Wave propagation in three-phase porous piezoelectric media: theory and simulation

Acoustic-wave propagation in multiphase porous piezoelectric media is closely related to optimizing piezoelectric composite transducers and to the remodelling and regeneration of cancellous bone. Most existing wave theories, however, treat a two-phase system of one piezoelectric skeleton and one fluid, whereas classical three-phase models seldom include both anisotropy and piezoelectric coupling. This study investigates, both theoretically and numerically, elastic wave propagation in a three-phase porous piezoelectric medium. The medium comprises a transversely isotropic piezoelectric skeleton, a compressible pore fluid, and a non-piezoelectric filling solid, with piezoelectricity confined to the skeleton. The governing equations couple a Biot-type three-phase momentum balance with the piezoelectric constitutive relations. Introducing a plane-wave ansatz reduces the problem to a complex Christoffel system, which yields the phase velocity, group velocity, and attenuation of seven body-wave modes: three quasi-compressional, two quasi-shear, and two horizontally polarized shear (SH) waves. For wavefield simulation, a two-dimensional electromechanically coupled staggered-grid finite-difference time-domain algorithm is developed, together with a coupled electromechanical perfectly matched layer for the poroelastic and piezoelectric fields. The simulated arrival times agree with the Christoffel predictions to within 1%, and, in the reduced single-solid, single-fluid limit, the scheme is benchmarked against a commercial finite-element poroelastic solution. The results show that piezoelectric coupling anisotropically stiffens the skeleton-borne quasi-compressional and quasi-shear waves, while leaving the SH waves unchanged. Interphase friction selectively attenuates the slow modes: the friction between skeleton and pore fluid predominantly damps the fluid-borne slow wave, whereas that between the two solids attenuates the filling-solid-borne waves.