Flexoelectricity is the phenomenon that allows some materials to convert mechanical strain gradients to electrical polarizations and vice versa. As flexoelectricity is a ferroelectric phenomenon, its applications are of maximum importance and should be studied thoroughly. The polarization magnitude is connected to the strain gradients and so situations that produce large strain gradients are interesting. The cracking seems to be very promising. The mode III crack is an anti-plane problem that can be solved also considering the flexoelectric effect. As known from classic elasticity, the anti-plane problem is a sub-case of 3D-elasticity. The mode III crack, is also a dynamic problem.

By considering the contribution of the flexoelectric phenomenon to the total energy density, a solution of the anti-plane flexoelectric problem can be formed. A direct analogue is presented between the anti-plane flexoelectric problem and the anti-plane couple stress elasticity problem, which allows the distinction of the flexoelectric problem into three regions: the elliptic, the hyperbolic and the intermediate.

The hyperbolic region is studied further. The characteristic lines, a method of solving hyperbolic equations, allows some simplifications of the differential equation and thus a full field analytical solution is presented. Mach cones are visible as the displacement is concerned. For this displacement, the polarization can be calculated. The crack tip and the end of the cohesive zone are the positions of maximum polarization and thus positions of possible electrical yielding (abrupt change of the polarization vector). Also, the polarization of a screw-like dislocation is calculated. In this case, the polarization is described with a “δ function” – like distribution. 

The anti-plane dynamic problem is responsible for the propagation of waves. Because of the microstructure (for the couple stress elasticity problem, or the flexoelectric properties on the normal anti-plane problem), those waves are dispersive, a fact that signifies the possibilities of a lot more applications. The dispersion is the next thing studied. The dispersion relations show great similarity with viscoelastic materials, as the flexoelectric metamaterials are concerned.

Lastly, through another analogue between the anti-plane problems and the plate problems, numerical calculations are possible for a great number of cases. The analogue is modified in order to be able to solve also hyperbolic problems. This is the first time the Analogue Equation Method is used in a Finite Element Code. Through a standard Finite Element Method (FEM) code (ABAQUS), the Mach cone - like displacement is proved, in the hyperbolic problem. Also, the angle of the cones, is in agreement with the previous bibliographic suggestions and depends on the microstructure and the velocity of the problem. .