Large electrostrictive actuation of barium titanate single crystals

An experimental investigation of the electromechanical behavior of single crystals of the ferroelectric perovskite barium titanate is presented. An experimental setup has been designed to investigate large strain actuation in single crystal ferroelectrics subjected to combined electrical and mechanical loading. Experiments have been performed on initially single domain crystals of barium titanate with (100) and (001) orientation at compressive stresses between 0 and . Global strain and polarization histories have been recorded. The electrostrictive response is shown to be highly dependent on the level of applied stress with a maximum strain of 0.9% measured at a compressive stress of about and electric field of about . This level of strain is about 5 times higher than in typical commercial piezoelectric PZT. Polarized light microscopy has been used to observe the evolution of the domain pattern simultaneously with the strain and polarization measurement. The observations reveal that the observed large strain behavior is the result of 90° domain switching.     

Two-scale micromechanics-based probabilistic modeling of domain switching in ferroelectric ceramics

A two-scale micromechanics model is developed in this paper to analyze domain switching in ferroelectric ceramics, using a probabilistic domain switching criterion based on energetic analysis. The microstructure of ferroelectric ceramics at two distinct length scales, domains and grains, has been carefully analyzed. The interaction at domain level is accounted for by energy minimization theory, while the fluctuation at grain level is analyzed using ellipsoidal two-point correlation function. The model has been implemented by Monte Carlo method, and applied to simulate the electric poling and mechanical depoling of Pb(Zr_xTi_1-x)O₃ (PZT) ceramics across morphotropic phase boundary (MPB). The drastically different switching characteristics of PZT ceramics across MPB has been captured, and good agreement with experiments has been observed. The effects of the transformation strains and spontaneous polarizations are highlighted, confirming the proposition of Li et al. [2005. Domain switching in polycrystalline ferroelectric ceramics. Nature Materials 4, 776–781] that the strain compatibility plays a dominant role in domain switching in ferroelectric ceramics.     

Energy principle of ferroelectric ceramics and single domain mechanical model

Many physical experiments have shown that the domain switching in a ferroelectric material is a complicated evolution process of the domain wall with the variation of stress and electric field. According to this mechanism, the volume fraction of the domain switching is introduced in the constitutive law of ferroelectric ceramic and used to study the nonlinear constitutive behavior of ferroelectric body in this paper. The principle of stationary total energy is put forward in which the basic unknown quantities are the displacement u _i , electric displacement D _i and volume fraction ρ _I of the domain switching for the variant I. Mechanical field equation and a new domain switching criterion are obtained from the principle of stationary total energy. The domain switching criterion proposed in this paper is an expansion and development of the energy criterion. On the basis of the domain switching criterion, a set of linear algebraic equations for the volume fraction ρ _I of domain switching is obtained, in which the coefficients of the linear algebraic equations only contain the unknown strain and electric fields. Then a single domain mechanical model is proposed in this paper. The poled ferroelectric specimen is considered as a transversely isotropic single domain. By using the partial experimental results, the hardening relation between the driving force of domain switching and the volume fraction of domain switching can be calibrated. Then the electromechanical response can be calculated on the basis of the calibrated hardening relation. The results involve the electric butterfly shaped curves of axial strain versus axial electric field, the hysteresis loops of electric displacement versus electric filed and the evolution process of the domain switching in the ferroelectric specimens under uniaxial coupled stress and electric field loading. The present theoretic prediction agrees reasonably with the experimental results given by Lynch. The project supported by the National Natural Science Foundation of China (10572138).     

Analysis of the electromechanical behavior of ferroelectric ceramics based on a nonlinear finite element model

A nonlinear finite element (FE) model based on domain switching was proposed to study the electromechanical behavior of ferroelectric ceramics. The incremental FE formulation was improved to avoid any calculation instability. The problems of mesh sensitivity and convergence, and the efficiency of the proposed nonlinear FE technique have been assessed to illustrate the versatility and potential accuracy of the said technique. The nonlinear electromechanical behavior, such as the hysteresis loops and butterfly curves, of ferroelectric ceramics subjected to both a uniform electric field and a point electric potential has been studied numerically. The results obtained are in good agreement with those of the corresponding theoretical and experimental analyses. Furthermore, the electromechanical coupling fields near (a) the boundary of a circular hole, (b) the boundary of an elliptic hole and (c) the tip of a crack, have been analyzed using the proposed nonlinear finite element method (FEM). The proposed nonlinear electromechanically coupled FEM is useful for the analysis of domain switching, deformation and fracture of ferroelectric ceramics.The project supported by the National Natural Science Foundation of China (10025209, 10132010 and 90208002), the Research Grants of the Council of the Hong Kong Special Administrative Region, China (HKU7086/02E) and the Key Grant Project of the Chinese Ministry of Education (0306)     

