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.     

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.     

The fracture behavior of multiple cracks emanating from a circular hole in piezoelectric materials

Using the complex variable method and a new conformal mapping, the fracture behavior of multiple cracks emanating from a circular hole in piezoelectric materials is considered under remotely uniform in-plane electric and anti-plane mechanical loadings. The analytic solutions of the field intensity factors and the energy release rate are presented by taking the effect of dielectric permittivity into consideration. Known results can be derived as special cases from the general solutions. The results are illustrated with plots, showing that the piezoelectric material containing three radial cracks spaced equally at 120° apart originating from a circular hole is the easiest to fail for all cases of cracks originating from a circular hole under in-plane electric and anti-plane mechanical loadings. Moreover, if there exist multiple cracks (n ≥ 3) originating from a circular hole in piezoelectric materials, an increase in the number of cracks can enhance the reliability of these materials. The star-shaped cracks possess similar characteristics.     

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.     

The Interaction between a Piezoelectric Screw Dislocation Dipole and an Elliptic Blunt Crack in Elliptical Inhomogeneity

The interaction problem of a piezoelectric screw dislocation dipole with a confocal elliptic blunt crack in elliptical inhomogeneity subjected to remote anti-plane stress field and in-plane electric field is investigated by using the complex method of elasticity. The exact closed-form solutions of a series of quantities, such as singularity stress field, image force and image torque acting on the center of screw dislocation dipole, stress intensity factor and electric displacement intensity factor of crack tip, energy release rate, and generalized strain energy density are obtained. Then the influence laws of remote load, the dip angle of dislocation dipole, the size of blunt crack, and the material constants on the quantities are analyzed. The numerical results show that the image force, image torque, stress intensity factor, and electric displacement intensity factor make periodic variation as the dip angle of dislocation dipole; the energy release rate of crack tip is negative when subjected to pure electric field, however, it can be positive or negative when subjected to the combined action of mechanical field and electric field; the sharp crack is not easy to expand in some combined action of mechanical field and electric field.     

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.     

Elliptical inclusion problem in antiplane piezoelectricity: Implications for fracture mechanics

An elliptical piezoelectric inclusion embedded in an infinite piezoelectric matrix is analyzed in the framework of linear piezoelectricity. Using the conformal mapping technique, a closed-form solution is obtained for the case of a far-field antiplane mechanical load and an inplane electrical load. The solution to a permeable elliptical hole problem is obtained as a limiting case of vanishing elastic modulus of the inclusion. This enables the study of the nature of crack tip electric field singularity which is shown to depend on the electrical boundary condition imposed on the crack faces. The energy release rate of a self-similarly expanding slender crack in the presence of electric fields is obtained by using the generalized M-integral. The energy release rate expression indicates that the electric field has a crack-arresting influence. This effect is inferred to have a more fundamental physical origin in the interaction between the applied electric field and the induced surface charges on the crack faces. An experimental result contradicting the theoretical prediction on the crack-arresting effect is also discussed.     

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。     

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.     

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.     

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。     

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 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.     

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.     

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.     

Edge cracked piezoelectric ceramic block under electromechanical impact loading

The dynamic field intensity factors and energy release rates in a piezoelectric ceramic block containing an edge crack with the condition of continuous electric crack faces under electromechanical impact loading are obtained. Integral transform method is used to reduce the problem to two pairs of dual integral equations, which are then expressed to an Fredholm integral equation of the second kind. Numerical values on the dynamic stress intensity factor and dynamic energy release rate are obtained to show the influence of the geometry and electric field.     

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.     

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.     

Fracture analysis of a piezoelectric layer with a penny-shaped and energetically consistent crack

The problem of an eccentric penny-shaped crack embedded in a piezoelectric layer is addressed by using the energetically consistent boundary conditions. The Hankel transform technique is applied to solve the boundary-value problem. Then two coupling Fredholm integral equations are derived and solved by using the composite Simpson’s rule. The intensity factors of stress, electric displacement, crack opening displacement and electric potential together with the energy release rate are further given. The effects of the thickness of a piezoelectric layer and the discharge field inside the penny-shaped crack on the fracture parameters of concern are discussed through numerical computations. The observations reveal that an increase of the discharge field decreases the stress intensity factor and the energy release rate. An eccentric penny-shaped crack is easier to propagate than a mid-plane one in a piezoelectric layer, and the geometry of the crack along with the layer thickness have significant influences on the electrostatic traction acting on the crack faces. The solutions for a penny-shaped dielectric crack in an infinite or a semi-infinite piezoelectric material can be obtained easily.     

压电材料中反平面渗透裂纹的解

用复变函数的保角映射法,采用可渗透边界条件,研究了含裂纹的无限大压电材料在平面内电场和反平面荷载作用下的耦合场,得到了精确的解和场强度因子以及能量释放率。结果表明,电场强度在裂尖没有奇异性,应变、应力、电位移具有1/2阶的奇异性,能量释放率总是正的。