A constitutive model for ferroelectric polycrystals
01 Aug 1999-Journal of The Mechanics and Physics of Solids (Elsevier Limited)-Vol. 47, Iss: 8, pp 1663-1697
TL;DR: In this paper, a constitutive model for the non-linear switching of ferroelectric polycrystals under a combination of mechanical stress and electric field is developed for nonlinear switching, where the switching event, which converts one crystal variant into another, gives rise to a progressive change in remanent strain and polarisation.
Abstract: A constitutive model is developed for the non-linear switching of ferroelectric polycrystals under a combination of mechanical stress and electric field. It is envisaged that the polycrystal consists of a set of bonded crystals and that each crystal comprises a set of distinct crystal variants. Within each crystal the switching event, which converts one crystal variant into another, gives rise to a progressive change in remanent strain and polarisation and to a change in the average linear electromechanical properties. It is further assumed that switching is resisted by the dissipative motion of domain walls. The constitutive model for the progressive switching of each crystal draws upon elastic–plastic crystal plasticity theory, and a prescription is given for the tangent moduli of the crystal, for any assumed set of potentially active transformation systems. A self-consistent analysis is used to estimate the macroscopic response of tetragonal crystals (representative of lead titanate) under a variety of loading paths. Also, the evolution of the switching surface in stress-electric field space is calculated. Many of the qualitative features of ferroelectric switching, such as butterfly hysteresis loops, are predicted by the analysis.
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TL;DR: In this article, the fracture behavior of piezoelectric ceramics under combined electrical and mechanical loading has been among the most prevalent research topics, and four types of nonlinear approaches considered are: electrostriction, domain switching, domain wall kinetics, and polarization saturation at a crack tip.
Abstract: Publisher Summary Piezoelectric ceramics can sense and actuate by rapidly converting mechanical and thermal signals into electrical ones, the reverse also being true. The piezoelectric properties and quick response characteristics have made piezoelectric ceramics one of the most commonly used smart materials. The intrinsic brittleness of piezoelectric ceramics and damageability of the materials under electric field — making the materials prone to fracture — are of major concern for product reliability. The fracture of piezoelectric ceramics under combined electrical and mechanical loading has been among the most prevalent research topics. The chapter describes piezoelectricity, ferroelectrics, spontaneous polarization, and electric domains; and discusses the poling process, the hysteresis loop of polarization versus the electric field strength, and the butterfly loop of strain versus the electric field. The chapter focuses on the basic equations commonly used in the study of the fracture behavior of piezoelectric ceramics within the thermodynamics framework, the general solution based on Stroh's formalism (a powerful tool for solving two-dimensional electroelastic problems), analysis of Green's functions for insulating elliptical cavities and cracks, study of conductive elliptical cavities and cracks, study of piezoelectric interface cracks, and three-dimensional electroelastic problems. The four types of nonlinear approaches considered are: electrostriction, domain switching, domain wall kinetics, and polarization saturation at a crack tip. The polarization saturation model that treats piezoelectric ceramics as mechanically brittle and electrically ductile materials is also discussed. The chapter provides an overview of experimental observations whose results show that microstructure and temperature have a profound influence on the fracture behaviors of piezoelectric ceramics under purely mechanical loads, and also discusses the commonly used failure criteria, the electric saturation model, the stress intensity factor criterion and the stress criterion.
330 citations
TL;DR: In this paper, a phase-field model based on the time-dependent Ginzburg-Landau equation was used to simulate the polarization switching in a ferroelectric subjected to an electric field or a stress field.
Abstract: Polarization switching in a ferroelectric subjected to an electric field or a stress field is simulated using a phase-field model based on the time-dependent Ginzburg–Landau equation, which takes both multiple-dipole–dipole–electric and multiple-dipole–dipole–elastic interactions into account. The temporal evolution of the polarization switching shows that the switching is a process of nucleation, if needed, and growth of energy-favorite domains. Macroscopic polarization and strain are obtained by averaging polarizations and strains over the entire simulated ferroelectric. The simulation results successfully reveal the hysteresis loop of macroscopic polarization versus the applied electric field, the butterfly curve of macroscopic strain versus the applied electric field, and the macroscopically nonlinear strain response to applied compressive stresses.
288 citations
TL;DR: Theoretical analyses and experimental observations of the failure and fracture behaviors of piezoelectric materials are presented in this paper, where the theoretical analyses are based on the Stroh formalism.
Abstract: Theoretical analyses and experimental observations of the failure and fracture behaviors of piezoelectric materials are presented. The theoretical analyses are based on the Stroh formalism. A strip dielectric breakdown model is proposed to estimate the effect of electrical non-linearity on the piezoelectric fracture of electrically insulated cracks. The reviewed experiments include the indentation fracture test, the bending test on smooth samples, the fracture test on pre-notched or pre-cracked samples, the environment-assisted fracture test, etc. For electrically insulated cracks, the experimental results show a complicated fracture behavior under combined electrical and mechanical loading. Fracture data are greatly scattered when a static electric field is applied. For electrically conducting cracks, the experimental results demonstrate that static electric fields can fracture poled and depoled lead zirconate titanate (PZT) ceramics. A charge-free zone model is introduced to understand the failure behavior of conducting cracks in the depoled lead zirconate titanate ceramics under electrical and/or mechanical loading. These theoretical and experimental results indicate that fracture mechanics concepts are useful in the study of the failure behaviors of piezoelectric materials.
