The strip dielectric breakdown model

This paper reports on the analysis of the strip dielectric breakdown (DB) model for an electrically impermeable crack in a piezoelectric medium based on the general linear constitutive equations. The DB model assumes that the electric field in a strip ahead of the crack tip is equal to the dielectric breakdown strength, which is in analogy with the classical Dugdale model for plastic yielding. Using the Stroh formalism and the dislocation modeling of a crack, we derived the relationship between the DB strip size and applied mechanical and electrical loads, the intensity factors of stresses and electric displacement, and the local energy release rate. Based on the results, we discussed the effect of electric fields on fracture of a transversely isotropic piezoelectric ceramic by applying the local energy release rate as a failure criterion. It is shown that for an impermeable crack perpendicular to the poling direction, a positive electric field will assist an applied mechanical stress to propagate the crack, while a negative electric field will retard crack propagation. However, for an impermeable crack parallel to the poling direction, it is found that the applied electric field does not change the mode I stress intensity factor and the local energy release rate, i.e., the applied electric field has no effect on the crack growth.     

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 fracture testing of piezoelectric ceramics

Fracture tests carried out on unpoled and poled PZT-5H four-point bend specimens are presented in this paper. The crack faces were parallel to the poling direction. Both mechanical loads and electric fields were applied to the poled specimens. The experimental results were analyzed by means of the finite element method and a conservative M-integral including the crack face boundary conditions. Fracture tests on four-point bend PIC-151 specimens with the crack faces perpendicular to the poling directions were also analyzed here; the experimental results were taken from the literature. A mixed mode fracture criterion is proposed for piezoelectric ceramics. This criterion is based upon the energy release rate and two phase angles. This criterion was implemented with experimental results from the literature and from this investigation. Excellent agrement was found between the fracture curve and the experimental results of the specimens with the crack faces perpendicular to the poling direction. With some scatter, reasonable agreement was observed between the fracture curve and the experimental results of the specimens with crack faces parallel to the poling direction.     

The influence of Maxwell stresses on the fracture mechanics of piezoelectric materials

The influence of Maxwell stresses on the generalized 2D fracture mechanics problem of piezoelectric materials under combined mechanical and electric loads at infinity is studied. The electrically semi-permeable crack boundary condition is adopted in this paper. Based on the Stroh’s formalism, explicit and closed-form solutions of electric displacement inside the crack, stress and electric intensity factors are obtained. Numerical results are also given to discuss the effects of Maxwell stresses on the stress and electric displacement intensity factors when the interior of the crack and the surrounding space at infinity are filled with different dielectric medium. It is found that the stress intensity factor increases rapidly with increasing value of the applied electric displacement load for the case of the dielectric constant of the surrounding at infinity is smaller than that inside the crack. The electric displacement intensity factor always increases as the applied electric loads or the applied mechanical loads increase.     

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.     

Micro-crack initiation at the tip of a semi-infinite rigid line inhomogeneity in piezoelectric solids

A dislocation emission mechanism for micro-crack initiation at the tip of a semi-infinite rigid line inhomogeneity in a piezoelectric solid is proposed in the present paper. For a rigid line inhomogeneity embedded in a piezoelectric matrix, dislocations of one sign are driven away from the tip due to high stress level, while the stationary dislocations of the opposite sign are left behind near the tip of the inhomogeneity. As a result, a micro-Zener–Stroh crack is initiated ahead of the line inhomogeneity. In the current study, a dislocation pileup mechanism for micro-crack initiation at the inhomogeneity tip is proposed. An interesting result is that the critical stress intensity factors for a line inhomogeneity perpendicular to the poling direction can be related to the fracture toughness of a conventional crack in the same material. Analytical solutions show that the critical plane shear stress intensity factor depends on the plane shear mechanical and displacement loadings, and the critical opening stress and electric displacement intensity factors depend on not only the mechanical and displacement loadings, but the electric field and displacement loadings as well.     

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.     

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.     

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.     

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.     

