An I-integral method for crack-tip intensity factor variation due to domain switching in ferroelectric single-crystals

In the present study, an I-integral method is established for solving the crack-tip intensity factors of ferroelectric single-crystals. The I-integral combined with the phase field model is successfully used to investigate crack-tip intensity factor variations due to domain switching in ferroelectricity subjected to electromechanical loadings, which exhibits several advantages over previous methods based on small-scale switching. First, the shape of the switching zone around a crack tip is predicted by the time-dependent Ginzburg–Landau equation, which does not require preset energy-based switching criterion. Second, the I-integral can directly solve the crack-tip intensity factors and decouple the crack-tip intensity factors of different modes based on superimposing an auxiliary state onto an actual state. Third, the I-integral is area-independent, namely, the I-integral is not affected by the integral area size, the polarization distributions, or domain walls. This makes the I-integral applicable to large-scale domain switching. To this end, the electro-elastic field intensity factors of an impermeable crack in PbTiO₃ ferroelectric single crystals are evaluated under electrical, mechanical, and combined loading. The intensity factors obtained by the I-integral agree well with those obtained by the extrapolation technique. From numerical results, the following conclusions can be drawn with respect to fracture behavior of ferroelectrics under large-scale switching. Under displacement controlled mechanical loading, the stress intensity factors (SIFs) decrease monotonically due to the domain switching process, which means a crack tip shielding or effective switching-induced toughening occurs. If an external electric field is applied, the electric displacement intensity factor (EDIF) increases in all cases, i.e., the formed domain patterns enhance the electric crack tip loading. The energy release rate, expressed by the crack-tip J-integral, is reduced by the domain switching in all examples, which underlines the switching-induced-toughening effect. In contrast, under stress controlled load, the SIF evolves due to large-scale switching to a stable value, which is higher than the non-switching initial value, i.e., fracture is promoted in this case.     

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.     

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 accompa-nied by domain switching, we propose a micromechanical model in which both crack propagation and domain switch-ing 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 compari-son, the electric-field-induced switching may have either a toughening or detoughening effect. The proposed model can qualitatively agree with the existing experimental results.     

Time-harmonic P-waves engulfing a rectangular limited-permeable crack in piezoelectric medium: Energy density and energy release

The dynamic behavior of a limited-permeable rectangular crack in a transversely isotropic piezoelectric material is impinged by to a P-wave. The generalized Almansi theorem and the Schmidt method are used to determine the stress intensity factor and energy density factor as the primary fracture criterion of failure. The mixed boundary value problem entails the evaluation of the appropriate crack edge stress singularities that are characteristics of the fundamental functions. The stress and electric displacement intensity factors are also used to find the energy release rate that can be computed numerically and compared with the results corresponding to those of the stress intensity factor, and energy density factor. Graphical presentation shows that the energy release rate is always negative for the boundary conditions considered while the energy density factors always remain positive. Under certain conditions, the stress and electric displacement intensity factors can be negative and subject to physical limitations. Piezoelectric material boundary value problem solutions should therefore be qualified by the application of failure criteria by fracture of otherwise, particularly when the mechanical and electrical energy can release by creating free surface at the macroscopic and microscopic scales. Negative energy release rate found for the piezoelectric medium in this work can be a case in point.Positive definiteness of the energy density factor can be applied to mutliscale fracture. This is not true for the stress intensity factor nor the energy release rate. Hence, crack initiation behavior for the permittivity of a rectangular crack due to the wave propagation effects may be studied. In particular, the initiation of micro-cracks may be identified with certain critical stress wave frequency band. Negative stress intensity factor may not enhance macrocracking but it does not exclude microcrack initiation.     

Mixed mode crack initiation in piezoelectric ceramic strip

When piezoelectric ceramics are subjected to mechanical and electrical load, they can fracture prematurely due to their brittle behavior. Hence, it is important to know the electro–elastic interaction and fracture behavior of piezoelectric materials. The problem of a through crack in a piezoelectric strip of finite thickness is studied in this paper. Fourier transforms are used to reduce the problem to the solution of singular integral equations. The model technique can solve for polarization in an arbitrary direction and material anisotropy. Numerical values of the crack-tip field amplification for a piezoelectric strip under in-plane electromechanical loading are obtained. Energy density factor criterion is applied to obtain the maximum of the minimum energy density and direction of crack initiation. The influence of crack length and crack position on stress intensity and energy density factors is discussed.     

Effect of transverse stress on switch-toughening of ferroelectrics

Testing data indicated that T stress significantly altersR-curves of ferroelectrics. The model of stress-induced polarization switching is adopted to evaluate the fracture toughness of ferroelectrics under K field and T stress. Analytical solution is obtained to estimate the steady state fracture resistance of mono-domain ferroelectrics. The result is generalized to multi-domain ferroelectric ceramics via Reuss approximation, which enables us to explain quantitatively R-curves under two testing configurations.     

