Macroscopic description of the non-linear electro-mechanical coupling in ferroelectrics

As in many cases piezoelectric devices are made of PZT, it is our objective to incorporate the non-linearly coupled electro-mechanical behavior of ferroelectric PZT-ceramics into a finite element tool for reliability analyses. In this paper, we will present a phenomenological model for ferroelectrics which is conceived to be valid for complex electro-mechanical loading histories and simple enough to be implemented with acceptable effort in an FE-code.(1) This has been achieved by introducing irreversible polarization and irreversible strain as internal variables besides stress, strain, electric field and polarization. The internal variables represent the loading history and are governed by ordinary differential equations. Each of these evolution equations is subjected to two loading conditions differing in nature. The first indicates the onset of irreversible change by domain switching, while the second characterizes the maximum amount of possible irreversible change corresponding to a totally switched domain structure. Polarization induced anisotropy is taken into account to the extent deemed necessary. For sake of simplicity, no rate effects are included. The model response to uniaxial electro-mechanical loading histories will be discussed and compared to known experimental results. By bilinear approximation the following characteristic macroscopic phenomena of PZT-ceramics can be represented: dielectric hysteresis, polarization induced piezoelectricity, butterfly hysteresis, ferroelastic hysteresis, mechanical depolarization, field dependent coercitive stress.