The antiplane dynamic flexoelectric problem is stated as a dielectric solid that incorporates gradients of electric polarization and flexoelectricity due to strain gradients. 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 elasto-dynamics. We investigated the influence of various parameters such as the shear wave velocity and two naturally emerging micro-structural and micro-inertia lengths. In the context of flexoelectricity, the two lengths are due to the interplay of the elastic and the flexoelectric parameters. We also investigated the connection of the electric boundary conditions with boundary conditions of the dynamic couple stress elasticity. Furthermore, we investigated the subsonic and the supersonic steady-state crack rupture and showed that the Mach cones depend on the micro-structural as well as the micro-inertial lengths. The results are important for all dielectrics such as ceramics, ice, perovskites and polymers that exhibit strong flexoelectric effects, often uncoupled from piezoelectricity (centrosymmetric materials). Moreover, the results can be useful for other dispersive materials, provided we identify the pertinent micro-structural and micro-inertial lengths in accord with the behaviour 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。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，例如剪切波速度和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。我们还研究了电边界条件与动态耦合应力弹性边界条件的联系。此外，我们研究了亚音速和超音速稳态裂纹的破裂情况，结果表明，马赫锥取决于微观结构以及微观惯性长度。结果对于所有电介质（例如陶瓷，冰，表现出强柔电效应的钙钛矿和聚合物，通常与压电性无关（中心对称材料）。此外，如果我们确定与材料在高频下的行为相符的相关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。