Exact solutions for piezoelectric materials with an elliptic hole or a crack under uniform internal pressure

The existing investigations on piezoelectric materials containing an elliptic hole or a crack mainly focus on remote uniform tensile loads.In order to have a better understanding for the fracture behavior of piezoelectric materials under different loading conditions,theoretical and numerical solutions are presented for an elliptic hole or a crack in transversely isotropic piezoelectric materials subjected to uniform internal pressure and remote electro-mechanical loads.On the basis of the complex variable approach,analytical solutions of the elastic and electric fields inside and outside the defect are derived by satisfying permeable electric boundary condition at the surface of the elliptical hole.As an example of PZT-4 ceramics,numerical results of electro-elastic fields inside and outside the crack under various electric boundary conditions and electro-mechanical loads are given,and graphs of the electro-elastic fields in the vicinity of the crack tip are presented.The non-singular term is compared to the asymptotic one in the figures.It is shown that the dielectric constant of the air in the crack has no effect on the electric displacement component perpendicular to the crack,and the stresses in the piezoelectric material depend on the material properties and the mechanical loads on the crack surface and at infinity,but not on the electric loads at infinity.The figures obtained are strikingly similar to the available results.Unlike the existing work,the existence of electric fields inside an elliptic hole or a crack is considered,and the piezoelectric solid is subjected to complicated electro-mechanical loads.     

稳态温度场中螺旋位错与圆弧裂纹的交互作用

研究压电材料在稳态温度场下螺旋位错与圆弧裂纹的交互作用.综合运用复变函数分区全纯理论、解析函数奇性主部分析方法、解析延拓原理、Cauchy型积分以及Riemann边值问题求解方法,导出各复势函数之间的解析关系,进一步得到特殊情况下所讨论问题的封闭解,并解出像力随温度梯度和位错位置变化的表达式.数值分析结果表明,温度梯度越大,像力变化越明显.     

Repair of a cracked Timoshenko beam subjected to a moving mass using piezoelectric patches

This paper presents an analytical method for the application of piezoelectric patches for the repair of cracked beams subjected to a moving mass. The beam equations of motion are obtained based on the Timoshenko beam theory by including the dynamic effect of a moving mass traveling along a vibrating path. The criterion used for the repair is altering the first natural frequency of the cracked beam towards that of the healthy beam using a piezoelectric patch. Conceptually, an external voltage is applied to actuate a piezoelectric patch bonded on the beam. This affects the closure of the crack so that the singularity induced by the crack tip will be decreased. The equations of motion are discretized by using the assumed modes method. The cracked beam is modeled as number of segments connected by two massless springs at the crack locations (one, extensional and the other, rotational). The relationships between any two spans can be obtained by considering the compatibility requirements on the crack section and on the ends of the piezoelectric patch. Using the analytical transfer matrix method, eigensolutions of the system can be calculated explicitly. Finally, numerical simulations are performed with respect to different conditions such as the moving load velocity. It is seen that when the piezoelectric patch is used, the maximum deflection of the cracked beam approaches maximum deflection of the healthy beam.     

压电材料中螺型位错和偶极子与半椭圆槽表面裂纹的干涉效应

运用弹性复势方法,研究纵向剪切和面内电场共同作用下无限半平面压电材料中螺型位错和偶极子与半椭圆槽表面裂纹的电弹干涉效应,得到该问题复势函数的封闭形式解答,并由此导出广义应力场、裂纹尖端的广义应力强度因子以及作用在螺型位错上的位错力.算例结果表明:增大材料压电常数会相应增大位错力、位错对裂纹尖端的屏蔽和反屏蔽效应;增大位错离裂纹尖端的距离,位错力会相应减小;变化φ(偶极子臂与x轴正半轴夹角)值会出现一个改变位错偶极子对应力强度因子作用方向的临界值.     

