will create axial, bending and shear deformations of the MFC attached to the plate's surface. The actuating force caused by the deformations will be transmitted through the layer sticking up between the MFC and the plate. To consider the influence of the shear deformation of the MFC laminated plate on the MFC actuating force, SSDT [32] is adopted to deduce the MFC actuating force formula. Considering that differences exist in material parameters between the MFC and the plate, deformation compatibility and stress balance should be ensured between them, global-local displacement distribution functions are introduced to establish the stress balance equation of MFC laminated plate, and then, according to the equal shear force between the MFC and the plate, the system of stress partial differential equations is simplified and solved to obtain the interfacial shear force between the MFC and the plate structure and the actuating force and actuating bending moment of the MFC. The following hypothesis is assumed for the MFC actuating force calculation: (1) the MFC and plate structure are entirely adhered without considering the thickness of the stickup layer [21]; (2) the MFC curvature along the directions of the \(x\) and \(y\) axes can be viewed as constant [19]; (3) no shear deformation exists at the midline position of the \(xy\) in-plane MFC laminated plate [33]. According to the SSDT proposed by [PERSON] [25], displacement of the MFC laminated plate is composed of axial displacement, displacement created by bending deformation and by shear deformation. Suppose that the displacement \(w(x,y,z)\) in the normal direction of the structure has nothing to do with \(z\), then the displacement and strain on the MFC laminated plate can be indicated as:  
\[ \begin{array}{l}u(x,y,z)=u_{0}(x,y)-z\frac{\partial w(x,y)}{ \partial x}+\psi_{x}(x,y)f_{x}(z)\\ v(x,y,z)=v_{0}(x,y)-z\frac{\partial w(x,y)}{\partial y}+\psi_{y}(x,y)f_{y}(z )\\ \qquad\qquad w(x,y,z)=w(x,y)\end{array} , \tag{5}\]  
where \(u_{0}(x,y)\) and \(v_{0}(x,y)\) are the displacements in the mid-place position along the \(x\) and \(y\) axes; \(w(x,y)\) is the lateral displacement of the plate; \(\psi_{x}(x,y)\) and \(\psi_{y}(x,y)\) are the shear deformations of the cross section along the \(x\) and \(y\) axes; \(f_{x}(z)\) and \(f_{y}(z)\) are the displacement distribution functions along the thickness in the \(x\) and \(y\) directions. Due to different material properties of the MFC and the plate structure, global-local displacement distribution functions are introduced to ensure deformation compatibility and stress balance between them, among which \(f_{g}(z)\) is the global displacement distribution function and \(f_{l}(z)\) is the local displacement distribution function. Consequently, the sectional global-local displacement distribution functions of MFC laminated plate are:  
\[ \begin{array}{l}f_{x}(z)=f_{y}(z)=f_{g}(z)+f_{l}(z),f_{g}(z)=\Big{(}\frac{R +\tau}{\pi}\Big{)}sin\Big{(}\frac{\pi\pi}{R+\tau}\Big{)}\\ f_{l}(z)=\left\{\begin{array}{l}a_{1}\xi_{1}+b_{1}\xi_{1}{}^{2},z\in[R,r]\\ a_{2}\xi_{2}+b_{2}\xi_{2}{}^{2},z\in[r,-r]\end{array}\right.,\end{array}  \tag{6}\]  
Figure 2: MFC laminated plate structure. where \(\xi_{1}\) and \(\xi_{2}\) are local coordinates of the section.