Dispersion phenomena in transversely isotropic piezo-electric plates with either short or open circuit boundary conditions

Abstract

The dispersion of small amplitude waves in a transversely isotropic, piezo-electric plate is discussed in respect of both short circuit and open circuit boundary conditions. In both cases the mechanical boundary conditions are taken as traction-free. In both cases, symmetric and anti-symmetric dispersion relations are derived, with long and short wave approximations then established, giving phase speed, and frequency, as functions of scaled wave number. It is shown that some particularly novel features occur within the vicinity of the associated cut-off frequencies. In particular, it is established that for some families the cut-off frequencies depend only on elastic terms, with others depending both on electrical and elastic terms. In each case, the appropriate asymptotic form of displacement is established. This reveals that for motion close to some frequencies, one of the scaled displacements is an order of magnitude larger than the electric potential, however for motion close to other frequencies the opposite situation arises. This information may have applications for the development and design of sensing and actuating devices. The paper also provides the necessary asymptotic framework for the derivation of asymptotically approximate models to fully elucidate the dynamic response of such plates near these resonance frequencies.