The nonlinear fracture behavior of an arbitrarily oriented dielectric crack in piezoelectric materials

Summary. This paper presents a comprehensive study on the plane problem of an arbitrarily oriented crack in a piezoelectric medium. Using a dielectric crack model, the electric boundary condition along the crack surfaces is assumed to be governed by the opening displacement of the crack. The formulation of this nonlinear problem is based on the use of Fourier transform and solving the resulting nonlinear singular integral equations. Multiple deformation modes are observed according to different geometric and loading conditions. The effects of the crack orientation and the applied loads upon the fracture behavior of cracked piezoelectric materials are studied. The relation between the current crack model and the commonly used permeable and impermeable models is discussed.     

Coupled behaviour of interacting dielectric cracks in piezoelectric materials

Existing studies indicate that the commonly used electrically impermeable and permeable crack models may be inadequate in evaluating the fracture behaviour of piezoelectric materials in some cases. In this paper, a dielectric crack model based on the real electric boundary condition is used to study the electromechanical behaviour of interacting cracks arbitrarily oriented in an infinite piezoelectric medium. The electric boundary condition along the crack surfaces is governed by the opening displacement of the cracks. The formulation of this nonlinear problem is based on modelling the cracks using distributed dislocations and solving the resulting nonlinear singular integral equations using Chebyshev polynomials. Numerical simulation is conducted to show the effect of crack orientation, crack interaction and electric boundary condition upon the fracture behaviour of cracked piezoelectric media.     

Inherent relations between the Bueckner integral and the J k -integral or the M-integral in piezoelectric materials containing multiple defects

This paper deals with the inherent relations between the Bueckner work conjugate integral and the J _k -integral (k = 1, 2) or the M-integral in piezoelectric materials with a number of arbitrarily oriented and distributed defects such as cracks, voids and inclusions. The explicit expression of the Bueckner integral is derived in piezoelectric materials by using its path-independent property and asymptotic feature for each pair of the complex potential functions in the series expansion forms. It is concluded that the J _k -integral or the M-integral are only three different special cases of the Bueckner integral when three different complementary fields are introduced, respectively, and when the closed integral contour encloses all the defects. In other words, there are universal relations between the Bueckner integral and the J _k -integral or M-integral whatever the detailed configurations of the multiple defects are. It is also concluded that both components of the J _k -integral vanish when the contour selected to calculate the invariant integral encloses all defects, providing that no resultant force acting on each defect exists. This leads to the independence of the M-integral from the global coordinate shifts, and its value is significantly influenced by the mechanical–electrical feature of the damaged piezoelectric materials, e.g., the material properties, the remote loading conditions, the defect configurations, etc.     

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.     

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.     

Modelling and analysis of the dynamic behaviour of piezoelectric materials containing interacting cracks

This study is concerned with the treatment of the dynamic behaviour of piezoelectric materials containing interacting cracks under antiplane mechanical and inplane electric loading. A general electrical boundary condition is used to enable the treatment of both permeable and impermeable conditions along the crack surfaces. The theoretical solution of the problem is formulated using integral transform techniques and an appropriate pseudo-incident wave method. The resulting singular integral equations are solved using Chebyshev polynomials to provide the dynamic stress and electric fields. Numerical examples are provided to show the effect of the geometry of the cracks, the piezoelectric constant of the material and the frequency of the incident wave upon the dynamic stress intensity factors. The results show the significant effect of electromechanical coupling upon local stress distribution.     

