Discussion of contributions of the direct flexoelectric effects and strain gradient effects to fracture criteria of flexoelectric solids

Abstract

It is known that the direct flexoelectric effect is a consequence of the polarization of the material, which is proportional to the strain gradients. The strain gradients are prominent near material defects, especially at the crack tips, where the flexoelectric effect redistributes the stress field and consequently influences the crack propagation. The flexoelectricity is a size dependent effect, i.e. it depends on an internal material length parameter as the additional material characteristic. This fact makes the equilibrium, constitutive, and boundary equations complicated as well as the asymptotic solution at the crack tip contrary to the asymptotic solution in the linear elastic fracture mechanics. In our recent work (Profant et al, 2023) we have applied the matched asymptotic expansion method known mainly from the fluid mechanics to derive the expressions for the amplitude factors that appear in the flexoelectric asymptotic solution for the crack as functions of the classical stress intensity factors of LEFM in the loadings of mode I or mode II. The application of the matched asymptotic expansion method is conditioned by the knowledge of the so-called boundary layer, which is evaluated from the energetic criteria at the crack tip. The principal advantage is that the amplitude factors in the flexoelectric asymptotic solution do not need to be calculated through finite element simulation of a finite crack.In this contribution, we will use the results of the forementioned matched asymptotic expansion analysis for the study of crack propagation by taking into account that the asymptotics is only valid within a region around the crack tip that is on the order of the flexoelectric length scale or the strain gradient elasticity (SGE) length scale. The classical Griffith postulate regarding a critical energy release rate Gc is applied. The aim is to estimate the contributions of direct flexoelectric effects and strain gradient effects for various combinations of flexoelectric material properties to the expected reduction of the energy release rate.