The antiplane dynamic flexoelectric problem is stated as a dielectric solid that incorporates gradients of electric polarization and flexoelectricity due to strain gradients. The work examines dielectric materials without piezoelectric coupling or nonlinear ferroelectric switching and considers the inverse flexoelectric effect. It is shown that the coupling of the mechanical with the electrical problem can be condensed in a single mechanical problem that falls in the area of dynamic couple stress elasticity. Moreover, static and steady state dynamic antiplane problems of flexoelectric and couple stress elastic materials can be modeled as anisotropic plates with a non-equal biaxial pre-stress. This analogy was materialized in a finite element code. In this work, we solved the steady-state problem of a semi-infinite antiplane crack located in the middle of an infinite flexoelectric material, with its crack-tip moving with constant velocity. The particular type of loading investigated serves to relate the present solutions with known results from classic elastodynamics. We investigated the influence of various parameters such as the shear wave velocity and two naturally emerging microstructural and micro-inertia lengths. In the context of flexoelectricity, the two lengths are due to the interplay of the elastic and the flexoelectric parameters. Furthermore, we investigated the subsonic and the supersonic steady state crack rupture and showed that the Mach cones depend on the microstructural as well as the micro-inertial lengths. An important finding of this work is the existence of surface waves of Bleustein–Gulyaev type that do not appear in classic elastodynamics, but have been found in piezoelectric materials. The case of dielectric metamaterials with negative electric susceptibility is examined for the first time. The results can be useful for other dispersive materials, provided we identify the pertinent microstructural and micro-inertial lengths in accord with the behavior of the material at high frequencies.

反平面动态柔电问题被描述为一种介电固体，由于应变梯度，该电介质包含了电极化和柔电的梯度。这项工作研究了没有压电耦合或非线性铁电切换的介电材料，并考虑了反挠电效应。结果表明，机械与电气问题的耦合可以集中在单个机械问题中，该问题属于动态耦合应力弹性区域。而且，可以将挠性电和耦合应力弹性材料的静态和稳态动态反平面问题建模为具有不相等的双轴预应力的各向异性板。在有限元代码中实现了这种类比。在这项工作中 我们解决了位于无限柔电材料中间的半无限反平面裂纹的稳态问题，其裂纹尖端以恒定速度运动。所研究的特定类型的载荷用于将本解决方案与经典弹性动力学的已知结果相关联。我们研究了各种参数的影响，例如剪切波速度以及两个自然出现的微结构和微惯性长度。在柔性电的情况下，这两个长度是由于弹性参数和柔性电参数的相互作用。此外，我们研究了亚音速和超音速稳态裂纹的破裂，并显示出马赫锥取决于微观结构以及微观惯性长度。这项工作的重要发现是存在不存在于经典弹性力学中但已在压电材料中发现的Bleustein–Gulyaev型表面波。首次检查了具有负电化率的介电超材料的情况。如果我们根据材料在高频下的行为确定相关的微结构和微惯性长度，则该结果对于其他分散材料也可能有用。