The symplectic method for plane problem of Functionally Graded Piezoelectric Materials

This paper applies the symplectic method to solve the plane problem of functional graded piezoelectric materials (FGPM) whose elastic stiffness, piezoelectric and dielectric constants vary exponentially with the axial coordinate. After introducing the displacements (the electrical potential function) and their conjugate stress (electric displacement), the problem is formulated within the frame of state space and it is solved using the method of separation of variables along with the eigenfunction expansion technique. Compared with that for homogeneous materials, the operator matrix is not in an exact Hamiltonian form, but it has similar properties. This operator matrix is called the shifted-Hamiltonian matrix since the eigenvalues are symmetric with respect to -alpha/2, rather than zero in the normal Hamilton matrix. In this case, the symplectic adjoint eigenvalue of zero isn´t itself but -alpha . In this paper the eigensolutions corresponding to zero and -alpha are gained which indicate certain physical essence of the problem that can not be revealed by other methods. These also can be degenerated to the ones for homogeneous materials after suppressing certain rigid motions.