Dynamic equations for an orthotropic piezoelectric plate

Paper in proceeding, 2010

Piezoelectric materials have been used widely in applications for sensing and actuation purposes in recent years. As piezoelectric sensors and actuators usually are thin in comparison to the relevant wavelengths, the analysis of thin piezoelectric layers has attracted considerable research interest. The present work considers the derivation of new plate equations and the corresponding edge boundary conditions for an orthotropic piezoelectric plate. Here, power series expansions in the thickness coordinate for the displacements and the electric potential are stated, that lead to recursion relations among the expansion functions. These recursion relations are important as all fields hereby can be expressed in a finite number of expansions functions without performing any truncations. Moreover, the recursion formulas involve no approximations since they stem from the definition of the series expansions and are as such exact. Using the recursion relations in the boundary conditions on the surfaces and along the edges, a set of plate equations and the corresponding edge boundary conditions are obtained in a systematic fashion. Hence the surface boundary conditions constitute the plate equations of motion, and as such the lateral boundary conditions are exactly fulfilled. These plate equations can be truncated to any order in the thickness and it is believed that they are asymptotically correct, based on experience for other structures. Contrary to most traditional theories, there is no need to assume kinematical simplifications based on various engineering assumptions; neither for the equations of motion nor for the corresponding edge boundary conditions. Note that various sorts of series expansion procedure have been developed by others, but there are several differences in the derivation procedure among these works, such as in the series expansion formulation, the development of recursion relations or in the truncation process. The plate theory developed is given explicitly for flexural motion using a low order truncation. These equations of motion resemble the Mindlin type plate equations, albeit the present theory involves some further terms as well as no correction factors. The accuracy of the plate theory is displayed for the displacement, electric and stress fields, where comparisons are made with exact and traditional approximate theories. These results show that the present asymptotic plate theory renders considerably more accurate results than the Kirchhoff and the Mindlin theories.