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

Abstract

Abstract. The one-dimensional free energy model for ferroelectric materials developed by is generalized to two dimensions. The two-dimensional free energy potential proposed in this paper consists of four energy wells that correspond to four variants of the material. The wells are separated by four saddle points, representing the barriers for 90 • -switching processes, and a local maximum, across which 180 • -switching processes take place. The free energy potential is combined with evolution equations for the variant fractions based on the theory of thermally activated processes. The model is compared to recent measurements on BaTiO 3 single crystals by Burcsu et al. [8], and predicitions are made concerning the response to the application of in-plane multi-axial electric fields at various frequencies and loading directions. The kinetics of the 90 • -and 180 • -switching processes are discussed in detail.

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

“…In addition, the proposed three-dimensional model fully takes into account the effects of electric field as well as mechanical stress, which had also been addressed recently by Ball et al (in press) using concepts of partial equilibria. The effects of mechanical stress had not been included in the previous one-and two-dimensional models of Smith et al (2002Smith et al ( , 2003Smith et al ( , 2005 and Seelecke et al (2005). The responses of the present model are finally compared with recent observations of Burcsu et al (2004) for BaTiO 3 single crystals, and model responses to multi-axial electrical loading at various rates and compressive stress loading are discussed.…”

“…Thermodynamic driving forces are calculated from a nine-dimensional free energy potential given as a piecewise quadratic function of polarization vector and strain tensor for each variant of a tetragonal ferroelectric single crystal. In the proposed model, contrary to the one-dimensional model of Smith et al (2002Smith et al ( , 2003Smith et al ( , 2005 and the two-dimensional model of Seelecke et al (2005), it is not necessary to describe the free energy potential over the entire nine-dimensional space of polarization vector and strain tensor, because the energy barriers are directly linked to the minima of the energy wells. In particular, it is not necessary to construct the non-convex, unstable regimes between the wells and to determine the locations and the energy values at saddle points in a high-dimensional space in order to find the energy barriers.…”

“…However, the evaluation of energy barriers is not a trivial job in multi-dimensional space. In the two-dimensional model recently proposed by Seelecke et al (2005), the locations of critical points across which lattice elements jump during a switch have to be determined using given free energy potential and then energy barriers are evaluated at those locations. In the general case of nine-dimensional space, this job is almost impossible to do.…”

“…As a result, the two types of switchings are modeled to have different material constants in associated constitutive equations. For example, different parameters are assigned in the activation energy equations of Belov and Kreher (2005) for different types of switchings, and different values of energy barriers are used for different types of switchings in the two-dimensional modeling of Seelecke et al (2005), which is a generalization of the one-dimensional model of Smith et al (2002Smith et al ( , 2003Smith et al ( , 2005.…”

Contraction Induced Muscle Injury: Towards Personalized Training and Recovery Programs

“…Thus, from the mechanical perspective, each sarcomere (or more strictly each half sarcomere) is a bistable mechanism 124 and a single myofibril can be thought of as a chain of bistable elements connected in series. 8 Interestingly, a large body of works showed that the bistable chain model is prototypical for a range of nonlinear phenomena, from materials physics 6,8,98,120,129,135,145 to biophysics of macromolecules and biomimetics. 7,20,24,47,112,122 The force-velocity (F-V) relation of the sarcomere, Fig.…”