An Analysis Method Of Symplectic System For Mechanics In Piezoelectric Materials

Piezoelectric materials can change electric effect and mechanical deformation to eachother. Because piezoelectric materials are very fast in reacting, easy to control and verysimple in structure, they are widely used in many fields like electrics, machine and so on.With the development of science and technology, more and more questions emerged. So theresearches of piezoelectricity become very important. But the coupling characteristic ofpiezoelectric materials makes the problem very complicated. A new method in piezoelectricmaterials is quite necessary in the research work.This paper presents a symplectic method of transversely isotropic piezoelectric materialin plane and 3D cylinder. A duality system is established directly by introducing dualvariables and a complete space of eigensolutions is obtained. All solutions of the problem areboiled down to zero-eigenvalue solutions and all their Jordan normal form of thecorresponding Hamiltonian operator matrix and non-zero eigenvalue solutions. The problemchanges into getting zero eigensolutions and non-zero eigensolutions.The classical solutions are described by zero eigensolutions, which express the simpleextension, pure bending, shearing-bending, equipotential field, uniform electric displacementfield and so on. On the other hand, non-zero eigensolutions show localized solutions, whichare ignored in San-Vienna solutions. Non-zero eigensolutions stand for the boundary effectscaused by some special boundary conditions, and these effects are related with the boundaryconditions of ends and are attenuated with the distance to the end.Formulations and numerical examples are listed in the paper for both plane and 3Dcylinder problems. The adjoint symplectic orthogonal relationship of eigensolutions are usedin equations. The figures of numerical results show effects of different stress anddisplacement loaded on transversely isotropic piezoelectric materials.A conclusion can be made from the numerical results that the symplectic method fits thepiezoelectric problems well. It is especially good in describing the localized effects and theirreduction, which are seldom mentioned in former researches. The duality system provides aneffective way in solving piezoelectric problems.