Dispersion and attenuation relations in flexoelectricity

The dispersion relations in flexoelectricity are examined for plane time-harmonic waves that propagate in the flexoelectric materials. In contrast to classic elastodynamics, dispersion is observed in the displacement field due to two micro-structural and two micro-inertial lengths that emerge from the electromechanical coupling. In the absence of such coupling, we return to the classic elastodynamic results. The problem dissociates in longitudinal and transverse waves, as is the case in classic elastodynamics. The group velocity of the mechanical field is also the velocity of the energy transfer across the planes of the waves. An optical branch of the dispersion relation appears due to the polarization field that follows the mechanical field. The longitudinal and transverse velocities of the plane waves was found to depend on the corresponding microstructural lengths and are less than or equal to the classic plane wave velocities because the micro-inertial lengths are greater than or equal to the micro-structural length. The opposite effect is expected when we encounter flexoelectric metamaterials in which case the micro-inertial lengths are less than the micro-structural length.