Uniformly moving antiplane crack in flexoelectric materials

Abstract

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.