Dielectric Breakdown Model For An Electrically Impermeable Crack In A Piezoelectric Material

The present work presents a strip Dielectric Breakdown (DB) model for an electrically impermeable crack in a piezoelectric material. In the DB model, the dielectric breakdown region is assumed to be a strip along the crack's front line. Along the DB strip, the electric field strength is equal to the dielectric breakdown strength. The DB model is exactly in analogy with the mechanical Dugdale model. Two energy release rates emerge from the analysis. An applied energy release rate appears when evaluating J-integral along a contour surrounding both the dielectric breakdown strip and the crack tip, whereas a local energy release rate appears when evaluating J-integral along an infinitesimal contour surrounding only the crack tip. Under small yielding conditions, the local energy release rate, if used as a failure criterion, gives a linear relationship between the applied stress intensity factor and the applied electric intensity factor.     

Dielectric breakdown model for a conductive crack and electrode in piezoelectric materials

The strip dielectric breakdown (DB) model introduced by Zhang and Gao [T.Y. Zhang, C.F. Gao, Fracture behavior of piezoelectric materials, Thero. Appl. Fract. Mech. 41 (2004) 339–379] is used to study the generalized 2D problem of a conductive crack and an electrode in an infinite piezoelectric material. The energy release rate and stress intensity factors are derived based on the Stroh formalism, and then they are applied as failure criteria to predict the critical fracture loads. It is found that the DB strip may take the shielding effect on a conductive crack or electrode. For the case of an electrode, the local energy release rate and stress intensity factor become zero when DB happens ahead of the electrode tip. For the case of a mode-I conductive crack in a transversely isotropic piezoelectric solid, the results based on the DB model show that the critical stress intensity factor linearly increases as the applied electric field parallel to the poling direction increases, while it linearly decreases as the applied electric field anti-parallel to the poling direction increases. Finally, the upper and lower bounds of the actual critical fracture loads are proposed for a conductive crack in a piezoelectric material under combined mechanical–electrical loads.     

Electric field dependence of the mode I energy release rate in single-edge cracked piezoelectric ceramics: effect due to polarization switching/dielectric breakdown

This study presents the results of the mode I energy release rate of a rectangular piezoelectric material with a single-edge crack under electromechanical loading. A crack was created normal or parallel to the poling direction, and electric fields were applied parallel and antiparallel to the poling. A nonlinear plane strain finite element analysis was carried out, and the effect of localized polarization switching on the energy release rate was discussed for the permeable, impermeable, open, and discharging cracks under a high-negative electric field. The effect of dielectric breakdown on the energy release rate was also examined under a high-positive electric field.     

Effects of electric field and poling on the mode I energy release rate in cracked piezoelectric ceramics at cryogenic temperatures

We discuss the mode I energy release rate of a rectangular piezoelectric material with a crack under electromechanical loading at cryogenic temperatures. A crack was created normal or parallel to the poling direction, and electric fields were applied parallel or normal to the poling. A plane strain finite element analysis was carried out, and the effects of electric field and localized polarization switching on the energy release rate were discussed for the piezoelectric ceramics at cryogenic temperatures.     

On saturation-strip model of a permeable crack in a piezoelectric ceramic

Summary. The saturation-strip model for piezoelectric crack is re-examined in a permeable environment to analyze fracture toughness of a piezoelectric ceramic. In this study, a permeable crack is modeled as a vanishing thin but finite rectangular slit with surface charge deposited along crack surfaces. This permeable saturation crack model reveals that there exists a possible leaky mode for electrical field, which allows applied electric field passing through the dielectric medium inside a crack. By taking into account the leaky mode effect, a first-order approximated solution is obtained with respect to slit height, h ₀, in the analysis of electrical and mechanical fields in the vicinity of a permeable crack tip. The permeable saturation crack model presented here also considers the effect of charge distribution on crack surfaces, which may be caused by any possible charge-discharge process in the dielectric medium inside the crack. A closed form solution is obtained for the permeable crack perpendicular to the poling direction under both mechanical as well electrical loads. Both local and global energy release rates are calculated. Remarkably, the global energy release rate for a permeable crack has an expression, where M is elastic modulus, a is the half crack length, is permittivity constant, and e is piezoelectric constant. This result is in a broad agreement with some experimental observations and may be served as the fracture criterion for piezoelectric materials. This contribution elucidates how an applied electric field affects crack growth in piezoelectric ceramic through its interaction with permeable environment surrounding a crack. The author would like to acknowledge the support from the Academic Senate Committee on Research at University of California (Berkeley) through the fund of BURNL-07427-11503-EGSLI.     