三维压电介质界面裂纹的边界积分-微分方程

基于三维两相横观各向同性压电介质的基本解和压电介质的Somigliana恒等式，利用发散积分的有限部理论，建立以裂纹面上的不连续位移和不连续电势为基本未知量的三维压电介质界面裂纹问题的超奇异积分－微分方程组，其中的积分核具有O（1/r^2）阶的奇异性。当两相材料退化为均质材料或单相材料时，方程组中的微分项的系数为零，从而积分－微分方程组退化为已有的均质压电材料的超奇异边界积分方程。     

压电材料平面渗透裂纹的机电耦合场

为提高压电传感器和换能器的品质,针对材料缺陷导致的压电元器件失效或不稳定,用复变函数的方法,结合椭圆形夹杂内的电场强度和电位移为常量这一早期研究结果,研究了压电材料平面电渗透裂纹的机电耦合场及其奇异性。解答表明,切向电场强度和法向电位移在裂纹尖端有由机械载荷引起的奇异,而与电载荷无关;应力强度因子与纯弹性材料结果一致。     

A piezoelectric screw dislocation interacting with an elliptical inclusion containing electrically conductive interfacial rigid lines

The problem of the electroelastic interaction between a piezoelectric screw dislocation and an elliptical inclusion with electrically conductive interfacial rigid lines under antiplane shear and inplane electrical loads is dealt with. The screw dislocation (antiplane singularity) is located inside either the elliptical inclusion or the infinite matrix. An efficient complex variable method for multiply connected regions is developed, in terms of which explicit series solutions for the electroelastic fields in the matrix and the inclusion regions are derived. The image force acting on the piezoelectric screw dislocation is then calculated by using the generalized Peach–Koehler formula. The influence of the aspect ratio of the ellipse, rigid line geometry and material electroelastic dissimilarity on the image force is discussed. It is shown that the effect of the interfacial rigid line on the equilibrium position of the dislocation near an elliptical inclusion is significant. A stable equilibrium position of the screw dislocation in the matrix can exist when the dislocation near the soft elliptical inclusion and the interfacial rigid line. An important founding is that there exists a same critical value of the interfacial rigid line angle to change the direction of the image force acting on the dislocation regardless of the shape of the inclusion as the area of the elliptical inclusion taking a constant.     

Fracture behavior of two non-symmetrical collinear cracks emanating from an elliptical hole in a piezoelectric material

Based on the Stroh-type formalism and the technique of conformal mapping, the problem of two non-symmetrical collinear cracks emanating from an elliptical hole in a piezoelectric solid is investigated under remotely uniform in-plane electric loading and anti-plane mechanical loading, which allowed us to take the electric field inside the hole and cracks into account. The analytical solutions of the field intensity factors and the energy release rate are presented in closed-form, which includes the extreme cases for an impermeable crack and a permeable crack. Numerical results are then presented to reveal the effects of geometrical parameters, crack permeability and combined mechanical and electric loadings on the energy release rate.     

回传射线矩阵法含裂纹框架的电阻抗模型

基于回传射线矩阵法(RMM)提出了修正的电阻抗理论模型,使得电阻抗信号能与含裂纹结构的各物理参数定量联系起来.分别用Timoshenko梁理论和经典杆理论描述门式框架各结构元件的弯曲振动和轴向振动,结构裂纹模拟为具有一定刚度的转动弹簧.对于粘贴有压电片的结构元件,把它作为压电片-粘结剂-主体结构这一耦合结构系统加以考察,最后推导出包含了结构裂纹信息的电阻抗的解析表达式.基于该模型,从压电片中提取的电阻抗信号可以有效识别门式框架中的裂纹.     

