Thickness resonances dispersion characteristics of a lossy piezoceramic plate with electrodes of arbitrary conductivity

A theoretical description of the dissipative phenomena in the wave dispersion related to the "energy-trap" effect in a thickness-vibrating, infinite thickness-polarized piezoceramic plate with resistive electrodes is presented. The three-dimensional (3-D) equations of linear piezoelectricity were used to obtain symmetric and antisymmetric solutions of plane harmonic waves and investigate the eigen-modes of thickness longitudinal (TL) up to third harmonic and shear (TSh) up to ninth harmonic vibrations of odd-and even-orders. The effects of internal and electrode energy dissipation parameters on the wave propagation under regimes ranging from a short-circuit (sc) condition through RC-type relaxation dispersion to an open-circuit (oc) condition are examined in detail for PZT piezo-ceramics with three characteristic TL-mode energy-trap figure-of-merit c dot 33 D /c dot 44 E values - less, near equal and higher 4 - when the second harmonic spurious TSh resonance lies below, inside, and above the fundamental TL resonance-antiresonance frequency interval. Calculated complex lateral wave number dispersion dependences on frequency and electrode resistance are found to follow the universal scaling formula similar to those for dielectrics characterization. Formally represented as a Cole-Cole diagram, the dispersion branches basically exhibit Debye-like and modified Davidson-Cole dependences. Varying the dissipation parameters of internal loss and electrode conductivity, the interaction of different branches was demonstrated by analytical and numerical analysis. For the purposes of dispersion characterization of at least any thickness resonance, the following theorem was stated: the ratio of two characteristic determinants, specifically constructed from the oc and sc boundary conditions, in the limit of zero lateral wave number, is equal to the basic elementary-mode normalized admittance. As was found based on the theorem, the dispersion near the basic and nonbasic TL and TSh resonances reveal some simple representations related to the respective elementary admittance and showing the connection between the propagation and excitation problems in a continuous piezoactive medium.