Mixed-mode fracture analysis of orthotropic functionally graded materials under mechanical and thermal loads

Mixed-mode fracture problems of orthotropic functionally graded materials (FGMs) are examined under mechanical and thermal loading conditions. In the case of mechanical loading, an embedded crack in an orthotropic FGM layer is considered. The crack is assumed to be loaded by arbitrary normal and shear tractions that are applied to its surfaces. An analytical solution based on the singular integral equations and a numerical approach based on the enriched finite elements are developed to evaluate the mixed-mode stress intensity factors and the energy release rate under the given mechanical loading conditions. The use of this dual approach methodology allowed the verifications of both methods leading to a highly accurate numerical predictive capability to assess the effects of material orthotropy and nonhomogeneity constants on the crack tip parameters. In the case of thermal loading, the response of periodic cracks in an orthotropic FGM layer subjected to transient thermal stresses is examined by means of the developed enriched finite element method. The results presented for the thermally loaded layer illustrate the influences of the material property gradation profiles and crack periodicity on the transient fracture mechanics parameters.     

Toughening of ferroelectrics by the out-of-plane poling

Subjected to the prior out-of-plane poling, the ferroelectrics can be toughened considerably. The present paper describes the variation of the stress intensity factor (SIF) by 90° switching in ferroelectrics. The analysis is carried out for the combined mechanical and electrical loading, with simple relations obtained for the case of the purely electrical loading. The out-of-plane poling is found to raise the SIF for the crack initiation, but appreciably reduces the SIF for the crack growth in a steady state. More stable fracture resistance curves can be achieved by the out-of-plane poling. This prediction is supported quantitatively by the testing data of SENB specimens of PZT-5 samples, when the toughening effects of polings in three orthogonal directions are compared. The project supported by Sino-Germany Center for Science and Technology (G2050/8)     

热电材料孔边周期裂纹问题的热电弹分析

论文研究了均匀电流密度和能量流作用下,热电材料中带4k个周期径向裂纹的圆形孔口问题.考虑非渗透型电和热边界条件,运用复变函数理论和保形映射方法,得到了热电材料中电流密度、能量密度和应力场的精确解.依据断裂力学理论,运用Cauchy积分公式得到了周期裂纹的电流、能量和应力强度因子.数值结果分析了场强度因子随各个参数的变化...     

Macroscopical non-linear material model for ferroelectric materials inside a hybrid finite element formulation

A new approach for modeling hysteretic non-linear ferroelectric ceramics is presented, based on a fully ferroelectric/ferroelastic coupled macroscopic material model. The material behavior is described by a set of yield functions and the history dependence is stored in internal state variables representing the remanent polarization and the remanent strain. For the solution of the electromechanical coupled boundary value problem, a hybrid finite element formulation is used. Inside this formulation the electric displacement is available as nodal quantity (i.e. degree of freedom) which is used instead of the electric field to determine the evolution of remanent polarization. This involves naturally the electromechanical coupling. A highly efficient integration technique of the constitutive equations, defining a system of ordinary differential equations, is obtained by a customized return mapping algorithm. Due to some simplifications of the algorithm, an analytical solution can be calculated. The automatic differentiation technique is used to obtain the consistent tangent operator. Altogether this has been implemented into the finite element code FEAP via a user element. Extensive verification tests are performed in this work to evaluate the behavior of the material model under pure electrical and mechanical as well as coupled and multi-axial loading conditions.     

Fracture mechanics for a mode III crack in a magnetoelectroelastic solid

In this paper, we developed a Stroh-type formalism for anti-plane deformation and then investigated the fracture mechanics for an elliptical cavity in a magnetoelectroelastic solid under remotely uniform in-plane electromagnetic and/or anti-plane mechanical loading, which allowed us to take the electromagnetic field inside the cavity into account. Reducing the cavity into a crack, we had explicit solutions in closed forms for a mode III crack, which included the extreme cases for an impermeable crack and a permeable crack. The results were illustrated with plots, showing that in the absence of mechanical loads, an applied electric or magnetic field, positive or negative, always tended to close the crack. On the other hand, in the presence of a mechanical load, a negative electric or magnetic field retarded crack growth, while a positive field could either enhance or retard crack propagation, depending on the strengths of the applied electric/magnetic fields and the level of the mechanical load as well. In other words, the effect of electric/magnetic fields on the fracture behavior is mechanical load-dependent.     

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.     

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.     

An exact solution of crack problems in piezoelectric materials

IntroductionInrecentyearscrackproblemsinpiezoelectricmaterialhavereceivedmuchattention.Manytheoreticalanalyseshavebeengivenby[1～16].Itshouldbe,however,notedthatalltheaboveanalysesarebasedonaso-calledimpermeablecrackassumphon,i.e.thecrackfacesareassumedtobeimpermeabletoelectricfield,sotheelectricdisplacementvanishesinsidethecrack.Usingthisassumption,onewillobtainthefollowingresultS[2'3'5,6'9'16]=whentheelectricloadsaresolelyaPPliedatinLfinity,theelectricdisplacementissquare-rootsingularatthe…     

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.     