Thermodynamic driving force in ferroelectric crystals with a rank-2 laminated domain pattern, and a study of enhanced electrostriction

The thermodynamic driving force for domain growth in a rank-2 laminated ferroelectric crystal is derived in this article, and we used it, together with a homogenization theory, to study the issue of enhanced electrostrictive actuation recently reported by Burcsu et al. [2004. Large electrostrictive actuation of barium titanate single crystals. J. Mech. Phys. Solids 52, 823-846]. We derived this force from the reduction of Gibbs free energy with respect to the increase of domain concentration. It is shown that both the free energy and the thermodynamic force consist of three parts: the first arises from the difference in M₀ and M₁, the linear electromechanical compliances of the parent and product domains, respectively, at a given level of applied stress and electric field, the second stems from the electromechanical work associated with the change of spontaneous strain and spontaneous polarization during domain switch, and the third from the internal energy due to the distribution of polarizations strain and electric polarization inside the crystal. We prove that the first term is substantially lower than the second one, and the third one is identically zero with compatible domain pattern. The second one is, however, not exactly equal to the commonly written sum of the products of stress with strain, and electric field with polarization during switch, unless both domains have identical moduli in the common global axes. We also show that, with compatible domain patterns and when M₁=M₀, this driving force is identical to Eshelby's driving force acting on a flat interface due to the jump of energy-momentum tensor. Applications of the theory to a BaTiO₃ crystal subjected to a fixed axial compression and decreasing electric field from the [0 0 1] state reveal that the crystal undergoes a three-stage switching process: (i) the 0→90° switch to form a rank-1 laminate, (ii) the 0→180° switch inside the 0° domain to form a laminate I with a concurrent 90°→−90° switch inside the 90° domain to form laminate II, creating a rank-2-laminated domain pattern, and (iii) finally the 90→180° switch. It is the exchange of stability between the 0, 90°, and 180° domains under compression and electric field that is the origin of the enhanced actuation. We illustrate these intrinsic features by showing the evolution of these domains, and demonstrate how the reported large actuation strain can be attained with a rank-2 laminate.     

The shift of Curie temperature and evolution of ferroelectric domain in ferroelectric crystals

A micromechanics-based thermodynamic model for the phase transition of ferroelectric crystals is developed and, with it, the shift of Curie temperature and evolution of ferroelectric phase upon cooling are examined. This approach differs from the classical phenomenological one in that the evolution of new domain concentration can be predicted. We start out by formulating the Gibbs free energy of a generic, two-phase crystal consisting of the parent paraelectric phase and the transformed ferroelectric phase, at a given level of temperature, stress, and electric field. The thermodynamic driving force for domain growth is then derived and, together with the resistance force, a kinetic equation is established. The derived driving force is found to arise from three different sources of Gibbs free energy: (i) the interaction energy due to the heterogeneity of electromechanical moduli of the parent and product phases, (ii) the energy dissipation due to spontaneous polarization, and (iii) the self-energy of the dual-phase system due to the existence of polarization strain and electric polarization. For a BaTiO₃ crystal the electromechanical heterogeneity is found to play a rather significant role that seems not to have been recognized before. The derived shift recovers to the Clausius-Clapeyron relation if such heterogeneity disappears. We have examined in detail several factors that affect the shift of Curie temperature, and calculated the evolution of overall polarization and dielectric constant of a BaTiO₃ crystal. The results are found to be consistent with available test data.     

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

The one-dimensional free energy model for ferroelectric materials developed by Smith et al. [29–31] 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 BaTiO₃ 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.     

An optical method to assess electromechanical coupling in ferroelectric ceramics

A new technique is described, which allows the assessment of elastic and inelastic regions around a macroscopic defect in ferroelectric-ferroelastic ceramics. The accuracy and robustness of the method are demonstrated on a PZT plate with a centered hole subjected to uni-axial compressive stresses. From the electrical potential distribution on the sample surface, the mechanical response of the material is obtained at different load levels.     

