Effect of electric boundary conditions on crack energy density and its derivatives for piezoelectric material

This paper deals with crack energy density (hereafter CED) as a possible fracture parameter in piezoelectricity under arbitrary electric boundary conditions on a notch-like crack surface. The definitions of CED and its derivatives are given first under exact boundary condition. Next, their path independent integrals are also derived and their approximate expressions are discussed under some restrictions on the crack surfaces. It is found that electrical terms along the notch-like crack surface do not vanish unlike in the case of impermeable crack. Then, we introduce evaluation methods of CED, and, with the help of the results of finite element analyses (FEA), we closely examine how electric boundary conditions along the notch surface and initial notch width influence CED and its derivatives. It is shown from the FEA results that because of the difficulties of computing path integral terms along the notch-like crack tip in the path independent expressions, the evaluation by the definitions of CED and its derivatives is preferable and more convenient than the evaluation of their path independent expressions. It is also found that all the parameters are significantly affected by both permittivity inside the electric inclusion and root radius of the notch. Finally, the possibility of mechanical CED as a governing fracture parameter is discussed.