284 citations
TL;DR: In this paper, a continuum thermodynamics framework is devised to model the evolution of ferroelectric domain structures, which falls into the class of phase-field or diffuse-interface modeling approaches.
Abstract: A continuum thermodynamics framework is devised to model the evolution of ferroelectric domain structures. The theory falls into the class of phase-field or diffuse-interface modeling approaches. Here a set of micro-forces and governing balance laws are postulated and applied within the second law of thermodynamics to identify the appropriate material constitutive relationships. The approach is shown to yield the commonly accepted Ginzburg–Landau equation for the evolution of the polarization order parameter. Within the theory a form for the free energy is postulated that can be applied to fit the general elastic, piezoelectric and dielectric properties of a ferroelectric material near its spontaneously polarized state. Thereafter, a principle of virtual work is specified for the theory and is implemented to devise a finite element formulation. The theory and numerical methods are used to investigate the fields near straight 180° and 90° domain walls and to determine the electromechanical pinning strength of an array of line charges on 180° and 90° domain walls.
275 citations
Cites background or methods from "A constitutive model for ferroelect..."
...Sharp interface theory Kessler and Balke (2001), single crystal continuum slip theory Huber et al....
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...It may be possible (and valuable) to construct a rigorous normality framework, but such a theory will differ from that already established in other phenomenological theories (Kessler and Balke, 2001; Huber et al., 1999; Landis 2002a, b)....
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...Sharp interface theory Kessler and Balke (2001), single crystal continuum slip theory Huber et al. (1999), and polycrystalline flow theory Landis (2002b) have each identified the contribution to the driving force for switching associated with the changes in material properties due to switching....
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TL;DR: In this article, a general form for multiaxial constitutive laws for ferroelectric ceramics is constructed, where switching surfaces and associated flow rules are postulated in a modified stress and electric field space such that a positive dissipation rate during switching is guaranteed.
Abstract: In this paper, a general form for multi-axial constitutive laws for ferroelectric ceramics is constructed. The foundation of the theory is an assumed form for the Helmholtz free energy of the material. Switching surfaces and associated flow rules are postulated in a modified stress and electric field space such that a positive dissipation rate during switching is guaranteed. The resulting tangent moduli relating increments of stress and electric field to increments of strain and electric displacement are symmetric since changes in the linear elastic, dielectric and piezoelectric properties of the material are included in the switching surface. Finally, parameters of the model are determined for two uncoupled cases, namely non-remanent straining ferroelectrics and purely ferroelastic switching, and then for the fully coupled ferroelectric case.
247 citations
Cites background from "A constitutive model for ferroelect..."
...Kamlah and Tsakmakis (1999) and Huber and Fleck (2001) have also used Eq....
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...The most notable is the self-consistent model of Huber et al. (1999)....
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...Landis and McMeeking (1999) and Huber and Fleck (2001) have already discussed lock-up conditions for ferroelastic switching....
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References
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06 Oct 1977
TL;DR: In this paper, the theory of ferroelectricity in terms of soft modes and lattice dynamics is developed and modern techniques of measurement, including X-ray, optic, and neutron scattering, infra-red absorption, and magnetic resonance.
Abstract: The book develops the modern theory of ferroelectricity in terms of soft modes and lattice dynamics and also describes modern techniques of measurement, including X-ray, optic, and neutron scattering, infra-red absorption, and magnetic resonance. It includes a discussion of the related phenomena of antiferroelectricity, pyroelectricity, and ferroelasticity and seconds on domains, thin films, ceramics, and polymers, leading on to a comprehensive survey of potential and actual device capabilities for pyroelectric detection, memories, display, and modulation. It should provide an authoritative account for those engaged in research or graduate ferroelectric or pyroelectric devices.
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TL;DR: In this article, the elastic moduli of two-phase composites are estimated by a method that takes account of the inhomogeneity of stress and strain in a way similar to the Hershey-Kroner theory of crystalline aggregates.
Abstract: T he macroscopic elastic moduli of two-phase composites are estimated by a method that takes account of the inhomogeneity of stress and strain in a way similar to the Hershey-Kroner theory of crystalline aggregates. The phases may be arbitrarily aeolotropic and in any concentrations, but are required to have the character of a matrix and effectively ellipsoidal inclusions. Detailed results arc given for an isotropic dispersion of spheres.
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TL;DR: In this article, the authors explore a theoretical approach to these fine phase mixtures based on the minimization of free energy and show that the α-phase breaks up into triangular domains called Dauphine twins which become finer and finer in the direction of increasing temperature.
Abstract: Solid-solid phase transformations often lead to certain characteristic microstructural features involving fine mixtures of the phases. In martensitic transformations one such feature is a plane interface which separates one homogeneous phase, austenite, from a very fine mixture of twins of the other phase, martensite. In quartz crystals held in a temperature gradient near the α-β transformation temperature, the α-phase breaks up into triangular domains called Dauphine twins which become finer and finer in the direction of increasing temperature. In this paper we explore a theoretical approach to these fine phase mixtures based on the minimization of free energy.
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