A moving mode-III crack in functionally graded piezoelectric material: permeable problem

In this paper a moving mode-III crack in functionally graded piezoelectric materials (FGPM) is studied. The crack surfaces are assumed to be permeable. The governing equations for FGPM are solved by means of Fourier cosine transform. The mathematical formulation for the permeable crack condition is derived as a set of dual integral equations, which, in turn, are reduced to a Fredholm integral equation of the second kind. The results obtained indicate that the stress intensity factor of moving crack in FGPM depends only on the mechanical loading. The gradient parameter of the FGPM and the moving velocity of the crack do have significant influence on the dynamic stress intensity factor.     

Crack in functionally graded piezoelectric strip bonded to elastic surface layers under electromechanical loading

Solved is the problem of a crack in a functionally graded piezoelectric material (FGPM) bonded to two elastic surface layers. It is assumed that the elastic stiffness, piezoelectric constant, and dielectric permittivity of the FGPM vary continuously along the thickness of the strip. The outside layers are under antiplane mechanical loading and in-plane electric loading. The solution involves solving singular integral equations by application of the Gauss–Jacobi integration formula. Numerical calculations are carried out to obtain the energy density factors. Their variations with the geometric, loading and material parameters are shown graphically.     

A finite crack in a semi-infinite strip of a graded piezoelectric material under electric loading

In this paper, the mixed-mode crack problem for a functionally graded piezoelectric material (FGPM) strip is considered. It is assumed that the electroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under in-plane electric loading. The problem is formulated in terms of a system of singular integral equations. The stress and electric displacement intensity factors are presented for various values of dimensionless parameters representing the crack size, the crack location, and the material nonhomogeneity.     

SCATTERING OF THE HARMONIC STRESS WAVE BY CRACKS IN FUNCTIONALLY GRADED PIEZOELECTRIC MATERIALS

The present paper considers the scattering of the time harmonic stress wave by a single crack and two collinear cracks in functionally graded piezoelectric material （FGPM）. It is assumed that the properties of the FGPM vary continuously as an exponential function. By using the Fourier transform and defining the jumps of displacements and electric potential components across the crack surface as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement and electric potential components across the crack surface are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the influences of material properties on the dynamic stress and the electric displacement intensity factors.     

Electromechanical impact of an impermeable parallel crack in a functionally graded piezoelectric strip

The dynamic fracture problem for a functionally graded piezoelectric material (FGPM) strip containing a crack parallel to the free boundaries is considered in this study. It is assumed that the electroelastic properties of the strip vary continuously along the thickness direction of the strip, and that the strip is under the in-plane mechanical and electric impact. Integral transform techniques and dislocation density functions are employed to reduce the problem to the solutions of a system of singular integral equations. The dynamic stress and electric displacement intensity factors versus time are presented for various values of dimensionless parameters representing the crack size, the crack location, the material nonhomogeneity and the loading combination.     

Functionally graded piezoelectric strip with a penny-shaped crack under electromechanical loadings

In this paper, the mixed-mode penny-shaped crack problem for a functionally graded piezoelectric material (FGPM) strip is considered. It is assumed that the electroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under in-plane electromechanical loadings. The problem is formulated in terms of a system of singular integral equations. The stress and electric displacement intensity factors are presented for various values of dimensionless parameters representing the crack size, the crack location, and the material nonhomogeneity.     

Electro-mechanical frictionless contact behavior of a functionally graded piezoelectric layered half-plane under a rigid punch

The frictionless contact problem of a functionally graded piezoelectric layered half-plane in-plane strain state under the action of a rigid flat or cylindrical punch is investigated in this paper. It is assumed that the punch is a perfect electrical conductor with a constant potential. The electro-elastic properties of the functionally graded piezoelectric materials (FGPMs) vary exponentially along the thickness direction. The problem is reduced to a pair of coupled Cauchy singular integral equations by using the Fourier integral transform technique and then is numerically solved to determine the contact pressure, surface electric charge distribution, normal stress and electric displacement fields. For a flat punch, the normal stress intensity factor and electric displacement intensity factor are also given to quantitatively characterize the singularity behavior at the punch ends. Numerical results show that both material property gradient of the FGPM layer and punch geometry have a significant influence on the contact performance of the FGPM layered half-plane.     

Propagation of an anti-plane moving crack in a functionally graded piezoelectric strip

Summary The propagation of an anti-plane moving crack in a functionally graded piezoelectric strip (FGPS) is studied in this paper. The governing equations for the proposed analysis are solved using Fourier cosine transform. The mixed boundary value problems of the anti-plane moving crack, which is assumed to be either impermeable or permeable, are formulated as dual integral equations. By appropriate transformations, the dual integral equations are reduced to Fredholm integral equations of the second kind. For the impermeable crack, the stress intensity factor (SIF) of the crack in the FGPS depends on both the mechanical and electric loading, whereas, the SIF for the permeable crack depends only on the mechanical loading. The results obtained show that the gradient parameter of the FGPS and the velocity of the crack have significant influence on the dynamic SIF.Support from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7081/00E) is acknowledged. Support from the National Natural Science Foundation of China (Project No. 10072041) is also acknowledged.     

