Crack tip electric polarization saturation of a thermally loaded penny-shaped crack in an infinite thermo-piezo-elastic medium

In the present paper, the electric polarization saturation at the front of a thermally loaded penny-shaped crack in an infinite thermo-piezo-elastic medium is studied in both analytical and numerical manners. This crack is assumed to be electrically semi-permeable, and the electric field in the crack cavity is taken into account. An approximate analytical solution to the present non-linear crack problem is derived on the basis of the electric polarization saturation model together with the fundamental solutions for two concentrated loading crack problems. Further, the refined crack surface displacement, crack surface electric potential and electric displacement outside the crack are also presented in closed-forms. The validity of the analytical solution is checked via the numerical discrete method together with an iterative approach. A good agreement is observed. The influences of some material constants on the electric yielding zone are numerically revealed. An empirical formula is developed for the extent of electric yielding zone at the crack tip. In view of the wide applications of piezoelectric ceramics in various thermal environments, the present analysis could serve as a reference for the design of various smart structures of piezoelectric media.