Fully coupled, multi-axial, symmetric constitutive laws for polycrystalline ferroelectric ceramics

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.     

Effect of electric boundary conditions on crack propagation in ferroelectric ceramics

In this paper, the effect of electric boundary conditions on Mode I crack propagation in ferroelectric ceramics is studied by using both linear and nonlinear piezoelectric fracture mechanics. In linear analysis, impermeable cracks under open circuit and short circuit are analyzed using the Stroh formalism and a rescaling method. It is shown that the energy release rate in short circuit is larger than that in open circuit. In nonlinear analysis, permeable crack conditions are used and the nonlinear effect of domain switching near a crack tip is considered using an energy-based switching criterion proposed by Hwang et al.（Acta Metal. Mater.,1995）. In open circuit, a large depolarization field induced by domain switching makes switching much more diffcult than that in short circuit. Analysis shows that the energy release rate in short circuit is still larger than that in open circuit, and is also larger than the linear result. Consequently,whether using linear or nonlinear fracture analysis, a crack is found easier to propagate in short circuit than in open circuit, which is consistent with the experimental observations of Kounga Njiwa et al.（Eng. Fract. Mech., 2006）.     

Use of thin polyparaxylene films to study the plastic deformation bismuth single crystals

The effect of thin polyparaxylene films on the mechanical twinning of bismuth single crystals with the (111) surface subjected to local deformation. It is found that the number of twins formed near the stress concentrator increases in the presence of the film. Possible mechanisms are proposed to explain an increase in the mobility of twin dislocations in a deformable crystal whose surface is coated with a polyparaxylene film. Spalling of bismuth is found in the regions deformed by the indenter. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 162–166, July–August, 2006.     

Effect of electric displacement saturation on the stress intensity factor for a crack in a ferroelectric ceramic

A crack in a ferroelectric ceramic with perfect saturation under electric loading is analyzed. The boundary of the electric displacement saturation zone ahead of the crack tip is assumed to be ellipse in shape. The shape and size of ferroelectric domain switching zone near a crack tip is determined based on the nonlinear electric theory. The stress intensity factor induced by ferroelectric domain switching under small-scale conditions is numerically obtained as a function of the electric saturation zone parameter and the ratio of the coercive electric field to the yield electric field. It is found that the stress intensity factor increases as the ratio of the semi-axes of the saturation ellipse increases.     

A continuum analysis of the driving force of ferroelectric/ferroelastic domain wall motions

First, Eshelby's driving force for the motion of a sharp interface is rederived from general thermodynamic principles. Ferroelectric and ferroelastic domain walls represent a special class of such interfaces in mechanically stressed crystals subject also to an electric field. The corresponding bulk contributions to the driving force are caused by variations of the ferroelectric/ferroelastic anisotropy energy, whereas the interface contributions arise from variations of the intrinsic surface energy of the domain wall and domain wall bending. The general expressions for the local driving force per unit area of such domain walls are specialized then to domain walls in piezoelectric crystals.     

Domain Engineered Relaxor Ferroelectric Single Crystals

Single crystal relaxor ferroelectric materials exhibit extraordinary electromechanical properties. They are being applied in high performance sensors, actuators, and transducers. Field induced polarization switching and phase transitions of these crystals lead to complex nonlinear behavior. In recent years experimental investigations have been conducted to characterize the polarization switching and phase transition behavior as a function of crystallographic orientation, temperature, electric field, and stress. The results give insight into the mechanism underlying the observed large field hysteretic behavior. This review article describes the observed behavior and presents results of multiscale modeling that predicts the macroscopic behavior from the single domain single crystal behavior and evolution of crystal variants at the microscale.     