Transient response of an insulating crack between dissimilar piezoelectric layers under mechanical and electrical impacts

Summary  The dynamic response of an interface crack between two dissimilar piezoelectric layers subjected to mechanical and electrical impacts is investigated under the boundary condition of electrical insulation on the crack surface by using the integral transform and the Cauchy singular integral equation methods. The dynamic stress intensity factors, the dynamic electrical displacement intensity factor, and the dynamic energy release rate (DERR) are determined. The numerical calculation of the mode-I plane problem indicates that the DERR is more liable to be the token of the crack growth when an electrical load is applied. The dynamic response shows a significant dependence on the loading mode, the material combination parameters as well as the crack configuration. Under a given loading mode and a specified crack configuration, the DERR of an interface crack between piezoelectric media may be decreased or increased by adjusting the material combination parameters. It is also found that the intrinsic mechanical-electrical coupling plays a more significant role in the dynamic fracture response of in-plane problems than that in anti-plane problems. Received 4 September 2001; accepted for publication 23 July 2002 The work was supported by the National Natural Science Foundation under Grant Number 19891180, the Fundamental Research Foundation of Tsinghua University, and the Education Ministry of China.     

功能梯度压电板条中电绝缘型运动裂纹的电弹性场

基于三维弹性理论和压电理论，对材料系数按指数函数规律分布的功能梯度压电板条中的反平面运动裂纹问题进行了求解。利用Fourier积分变换方法将电绝缘型运动裂纹问题化为对偶积分方程，并进一步归结为易于求解的第二类Fredholm积分方程。通过渐近分析，获得了裂纹尖端应力、应变、电位移和电场的解析解，给出了裂纹尖端场各个变量的角分布函数，并求得了裂纹尖端场的强度因子，分析了压电材料物性梯度参数、几何尺寸及裂纹运动速度对它们的影响。结果表明，对于电绝缘型裂纹，功能梯度压电板条中运动裂纹尖端附近的各个场变量都具有-1／2阶的奇异性；当裂纹运动速度增大时，裂纹扩展的方向会偏离裂纹面。     

Propagation of Love waves in a functionally graded piezoelectric material (FGPM) layered composite system

In this theoretical study, we investigate the propagation of Love waves in a layered structure consisting of two different homogenous piezoelectric materials, an upper layer and a substrate. A functionally graded piezoelectric material (FGPM) buffer layer is in between the upper layer and the substrate. We employ the power series technique to solve the governing differential equations with variable coefficients. The influence of the gradient coefficients of FGPM and the layer thicknesses on the dispersion relations, the electro-mechanical coupling factor, and the stress distributions of Love waves in this structure are investigated. We demonstrate that the low gradient coefficient raises the significant variation of the phase velocity within a certain range of ratios of upper layer thickness to equivalent thickness. The electro-mechanical coupling factor can be increased when the equivalent thickness equals one or two wavelengths, and the discontinuity of the interlaminar stress can be eliminated by the FGPM buffer layer. The theoretical results set guidelines not only for the design of high-performance surface acoustic wave (SAW) devices using the FGPM buffer layer, but also for the measurement of material properties in such FGPM layered structures using Love waves.     

动态载荷下功能梯度复合材料的圆币形裂纹问题

研究了动态载荷下功能梯度材料中的圆币形裂纹问题．假设材料为横观各向同性,并且含有多个垂直于厚度方向的裂纹,材料参数沿轴向（与裂纹面垂直的方向）为变化的,沿该方向将材料划分为许多单层,各单层材料参数为常数,利用Hankel变换祛,在Laplace域内推导出了控制问题的对偶积分方程组．利用Laplace数值反演,得出了裂纹尖端的动态应力强度因子和能量释放率．研究了含两个裂纹的功能梯度接头结构,分析了材料非均匀性参数对应力强度因子和能量释放率的影响．     

Electroelastic Analysis of an Internal Interface Crack in a Half-Plane Consisting of Two Bonded Dissimilar Piezoelectric Quarter-Planes

The problem of an interface crack in a half-plane consisting of two bonded dissimilar piezoelectric quarters is considered under antiplane shear and inplane electric loading. The problem is solved under the electrically permeable assumption for a crack. The integral transform technique is employed to reduce the problem to triple integral equations, which is further converted to a hypersingular integral equation for the crack sliding displacement. By solving the resulting equation analytically, the electroelastic field along the interface and the energy release rate are obtained in explicit form, respectively. Several examples are given to illustrate the influence of the material properties and the crack position on the energy release rate.     

Pre-kinking of a moving crack in a magnetoelectroelastic material under in-plane loading

A constant moving crack in a magnetoelectroelastic material under in-plane mechanical, electric and magnetic loading is studied for impermeable crack surface boundary conditions. Fourier transform is employed to reduce the mixed boundary value problem of the crack to dual integral equations, which are solved exactly. Steady-state asymptotic fields near the crack tip are obtained in closed form and the corresponding field intensity factors are expressed explicitly. The crack speed influences the singular field distribution around the crack tip and the effects of electric and magnetic loading on the crack tip fields are discussed. The crack kinking phenomena is investigated using the maximum hoop stress intensity factor criterion. The magnitude of the maximum hoop stress intensity factor tends to increase as the crack speed increases.     

