On approach of crack tip energy release rate for a semi-permeable crack when electromechanical loads become very large

This paper presents an investigation on the approach of the crack tip energy release rate (ERR) for a semi-permeable crack full with air/vacuum or Silicon oil when the electromechanical loads become very large. Numerical results for a central semi-permeable crack, respectively, in seven kinds of piezoelectric ceramics are compared with those for a central impermeable crack when the mechanical loads vary from 50 to 100 MPa and the electric loads are fixed to be 1 MV/m, 0, and -1 MV/m, respectively, within the range of practical interest. It is verified that McMeeking's statement (2004): 'as the electromechanical loads become very large, the crack tip ERR approaches the values associated with an impermeable crack' is actually valid under very large mechanical and positive electric loads. However, under very large mechanical and negative electric loads, the approach is quite different showing large discrepancies between the calculated values for the semi-permeable crack and those for an impermeable crack in all seven kinds of piezoelectric ceramics. This means that his statement is not valid when the electric loads are negative even though the mechanical loads still remain very large although, mathematically, MeMeeking's statement is correct if McMeeking's statement: "very large" is replaced by "infinitely large". Moreover, under purely mechanical loads his statement is uncertain, depending on which kind of piezoelectric ceramic is used. It is concluded that, generally speaking, the crack tip ERR for a semi-permeable crack does not approach the values associated with an impermeable crack, depending on the direction of the electric loads with respect to the poling axis. Physically, this is because of the inherent piezoelectric effect that yields the surface charges distributed on the crack surfaces for a semi-permeable crack under the mechanical loads, whereas on the surfaces of an impermeable crack the unphysical charge-free condition leads to incorrect estimations: the applied mechanical loads do not yield any surface charges on the crack surfaces. The influence of the permittivity of medium inside the semi-permeable crack gap on McMeeking's statement is discussed too. It is found that Silicon oil yields larger discrepancies than air from those for an impermeable crack.