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.     

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

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压电薄膜表面贯穿裂纹的有限元分析

基于有限元软件ANSYS数值模拟,计算了激光作用下的压电薄膜表面贯穿裂纹外场应力强度因子和电位移强度因子,并且研究了90°畴变所诱致的畴变增韧行为。首先,求解无裂纹压电薄膜在激光作用下的热-力-电响应,将求得的应力和电位移场反向作用于裂纹面,求解裂纹尖端处的外场应力和电位移强度因子,然后基于小范围畴变理论求解了90°畴变所致的屏蔽应力强度因子。讨论了薄膜表面裂纹的外场应力强度因子、电位移强度因子及屏蔽应力强度因子随激光作用时间和裂纹位置的变化关系,从而预测压电薄膜体系在加热工作状况下的裂纹扩展和断裂行为。     

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 and apply-ing the idea of multiscale singularity fields in piezoelectric fracture,we have obtained an empiricalcriterion for the crack closure.Based on the domain switching in the electric yield region,referringto Yang's results on the small scale yield model for the electrical fatigue crack,a model of thecrack closure during electric-field-induced fatigue is developed to analyze the crack growth.Interms of the model we have obtained the formula of the rate of the crack growth under cyclicelectric loading.Finally we compare the theoretical predictions with the results given by Cao andEvans experimentally.It should be pointed out that the model proposed is empirical and needsto be verified by more experimental results.     

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.     

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.     

On the fracture toughness of ferroelastic materials

The toughness enhancement due to domain switching near a steadily growing crack in a ferroelastic material is analyzed. The constitutive response of the material is taken to be characteristic of a polycrystalline sample assembled from randomly oriented tetragonal single crystal grains. The constitutive law accounts for the strain saturation, asymmetry in tension versus compression, Bauschinger effects, reverse switching, and strain reorientation that can occur in these materials due to the non-proportional loading that arises near a propagating crack. Crack growth is assumed to proceed at a critical level of the crack tip energy release rate. Detailed finite element calculations are carried out to determine the stress and strain fields near the growing tip, and the ratio of the far field applied energy release rate to the crack tip energy release rate. The results of the finite element calculations are then compared to analytical models that assume the linear isotropic K-field solution holds for either the near tip stress or strain field. Ultimately, the model is able to account for the experimentally observed toughness enhancement in ferroelastic ceramics.     

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.     

非均匀铁弹畴变对离面极化压电陶瓷应力强度因子的影响研究

基于文献中报道的试验结果，本文考虑非均匀的畴变过程区，它包含一个位于中心的饱和区和环绕饱和区的渐变区．为了描述外加应力引起的部分铁弹畴变，本文采用一个显式的基于最小能量原理的非均匀畴变准则．考虑离面极化的压电陶瓷，假设其初始极化矢量平行于离面方向．畴变后的电畴位于面内，具体方位由最大释放功来确定．基于非均匀畴变准则，本文给出了裂尖处非均匀畴变区几何及畴变体积分数的分布．并在静止裂纹和稳态扩展裂纹两种特殊情况下计算了铁弹畴变对裂尖处应力强度因子的影响．结果表明：(1) 静止裂纹尖端处的畴变减小了材料的起裂强度；(2) 准静态稳态裂纹附近的畴变增加了材料的断裂强度．且理论预测的R曲线渐近值与试验结果定量吻合．     

Electroelastic analysis of a penny-shaped crack in a piezoelectric ceramic under mode I loading

Following the theory of linear piezoelectricity, we consider the electroelastic problem for a piezoelectric ceramic with a penny-shaped crack under mode I loading. The problem is formulated by means of Hankel transform and the solution is solved exactly. The stress intensity factor, energy release rate and energy density factor for the exact and impermeable crack models are expressed in closed form and compared for a P-7 piezoelectric ceramic. Based on current findings, we suggest that the energy release rate and energy density factor criteria for the exact crack model are superior to fracture criteria for the impermeable crack model.     

A domain independent integral expression for the crack extension force of a curved crack in three dimensions

An integral expression that is domain independent in curvilinear coordinates and compatible with zero divergence of Eshelby's (Phil. Trans. Roy. Soc. (London) 244 (1951) 87.) energy momentum tensor was obtained from the principle of virtual work. By applying Eshelby's definition of the force on a material defect a general expression of the crack extension force for a curved crack in three dimensions, here called the F-integral, was derived from the domain independent integral expression. The F-integral is given explicitly for a number of curved cracks and found to be in agreement with previously known solutions, when available. The influence of crack surface and crack front curvature upon the various forms of the F-integral is discussed. The F-integral presented in this work is a generalisation of the J-integral (Rice, J. Appl. Mech. 35 (1968) 379.) to curved cracks in orthogonal curvilinear coordinates.     

