A multi-axial homogenized energy model is developed to account for nonlinear and hysteretic ferroelectric constitutive behavior induced by multi-axial electric field loading. The modeling approach extends a one-dimensional multi-scale modeling framework developed for ferroic materials [1, 2]. A three-dimensional energy function is introduced at the mesoscopic length scale and subsequently approximated as piecewise polynomial approximations to improve computational efficiency. Multi-scale field relations are then developed by introducing a distribution of effective electric fields and coercive fields that govern the nucleation of localized domain switching in polycrystalline ferroelectric materials. The distribution of field relations is used to relate the localized domain switching processes to observed macroscopic behavior by utilizing a stochastic homogenization technique. It is demonstrated that a simplified stochastic distribution of effective fields and coercive fields is sufficient to predict multi-axial ferroelectric switching in ferroelectric ceramics. Examples are given to validate the model in comparison to multi-axial loading experiments given in the literature. The model reduction provides a simple and efficient multi-scale modeling approach that is important for developing reliable piezoelectric actuator systems as well as implementation in model-based control of two and three dimensional structures.

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