Approximate theory and analytical solution for functionally graded piezoelectric rectangular plates

Functionally graded piezoelectric material (FGPM) has electric-mechanically coupled property. Furthermore, its material property parameters vary continuously in some direction so that the stress concentration effect due to temperature change can be greatly decreased. Probably, some prospective properties can also be obtained by adjusting the varying gradient of the material property. Thus, it has a good prospect in application. Recently, study of FGPM is focused on the analyses of thermal response, static and dynamic response of the material, buckling behavior, fracture behavior of structures, optimal design and parameter identification of structures, etc. The theory and approaches for FGPM plates and shells includes simplified model method, laminated model method, asymptotic method, exact solution, finite element method, etc. But, the work for FGPM plates and shells based on approximate theory is rarely found. Although some exact solutions and FEM solutions have been worked out, simplified approximate solutions based on some assumptions with satisfactory precision are also attractive. In this work, several assumptions, such as Kirchhoff assumption, Reissner-Mindlin assumption and some other assumptions proposed by the authors are introduced. The first order shearing theory for plates is employed. Exponential gradient for material properties across the thickness of the plates is prescribed. Using the governing equations of FGPM and the pertinent boundary conditions, the approximate theory for the FGPM plate is established. The solutions of deflection slopes and electric potential for simply supported rectangular plate with its periphery grounded and electric-mechanical transverse loading applied are obtained. The electric potential solution is expressed in the form of double Fourier series. The solution is typically much simpler than the exact solutions, and its numerical computation is proved to be quite easy. The numerical results of this solution are given and compared with the 3D finite element solution by ANSYS. It is shown that this solution is in good agreement with the 3D finite element solution. It is found that the present solution has high precision even for thick plates. Finally, the limitation of this theory and the analytical solution is discussed.