The Symplectic Solution for the Bi-Directional Functionally Graded Piezoelectric Materials

The symplectic method, is applied to study analytically the deflection field of bi-directional functionally graded piezoelectric materials in this paper. And the material properties varies exponentially both along the axial and transverse coordinates.The dual equations were presented by variation principle and introducing separation of variables used. Then in the symplectic space which consists of the original variables and their dual variables, the problem can be solved via symplectic expansion. This comparisons with experimental data were carried out to verify the validity of the symplectic method.