Electrical failure of piezoelectric ceramics with a conductive crack under electric fields

The failure behavior of piezoelectric ceramics with a conductive crack under purely electric loading is investigated. Electrical fracture tests are conducted to study the influence of the directions of poling and electric loading. Two failure modes of piezoelectric materials are observed: fracture that is accompanied with dielectric discharging and the formation of tubular channels without fracture. The critical J integrals at the onset of both fracture and breakdown are calculated numerically via finite element analysis. The effects of both the direction of the electric field and the poling direction on both fracture and breakdown resistance are discussed.     

On a moving dielectric crack in a piezoelectric interface with spatially varying properties

This paper provides a comprehensive theoretical analysis of a finite crack propagating with constant speed along an interface between two dissimilar piezoelectric media under inplane electromechanical loading. The interface is modeled as a graded piezoelectric layer with spatially varying properties (functionally graded piezoelectric materials, i.e., FGPMs). The analytical formulations are developed using Fourier transforms and the resulting singular integral equations are solved with Chebyshev polynomials. Using a dielectric crack model with deformation-dependent electric boundary condition, the dynamic stress intensity factors, electric displacement intensity factor, crack opening displacement (COD) intensity factor, and energy release rate are derived to fully understand this inherent mixed mode dynamic fracture problem. Numerical simulations are made to show the effects of the material mismatch, the thickness of the interfacial layer, the crack position, and the crack speed upon the dynamic fracture behavior. A critical state for the electromechanical loading applied to the medium is identified, which determines whether the traditional impermeable (or permeable) crack model serves as the upper or lower bound for the dielectric model considering the effect of dielectric medium crack filling.     

Periodically distributed parallel cracks in a functionally graded piezoelectric (FGP) strip bonded to a FGP substrate under static electromechanical load

In this paper, the Fourier integral transform–singular integral equation method is presented for the problem of a periodic array of cracks in a functionally graded piezoelectric strip bonded to a different functionally graded piezoelectric material. The properties of two materials, such as elastic modulus, piezoelectric constant and dielectric constant, are assumed in exponential forms and vary along the crack direction. The crack surface condition is assumed to be electrically impermeable or permeable. The mixed boundary value problem is reduced to a singular integral equation over crack by applying the Fourier transform and the singular integral equation is solved numerically by using the Lobatto–Chebyshev integration technique. The analytic expressions of the stress intensity factors and the electric displacement intensity factors are derived. The effects of the loading parameter λ, material constants and the geometry parameters on the stress intensity factor, the energy release ratio and the energy density factor are studied.     

Energy Analysis for Permeable and Impermeable Cracks in Piezoelectric Materials

This letter deals with an energy analysis for both permeable and impermeable cracks in piezoelectric materials. Computed numerical results are plotted in figures, which support Park-Sun's conclusion (1995a,b) that the total energy release rate (TERR) involving both mechanical and electric parts is not suitable to describe piezoelectric fracture for a plane impermeable crack because the two parts have different signs: the former is positive and the latter is always negative under any kinds of combined mechanical-electric loading. This provides the major reason as why the mechanical part (the mechanical strain energy release rate, MSERR) must be used as a fracture criterion empirically. Whereas the electric part of the TERR for a permeable crack does always vanish whatever the poling direction is oriented with respect to the remote electric loading direction. This finding supports McMeeking's (1990, 1999) conclusion that the TERR could be used as a fracture criterion for permeable cracks.     