A continuum analysis of charge induced ferroelectric domain wall motions

A continuum model is presented for the motion of a domain wall in a plane 90°-domain configuration subjected to an isolated extrinsic charge near the surface of a ferroelectric single crystal. Local pinning is postulated for the kinetic law. Before the appearance of the extrinsic charge, all polarization surface charges are taken to be neutralized by environmental charges. The domain wall motion after the appearance of the extrinsic charge is assumed to proceed sufficiently fast without any significant conductive currents on the surface or in the interior of the crystal such that new surface and interface polarization charges remain unscreened and contribute to the ferroelectric anisotropy energy. A non-admissible divergence of the electric field and consequently of the local thermodynamic driving force and of the domain wall velocity appears in the model if the domain wall charged by interface polarization charges intersects the crystal surface charged by surface polarization charges under an arbitrary angle. The physically possible domain wall angle is identified using the condition of a non-divergent driving force. The ferroelectric anisotropy energy and an intrinsic surface energy of the domain wall, however, do not provide stability of the domain wall trajectory against an unlimited increase of its curvature at the surface. The problem has been solved conceptually by proper account of the domain wall bending energy. Numerical and dimensional analysis explain also why domain walls driven by extrinsic charges remain almost straight in soft ferroelectrics.     

Features of evolution of wedge-shaped twins in bismuth single crystals subjected to polysynthetic twinning

Features of evolution of wedge-shaped twins in bismuth single crystals with polysynthetic twins are examined. Polysynthetic twins are found to promote an increase in number, and a decrease in length, of indentation-induced wedge-shaped twins. The latter quantities depend on the density of twins in a polysynthetic twin. Based on the dislocation model, stress fields in the vicinity of wedge-shaped and polysynthetic twins are calculated at a mesoscopic level. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 208–216, May–June, 2008.     

镁单晶变形行为的原子模拟

应用分子动力学方法模拟镁单晶在单向拉伸作用下的力学性能和微观结构演化过程.不同应变和不同温度下的模拟结果中都观测到{1011},{1012}型李晶.表明这两种孪晶是镁单晶拉伸变形的主要机制.其中{1012}型挛晶的产生伴随新晶粒生成,并且随应变增加孪晶的数量增加.此外,随着温度的增加,原子热激活效应显著.镁单晶的最大弹性形变减小,最大应力值亦变小.     

A micromechanics method to study the effect of domain switching on fracture behavior of polycrystalline ferroelectric ceramics

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Modeling of dielectric viscoelastomers with application to electromechanical instabilities

Soft dielectrics are electrically-insulating elastomeric materials, which are capable of large deformation and electrical polarization, and are used as smart transducers for converting between mechanical and electrical energy. While much theoretical and computational modeling effort has gone into describing the ideal, time-independent behavior of these materials, viscoelasticity is a crucial component of the observed mechanical response and hence has a significant effect on electromechanical actuation. In this paper, we report on a constitutive theory and numerical modeling capability for dielectric viscoelastomers, able to describe electromechanical coupling, large-deformations, large-stretch chain-locking, and a time-dependent mechanical response. Our approach is calibrated to the widely-used soft dielectric VHB 4910, and the finite-element implementation of the model is used to study the role of viscoelasticity in instabilities in soft dielectrics, namely (1) the pull-in instability, (2) electrocreasing, (3) electrocavitation, and (4) wrinkling of a pretensioned three-dimensional diaphragm actuator. Our results show that viscoelastic effects delay the onset of instability under monotonic electrical loading and can even suppress instabilities under cyclic loading. Furthermore, quantitative agreement is obtained between experimentally measured and numerically simulated instability thresholds. Our finite-element implementation will be useful as a modeling platform for further study of electromechanical instabilities and for harnessing them in design and is provided as online supplemental material to aid other researchers in the field.     

Three-dimensional finite element simulations of ferroelectric polycrystals under electrical and mechanical loading

Complex, non-linear, irreversible, hysteretic behavior of polycrystalline ferroelectric materials under a combined electro-mechanical loading is a result of domain wall motion, causing simultaneous expansion and contraction of unlike domains, grain sub-divisions that have distinct spontaneous polarization and strain. In this paper, a 3-dimensional finite element method is used to simulate such a polycrystalline ferroelectric under electrical and mechanical loading. A constitutive law due to Huber et al. [1999. A constitutive model for ferroelectric polycrystals. J. Mech. Phys. Solids 47, 1663-1697] for switching by domain wall motion in multidomain ferroelectric single crystals is employed in our model to represent each grain, and the finite element method is used to solve the governing conditions of mechanical equilibrium and Gauss's law. The results provide the average behavior for the polycrystalline ceramic. We compare the outcomes predicted by this model with the available experimental data for various electromechanical loading conditions. The qualitative features of ferroelectric switching are predicted well, including hysteresis and butterfly loops, the effect on them of mechanical compression, and the response of the polycrystal to non-proportional electrical loading.