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.

中文翻译：

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挠曲电中的色散和衰减关系


研究了在挠曲电材料中传播的平面时谐波波的挠曲电色散关系。与经典弹性动力学相反，由于机电耦合产生的两个微观结构长度和两个微观惯性长度，在位移场中观察到色散。在没有这种耦合的情况下，我们回到经典的弹性动力学结果。这个问题在纵波和横波中分离，就像经典弹性动力学的情况一样。机械场的群速度也是能量在波平面上传递的速度。由于跟随机械场的偏振场，出现了色散关系的光学分支。发现平面波的纵向和横向速度取决于相应的微观结构长度，并且小于或等于经典平面波速度，因为微惯性长度大于或等于微观结构长度。当我们遇到挠曲电超材料时，预计会出现相反的效果，在这种情况下，微惯性长度小于微结构长度。