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.     

Micromechanics of residual stress and texture development due to poling in polycrystalline ferroelectric ceramics

This paper presents micromechanics based analysis of elastic strain and changes in the texture of poled polycrystalline ferroelectric PZT ceramics for direct comparison with synchrotron X-ray measurements. The grains are modelled as spherical inclusions, to which transformation strains are assigned depending on the fractions of different ferroelectric domains. Eshelby's inclusion problem with the classical self-consistent method is applied to evaluate the elastic state of the grains. In particular, the elongation due to lattice elastic strain is calculated as a function of inclination Ψ relative to the polar axis. The ratio of diffraction peak intensities, corresponding to the domain fractions, is also expressed as a function of Ψ. This analysis identifies the special character of the reflection, for which the lattice strain along in the stress free state is independent of ferroelectric domain population and hence unaffected by poling. The elongation due to the lattice strain parallel to and peak intensity ratio are expressed in terms of the overall macroscopic strain of a poled specimen, each having a dependence.     

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

ＩｎｔｒｏｄｕｃｔｉｏｎＲｅｃｅｎｔｌｙ ,ｔｈｅｆｅｒｒｏｅｌｅｃｔｒｉｃｃｅｒａｍｉｃｓｈａｓｓｕｃｈｅｘｃｅｌｌｅｎｔｃｈａｒａｃｔｅｒｉｓｔｉｃｓｏｆｐｉｅｚｏｅｌｅｃｔｒｉｃｉｔｙａｎｄｐｙｒｏｅｌｅｃｔｒｉｃｉｔｙｅｔｃ .ｔｈａｔｉｔｂｅｃｏｍｅｓｏｎｅｏｆｔｈｅｍｏｓｔｉｍｐｏｒｔａｎｔｆｕｎｃｔｉｏｎａｌｍａｔｅｒｉａｌｓ.Ｆｏｒｉｎｓｔａｎｃｅ ,ｔｈｅｗｉｄｅｌｙａｐｐｌｉｅｄｓｅｎｓｏｒｓ,ｔｒａｎｓｄｕｃｅｒｓａｎｄａｃｔｕａｔｏｒｓｅｔｃ .ａｒｅｍａｄｅｏｆｔｈｅｆｅｒｒｏｅ…     

A one-dimensional continuum model for ferroelectric ceramics

In this article, we introduce a one-dimensional continuum model for ferroelectric ceramics within a thermodynamical framework. The model consists of a free energy potential, a switching criterion, and a kinetic relation. The free energy potential is given as a function of polarization, strain, and two internal variables – remanent polarization and remanent strain. A polarization switching is described by evolutions of the two internal variables and evolution laws called kinetics are proposed based on the second law of thermodynamics. The predictions of the model are compared with experimental observations. It is suggested to model unpoled domains in the fully poled state for improved model responses.     

Electric-field-induced fatigue crack growth in ferroelectric ceramics

Electric-field-induced fatigue crack growth in pre-cracked PZT ferroelectric ceramics is experimentally investigated in this work. It is found that the crack open and close under an alternating electric field is a major mechanism of crack propagation. The experimental results also show that the frequency, waveform, as well as the amplitude ratio, of the electric loading, play important roles in electric-field-induced fatigue cracking. Empirical formulations of fatigue crack propagation rates are obtained based on the experimental results. It is revealed that the crack grows at a nearly constant rate when the loading frequency is below 100 Hz. However, with the increase of the loading frequency over 125 Hz, the crack propagation rate diminishes rapidly.     

A phenomenological constitutive model for ferroelastic and ferroelectric hysteresis effects in ferroelectric ceramics

This paper is concerned with a macroscopic constitutive law for domain switching effects, which occur in ferroelectric ceramics. The three-dimensional model is thermodynamically consistent and is determined by two scalar valued functions: the Helmholtz free energy and a switching surface. In a kinematic hardening process the movement of the center of the switching surface is controlled by internal variables. In common usage, the remanent polarization and the irreversible strain are employed as internal variables. The novel aspect of the present work is to introduce an irreversible electric field, which serves instead of the remanent polarization as internal variable. The irreversible electric field has only theoretical meaning, but it makes the formulation very suitable for a finite element implementation, where displacements and the electric potential are the nodal degrees of freedom. The paper presents an appropriate implementation into a hexahedral finite brick element. The uni-axial constitutive model successfully reproduces the ferroelastic and the ferroelectric hysteresis as well as the butterfly hysteresis for ferroelectric ceramics. Furthermore it accounts for the mechanical depolarization effect, which occurs if the polarized ferroelectric ceramic is subjected to a compression stress.     

Effect of electric fields on fracture behavior of ferroelectric ceramics

The asymptotic problem of a semi-infinite crack perpendicular to the poling direction in a ferroelectric ceramic subjected to combined electric and mechanical loading is analyzed to investigate effect of electric fields on fracture behavior. Electromechanical coupling induced by the piezoelectric effect is neglected in this paper. The shape and size of the switching zone is shown to depend strongly on the relative magnitude between the applied electric field and stress field as well as on the ratio of the coercive electric field to the yield electric field. A universal relation between the crack tip stress intensity factor and the applied intensity factors of stress and electric field under small-scale conditions is obtained from the solution of the switching zone. It is found that the ratio of the coercive electric field to the yield electric field plays a significant role in determining the enhancement or reduction of the crack tip stress intensity factor. The fracture toughness variation of ferroelectrics under combined electric and mechanical loading is also discussed.     

