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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. [ABSTRACT FROM AUTHOR]

Fundamental spatiotemporal field properties and particle velocity waveform signatures of sub-Rayleigh and supershear ruptures were experimentally investigated through a series of laboratory earthquake experiments. We appeal to dynamic rupture theory to extract and highlight previously unnoticed aspects and results, which are of direct relevance to our new experiments. Kinematic relationships derived from both singular and non-singular solutions are applied to analyze and interpret various features observed in these experiments. A strong correspondence is demonstrated between particle velocity records obtained in lab experiments and synthetic particle velocity waveform profiles derived from theory. Predicted temporal profiles, sense of particle motion, and amplitude decay properties of sub-Rayleigh and supershear particle velocity waveforms are experimentally verified. In a particular set of supershear rupture experiments, the fault-normal (FN) and fault-parallel (FP) velocity waveforms were simultaneously recorded at fixed, off-fault field points as a shear Mach front swept these locations. Particle velocity records collected over a broad range of stable supershear rupture speeds validate the predicted scaling relationship δ u ̇ 1 s / δ u ̇ 2 s = V r 2 / C s 2 − 1 = β s , between the FP ( δ u 1 s ̇ ) and the FN ( δ u 2 s ̇ ) velocity jumps propagated by a shear Mach front. Additional experimental findings include detailed rupture speed measurements of sub-Rayleigh and supershear ruptures and the observation of a supershear daughter crack with vanishing shear Mach front. Previously unappreciated scaling relations between particle velocity field components, attributed to dilatational and shear waves, are also developed and experimentally verified. In particular, the FP velocity jump δ u 1 s ̇ ( x 1 , x 2 ) propagated by the shear Mach front, and the sliding speed δ u ̇ 1 ( x 1 , 0 + ) , measured at a field point positioned extremely close to the frictional fault plane, are shown to obey a speed-dependent scaling relationship given by δ u ̇ 1 s / δ u ̇ 1 + = 1 − 2 C s 2 V r 2 , which was gleaned from a non-singular, steady state velocity field solution. [ABSTRACT FROM AUTHOR]