Electromechanical influence of crack velocity at bifurcation for poled ferroelectric materials

Abstract

The piezoelastodynamic field equations are solved to determine the crack velocity at bifurcation for poled ferroelectric materials where the applied electrical field and mechanical stress can be varied. The underlying physical mechanism, however, may not correspond to that assumed in the analytical model. Bifurcation has been related to the occurrence of a pair of maximum circumferential stress oriented symmetrically about the moving crack path. The velocity at which this behavior prevails has been referred to as the limiting crack speed. Unlike the classical approach, bifurcation will be identified with finite distances ahead of a moving crack. Nucleation of microcracks can thus be modelled in a single formulation. This can be accomplished by using the energy density function where fracture initiation is identified with dominance of dilatation in relation to distortion. Poled ferroelectric materials are selected for this study because the microstructure effects for this class of materials can be readily reflected by the elastic, piezoelectic and dielectric permittivity constants at the macroscopic scale. Existing test data could also shed light on the trend of the analytical predictions. Numerical results are thus computed for PZT-4 and compared with those for PZT-6B in an effort to show whether the branching behavior would be affected by the difference in the material microstructures. A range of crack bifurcation speed v b is found for different r/ a and E/ σ ratios. Here, r and a stand for the radial distance and half crack length, respectively, while E and σ for the electric field and mechanical stress. For PZT-6B with v b in the range 100–1700 m/s, the bifurcation angles varied from ±6° to ±39°. This corresponds to E/ σ of −0.072 to 0.024 V m/N. At the same distance r/ a=0.1, PZT-4 gives v b values of 1100–2100 m/s; bifurcation angles of ±15° to ±49°; and E/ σ of −0.056 to 0.059 V m/N. In general, the bifurcation angles ± θ 0 are found to decrease with decreasing crack velocity as the distance r/ a is increased. Relatively speaking, the speed v b and angles ± θ 0 for PZT-4 are much greater than those for PZT-6B. This may be attributed to the high electromechanical coupling effect of PZT-4. Using v b 0 as a base reference, an equality relation v b −< v b 0< v b + can be established. The superscripts −, 0 and + refer, respectively, to negative, zero and positive electric field. This is reminiscent of the enhancement and retardation of crack growth behavior due to change in poling direction. Bifurcation characteristics are found to be somewhat erratic when r/ a approaches the range 10 −2–10 −1 where the kinetic energy densities would fluctuate and then rise as the distance from the moving crack is increased. This is an artifact introduced by the far away condition of non-vanishing particle velocity. A finite kinetic energy density prevails at infinity unless it is made to vanish in the boundary value problem. Future works are recommended to further clarify the physical mechanism(s) associated with bifurcation by means of analysis and experiment. Damage at the microscopic level needs to be addressed since it has been known to affect the macrocrack speeds and bifurcation characteristics.