Discontinuous crack model for piezoelectric materials and its application to CED evaluation

In order to study the influence of electric displacement saturation on fracture behavior of a piezoelectric material, the electric displacement strip-saturation model by [Gao H, Zhang TY, Tong P. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J Mech Phys Solids 1997;45:491-510] has been applied. However, this model is only applicable to problems such as that of a crack in an infinite plate for which it provides a singular solution. In order to overcome this situation, we developed a crack model for a piezoelectric material named discontinuous crack model that is presented in this paper, to evaluate crack energy density (CED) considering electric yielding and we studied its applicability through finite element analyses. The model is defined and methods to establish its constitutive equation are discussed. Moreover, it is shown that the model by Gao et al. and the ordinary crack model in continuum can be regarded as special cases of the discontinuous crack model. Subsequently, the CED and its derivatives for the discontinuous crack model are defined and their path independent expressions are also derived based on conservation laws. Finally, a finite element formulation is devised and the applicability of the model to the evaluations of CED and its derivatives is studied through finite element analyses of an example.     

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.     

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.     

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.     

加层压电条界面裂纹的稳态扩展

研究当压电条同时与两个不同材料的弹性条粘接在一起,在反平面机械载荷及面内电载荷联合作用下,长度不变的有限Griffith 界面裂纹沿加层压电条界面以常速稳态扩展时裂尖的动态断裂问题.应用Fourier积分变换将问题化为以第二类Fredholm积分方程表示的对偶积分方程,导出了相应的动应力强度因子表达式.给出了动应力强度因子与裂纹传播速度、裂纹长度、压电条及弹性条厚度、电荷载大小及方向的关系曲线.研究结果对结构设计及结构失效的预防具有理论和应用价值.     

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.     

Fracture mechanics of piezoelectric materials

This paper presents an analysis of crack problems in homogeneous piezoelectrics or on the interfaces between two dissimilar piezoelectric materials based on the continuity of normal electric displacement and electric potential across the crack faces. The explicit analytic solutions are obtained for a single crack in an infinite piezoelectric or on the interface of piezoelectric bimaterials. For homogeneous materials it is found that the normal electric displacement D₂, induced by the crack, is constant along the crack faces which depends only on the remote applied stress fields. Within the crack slit, the perturbed electric fields induced by the crack are also constant and not affected by the applied electric displacement fields. For bimaterials, generally speaking, an interface crack exhibits oscillatory behavior and the normal electric displacement D₂ is a complex function along the crack faces. However, for bimaterials, having certain symmetry, in which an interface crack displays no oscillatory behavior, it is observed that the normal electric displacement D₂ is also constant along the crack faces and the electric field E₂ has the singularity ahead of the crack tip and has a jump across the interface. Energy release rates are established for homogeneous materials and bimaterials having certain symmetry. Both the crack front parallel to the poling axis and perpendicular to the poling axis are discussed. It is revealed that the energy release rates are always positive for stable materials and the applied electric displacements have no contribution to the energy release rates. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

压电复合材料的共线周期性裂纹平面问题的电弹性场分析

利用复变函数知识、半逆解法及待定系数法, 研究了压电复合材料的共线周期性裂纹问题, 给出了在电不可渗透边界条件下的应力、电位移、应力强度因子、电位移强度因子和机械应变能释放率的解析解。当裂纹间距趋于无穷时, 共线周期性裂纹退化为一条单裂纹, 得到了压电复合材料一条单裂纹的结果。通过数值算例讨论了共线周期性裂纹的裂纹长度、裂纹间距和机电载荷对机械应变能释放率的影响规律。结果表明, 机械应变能释放率随着共线周期性裂纹的裂纹长度、共线周期性裂纹的裂纹间距、机械载荷和正电场的增大而增大, 随着负电场的增大而减小。     

An accumulation damage model for fatigue fracture of ferroelectric ceramics

The accumulated plastic displacement criterion for crack propagation in traditional materials is extended to develop equations to predict the fatigue crack growth of ferroelectric ceramics subjected to combined electromechanical loads. The crack-line is perpendicular to the poling direction of the medium. An electric saturation zone and a stress saturation zone are assumed to develop at the crack tips when the medium is subjected to external electromechanical loads. This assumption makes it possible to obtain the accumulated plastic deformation in closed form. A fatigue crack growth law, which is a fourth-power function of the effective stress intensity factor, similar to the well-known Paris law, is derived. Graphical results for the effect of electric load on the effective crack tip stress intensity factor and crack growth rate are provided.