A theory of ferroelectric hysteresis with a superimposed stress

Based on the mechanism of domain switch, a micromechanics-based model is developed to calculate the hysteresis loop of ferroelectric ceramics under combined electromechanical loading. The development makes use of an extension of the Eshelby-type elastic inclusion problem to a heterogeneous electromechanically coupled ferroelectric medium with distribution of eigenstrain and eigenpolarization generated by domain switch. The ferroelectric ceramic at a generic state is considered to consist of the parent domain and the switched domains whose volume fraction $fp,$ continues to evolve under an increasing electromechanical load. At a given level of applied stress and electric field, the volume fraction of the new domain is determined from a kinetic equation that is derived from consideration of the thermodynamic driving force caused by the reduction of Gibbs free energy and the resistance force associated with the domain wall movement. The theory developed is used first to simulate the hysteresis behavior of a PZT-51 without any superimposed stress, and then with the derived material constants, to independently predict the influence of a superimposed compression. It is found that, consistent with experimental observations, the hysteresis loops flatten out under the axial compression, but that the loops elongate under a transverse compression. Both the remanent polarization and coercive field also tend to decrease with the compressive force regardless of how the compression is applied.