Homogenized energy model for characterizing polarization and strains in hysteretic ferroelectric materials: Material properties and uniaxial model development

Ferroelectric materials, such as lead zirconate titanate, lanthanum-doped lead zirconate titanate, and BaTiO₃, are being considered, or are already being employed, for a large number of applications including nanopositioning, high-speed valves for fuel injectors, ultrasonic transducers, high-speed camera shutters and autofocusing mechanisms, energy harvesting, and pico air vehicle design. Their advantages include nanometer positioning resolution, broadband frequency responses, moderate power requirements, the capability for miniaturization, and complementary actuator and sensor capabilities. However, they also exhibit creep, rate-dependent hysteresis, and constitutive nonlinearities at essentially all drive levels due to their noncentrosymmetric nature. In this article, we model the hysteretic dependence of strains and polarizations on input fields and stresses using the homogenized energy model framework. At the domain level, the minimization of the Gibbs energy densities yields linear constitutive relations. Nonlinearities and hysteresis due to dipole switching are modeled at the grain level using the Boltzmann theory to specify the evolution of dipole fractions that serve as internal variables. In the final step of the development, stochastic homogenization, based on the assumption that interaction fields and driving forces are manifestations of underlying densities, is used to construct nonlinear constitutive relations for the bulk material. It is demonstrated that these relations are amenable to subsequent development of distributed system models. The article includes significant discussion regarding the mechanisms that produce hysteresis in ferroelectric materials. The capability of the framework for characterizing various hysteretic phenomena, including creep and various rate dependencies, is illustrated by validation with lead zirconate titanate and lanthanum-doped lead zirconate titanate data.