Dynamic propagation of a finite eccentric crack in a functionally graded piezoelectric ceramic strip

The dynamic propagation of an eccentric Griffith crack in a functionally graded piezoelectric ceramic strip under anti-plane shear is analyzed using the integral transform method. A constant velocity Yoffe-type moving crack is considered. Fourier transform is used to reduce the problem to a pair of dual integral equations, which is then expressed in a Fredholm integral equation of the second kind. We assume that the properties of the functionally graded piezoelectric material vary continuously along the thickness. The impermeable crack boundary condition is adopted. Numerical values on the dynamic stress intensity factors are presented for the functionally graded piezoelectric material to show the dependence of the gradient of material properties, crack moving velocity, and eccentricity. The dynamic stress intensity factors of a moving crack in functionally graded piezoelectric material increases when the crack moving velocity, eccentricity of crack location, material property gradient, and crack length increase. This paper was recommended for publication in revised form by Associate Editor Hyeon Gyu Beom Jeong Woo Shin received a B.S. and M.S. degree in Mechanical Engineering from Yonsei University in Seoul, Korea in 1998 and 2000, respectively. A major field of Mr. Shin is fracture mechanics. He is currently working on the KARI (Korea Aerospace Research Institute) as a senior researcher. He conducted load analysis of fixed wing aircraft and full scale airframe static test at the KARI. He is now developing landing gear in the KHP (Korea Helicopter Program) as a performance engineer.     

An electrically saturated anti-plane shear crack in a piezoelectric layer constrained between two elastic layers

Within the framework of linear elasticity and nonlinear electroelasticity, the electrical nonlinear behavior of an anti-plane shear crack in a piezoelectric ceramic layer constrained between two orthotropic layers is examined by using an electrical strip saturation model. The analysis is performed based on the electrical unified crack surface condition, which can describe all of the permeable, the impermeable, and the limited permeable ones. Fourier transforms are used to reduce the problem to the solutions of two pairs of dual integral equations, which are then expressed to two Fredholm integral equations of the second kind. Some numerical results for the energy release rates are presented to show the effects of the electrical crack condition parameter relating with elliptic flaw shape parameters and permittivity inside the crack, the electric field, the crack length, the electromechanical coupling coefficient, the crack position, and the bonded elastic materials.     

Central crack in a piezoelectric disc

This study is concerned with the general solution of the field intensity factors and energy release rate for a Griffith crack in a piezoelectric ceramic of finite radius under combined anti-plane mechanical and in-plane electrical loading. Both electrically continuous and impermeable crack surface conditions are considered. Employing Mellin transforms and Fourier series, the problem is reduced to dual integral forms. The solution to the resulting expressions is expressed in terms of Fredholm integral equation of the second kind. The solutions are provided to study the influence of the crack length, the crack surface boundary conditions on the intensity factors and the energy release rate.     

Dynamic characteristics of an eccentric crack in a functionally graded piezoelectric ceramic strip

The dynamic response of an eccentric Griffith crack in functionally graded piezoelectric ceramic strip under anti-plane shear impact loading is analysed using integral transform method. Laplace transform and Fourier transform are used to reduce the problem to two pairs of dual integral equations, which are then expressed to Fredholm integral equations of the second kind. We assume that the properties of the functionally graded piezoelectric material vary continuously along the thickness. The impermeable crack boundary condition is adopted. Numerical values on the dynamic stress intensity factors are presented for the functionally graded piezoelectric material to show the dependence of the gradient of material properties and electric loadings.     

Dynamic response of an anti-plane shear crack in a functionally graded piezoelectric strip

The dynamic response of a cracked functionally graded piezoelectric material (FGPM) under transient anti-plane shear mechanical and in-plane electrical loads is investigated in the present paper. It is assumed that the electroelastic material properties of the FGPM vary smoothly in the form of an exponential function along the thickness of the strip. The analysis is conducted on the basis of the unified (or natural) crack boundary condition which is related to the ellipsoidal crack parameters. By using the Laplace and Fourier transforms, the problem is reduced to the solutions of Fredholm integral equations of the second kind. Numerical results for the stress intensity factor and crack sliding displacement are presented to show the influences of the elliptic crack parameters, the electric field, FGPM gradation, crack length, and electromechanical coupling coefficient.     

