Dynamics of Plates with Thin Piezoelectric Layers

Licentiate thesis, 2007

The subject of this thesis is dynamics of plates with thin piezoelectric layers. Piezoelectric materials are often used in sensors and actuators and common applications for these are vibration control and ultrasonic transducers. In the first part of the thesis plate equations for a plate consisting of one anisotropic elastic layer and one piezoelectric layer with an applied electric voltage are derived. The displacements and the electric potential are expanded in power series in the thickness coordinate, which leads to recursion relations among the expansion functions. Using these in the boundary and interface conditions, a set of equations is obtained for some of the lowest-order expansion functions. This set is reduced to a system of six plate equations, where three of them are given to linear order in the thickness and correspond to the symmetric (in plane) motion, while the other three are given to quadratic order in the thickness and correspond to the antisymmetric (bending) motion. In principle it is possible to go to any order and it is believed that the equations are asymptotically correct. Some numerical comparisons are made with exact theory for infinite plates and very good agreement is obtained for low frequencies. In the second part of the thesis the plate equations are evaluated by investigating a vibration problem with a finite elastic layer and a shorter piezoelectric layer on top of it. The boundary conditions to combine with the plate equations are derived by inserting the power series expansions into the physical boundary conditions at the sides of the elastic layer and identifying equal powers of the thickness coordinate. Numerical comparisons of the displacement field are made with two other theories. The first one is another approximate theory based on the same type of power series expansions, where the piezoelectric layer is modeled as equivalent boundary conditions. The other one is exact three-dimensional theory. Both approximate theories yield accurate results for thin plates and low frequencies as long as the piezoelectric layer is thin in comparison to the total plate thickness.