In this paper, we discuss the development and implementation of a 3-D electromechanically coupled homogenized energy model (HEM) for ferroelectric materials. A stochastic-based methodology is introduced and applied to problems involving large scale switching of ferroelectric and ferroelastic materials. Switching criteria for polarization variants are developed using density distributions in three dimensions to accommodate both electrical and mechanical loading and their coupled response. The theory accommodates non-proportional loading and major/minor loop hysteresis. Such formulations are known to accelerate computations for real-time control of nonlinear and hysteretic actuators. The proposed formulation maintains superior computational efficiency in the three dimensional case through the application of density formulations that are based on internal distributions of stress and electric field to produce a distribution of polarization switching events over a range of applied fields and stresses.

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