Path-independent integrals in electrochemomechanical systems with flexoelectricity

Recently, a larger flexoelectric-like response in a bending semiconductor was found by Narvaez et al. This fact indicates that the electrochemical processes have a great influence on flexoelectricity. Thus, study on the coupling between the electrochemical process and the flexoelectric effect is crucial. In this paper, the classical Noether theorem is applied to the dissipative electrochemomechanical processes with consideration of the flexoelectric effect and the strain gradient. The necessary and sufficient conditions for obtaining the symmetry transformation are derived. Several conservation integrals, including J- and L-integrals, are obtained. Then, the relation of the J- and L-integrals with the energy release rate is verified. We also show that J-integral here can reduce to the result in previous work. Finally, a Mode III crack problem taking the flexoelectric effect into account is discussed to show the path-independence of the J-integral.