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.     

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。     

Electroelastic analysis of an interface anti-plane shear crack in a layered piezoelectric plate

The problem of an anti-plane interface crack in a layered piezoelectric plate composed of two bonded dissimilar piezoelectric ceramic layers subjected to applied voltage is considered. It is assumed that the crack is either impermeable or permeable. An integral transform technique is employed to reduce the problem considered to dual integral equations, then to a Fredholm integral equation by introducing an auxiliary function. Field intensity factors and energy release rate are obtained in explicit form in terms of the auxiliary function. In particular, by solving analytically a resulting singular integral equation, they are determined explicitly in terms of given electromechanical loadings for the case of two bonded layers of equal thickness. Some numerical results are presented graphically to show the influence of the geometric parameters on the field intensity factors and the energy release rate.     

含裂纹的矩形截面压电材料反平面问题的应力场和电场

研究了含裂纹的矩形截面的压电材料在平面内电场和反平面荷载作用下的问题。得到了满足拉普拉斯方程、电渗透裂纹面边界条件的位移函数解和电势函数解,从而得到了电场和弹性场的基本解。最后,用边界配置法计算了应力强度因子和能量释放率。结果表明,这种半解析半数值的方法计算简便,而且具有足够的精确性和广泛的应用性。     

Eccentric crack in a rectangular piezoelectric medium under electromechanical loadings

Summary The solutions of an eccentric crack problem in a rectangular piezoelectric ceramic medium under combined anti-plane shear and in-plane electrical loadings are obtained by the continuous electric crack face condition. Fourier transforms and Fourier series are used to reduce the problem to two pairs of dual integral equations, which are then expressed by a Fredholm integral equation of the second kind. Numerical values of the stress intensity factor and the energy release rate are obtained to show the influence of the electric field.     

Transient analysis of a piezoelectric strip with a permeable crack under anti-plane impact loads

The problem of a through permeable crack situated in the mid-plane of a piezoelectric strip is considered under anti-plane impact loads for two cases. The first is that the strip boundaries are free of stresses and of electric displacements, and the second is that the strip boundaries are clamped rigid electrodes. The method adopted is to reduce the mixed initial-boundary value problem, by using integral transform techniques, to dual integral equations, which are further transformed into a Fredholm integral equation of the second kind by introducing an auxiliary function. The dynamic stress intensity factor and energy release rate in the Laplace transform domain are obtained in explicit form in terms of the auxiliary function. Some numerical results for the dynamic stress intensity factor are presented graphically in the physical space by using numerical techniques for solving the resulting Fredholm integral equation and inverting Laplace transform.     

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.     

压电复合材料中幂函数型曲线裂纹的反平面问题

通过构造新保角映射, 利用Stroh公式研究了远场受反平面剪应力和面内电载荷共同作用下无限大压电复合材料中幂函数型曲线裂纹的断裂行为。给出了电不可渗透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解。该解析解在幂函数的幂次为零时, 可退化为已有文献中无限大压电复合材料含直线裂纹的结果, 证明了其合理性。由解析解可知, 裂纹几何形状一定时, 电场分布将不受机械载荷的影响。最后, 通过数值算例讨论了幂函数的幂次、 系数及其在 x₁轴上的投影长度对机械应变能释放率的影响。结果表明, 当压电体仅受 x₂方向载荷作用时, 对于给定幂次与开口的曲线裂纹, 在 x₁轴上的投影长度存在一临界值使其最容易开裂; 而对于给定投影长度与幂次的曲线裂纹, 开口越大裂纹越容易扩展。      

Transient response of a piezoelectric material with a semi-infinite mode-III crack under impact loads

The problem of a semi-infinite impermeable mode-III crack in a piezoelectric material is considered under the action of impact loads. For the case when a pair of concentrated anti-plane impact loads and electric displacements are exerted symmetrically on the upper and lower surfaces of the crack, the asymptotic electroelastic field ahead of the crack tip is determined in explicit form. The dynamic intensity factors of electroelastic field and dynamic mechanical strain energy release rate are obtained. The obtained results can be taken as fundamental solutions, from which general results may directly be evaluated by integration. The method adopted is to reduce the mixed initial-boundary value problem, by using the Laplace and Fourier transforms, into two simultaneous dual integral equations. One may be converted into an Abel's integral equation and the other into a singular integral equation with Cauchy kernel, and the solutions of both equations can be determined in closed-form, respectively. For some particular cases, the present results reduce to the previous results.     

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.     

TRANSIENT RESPONSE OF A CRACKED INFINITE PIEZOELECTRIC STRIP UNDER ANTI-PLANE IMPACT

By using the well-established integral transform methodology, the dynamic response of stress and electric displacement around a finite crack in an infinite piezoelectric strip are investigated under anti-plane impact. The dynamic intensity factors of stress and electric displacement are obtained analytically. The results show that the dynamic electric field will promote or retard the propagation of the crack at different stages of the loading process. On the other hand, the response of the electric field is coherent with the applied electric load and independent of the external mechanical load. The result obtained for the anti-plane impact of a cracked infinite piezoelectric ceramic can be regarded as a special case of the present work when the width of the strip tends to infinity.     