An anti-plane propagating crack in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric strip

Summary A finite crack propagating at constant speed in a functionally graded piezoelectric strip (FGPS) bonded to a homogeneous piezoelectric strip is considered. It is assumed that the electroelastic material properties of the FGPS vary exponentially across the thickness of the strip, and that the bimaterial strip is under combined anti-plane mechanical shear and in-plane electrical loads. The analysis is conducted for the electrically unified crack boundary condition, which includes both the traditional permeable and the impermeable ones. By using the Fourier transform, the problem is reduced to the solution of Fredholm integral equations of the second kind. Numerical results for the stress intensity factor and the crack sliding displacement are presented to show the influences of the crack propagation speed, electric loads, FGPS gradation, crack length, electromechanical coupling coefficient, properties of the bonded homogeneous piezoelectric strip and crack location.     

Mode-3 spontaneous crack propagation in unsymmetric functionally graded materials

The effects of the inhomogeneous material property variation on the mode-3 crack propagation characteristics have been analyzed. The spectral form of the elastodynamic boundary integral equations are derived for a functionally graded material, which is used as the numerical tool in our analysis. The material property gradient is assumed to vary unsymmetrically in the direction normal to the fracture plane. A parametric study has been performed by systematically varying the inhomogeneity length scale. In comparison with a homogeneous material, an unsymmetric functionally graded material offers lesser fracture resistance. The fracture resistance progressively decreases with increase in inhomogeneity, quantified by the increase in crack sliding displacement jumps, crack tip velocities and accelerations. The material property inhomogeneity affects only the transient crack propagation velocities, while the quasi-steady-state velocity remains unaltered.     

Angle cracks in two bonded functionally graded piezoelectric material under anti-plane shear

Consider two bonded functionally graded piezoelectric material (FGPM) with finite height. Each material contains an arbitrary oriented crack. The material properties are assumed in exponential forms in the direction normal to the interface. The crack surface condition is assumed to be electrically impermeable or permeable. Using the Fourier transform technique, the problem can be reduced to a system of singular integral equations, which are then solved numerically by applying the Gauss-Chebyshev integration formula to obtain the stress intensity factors at the crack tips. Numerical calculations are carried out to obtain the energy density factor S and the energy release rate G. In impermeable case, the energy release rate has been shown to be negative as the electric loads are applied. The positive definite characteristic of the energy density factor makes it possible for predicting the fracture behavior of the cracked structure. The influences of the non-homogeneous parameters and crack orientation on the energy density factors at the crack tips are discussed in detail. The results show that the energy density factor at the crack tip will be increased when the crack tip is located within the softer material.     

Study of a propagating finite crack in functionally graded piezoelectric materials considering dielectric medium effect

This paper provides a study of the problem of a propagating finite crack under in-plane loading in functionally graded piezoelectric materials (FGPMs). The analytical formulations are developed by Fourier transforms and the resulting singular integral equations are solved by using Chebyshev polynomials. By using a dielectric crack model with deformation-dependent electric boundary condition, numerical simulations are made to show the effects of the dielectric medium, the gradient of material properties and the speed of crack propagation on the fracture parameters, such as the stress, electric displacement and crack opening displacement intensity factors. A critical state for the electromechanical loading applied to the FGPMs is observed, which determines whether the traditionally impermeable (or permeable) crack model serves as the upper or lower bound for the dielectric model. The validity of this dielectric crack model is also examined by comparing the results of different existing crack models.     

Transient thermal response of functionally graded piezoelectric laminates with an infinite row of parallel cracks normal to the bimaterial interface

In this paper, the problem of a functionally graded piezoelectric material strip(FGPM strip)containing an infinite row of parallel cracks perpendicular to the interface between the FGPM strip and a homogeneous layer is analyzed under transient thermal loading condition. The crack faces are supposed to be completely insulated. Material properties are assumed to be exponentially dependent on the distance from the interface. Using the Fourier transforms, the electro-thermoelastic problem is reduced to a singular integral equation, which is solved numerically. The stress intensity factors are computed and presented as a function of the normalized time, the nonhomogeneous and geometric parameters.     

Dynamic fracture analysis of an annular interfacial crack between dissimilar magnetoelectroelastic layers

Transient response of an annular interfacial crack between dissimilar magnetoelectroelastic layers under impacts is investigated. On the crack surface, magnetoelectrically impermeable boundary condition is adopted. Using Laplace and Hankel transform techniques, the mixed boundary value problem is reduced to a system of singular integral equations. The integral equations are further reduced to a system of algebraic equations with the aid of Jacobi polynomials. The dynamic field intensity factor and dynamic energy release rate are determined. Numerical results reveal the effects of electric or magnetic loadings and material parameters of composite on crack propagation and growth.