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。     

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.     

Nonlinear ferro-electro-elastic beam theory

Motivated by the uniqueness and potential of the nonlinear range of piezoelectric and ferroelectric smart materials and structures, a static physically nonlinear ferro-electro-elastic beam theory which takes the effect of domain switching into account is developed. The kinematic assumptions adopt the geometrically linear Bernoulli–Euler form for the mechanical components and a first-order theory for the electrical potential, and lay the basis for further augmentation to higher order theories. The beam theory includes the field equations that correspond to the static case, the boundary conditions and the constitutive equations of ferro-electro-elasticity. The general 3-D constitutive equations are reduced to comply with the beam theory and formulated as ordinary differential equations by means of a set of generalized electro-mechanical stiffnesses. A micromechanical constitutive model that accounts for the loading history and for the domain switching phenomenon is adopted and an iterative solution procedure that incorporates the micromechanical approach is suggested. A numerical example that demonstrates the impact of the domain switching on the nonlinear electromechanical static response of a ferro-electro-elastic beam is presented and discussed. The quantitative assessment of this behavior takes a step towards new structural applications that cope with or even take advantage of the nonlinear ferro-electro-elastic range.     

循环电载下热效应对铁电材料畴变的影响

铁电材料在力、电载荷作用尤其是循环载荷作用下有明显的热效应.热效应会在铁电材料中引起应力场和电场.另外,在外载作用下铁电材料发生约束畴变时也会产生附加的应力场,这些附加的应力场和电场都会对铁电材料的畴变产生影响.而且在循环载荷作用的情况下,这种影响会逐渐累积.但在以往的研究中,很少涉及这种影响.该文就循环电载下热效应引起的力、电场和约束畴变时产生的附加应力场对铁电材料畴变的影响进行了初步研究.     

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.     

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).     

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.     

铁电陶瓷微观电畴演化的成核率模型

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.根据这些描述电畴运动的物理实验结果,提出了成核率模型来模拟铁电材料的非线性本构行为.进一步由模型建立了基于微观电畴运动的电畴翻转体积分数演化方程.应用该模型得到的理论计算结果与实验数据的比较表明,模型能够描述铁电材料的非线性本构行为.同时模型所揭示的微观电畴成核的演化行为可进一步的指导宏观唯象模型的建立与改进.     

A constitutive model for thermo-electro-mechanical behavior of ferroelectric polycrystals near room temperature

A constitutive model that can be used to predict thermo-electro-mechanical linear and nonlinear behavior of ferroelectric polycrystals near room temperature is proposed. A ferroelectric polycrystal is modeled by an agglomerate of 210 single crystallites that are distributed regularly over all directions. A variant in a single crystallite is characterized by a Gibbs free energy function whose coefficients have linear dependency on temperature. A dissipation inequality for domain switching is derived from the restriction of the second law of thermodynamics. Domain switching process is governed by a viscoplastic switching law with temperature-dependent switching parameters. The responses of the proposed model to electric field and mechanical stress loading at room and elevated temperatures are calculated and compared qualitatively with experimental observations available in literature.     

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.     

铁电陶瓷宏观单轴力电行为的双面模型

铁电陶瓷以其优越的力电耦合性作为新型的智能材料使用. 提出基于弹塑性双面理论的宏观铁电本构模型. 根据铁电陶瓷内部电畴在外电场和机械场作用下的微观运动，在宏观上除引入材料的畴变面外，还首次引入饱和面，并考虑以畴变面与饱和面之间的广义距离来表征铁电陶瓷的非线性行为. 数值计算结果与实验数据的比较表明所提出的初步理论可适当地反映力电加载下铁电陶瓷的宏观非线性行为.     

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)     

Influence of domain switching zones on the fracture toughness of ferroelectrics

The paper presents a fracture model for ferroelectric materials taking into account the hysteretic domain switching processes near to the tip of a macroscopic crack. The model is based on the balance of energy supplied by the driving forces, on the one hand, and the total of energies either dissipated by domain switching, stored in the crack wake region or consumed by the formation of new fracture surface, on the other hand. An internal variable theory describes the nonlinear coupled electromechanical material response within the framework of a three-dimensional continuum model. For simplicity, the complex orientation distribution function of domains in a polycrystalline ceramic is approximated by only six representative space orientations. The theory predicts certain dimensionless material parameter combinations which govern the change of fracture toughness under the application of different mechanical and electrical loadings. A comparison with data available in the literature for barium titanate ceramics yields a reasonable coincidence.     

