Variational‐based modeling of micro‐electro‐elasticity with electric field‐driven and stress‐driven domain evolutions

Recently, increasing interest in so‐called functional or smart materials with electromechanical coupling has been shown such as ferroelectric piezoceramics. These materials are characterized by microstructural properties, which can be changed by external stress and electric field stimuli, and hence find use as the active components in sensors and actuators. The electromechanical coupling effects result from the existence and rearrangement of microstructural domains with uniformly oriented electric polarization. The understanding and efficient simulation of these highly nonlinear and dissipative mechanisms, which occur on the microscale of ferroelectric piezoceramics, are a key challenge of the current research. This paper does not offer a substantially new physical model of these phenomena but a new mathematical modeling approach based on a rigorous exploitation of rate‐type variational principles. This provides a new insight in the structure of the coupled problem, where the governing field equations appear as the Euler equations of a variational statement. We outline a variational‐based micro‐electro‐elastic model for the microstructural evolution of both electrically and mechanically driven electric domains in ferroelectric ceramics, which also incorporates the surrounding free space. To this end, we extend recently developed multifield incremental variational principles of electromechanics from local to gradient‐extended dissipative response and specialize it by a Ginzburg–Landau‐type phase field model, where the thickness of the domain walls enters the formulation as a length scale. This serves as a natural starting point for a canonical compact, symmetric finite element implementation, considering the mechanical displacement, the microscopic polarization, and the electric potential induced by the polarization as the primary fields. The latter is defined on both the solid domain and a surrounding free space. Numerical simulations treat domain wall motions for electric field‐driven and stress‐driven loading processes, including the expansion of the electric potential into the free space. Copyright © 2012 John Wiley & Sons, Ltd.