Approximate theory and analytical solution for functionally graded piezoelectric rectangular plates

Functionally graded piezoelectric material (FGPM) haselectric-mechanically coupled property. Furthermore, its material propertyparameters vary continuously in some direction so that the stressconcentration effect due to temperature change can be greatly decreased.Probably, some prospective properties can also be obtained by adjusting thevarying gradient of the material property. Thus, it has a good prospect inapplication.Recently, study of FGPM is focused on the analyses of thermal response,static and dynamic response of the material, buckling behavior, fracturebehavior of structures, optimal design and parameter identification ofstructures, etc. The theory and approaches for FGPM plates and shellsincludes simplified model method, laminated model method, asymptotic method,exact solution, finite element method, etc. But, the work for FGPM platesand shells based on approximate theory is rarely found. Although some exactsolutions and FEM solutions have been worked out, simplified approximatesolutions based on some assumptions with satisfactory precision are alsoattractive. In this work, several assumptions, such as Kirchhoff assumption,Reissner-Mindlin assumption and some other assumptions proposed by theauthors are introduced. The first order shearing theory for plates isemployed. Exponential gradient for material properties across the thicknessof the plates is prescribed. Using the governing equations of FGPM and thepertinent boundary conditions, the approximate theory for the FGPM plate isestablished. The solutions of deflection slopes and electric potential forsimply supported rectangular plate with its periphery grounded andelectric-mechanical transverse loading applied are obtained. The electricpotential solution is expressed in the form of double Fourier series. Thesolution is typically much simpler than the exact solutions, and itsnumerical computation is proved to be quite easy. The numerical results ofthis solution are given and compared with the 3D finite element solution byANSYS. It is shown that this solution is in good agreement with the 3Dfinite element solution. It is found that the present solution has highprecision even for thick plates. Finally, the limitation of this theory andthe analytical solution is discussed.