A rate-dependent thermo-electro-mechanical free energy model for perovskite type single crystals

The three-dimensional electro-mechanical free energy potential developed by Kim and Seelecke [S.J. Kim and S. Seelecke, A rate-dependent three-dimensional free energy model for ferroelectric single crystals, Int. J. Solids Struct. 44 (2007) 1196–1209] is generalized to model various thermal aspects of perovskite type single crystals. A total of seven energy potentials are described in the 10-dimensional space of electric displacement vector, strain tensor and temperature, the first six of them representing the six distinct types of ferroelectric tetragonal variants and the seventh the paraelectric cubic phase of the materials. Energy barrier expressions given as functions of thermodynamic driving forces are combined with evolution equations to determine the phase fractions based on the theory of thermally activated processes, thus allowing for a natural treatment of rate-dependent effects. The thermodynamic Clausius–Clapeyron relation is derived from the energy potential and the double polarization hysteresis loops near Curie temperature observed by Merz [W.J. Merz, Double hysteresis loop of BaTiO3 at the Curie point, Phys. Rev. 91 (1953) 513–517] are predicted and compared. Besides, various nonlinear coupling behavior, such as variation of spontaneous polarization over temperature, mechanical depolarization, and rate-dependent hysteresis loops, are calculated and discussed.