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 ui, electric displacement Di and volume fraction pI 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 PI 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 evo- lution 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.

Numerical simulation of domain switching in multilayer ferroelectric actuators

Micromechanical finite element methods are developed based on a nonlinear constitutive model of ferroelectric polycrystals. Electromechanical behaviors ahead of an internal electrode tip are numerically simulated in multilayer ferroelectric actuators. Around the electrode edge, the nonuniform electric field generates a concentration of stress due to the incompatible strain as well as spontaneous strain. The preferred domain switching enhances the concentration of residual stress and may cause the actuators to crack. An electrically permeable crack emanating from an internal electrode is analyzed. A large scale domain switching zone is found in the vicinity of crack tips. The larger the actuating strain and electric field are, the larger the switching zone will be. The size of switching zone even reaches the scale of crack length with increasing electromechanical loading.

Coalescence of pore columns by domain switching

The present paper studies the coalescence of pore columns in ferroelectric ceramics driven by back and forth domain switching under cyclic electric field. A finite element method that incorporates mass transfer capacity is formulated to simulate the evolution of point defects subjected to the kinetics of pore surface diffusion and domain wall migration. The merge of point defects provides a mechanism for the vacancy agglomeration that leads to the formation of large pores or microcracks.

Phase field simulation of the 180°domain-switching process in PbTiO_3 single crystal under an antiparallel electric field

The process of 180°domain switching in PbTiO_3 single crystal under an antiparallel electric field was investigated by the three-dimensional phase field simulation,especially the effect of electric field on the type and duration of domain switching.It is found that the polarization reversal of domains takes place under an antiparallel electric field in PbTiO_3 single crystal.The results of the phase field simulation indicate that there is only 90°domain switching under a weak electric field.With the rise of the electric field,180°domain switching appears.If the electric field is strengthened further,90°domain switching disappears and the duration of domain switching is shortened.

A SIMPLIFIED MODEL FOR DOMAIN SWITCHING OF FERROELECTRIC CRYSTAL

Domain switching is the main source of nonlinear characteristics in ferroelectrics. According to crystal plasticity theory, the domains and domain switching systems for perovskite- type structure ferroelectrics are de?ned. Considering the traverse motion performance of domain wall, a rather simpli?ed form of evolution law about incremental of volume fraction during domain switching has been developed. The main factors, which exert an in?uence on domain switching, such as material parameters, domain wall motion history, kind of domain switching (180? or 90?) and volume fraction, could be addressed. The hysteresis loops of spontaneous electric polarization as a function of electric ?eld, the butter?y shaped strain versus electric ?eld curve and the platform relations between spontaneous polarization and stress, as well as the longitudinal strain and stress, are well simulated and discussed.

Tetragonal Domain Switching via Reversible tm Phase Transformation

Grinding-induced tm phase transformation and the resultant texture in ceria-yttria-doped tetragonal zirconia polycrystals with varied tetragonality have been studied by XRD. It is observed that the reversible tm phase transformation occurs during grinding and the intensity ratio of I(002)t/I(200)t increases with the transformability. The author proposes that the texture induced by grinding at low temperatures is due to the tetragonal variant reorientation via cyclic,reversible tm phase transformation, termed 'transformational domain switching', instead of the ferroelastic one

Nanoscale domain switching mechanism of Bi_(3.15)Eu_(0.85)Ti_3O_(12) thin film under the different mechanical forces

The switching process of ferroelectric thin films in electronic devices is one of the most important requirements for their application. Especially for the different external fields acting on the film surface, the mechanism of domain switching is more complicated. Here we observe the nanoscale domain switchings of Bi3.15Eu0.85Ti3O12 thin film under different mechanical forces at a fast scan rate. As the force increases from initial state to 247.5 n N, the original bright or grey contrasts within the selected grains are all changed into dark contrasts corresponding to the polarization vectors reversed from the up state to the down state, except for the clusters. As the mechanical force increases to 495 n N, the color contrasts in all of the selected grains further turn into grey contrasts and some are even changed into grey contrasts completely showing the typical 90° domain switching. When another stronger loading force 742.5 n N is applied, the phase image becomes unclear and it indicates that the piezoelectric signal can be suppressed under a sufficiently high force, which is coincident with previous experimental results. Furthermore, we adopt the domain switching criterion from the perspective of equilibrium state free energy of ferroelectric nanodomain to explain the mechanisms of force-generated domain switchings.

