Bending behaviors of the in-plane bidirectional functionally graded piezoelectric material plates

Abstract

The transverse bending behaviors of in-plane bidirectional functionally graded piezoelectric material (FGPM) plates are semi-analytically investigated by the scaled boundary finite element method (SBFEM) in association with the precise integration method (PIM). The proposed scheme is able to explore the structural characteristics of FGPM plates with the material coefficients obeying arbitrary form of mathematical functions according to the in-plane coordinates. The present methodology selects only four quantities consisting of three translational displacement components and the electric potential as the fundamental unknowns. Additionally, variations of the four primary variables across the thickness direction are expressed as an analytical exponential matrix. In the developed approach, plates are regarded as a kind of three dimensional structure. But, an arbitrary in-plane surface discretized with two-dimensional spectral elements is set as the research domain. The practice is conducive to cut down the computational effort and increase the calculation efficiency. The SBFEM governing equations are formulated from the three-dimensional basic equations of piezoelectric materials without any assumptions on the plate kinematics and distributions of electromechanical components. By means of the scaled boundary coordinate system and the dual vector methodology, the key partial differential equations of piezoelectric materials are conveniently converted into the easily-solved first order ordinary differential SBFEM governing equation. The stiffness matrix is constructed from the analytical exponential matrix aided by the highly accurate PIM to predict the changing patterns of mechanical and electric quantities. To further improve the precision, the technology of dividing the plate into two parts with equal thickness is exploited. Finally, numerical exercises of square, rectangular and triangular piezoelectric plates are provided to validate the accuracy and fast convergence of the developed technique and reveal the effect of geometrical shapes, gradient functions, types of external loadings and thickness-to-span ratios on the static flexure of FGPM plates owning the in-plane bidirectional stiffness.