On dispersion relations in piezoelectric coupled-plate structures

This paper presents the dispersion relations for wave propagation in piezoelectric coupled plates based on Kirchhoff and Mindlin plate theories. Two layers of piezoelectric actuators are surface bonded on the plate and poled in the transverse direction to induce flexural action. A half-cosine distribution for the electric potential in the transverse direction is assumed and the Maxwell static electricity equation is imposed for the piezoelectric layers. This assumption is verified numerically by finite-element analysis using a simple beam example. Based on Kirchhoff plate theory, a virtually linear relationship between the non-dimensional phase velocity and the non-dimensional wave number is obtained similar to that for pure plate structure. Based on Mindlin first-order plate theory, which accounts for the effects of both shear and rotary inertia, the phase velocity-wave number relationship deviates from linearity when the wave number is not small relative to the thickness of the beam. The phase velocity approaches a finite value, corresponding to the velocity of the Love wave, for large wave number in contrast to that based on Kirchhoff plate theory. The cut-off frequency is found to be a function of the ratio of the shear and flexural rigidities. The presence of the piezoelectric materials has the effect of reducing the phase velocity which is evident from the results of both Kirchhoff and Mindlin theories. Comparisons of the dispersive characteristics based on the two-plate models for different piezoelectric layers are also conducted by using a PZT actuator and a piezo-film sensor in the numerical simulations.