The size dependence of flexoelectricity at nanocracks

The flexoelectric effect is an electro-mechanical coupling between strain gradients and the electric polarization, and it is especially significant for nanoscale structures. Since the strain gradient scales up with the decrease in the sample's feature size, the flexoelectric effect is size dependent. Due to the stress concentration, large strain gradients can be found at the crack tip and result in significant flexoelectric effect. However, for micro- or nanoscale cracks, it is still not clear how the flexoelectric effect changes with the size of cracks. In practice, the crack tip has finite radius. So, in addition to the crack length, the crack tip radius is also one of the geometric parameters describing the size of nanocracks. In this work, using our collocation mixed finite element method (CMFEM), we study the size dependence of flexoelectricity around nanocracks through these two parameters. Numerical simulation results indicate that stronger flexoelectric field can be formed around the tip of cracks with either larger crack length or smaller tip radius. We also analyze the interplay of the crack length and the tip radius and show how the crack tip flexoelectric field varies when both of these two parameters are changing.