State dependent linear moduli in ferroelectrics

A novel experimental method is used to measure the evolution of the linear elastic, dielectric and piezoelectric moduli of a soft ferroelectric ceramic during loading. The applied loading states are combinations of uniaxial compressive stress and electric field. Short pulses of electric field and stress are used to increment the remanent strain and polarization state of the material, while the rates of change of electric displacement and strain during unloading are used to assess the moduli. The remanent quantities are treated as state variables, with a view to expressing the moduli as functions of the material state. The piezoelectric moduli are found to vary approximately linearly with polarization, regardless of the remanent strain state, whilst the dielectric moduli and elastic compliances show more complex behaviour. A simple model of the state dependence of the moduli, based on varying the volume fractions of six crystal variants in the tetragonal system, is used to interpret the results.