Steady-state antiplane crack considering the flexoelectrics effect: surface waves and flexoelectric metamaterials

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.