Symplectic analysis of plane problems of functionally graded piezoelectric materials

This paper develops an analytical method to investigate the plane problems of functionally graded piezoelectric materials within the symplectic framework. The material constants, including the elastic, piezoelectric and dielectric constants are assumed to vary along the length in an identical exponential form. A matrix state equation is derived by introducing new stress and electric displacement components, and is solved using the method of separation of variables. The operator matrix in the state equation is found to have similar properties as Hamiltonian matrix for homogeneous materials. Its eigenvectors (and hence eigensolutions) corresponding to particular eigenvalues (0 and −α) are derived; while the former also present in the homogeneous materials, the latter bear complete different forms. A detailed analysis shows, however, that the −α-group eigensolutions can degenerate to the ones for the homogeneous materials after eliminating certain rigid motions. Numerical results are given to show the effect of material inhomogeneity on these eigensolutions.