压电复合材料中圆孔边Ⅲ型非对称裂纹的场强度因子

研究了压电复合材料中圆孔边4个非对称裂纹在远处受面内电载荷和面外力载荷共同作用下的断裂行为。利用复变函数方法和新映射函数将问题转化为Cauchy积分方程组。通过求解Cauchy积分方程组,得到了电非渗透型和电渗透型两种边界条件下裂纹尖端电弹性场和场强度因子的解析解。所得结果不仅可退化为已有解,而且可模拟出若干新的缺陷构型,如压电复合材料中圆孔边三裂纹、半无限压电复合材料中半圆孔边单裂纹及半无限压电体中边界裂纹。将所得结果与有限元结果进行比较,吻合很好,证实了文中方法的正确性和有效性。数值算例分析了缺陷的几何参数对场强度因子的影响规律。     

Discussion of the Crack Face Electric Boundary Condition in Piezoelectric Fracture Mechanics

Selection of the electric boundary condition on crack faces in piezoelectric fracture mechanics has concerned researchers for a long time, and this may be related to discrepancies between the experimental measurements and theoretical predictions based on linear piezoelectric crack models that have been reported. In this letter, three well known electric boundary conditions, namely, impermeable, permeable and the boundary condition suggested by Parton en Kudryavtsev and Hao and Shen (PKHS condition) are discussed. We demonstrate that the impermeable condition is invalid for treating cracking problems, due to its physically unrealistic features and a mathematical singularity. Also, we show that the permeable condition can be used only in conjunction with nonzero mechanical traction boundary condition along the crack faces. Finally, it is shown that the PKHS condition is a reasonable one, but that the permittivity of the media inside the crack gap must be nonzero.     

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.     

The effective electroelastic property of cracked piezoelectric media

Summary This paper presents a study on the effective electroelastic property of piezoelectric media with parallel or randomly distributed cracks. The theoretical formulation is derived using the dilute model of distributed cracks and the solution of a single dielectric crack problem, in which the electric boundary condition along the crack surfaces is governed by the crack opening displacement. It is observed that the effective electroelastic property of such cracked piezoelectric media is nonlinear and sensitive to loading conditions. Numerical simulations are conducted to show the effects of crack distribution and electric boundary condition upon the effective electroelastic property. The transition between the commonly used electrically permeable and impermeable crack models is studied.     

Transient anti-plane crack problem of a functionally graded piezoelectric strip bonded to elastic layers

Summary. This paper examined the dynamic electromechanical behavior of a crack in a functionally graded piezoelectric layer bonded between two elastic layers under the combined anti-plane mechanical shear and in-plane electric impacts. Fourier cosine transforms are used to reduce the problem to the solution of a set of singular integral equations. It is found that the impermeable crack condition is more reasonable than the permeable crack condition to analyze the influence of electric loading, and the energy density factor is more acceptable than the energy release rate to be used as the fracture criterion. In addition, numerical results are also presented to show the influences of the crack position, electromechanical combination factor and material gradient parameter on the fracture behavior.     

The fracture behavior of functionally graded piezoelectric materials with dielectric cracks

This article provides a comprehensive investigation on the fracture behavior of cracked functionally graded piezoelectric materials (FGPMs). To account for the effect of dielectric medium inside the crack upon the fracture behavior of FGPMs, a dielectric crack model is used in this work, in which the electric boundary condition along crack surfaces is deformation-dependent and is nonlinear. The analytical formulations are developed using Fourier transform technique and solving the nonlinear singular equations using Chebyshev polynomials. A solution technique is developed to determine the desired deformation mode of the crack. Numerical simulations are given to show the effects of material gradient and the dielectric medium filling the crack upon the fracture behavior of FGPMs. The results obtained from this dielectric crack model clearly demonstrate how the transition between electrically impermeable and permeable crack models occurs with the change of crack opening displacement in response to the applied electromechanical loads. It is also observed that a critical state for the applied electromechanical loading exists for FGPMs that determines whether the impermeable (or permeable) crack model serves as the upper or lower bound for the dielectric crack model considering the effect of dielectric medium filling the crack.     