A MICROMECHANICS METHOD TO STUDY THE EFFECT OF DOMAIN SWITCHING ON FRACTURE BEHAVIOR OF POLYCRYSTALLINE FERROELECTRIC CERAMICS

The effect of domain switching on anisotropic fracture behavior of polycrystalline ferroelectric ceramics was revealed on the basis of the micromechanics method. Firstly, the electroelastic field inside and outside an inclusion in an infinite ferroelectric ceramics is carried out by the way of Eshelby-Mori-Tanaka's theory and a statistical model, which accounts for the influence of domain switching. Further, the crack extension force (energy-release rate) G(ext) for a penny-shape crack inside an effective polycrystalline ferroelectric ceramics is derived to estimate the averaged effect of domain switching on the fracture behavior of polycrystalline ferroelectric ceramics. The simulations of the crack extension force for a crack in a BaTiO3 ceramics are shown that the effect of domain switching must be taken into consideration while analyzing the fracture behavior of polycrystalline ferroelectric ceramics. These results also demonstrate that the influence of the applied electric field on the crack propagation is more profound at smaller mechanical loading and the applied electric field may enhance the crack extension in a sense, which are consistent with the experimental results.

Large and Recoverable Electrostrain in Strained Ferroelectric Superlattices due to Domain Switching

The ferroelectric superlattices have been widely studied due to their distinguished electromechanical coupling properties.Under different biaxial mismatch strains,ferroelectric superlattices exhibit different domain structures and electromechanical coupling properties.A three-dimensional phase field model is employed to investigate the detailed domain evolution and electromechanical properties of the PbTiO_(3)/SrTiO_(3)(PTO/STO)superlattices with different biaxial mismatch strains.The phase field simulations show that the ferroelectric superlattice exhibits large electrostrain in the stacking direction when an external field is applied.Under a large compressive mismatch strain,vortex domains appear in ferroelectric layers with the thickness of 4 nm.The vortex domains become stable cdomain under a large external electric field,which remains when the electric field is removed.When the initial compressive mismatch strain decreases gradually,the waved or a1/a2 domains replaces the initial vortex domains in the absence of electric field.The fully polarized c-domain by a large electric field switches to diagonal direction domain or a/c domain when the electric field is small.Furthermore,when a biaxial tensile strain is applied to the superlattice,ferroelectric domains switch back to the initial a1/a2 twin-like domain structure,resulting in the recoverable and large electrostrain.This provides an effective way to obtain the large and recoverable electrostrain for the engineering application.

In-situ Raman observation on crack tips of Vickers indent in PLZT ceramics

Vickers indentation was introduced into the originally in-plane and out-of-plane poled PLZT ceramics.The Raman spectra were in-situ recorded at selected crack tips before and after the indentations,as well as after the applications of external electric field.The results show that the changes in Raman intensities of optical modes could be sensitively related to 90° domain switching around the crack tips which are strongly dependent on the directions of original polarization and geometric locations.When the direction of electric field was perpendicular to the direction of original polarization,the 90° domain switching at crack tips of the Vickers indentation on the originally in-plane poled PLZT ceramics caused most significant change in the Raman intensity,which inhibited the crack growth.However,when the direction of electric field was parallel to the direction of original polarization,the growth of crack tips became predominantly without the 90° domain switching,which led to the crack growth.

A SIMPLE CONSTITUTIVE MODEL FOR FERROELECTRIC CERAMICS UNDER ELECTRICAL/MECHANICAL LOADING

A simple phenomenological model is developed for describing the macroscopic constitutive response of ferroelectric materials based on consideration of the fact that domain switching is a progressive evolution process with loading. The volume fraction of domain switching is taken as an internal variable, which is derived from the domain nucleation theory. The proposed theory can simulate the dielectric hysteresis, reversed butterfly hysteresis, nonlinear strain-stress hysteresis, as well as electric displacement-stress relation of ferroelectric materials. Its compaxison with experimental results and two other theoretical models reveals that the model presented can well predict the nonlinear hysteresis of ferroelectrics under electrical or mechanical loading.

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.

A micromechanics-based finite element model for the constitutive behavior of polycrystalline ferromagnets

A micromechanics-based finite element model for the constitutive behavior of polycrystalline ferromagnets is developed. In the model, the polycrystalline solid is assumed to comprise numerous single crystals with randomly distributed crystallographic orientations, and the single crystals, in turn, consist of ferromagnetic domains, each of which is represented by a cubic element. The dipole directions of the domains are randomly assigned to simulate the crystallographic nature of ferromagnetic polycrystals. A switching criterion for the domains is specified at the microscopic level. The macroscopic constitutive behavior is obtained by averaging the microscopic/local behavior of each domain. The developed model has been applied to the simulation of a ferromagnetic material. With appropriate material parameters adopted, hysteresis loops of the predicted magnetic induction versus magnetic field and those of the strain versus magnetic field are shown to agree well with experimental observations.

