Dynamic crack propagation in piezoelectric materials—Part II. Vacuum solution

TLDR: In this article, an alternative assumption is made that between the growing crack surfaces there is a permeable vacuum free space, in which the electrostatic potential is nonzero, in contrast to the electrode case of Part I, this case allows both applied charge and applied traction loading.

Abstract: In Part I of this work, antiplane dynamic crack propagation in piezoelectric materials was studied under the condition that crack surfaces behaved as though covered with a conducting electrode. Piezoelectric surface wave phenomena were clearly seen to be critical to the behavior of the moving crack. Closed form results were obtained for stress and electric displacement intensities at the crack tip in the subsonic crack speed range; the major result is that the energy release rate vanishes as the crack speed approaches the surface (Bleustein-Gulyaev) wave speed. In this paper, an alternative assumption is made that between the growing crack surfaces there is a permeable vacuum free space, in which the electrostatic potential is nonzero. By coupling the piezoelectric equations of the solid phase with the electric charge equation in the vacuum region, a closed form solution is again obtained. In contrast to the electrode case of Part I, this case allows both applied charge and applied traction loading. In addition, the work of Part I is extended to examine piezoelectric crack propagation over the full velocity range of subsonic, transonic and supersonic speeds. Several aspects of the results are explored. The energy release rate in this case does not go to zero when the crack propagating velocity approaches the surface wave speed, even if there is only applied traction loading. When the crack exceeds the Bleustein-Gulyaev wave speed, the character of the crack-tip singularities of the physical fields depends on both speed regime and type of loading. At the other extreme, the quasi-static limit of the dynamic solution furnishes a set of new static solutions. The general permeability assumptions made here allow for fully coupled conditions that are ruled out by the a priori interfacial assumptions made in previously published solutions.