A penny-shaped interface crack between a functionally graded piezoelectric layer and a homogeneous piezoelectric layer

The problem of a penny-shaped interface crack between a functionally graded piezoelectric layer and a homogeneous piezoelectric layer is investigated. The surfaces of the composite structure are subjected to both mechanical and electrical loads. The crack surfaces are assumed to be electrically impermeable. Integral transform method is employed to reduce the problem to a Fredholm integral equation of the second kind. The stress intensity factor, electric displacement intensity factor and energy release rate are derived, some typical numerical results are plotted graphically. The effects of electrical loads, material nonhomogeneity and crack configuration on the fracture behaviors of the cracked composite structure are analyzed in detail.     

Strip-electromechanical model solution for piezoelectric plate cut along two semi-permeable collinear cracks

A strip electric saturation and mechanical yielding model solution is proposed for a piezoelectric plate cut along two equal collinear semi-permeable mode-I cracks with electrical polarization reaching a saturation limit and normal stress reaching a yield stress along a line segment in front of the cracks. By using Stroh formalism and complex variable technique, we derived the analytical solution for the field quantities. Three different situations are investigated when developed electrical saturation zone is bigger/smaller or equal to the developed mechanical yield zone. Numerical results show that the effect of different electric boundary conditions on the crack opening displacement and crack opening potential drop is significant and should not be ignored. The influence of electric load displacement on the energy release rate is also investigated for PZT-4, PZT-5H and BaTiO₃ ceramics, and it may assists for the correct choosing of ceramic for specific job.     

Limited permeable crack moving along the interface of a piezoelectric bi-material

A plane problem for a crack moving with a subsonic speed along the interface of two piezoelectric semi-infinite spaces is considered. The crack is assumed to be free from mechanical loading. The limited permeable electric condition with an account of electric traction is adopted at its faces. A uniformly distributed mixed mode mechanical loading and an electric flux are prescribed at infinity. The problem is reduced to the Riemann–Hilbert problem by means of introducing a moving coordinate system and assuming that the electric flux is uniformly distributed along the crack region. An exact solution of this problem is proposed. It permits to find in closed form all necessary electromechanical characteristics at the interface and to formulate the equation for the determination of the electric flux. Analysis of this equation confirms the correctness of the assumption concerning the uniform distribution of the electric flux in the crack region. The values of the electric flux are determined by solving the obtained equation. Thereafter, the stress and electric intensity factors as well as their asymptotic fields at the crack tip are also found. The particular case of a crack moving in a homogeneous piezoelectric material is considered. The values of the electric flux and the fracture parameters are found exactly in a simple form for this case. Also, a numerical analysis is performed for a crack propagating with a subsonic speed between PZT4 and PZT5 materials and for a crack moving in PZT4 material. The electric flux in the crack region, stress and electric intensity factors, crack opening and the energy release rate (ERR) are found as functions of the crack speed, loading and electric permeability of the crack medium. The influence of the electric traction on the crack faces upon the mentioned parameters is demonstrated.     

Transient response of an insulating crack between dissimilar piezoelectric layers under mechanical and electrical impacts

Summary  The dynamic response of an interface crack between two dissimilar piezoelectric layers subjected to mechanical and electrical impacts is investigated under the boundary condition of electrical insulation on the crack surface by using the integral transform and the Cauchy singular integral equation methods. The dynamic stress intensity factors, the dynamic electrical displacement intensity factor, and the dynamic energy release rate (DERR) are determined. The numerical calculation of the mode-I plane problem indicates that the DERR is more liable to be the token of the crack growth when an electrical load is applied. The dynamic response shows a significant dependence on the loading mode, the material combination parameters as well as the crack configuration. Under a given loading mode and a specified crack configuration, the DERR of an interface crack between piezoelectric media may be decreased or increased by adjusting the material combination parameters. It is also found that the intrinsic mechanical-electrical coupling plays a more significant role in the dynamic fracture response of in-plane problems than that in anti-plane problems. Received 4 September 2001; accepted for publication 23 July 2002 The work was supported by the National Natural Science Foundation under Grant Number 19891180, the Fundamental Research Foundation of Tsinghua University, and the Education Ministry of China.     

电可通纳米尺度孔边均布多裂纹的电弹断裂分析

解析研究了面内电载荷和反平面机械载荷作用下压电体中纳米尺度圆孔边均布电可通多裂纹问题的断裂性能。基于Gurtin-Murdoch表面弹性理论,利用保角映射方法和复变弹性理论给出了裂纹尖端电弹场分布、电弹场强度因子及能量释放率的解析结果。阐述了无量纲电弹场强度因子、无量纲能量释放率的尺寸依赖效应,讨论了裂纹数量和缺陷几何参数对无量纲场强度因子和无量纲能量释放率的影响。结果表明：无量纲电弹场强度因子和无量纲能量释放率具有显著的尺寸依赖效应;考虑表面效应,孔径和裂纹长度相当时,电弹场强度因子达到最大;裂纹/孔径比对电弹场强度因子随裂纹数量变化的制约会随着裂纹数量的增加而逐渐消失;过大或过小的裂纹孔径比会削弱裂纹长度对能量释放率的影响。