On a non-linear finite element method for piezoelectric structures made of hysteretic ferroelectric ceramics

A simple macroscopic constitutive law for ferroelectric and ferroelastic hysteresis effects of piezoceramics is presented. After summarizing the uniaxial formulation, it is generalized to a three dimensional tensorial formulation. This constitutive model has been implemented in the public domain finite element code PSU of Stuttgart University (Germany). The fully coupled electro-mechanical boundary value problem for our hysteresis model is solved by incrementation of the loading history. In order to verify the capabilities of our tool, a multilayer like actuator geometry is analysed. It is shown that the remanent polarization remaining after poling gives rise to a non-vanishing distribution of the electric potential even if the voltage is reduced to zero at all electrodes. Concerning the residual stresses present after poling, a tensile stress field perpendicular to the direction of the electrodes can be found in the passive region of the actuator where so-called poling cracks are known to occur. It is the key feature of our finite element tool that it allows to consider in structural mechanical analyses such phenomena as they are induced by the remanent hysteresis properties of piezoceramics.