The potential theory method for a half-plane crack and contact problems of piezoelectric materials

The half-plane crack and contact problems for transversely isotropic piezoelectric materials are exactly analyzed. The potential theory method is employed with the resulting integro-differential (for crack problem) and integral (for contact problem) equations having identical structures with those reported earlier in the literature. Existing results in potential theory are thus utilized to obtain complete solutions of the problems under consideration. In particular, for the half-plane crack, both the permeable and impermeable electric conditions at the crack surfaces are considered. The solutions for the permeable crack and half-plane contact are entirely new to the literature.     

The central crack problem for a functionally graded piezoelectric strip

In this paper the dynamic anti-plane problem for a functionally graded piezoelectric strip containing a central crack vertical to the boundary is considered. The crack is assumed to be electrically impermeable or permeable. Integral transforms and dislocation density functions are employed to reduce the problem to Cauchy singular integral equations. Numerical results show the effects of loading combination parameter, material gradient parameter and crack configuration on the dynamic response. With the permeable assumption, the electric impact has no contribution to the crack tip field singularity. With the impermeable assumption, the direction of applied electric impact loading plays a great role in the behavior of dynamic stress intensity factor, and the existence of electric load always enhances the crack propagation. However, the crack is easier to propagate under the negative electric load than that under the positive electric load.     

The transient response of bonded piezoelectric and elastic half space with multiple interfacial collinear cracks

Summary This paper investigates the dynamic behavior of a bonded piezoelectric and elastic half space containing multiple interfacial collinear cracks subjected to transient electro-mechanical loads. Both the permeable and impermeable boundary conditions are examined and discussed. Based on the use of integral transform techniques, the problem is reduced to a set of singular integral equations, which can be solved using Chebyshev polynomial expansions. Numerical results are provided to show the effect of the geometry of interacting collinear cracks, the applied electric fields, and the electric boundary conditions along the crack faces on the resulting dynamic stress intensity and electric displacement intensity factors.     

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.     

Numerical Green's functions for some electroelastic crack problems

A plane electroelastic problem involving planar cracks in a piezoelectric body is considered. The deformation of the body is assumed to be independent of time and one of the Cartesian coordinates. The cracks are traction free and are electrically either permeable or impermeable. Numerical Green's functions which satisfy the boundary conditions on the cracks are derived using the hypersingular integral approach and applied to obtain a boundary integral solution for the electroelastic crack problem considered here. As the conditions on the cracks are built into the Green's functions, the boundary integral solution does not contain integrals over the cracks. It is used to derive a boundary element procedure for computing the crack tip stress and electrical displacement intensity factors.     

一维六方压电准晶中带三条不对称裂纹的圆形孔口的反平面问题

本文利用复变函数方法和保角映射,研究一维六方压电准晶材料中带不对称三裂纹的圆形孔口的的断裂问题。根据准晶压电材料基本方程的基础上,利用点群的对称性和一维六方准晶的线性压电效应,导出了一维六方准晶压电材料反平面问题的控制方程,并结合Cauchy积分公式,得到电非渗透与电渗透边界条件下的裂纹尖端场强度因子的解析表达式。当改变裂纹长度和孔口半径时,所得结果可以模拟出一些新裂纹模型。在不考虑电载荷作用时,所得结果和原有结果是一致的。通过数值算例讨论了材料的几何参数对场强度因子的影响,得出水平裂纹长度和圆半径可以促进裂纹增长。本研究为工程中材料的制备与应用将提供可靠的理论价值。     

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.     

Electro-elastic behavior induced by an external circular crack in a piezoelectric material

The electro-elastic problem of a transversely isotropic piezoelectric material with a flat crack occupying the outside of a circle perpendicular to the poling axis is considered in this paper. By using the Hankel transform technique, a mixed boundary value problem associated with the considered problem is solved analytically. The results are presented in closed form both for impermeable crack and for permeable crack. A full field solution is given, i.e., explicit expressions for electro-elastic field at any point in the entire piezoelectric space, as well as field intensity factors near the crack front, are determined. A numerical example for a cracked PZT-5H ceramic is given, and the effects of applied electric fields on elastic and electric behaviors are presented graphically.