A rate-dependent three-dimensional free energy model for ferroelectric single crystals

Abstract

The one-dimensional free energy model for ferroelectric materials developed by Smith et al. A unified framework for modeling hysteresis in ferroic materials. J. Mech. Phys. Solids 54, 46-85] is generalized to three space dimensions including both polarization and strain. In the resulting nine-dimensional energy function, six free energy potentials representing the six distinct types of tetragonal variants of perovskite lattice structures are given as quadratic functions of polarization vector and strain tensor. Energy barrier expressions as functions of thermodynamic driving forces are obtained through a generalization of the one-dimensional equations derived from the model of Smith et al. This approach presents an alternative to the cumbersome determination of higher-dimensional saddle points and is attractive for a computationally efficient implementation. The energy barrier expressions are combined with evolution equations for the variant fractions based on the theory of thermally activated processes and thus allow for a natural treatment of rate-dependent effects. The predictions of the model are compared with recent measurements on BaTiO 3 single crystals by Burcsu et al. [Burcsu, E., Ravichandran, G., Bhattacharya, K., 2004. Large electrostrictive actuation of barium titanate single crystals. J. Mech. Phys. 52,. The effects of applied stress and 90°-and 180°-switching processes are discussed in detail.

A Stress-dependent Hysteresis Model for Ferroelectric Materials

“…The incorporation of stochastic homogenization techniques and extension to ferroelectric [27,31], ferromagnetic [25,26], and polycrystalline SMA compounds [14,15,22] has led to the formulation of a general framework for characterizing hysteresis in ferroic compounds [24,29,30]. The theory presented here extends this framework and the work presented in [9] by incorporating stress and field-induced 90 • switching as motivated by both fundamental material considerations and design criteria associated with stress-dependence as illustrated in Figure 1(b).…”

“…The coefficients are chosen so that α ij > 0 and α i < 0 below the Curie point and can be related to physical properties of ferroelectric compounds such as the remanence polarization P R and coercive field E c -e.g., see (8) and (9). The electromechanical coupling component is given by…”

Thermodynamic Modelling of Ferromagnetic Shape Memory Actuators

“…Starting point is the Müller-Achenbach-Seelecke model for the transition (reorientation) between two martensite variants M -and M + with short and long axes c and a, respectively, being oriented along the beam axis [5]. The difference between the crystallographic axis c-a is due to a negative and positive eigenstrain .…”

“…The difference between the crystallographic axis c-a is due to a negative and positive eigenstrain . In this model, the transition is considered as a thermally activated process, which describes the switching of the variants in an energy landscape with two energy wells and one maximum between the wells [5]. From the relative phase fractions x -and x + of the variants under different loading conditions the stress-strain relation can be derived.…”

“…Different modeling approaches have been developed up to now, which either focus on a thermodynamic analysis [2,3] or on a phenomenological description [4]. Recently, a free energy model of the shape memory effect given in [5] has been combined with a Stoner-Wohlfarth model of magnetic anisotropy to describe the evolution of martensite variants as a function of stress and magnetic field [6]. The model has been implemented in a finite element (FEM) program for the simulation of coupled magneto-mechanical field problems.…”