Interface crack between two interface deformable piezoelectric layers

Based on an interface deformable piezoelectric bi-layer beam model, a bonded piezoelectric bi-material beam with an interface crack perpendicular to the poling axis is analyzed within the framework of the theory of linear piezoelectricity. The layer-wise approximations of both the elastic displacements and electric potential are employed, and each sub-layer is modeled as a single linearly elastic Timoshenko beam perfectly bonded together through a deformable interface. Using the impermeable crack assumption, the closed form solutions for the energy release rate (ERR) and crack energy density (CED) are derived for the layered piezoelectric beam subjected to combined uniformly distributed electromechanical loading. Based on superposition principle, both the ERR and CED and their components are all reduced to the functions of the crack tip loading parameters. Loading dependence of the total CED with respect to the applied electric field is manifested with the analytical results, showing that there is a transformation from an even dependence to an odd dependence for the normalized CED when the applied mechanical loading increases. Compared with the commonly used equivalent single layer model, the proposed analysis augments the crack driving force by alleviating the stress concentration along the interface and thus increases the loading parameters at the crack tip. The proposed model provides improved solutions for fracture analysis of piezoelectric layered structures and sheds light on the loading dependence of the fracture parameters (i.e., the ERR and CED) with respect to the applied electromechanical loadings.     

Dynamic fracture mechanics study of an electrically impermeable mode III crack in a transversely isotropic piezoelectric material under pure electric load

The dynamic response of an electrically impermeable Mode III crack in a transversely isotropic piezoelectric material under pure electric load is investigated by treating the electric loading process as a transient impact load, which may be more appropriate to mimic the real service environment of piezoelectric materials. The stress intensity factor, the mechanical energy release rate, and the total energy release rate are derived and expressed as a function of time for a given applied electric load. The theoretical results indicate that a purely electric load can fracture the piezoelectric material if the stress intensity factor or the mechanical energy release rate is used as a failure criterion.     

Effects of remanent field on an elliptical flaw and a crack in a poled piezoelectric ceramic

Commonly used piezoelectric ceramics such as PZT and PLZT are polarized ferroelectric polycrystals. After poling, remanent strains and a remanent polarization exist in a ceramic material. Remanent field can affect the electroelastic field and consequently plays a critical role in fracture of poled ceramics. Based on a linear constitutive law, the electroelastic field and the energy release rate of an elliptical cavity (or a crack) in a poled piezoelectric are re-examined in this study by including the effects of remanent field. It is noted that the remanent field generally has a minor effect on the stress field and a pronounced effect on the electric field at the apex of the major axis of an elliptical flaw. When the permittivity of the cavity is small, the effect of remanent polarization is similar to that of a very strong electric field applied along the poling direction. However, for the case of a conducting flaw, the remanent field does not influence the electroelastic field and energy release rate. Energy release rate of a flaw in a poled ferroelectric ceramic with and without the remanent polarization is generally different.     

Antiplane Permeable Edge Cracks in a Piezoelectric Strip of Finite Width

A piezoelectric strip with permeable edge cracks normal to the strip boundaries is analyzed. Under uniform antiplane mechanical shear and inplane electric loading, the distribution of the entire electroelastic field in a cracked piezoelectric strip is determined in explicit analytic form via the conformal mapping technique. It is found that the strain and the electric displacement exhibit the same singularity as the stress near the crack tips, while the electric field is always uniform. The field intensity factors and the energy release rate are independent of the applied electric load for prescribed stress, and related to the applied electric load for prescribed strain.     

压电复合材料中III型唇形裂纹问题的解析解

采用复变函数方法和保角映射技术,研究了压电复合材料中含唇形裂纹的无限大体远场受反平面机械载荷和面内电载荷作用下的反平面问题,利用复变函数中的留数定理和Cauchy积分公式,分别获得了电不可通和电可通两种边界条件下裂纹尖端场强度因子和机械应变能释放率的解析表达式。当唇形裂纹的高度趋于零时,可得到无限大压电复合材料中Griffith裂纹的解析解。若不考虑电场作用,所得解退化为经典材料的已知结果。数值算例显示了裂纹的几何尺寸和机电载荷对机械应变能释放率的影响规律。结果表明: 唇形裂纹高度的增加会阻碍裂纹的扩展;机械载荷总是促进裂纹的扩展;电载荷对裂纹扩展的影响与裂纹面电边界条件有关。     

