Modelling of thin piezoelectric layers on plates

The derivation of plate equations for a plate consisting of two layers, one anisotropic elastic and one piezoelectric, is considered. Power series expansions in the thickness coordinate for the displacement components and the electric potential lead to recursion relations among the expansion functions. Using these in the boundary and interface conditions, a set of equations are obtained for some of the lowest-order expansion functions. This set is reduced to six equations corresponding to the symmetric (in-plane) and antisymmetric (bending) motions of the elastic layer. These equations are given to linear (for the symmetric equations) or quadratic (for the antisymmetric equations) order in the thickness. It is noted that it is, in principle, possible to go to any order, and that it is believed that the corresponding equations are asymptotically correct. A few numerical results for guided waves along the plate and a 1D actuator case illustrate the accuracy.