A STUDY OF CRACK CLOSURE IN ELECTRIC-FIELD-INDUCED FATIGUE IN FERROELECTRIC CERAMICS

Considering the influence of the domain switching near the tip of a crack andapplying the idea of multiscale singularity fields in piezoelectric fracture, we have obtained anempirical criterion for the crack closure. Based on the domain switching in the electric yieldregion, referring to Yang's results on the small scale yield model for the electrical fatigue crack,a model of the crack closure during electric-field-induced fatigue is developed to analyze thecrack growth. In terms of the model we have obtained the formula of the rate of the crack growthunder cyclic electric loading. Finally we compare the theoretical predictions with the results givenby Cao and Evans experimentally. It should be pointed out that the model proposed is empirical andneeds to be verified by more experimental results.

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）.

Tuning of non-uniform switch toughening in ferroelectric composites by an electric field

This paper deals with a mode III interfacial crack subject to anti-plane stress and in-plane electric fields. The analysis concentrates on the tuning of fracture toughness from non-uniform ferroelectric-ferroelastic domain switching by an electric field. The electric loading changes the size of the asymmetric switching zone. Employing the weight function method, we obtain the electrically-dependent switch toughening for stationary and quasi-static growing interfacial cracks, respectively. Multi-domain solutions are derived for non-poled and fully-poled ferroelectric composites. Numerical results are presented on the electric field tuning of the critical applied stress intensity factor. The research provides ways to optimize fracture properties of ferroelectric composites by altering the electric field.

An energy-consistent fracture model for ferroelectrics

The fracture behavior of ferroelectrics has been intensively studied in recent decades, though currently a widely accepted fracture mechanism is still lacking. In this work, enlightened by previous experimental observations that crack propagation in ferroelectrics is always accompanied by domain switching, we propose a micromechanical model in which both crack propagation and domain switching are controlled by energy-based criteria. Both electric energy and mechanical energy can induce domain switching, while only mechanical energy can drive crack propagation. Furthermore, constrained domain switching is considered in this model, leading to the gradient domain switching zone near the crack tip. Analysis results show that stress-induced ferroelastic switching always has a toughening effect as the mechanical energy release rate serves as the driving force for both fracture and domain switching. In comparison, the electric-field-induced switching may have either a toughening or detoughening effect. The proposed model can qualitatively agree with the existing experimental results.

Temperature-Dependence of Microstructure Evolution in a Ferroelectric Single Crystal with Conducting Crack

The different temperature-induced nonlinear behavior near a conducting crack tip in a ferroelectric single crystal is studied based on a phase field approach containing the time-dependent Ginzburg-Landau equation.Since domain switching in a crack tip plays an important role in the fracture behavior,by using three-dimensional nonlinear finite element method,the temperature-induced domain switching behavior of a ferroelectric single crystal is simulated under applied electrical and mechanical loads.The simulations show that increasing the temperature will enhance the crack propagation under a strong electric field,which results in switching-weakening.In particular,increasing the temperature from 300°C to 600°C will impede the crack propagation under combined mechanical and electric field loading,which results in switching-toughening.Salient features of the results are consistent with many experimental observations.

Large scale domain switching around the tip of an impermeable stationary crack in ferroelectric ceramics driven by near-coercive electric field

It is widely accepted that the singular term plays a leading role in driving domain switching around the crack tip of ferroelectric ceramics.When an applied electric field approaches or even exceeds the coercive one,however,non-singular terms are no longer negligible and the switching of a large or global scale takes place.To analyze the large scale switching,one has to get a full asymptotic solution to the electric field in the vicinity of the crack tip.Take a double cantilever beam specimen as an example.The derivation of the full electric field is simplified as a mixed boundary value problem of an infinite strip containing a semi-infinite impermeable crack.The boundary value problem is solved by an analytic function and a conformal mapping to yield a full electric field solution in a closed form.Based on the full field solution,the large scale domain switching is examined.The switching zones predicted by the large and small scale switching models are illustrated and compared with each other near the tip of a stationary crack.

Electromechanical cracking in ferroelectrics driven by large scale domain switching

Experimental results indicate three regimes for cracking in a ferroelectric double cantilever beam (DCB) under combined electromechanical loading. In the loading, the maximum amplitude of the applied electric field reaches almost twice the coercive field of ferroelectrics. Thus, the model of small scale domain switching is not applicable any more, which is dictated only by the singular term of the crack tip field. In the DCB test, a large or global scale domain switching takes place instead, which is driven jointly by both the singular and non-singular terms of the crack-tip electric field. Combining a full field solution with an energy based switching criterion, we obtain the switching zone by the large scale model around the tip of a stationary impermeable crack. It is observed that the switching zone by the large scale model is significantly different from that by the small scale model. According to the large scale switching zone, the switch-induced stress intensity factor (SIF) and the transverse stress (T-stress) are evaluated numerically. Via the SIF and T-stress induced by the combined loading and corresponding criteria, we address the crack initiation and crack growth stability simultaneously. The two theoretical predictions roughly coincide with the experimental observations.