Energetically consistent boundary conditions for electromechanical fracture

Energetically consistent crack face boundary conditions are formulated for cracks in electromechanical materials. The model assumes that the energy of the solid can be computed from standard infinitesimal deformation theory and that the opening of the crack faces creates a capacitive gap that can store electrical energy. The general derivation of the crack face boundary conditions is carried out for non-linear but reversible constitutive behavior of both the solid material and the space filling the gap. It is shown that a simple augmentation of the J-integral can be used to determine the energy release rate for crack advance with these boundary conditions. The energetically consistent boundary conditions are then applied to the Griffith crack problem in a polar linear piezoelectric solid and used to demonstrate that the energy release rate computed near the crack tip is equivalent to the total energy release rate for the solid-gap system as computed from global energy changes. A non-linear constitutive law is postulated for the crack gap as a model for electrical discharge and the effects of the breakdown field on the energy release rate are ascertained.