Fatigue crack growth induced by domain switching under electromechanical load in ferroelectrics

Reliability calls for a better understanding of the failure of ferroelectric ceramics. The fracture and fatigue of ferroelectric ceramics under an electric field or a combined electric and mechanical loading are investigated. The small-scale domain-switching model is modified to analyze failure due to fracture and fatigue. Effects of anisotropy and electromechanical load coupling are taken into account. Analytical expressions are obtained for domain-switching regions near the crack tip such that of 90° domain switching can be distinguished from 180° domain switching in addition to different initial poling directions. The crack tip stress intensity variation of ferroelectric ceramics due to the domain switching is analyzed. A positive electric field tends to enhance the propagation of an insulating crack perpendicular to the poling direction, while a negative field impedes it. Fatigue crack growth under various coupling loads and effects of the stress field and electric field on near field stress intensity variation are analyzed. Predicted crack growth versus cyclic electric field agrees well with experiment.     

A constitutive model for magnetostrictive and piezoelectric materials

This paper is concerned with a macroscopic nonlinear constitutive law for magnetostrictive alloys and ferroelectric ceramics. It accounts for the hysteresis effects which occur in the considered class of materials. The uniaxial model is thermodynamically motivated and based on the definition of a specific free energy function and a switching criterion. Furthermore, an additive split of the strains and the magnetic or electric field strength into a reversible and an irreversible part is suggested. Analog to plasticity, the irreversible quantities serve as internal variables. A one-to-one-relation between the two internal variables provides conservation of volume for the irreversible strains. The material model is able to approximate the ferromagnetic or ferroelectric hysteresis curves and the related butterfly hysteresis curves. Furthermore, an extended approach for ferrimagnetic behavior which occurs in magnetostrictive materials is presented. A main aspect of the constitutive model is its numerical treatment. The finite element method is employed to solve the coupled field problem. Here the usage of the irreversible field strength permits the application of algorithms of computational inelasticity. The algorithmic consistent tangent moduli are developed in closed form. Hence, quadratic convergence in the iterative solution scheme of governing balance equations is obtained.     

Computational modeling of rate-dependent domain switching in piezoelectric materials

In this contribution a micromechanically motivated model for rate-dependent switching effects in piezoelectric materials is developed. The proposed framework is embedded into a three-dimensional finite element setting whereby each element is assumed to represent an individual grain. Related dipole (polarization) directions are thereby initially randomly oriented at the element level to realistically capture the originally un-poled state of grains in the bulk ceramics. The onset of domain switching processes is based on a representative energy criterion and combined with a linear kinetics theory accounting for time-dependent propagation of domain walls during switching processes. In addition, grain boundary effects are incorporated by making use of a macromechanically motivated probabilistic approach. Standard volume-averaging techniques with respect to the response on individual grains in the bulk ceramics are later on applied to obtain representative hysteresis and butterfly curves under macroscopically uniaxial loading conditions at different loading frequencies. It turns out that the simulations based on the developed finite element formulation nicely match experimental data reported in the literature.     

Electrical body forces and electrical tractions in the nonlinear response of ferroelectric actuators

The influence of the electrical body forces and electrical tractions on the nonlinear response of ferroelectric stack actuators is analytically investigated. While the role of the electrical body forces and tractions in the response of piezoelectric actuators is well documented (and in many cases is not significant), the questions of their effect on ferroelectric active materials is still of interest. To examine this influence, the analytical model for the electro-mechanical behavior of a ferroelectric stack actuator is augmented to account for the electrical body forces along the actuator and the electrical tractions at the material–electrode interfaces. Focusing on the effect of the electrical forces and tractions on the ferroelectric domain switching phenomenon, the model is used for the numerical analysis of a ferroelectric layer and for the comparison with the case that neglects the electrical body forces and traction. The comparison theoretically designates cases in which the effect of the electrical body forces and tractions may be prominent and other cases where the classical approach that neglects these effects can be adopted.     

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

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Three dimensional dynamic electromechanical constitutive relations for ferroelectric materials

We propose, in this paper, three dimensional constitutive relations suitable for describing the dynamic electromechanical responses of ferroelectric materials. Our approach is phenomenological and is based on the assumptions that the macroscopic electrical properties can be represented by electric dipoles whose magnitude and direction can be altered by external stimuli. The constitutive relations for the stress and the electric displacement take into account the transient and instantaneous responses of the dipole moments and the transient response of domain switching. A specific respesentation of the theory is also derived; this representation is suitable for studying the responses of ferroelectic materials to dynamic mechanical loading and rapidly applied electric field.     

A SIMPLIFIED MODEL FOR DOMAIN SWITCHING OF FERROELECTRIC CRYSTAL

I. INTRODUCTION Domain switching is the main source of nonlinear characteristics of ferroelectric materials. The trans-formation performance of domain is the basis for ferroelectric constitutive research. In many literatures[1??10], complete switching models were adopted, in which ferroelectric materials are considered tobe consist of several basic sorts of domains, which are independent of each other. Under the actionof su?ciently strong electric ?elds or mechanical stress, the orienta…