On the electrical boundary conditions on the crack surfaces in piezoelectric ceramics

This paper investigates a cracked piezoelectric ceramic under remote electro-mechanical loads. The ideal crack boundary conditions for electrically impermeable and permeable crack assumptions, and the deformed crack with a yet-to-be-determined crack shape are considered. The last is referred to as the “natural boundary condition (NBC)”. Closed-form solutions to the crack-tip field intensity factors are obtained. The analysis shows that traditional approaches to the electric boundary conditions on the crack faces, that is, either the impermeable crack assumption or the permeable crack assumption, produce significantly different results for the crack-tip quantities such as electric displacement intensity factor, energy release rate and crack opening displacement. There are also considerable differences between the results obtained from traditional impermeable and permeable crack analyses and those obtained from the proposed NBC. The difference increases with applied electric loads.     

加层压电层硬币型裂纹断裂分析

研究粘接着弹性层的压电层内硬币型裂纹的断裂问题。压电层与弹性层均为横观各向同性材料,r轴方向无限长,z轴方向有限厚度。压电层沿z轴方向极化。考虑电不导通裂纹表面条件,利用Hankel积分变换将问题化为求解积分方程组,导出了场强度因子与能量释放率的表达式。给出了数值计算结果,并分析了弹性层厚度对场强度因子与能量释放率的影响。     

Interaction of a dipole with an interfacial crack in piezoelectric media

We have studied the interaction of an electric dipole with an interface crack between two dissimilar piezoelectric materials. The solutions are derived based on the Stroh formalism. Three crack models studied here are the electrically impermeable, permeable and conducting cracks. The Green functions, the intensity factors of fields and the energy release rate are presented for the three cases. Numerical calculations have also performed to investigate the effect of the dipole’s rotation on the energy release rate at the crack tip. It is shown that the energy release rates for the three crack models all reach their maximum values when the direction of the dipole is perpendicular to the crack. Moreover, the energy release rate for an electrically conducting crack is greater than that for an electrically impermeable or permeable crack having the same size under the same loading conditions.     

A study on influence of poling direction on piezoelectric plate weakened by two collinear semi-permeable cracks

A study on the influence of change in poling direction is carried out for mechanical and electric strip-yield model for a transversely isotropic piezoelectric plate cut along two equal collinear semi-permeable straight hairline cracks. Two cases are considered, when the developed saturation zone is bigger or smaller than mechanical yield zone. The Stroh formalism and complex variable technique are employed to obtain the solution. The effect of change in poling direction is studied on crack opening displacement, crack opening potential drop and energy release rate. Also, a study is carried out on the suitability of crack closure for different piezoelectric ceramics. The obtained results are presented graphically and discussed.     

Antiplane permeable edge cracks in a piezoelectric strip of finite width

A priezoelectric strip with permeable edge cracks normal to the strip boundaries is analyzed. Under uniform antiplane mechanical shear and inplane electric loading, the distribution of the entire electroelastic field in a cracked piezoelectric strip is determined in explicit analytic form via the conformal mapping technique. It is found that the strain and the electric displacement exhibit the same singularity as the stress near the crack tips, while the electric field is always uniform. The field intensity factors and the energy release rate are independent of the applied electric load for prescribed stress, and related to the applied electric load for prescribed strain.     

An analytical solution for a cracked piezoelectric plate according to the PKHS electric boundary condition

Summary Based on the Stroh formalism, an analytical solution for a central crack normal to the poling direction in an infinite plane piezoelectric solid subjected to electromechanical loads at infinity is presented according to the crack model suggested by Parton and Kudryavtsev [1] and used by Hao and Shen [2] (abbreviated as the PKHS type crack [3]) without any treatment of the mapping technique. Under generalized plane deformation, the variable tendency of the energy release rate (ERR) is found to be load-dependent, but the variable tendency of the mechanical strain energy release rate (MSERR) is completely independent of the applied electric loads. It is concluded that the present numerical results under high applied mechanical and electrical loads provide a better understanding for the PKHS type crack, which will benefit to its practical applications.