Thermo‐hydrodynamic‐mechanical multiphase flow model for CO2 sequestration in fracturing porous media

In this paper, we introduce a fully coupled thermo‐hydrodynamic‐mechanical computational model for multiphase flow in a deformable porous solid, exhibiting crack propagation due to fluid dynamics, with focus on CO₂ geosequestration. The geometry is described by a matrix domain, a fracture domain, and a matrix‐fracture domain. The fluid flow in the matrix domain is governed by Darcy's law and that in the crack is governed by the Navier–Stokes equations. At the matrix‐fracture domain, the fluid flow is governed by a leakage term derived from Darcy's law. Upon crack propagation, the conservation of mass and energy of the crack fluid is constrained by the isentropic process. We utilize the representative elementary volume‐averaging theory to formulate the mathematical model of the porous matrix, and the drift flux model to formulate the fluid dynamics in the fracture. The numerical solution is conducted using a mixed finite element discretization scheme. The standard Galerkin finite element method is utilized to discretize the diffusive dominant field equations, and the extended finite element method is utilized to discretize the crack propagation, and the fluid leakage at the boundaries between layers of different physical properties. A numerical example is given to demonstrate the computational capability of the model. It shows that the model, despite the relatively large number of degrees of freedom of different physical nature per node, is computationally efficient, and geometry and effectively mesh independent. Copyright © 2017 John Wiley & Sons, Ltd.     

Dynamic crack deflection and penetration at interfaces in homogeneous materials: experimental studies and model predictions

We examine the deflection/penetration behavior of dynamic mode-I cracks propagating at various speeds towards inclined weak planes/interfaces of various strengths in otherwise homogeneous isotropic plates. A dynamic wedge-loading mechanism is used to control the incoming crack speeds, and high-speed photography and dynamic photoelasticity are used to observe, in real-time, the failure mode transition mechanism at the interfaces. Simple dynamic fracture mechanics concepts used in conjunction with a postulated energy criterion are applied to examine the crack deflection/penetration behavior and, for the case of interfacial deflection, to predict the crack tip speed of the deflected crack. It is found that if the interfacial angle and strength are such as to trap an incident dynamic mode-I crack within the interface, a failure mode transition occurs. This transition is characterized by a distinct, observable and predicted speed jump as well as a dramatic crack speed increase as the crack transitions from a purely mode-I crack to an unstable mixed-mode interfacial crack.     

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.     

Sudden deceleration or acceleration of an intersonic shear crack

Sudden jumps in the crack tip velocity were revealed by numerical simulation (in both continuum/cohesive element and molecular dynamics approaches) and experiments for rapid shear cracking. The cracking velocity may accelerate from a sub-Rayleigh speed to the intersonic range, or from an intersonic speed to a higher one, when the reflected impact wave reloads the crack tip. On the other hand, the cracking velocity may decelerate from an intersonic speed to a lower one or recede to the sub-Rayleigh range when the fracture driving force declines. The velocity change encountered during intersonic cracking plays a different role from that in the acceleration or deceleration of a subsonic crack. A crack propagating at an intersonic speed would leave a shear wave trailing behind. When the crack decelerates or accelerates, the effect of the trailing wave will lead to a transition period from one steady-state solution of crack tip singularity to another. This investigation aims at quantifying these processes. The full field solution of an intersonic mode II crack whose speed changed suddenly from one velocity (intersonic or subsonic) to another (intersonic or subsonic) is given in closed form. The solution is facilitated via superposing a series of propagating crack problems that are loaded by dislocations to seal the unwanted crack-face sliding or by concentrated forces moving at various speeds to negate the crack-face traction. In contrast to the subsonic solution, the results in the intersonic case indicate that the elastic fields around the crack tip depend on the deceleration or acceleration history that is traced back over a long time. Singularity matching dictates the jump that may actually take place.     

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.     

Can crack front waves explain the roughness of cracks?

We review recent theoretical progress on the dynamics of brittle crack fronts and its relationship to the roughness of fracture surfaces. We discuss the possibility that the small scale roughness of cracks, which is characterized by a roughness exponent ?0.5, could be caused by the generation, during local instabilities by depinning, of diffusively broadened corrugation waves, which have recently been observed to propagate elastically along moving crack fronts. We find that the theory agrees plausibly with the orders of magnitude observed. Various consequences and limitations, as well as alternative explanations, are discussed. We argue that another mechanism, possibly related to damage cavity coalescence, is needed to account for the observed large scale roughness of cracks that is characterized by a roughness exponent ?0.8.     

PHASE FIELD SIMULATION OF DOMAIN SWITCHING IN FERROELECTRIC SINGLE CRYSTAL WITH ELECTRICALLY PERMEABLE AND IMPERMEABLE CRACKS

Domain switching around electrically permeable and impermeable cracks in ferroelectric single crystals subjected to a mechanical load is investigated by using a phase field model.It is found that the e...     

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）.     

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.