A finite element model for rate-dependent behavior of ferroelectric ceramics

In this article, materials within a crystallite are modeled by continuum particles consisting of various types of ferroelectric variants which are characterized by their mass fractions. The constitutive behavior of each type of variant is characterized by a proposed Helmholtz free energy potential. Polarization switching is modeled by continuous changes of mass fractions which are governed by a onset criterion and a kinetic relation. A finite element algorithm is developed using the virtual work principle. The simulated results on the rate dependence in the polarization and strain responses to applied alternating electric field of different frequencies are in qualitative consistence with experimental observations. The rate-dependent behavior is explained in terms of changes of mass fractions of the variants that polarization switching involves, in response to the loading programs of different loading rates.     

Phenomenological model for the macroscopical material behavior of ferroelectric ceramics

A thermodynamically consistent phenomenological model for the simulation of the macroscopic behavior of ferroelectric polycrystalline ceramics is presented. It is based on the choice of microscopically motivated internal state variables, which describe the texture and the polarization state of the polycrystal. Saturation states are defined for the internal state variables. The linear material behavior is modelled by a transversely isotropic piezoelectric constitutive law, where the anisotropy is history dependent. For non-linear irreversible processes, a switching function and associated evolution rules are applied, satisfying the principle of maximum ferroelectric dissipation. Saturation is modelled by the use of energy-barrier functions in the electric enthalpy density function. Numerical examples demonstrate the capability of the proposed model, to predict the typical experimental results.     

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).     

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.     

铁电陶瓷微观电畴的成核率模型

大量的实验已经证实电畴翻转是铁电材料非线性和迟滞性本构曲线的根本原因。研究者已经对铁电陶瓷的微观电畴翻转行为进行了大量详细的研究。针对描述电畴成核的物理实验结果和经典的成核率实验数据,为了建立电畴翻转体积分数的演化方程提出了反应微观电畴翻转的成核率模型。针对铁电试样电畴随机分布的情况,应用该模型对铁电陶瓷的非线性本构行为进行了研究。理论结果与实验数据的比较表明,模型能较好的描述铁电材料的非线性本构行为。同时模型所揭示的微观反转的物理本质可进一步的指导宏观唯象模型的改进。     

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）.     

Time dependent behavior of ferroelectric materials undergoing changes in their material properties with electric field and temperature

This paper presents a time dependent polarization constitutive model suitable for predicting nonlinear polarization and electro-mechanical strain responses of ferroelectric materials subject to various histories of electric fields. The constitutive model is derived based on a single integral form with nonlinear (electric field and temperature dependent) integrand. The total polarization consists of the time-dependent and residual components. The residual component of the polarization is due to polarization switching in the ferroelectric materials. We use an ‘internal clock’ concept to incorporate the effect of electric field on the rate of polarization. The corresponding strain response is determined through the use of third order piezoelectric constant and/or fourth order electrostrictive constant that vary with polarization stage. It is assumed that in absence of polarization, both piezoelectric and electrostrictive constants are zero. To incorporate the effect of temperature on the overall polarization behavior all material parameters in the constitutive model are allowed to change with the ambient temperature. We present numerical studies on the effect of time, temperature, and electric field on the response of ferroelectric material followed by verification of the constitutive model. Experimental data on lead zirconate titanate (PZT) materials available in the literature are used to verify the model.     

Unconventional domain band structure at the crack tip in ferroelectric ceramics

Domain polarization switch near the tip of a crack or an electrode plays a critical role in the fracture or toughening of ferroelectric ceramics. The intensive electric field near a crack tip stimulates local domain switching. Experiment indicates that the domain band structure in front of an indentation crack under lateral electric loading is unconventional, attributed to the highly localizing crack tip electric field. The partially switched ferroelectric grain resembles a banded Eshelby inclusion embedded in a polycrystalline ferroelectric matrix. The domain wall energy for unconventional domain structures is estimated via arrays of misfit dislocations. Mesomechanics analysis quantifies the unconventional domain band structures. The predicted parameters include the volume fraction, the thickness, and the orientation of switched domain bands.     

Finite element simulation of a polycrystalline ferroelectric based on a multidomain single crystal switching model

In this paper, we compute the constitutive behavior of a ferroelectric ceramic by a plane strain finite element model, where each element represents a single grain in the polycrystal. The properties of a grain are described by the microscopic model for switching in multidomain single crystals of ferroelectric materials presented by Huber et al. [J. Mech. Phys. Solids 47 (1999) 1663]. The poling behavior of the polycrystal is obtained by employing the finite element formulation for electromechanical boundary value problems developed by Landis [Int. J. Numer. Meth. Eng. 55 (2002) 613]. In particular, we address the influence of the single grain properties and the interaction between grains, respectively.     

爆轰驱动飞片撞击铁电体爆电换能的放电特性

为了研究铁电体作小型电磁脉冲弹高功率脉冲电源的放电特性,提出了爆轰驱动飞片撞击铁电体的冲击波数值算法,计算了不同装药长度下飞片对铁电体的冲击载荷,建立了铁电体负载为高压储能电容的去极化放电模型,设计了铁电体去极化放电装置,并进行了实验。实验表明:5mm 的装药柱长度能够满足铁电体相变的压强条件;负载为0.2、18H 时,储能电容能够将放电脉宽拉伸至20s,去极化放电能量随着铁电体并联数目增加而增加,放电损耗率大约15%。