Impact response of a finite crack in an orthotropic piezoelectric ceramic

Summary The transient dynamic stress intensity factor and dynamic energy release rate were determined for a cracked piezoelectric ceramic under normal impact in this study. A plane step pulse strikes the crack and stress wave diffraction takes place. Laplace and Fourier transforms are employed to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform plane. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. A numerical Laplace inversion technique is used to compute the values of the dynamic stress intensity factor and the dynamic energy release rate for some piezoelectric ceramics, and the results are graphed to display the electroelastic interactions.     

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.     

Griffith crack moving along the interface of two dissimilar piezoelectric materials

The problem of an anti-plane Griffith crack moving along the interface of dissimilar piezoelectric materials is solved by using the integral transform technique. It is shown from the result that the intensity factors of anti-plane stress and electric displacement are dependent on the speed of the Griffith crack as well as the material coefficients. When the two piezoelectric materials are identical, the present result will reduce to the result for the problem of an anti-plane moving Griffith crack in homogeneous piezoelectric materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

Non-symmetric extension of a crack in an infinite elastic medium under anti-plane shear stress

The dynamic problem of non-symmetric extension of a crack in an infinite elastic medium, which is initially in a state of uniform anti-plane shear, has been considered. The problem of non-symmetric extension of a crack due to cohesive traction has also been treated. The method of analysis is based on the observation that certain field quantities show dynamic similarity. The results include expressions for the stress intensity factors at the crack tips and the rate of energy flux into the crack edges for problem I. Numerical calculations are carried out to obtain stress intensity factors and the rate of energy flux into the crack tips for problem I.     

Dynamic fracture behavior of an internal interfacial crack between two dissimilar magneto-electro-elastic plates

In this paper, the anti-plane problem for an interfacial crack between two dissimilar magneto-electro-elastic plates subjected to anti-plane mechanical and in-plane magneto-electrical impact loadings is investigated. Four kinds of crack surface conditions are adopted: magneto-electrically impermeable (Case 1), magnetically impermeable and electrically permeable (Case 2), magnetically permeable and electrically impermeable (Case 3), and magneto-electrically permeable (Case 4). The position of the interfacial crack is arbitrary. The Laplace transform and finite Fourier transform techniques are employed to reduce the mixed boundary-value problem to triple trigonometric series equations in the Laplace transform domain. Then the dislocation density functions and proper replacements of the variables are introduced to reduce the series equations to a standard Cauchy singular integral equation of the first kind. The resulting integral equation together with the corresponding single-valued condition is approximated as a system of linear algebra equations, which can easily be solved. Field intensity factors and energy release rates are determined and discussed. The effects of loading combination parameters on dynamic energy release rate are plotted for Cases 1-3. On the other hand, since the magneto-electrically permeable condition is perhaps more physically reasonable for type III crack, the effect of the crack configuration on the dynamic fracture behavior of the crack tips is studied in detail for Case 4. The results could be useful for the design of multilayered magneto-electro-elastic structures and devices.     

Dynamic J integral, separated dynamic J integral and component separation method for dynamic interfacial cracks in piezoelectric bimaterials

First, the near-tip stress and electric displacement fields are analytically solved for a dynamically propagating interfacial crack in a piezoelectric bimaterial. Second, from the rate formulation of the energy balance in a piezoelectric material, the path independent dynamic J integral is derived, which has the physical significance of the energy release rate. Using the present near-tip analytical solutions, the relationships between the dynamic J integral and the stress and electric displacement intensity factors are also obtained. It is shown that the path independent dynamic J integral contains the static J integral and the dynamic J integral for elastic materials, and static J integral for piezoelectric materials as special cases. Third, for an interfacial crack in a piezoelectric bimaterial, the path independent separated dynamic J integrals are derived, which have the physical significance of energy flow rates into the propagating interfacial crack tip from the individual material sides or, equivalently, the separated dynamic energy release rates. Fourth, to accurately evaluate mixed-mode stress and electric displacement intensity factors, the component separation method of the dynamic J integral is developed. Finally, the finite element analyses of a static stationary interfacial crack in a piezoelectric bimaterial subject to mechanical, electrical and combined loading are carried out to demonstrate the applicability of the generalized (dynamic) J integral and the separated J integral, and the component separation method.     

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.     

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。     

有限高磁电弹性体中双半动态与静态裂纹分析

狭长体中的裂纹是断裂力学中经常采用的研究模型。含有共线无限长裂纹的条形磁电弹性体,当面内的力电磁和反平面的剪应力作用在左边裂纹尖端附近的一段裂纹面上时,往往会产生动态断裂。利用复变函数法中的拱形变换公式,导出了磁电全非渗透型边界条件下左裂纹尖端动态的应力强度因子以及机械应变能释放率的解析解。当运动速度趋于零时退化为静止状态下的解。通过数值算例分析了断裂机理,讨论了静止状态下狭长体和裂纹的几何尺寸、外力、电场和磁场分别对能量释放率的影响,为相关器件的设计与制造提供了帮助。