Analysis of three collinear antiplane interfacial cracks in dissimilar piezoelectric materials under non-self equilibrated electromechanical loadings on a center crack

The problem of three collinear interfacial cracks between two dissimilar transversely isotropic piezoelectric materials is considered under electromechanical loadings. The crack surfaces are assumed to be impermeable to the electric field. A single antiplane mechanical and inplane electrical loads are applied at a point on centred crack surface. The problem is formulated by the complex function method, and reduced to the vector Hilbert problem. By solving the problem, a closed form solution for the stress intensity and electric displacement intensity factor is obtained. This solution can be used as a Green’s function for different loading conditions.     

Analysis of a radial crack in cross-ply laminates under uniaxial tension

The problem of a radial crack in cross-ply laminates under uniaxial tension is investigated in this paper. The normalized stress intensity factors are obtained by the modified mapping collocation method which is based on analytic complex function theory of complex variables. The present results for an isotropic infinite plate show good agreement with existing solutions. In the range of small crack length, the stress intensity factor for a radial crack in cross-ply laminates under uniaxial tension becomes larger as the percentage of 0° plies increases. However in the range of large crack length, it is insensitive to the percentage of 0° plies.     

三维压电介质中不同电边界条件的裂纹分析

在不可导通、可导通和半可导通等三种电边界条件下，系统研究并给出三维无限横观各向同性压电介质中平行于各向同性面的、任意形状的平片裂纹在任意载荷作用下的不连续位移和不连续电势边界积分方程方法。无论平片裂纹的形状如何，结果都表明：对不可导通裂纹，应力强度因子只与机械载荷有关，电位移强度因子只与电载荷有关；而可导通裂纹的应力强度因子和电位移强度因子只与机械载荷有关，电位移强度因子由机械载荷通过压电效应产生。半可导通裂纹所对应的边界积分方程组为非线性方程组，文中给出一种迭代解法。在均布的机械载荷和电载荷作用下。裂纹腔内的电位移为一常数，只与外加载荷有关，而与裂纹形状无关。     

Transient response of a permeable crack normal to a piezoelectric-elastic interface: Anti-plane problem

In this paper, the anti-plane transient response of a central crack normal to the interface between a piezoelectric ceramics and two same elastic materials is considered. The assumed crack surfaces are permeable. By virtue of integral transform methods, the electroelastic mixed boundary problems are formulated as two set of dual integral equations, which, in turn, are reduced to a Fredholm integral equation of the second kind in the Laplace transform domain. Time domain solutions are obtained by inverting Laplace domain solutions using a numerical scheme. Numerical values on the quasi-static stress intensity factor and the dynamic energy release rate are presented to show the dependences upon the geometry, material combination, electromechanical coupling coefficient and electric field.     

Anti-plane shear crack in a functionally gradient piezoelectric layer bonded to dissimilar half spaces

In this paper, the problem of a crack located in a functionally gradient piezoelectric interlayer between two dissimilar homogeneous piezoelectric half-planes being subjected to an anti-plane mechanical loading and an in-plane electric loading is considered. The material properties of the interlayer, such as the elastic stiffness, piezoelectric constant and dielectric constant, are assumed to vary continuously along the thickness of the interlayer, and the crack surface condition is assumed to be impermeable or permeable. By using the Fourier transform, the problem is first reduced to two pairs of dual integral equations and then into a Fredholm integral equation of the second kind. Numerical calculations are carried out, and the effects of crack geometric parameters on the stress intensity factor and the energy release rate are shown graphically.     

压电柔性梁裂缝损伤识别实验

针对压电柔性悬臂梁裂缝损伤检测与损伤程度识别问题,采用小波包分析和小波神经网络相结合的方法进行裂缝深度识别实验研究.利用小波包频带能量谱构造柔性悬臂梁裂缝损伤指标,即能量比相对变化量的H2范数,并建立压电柔性梁裂缝损伤实验装置.激励柔性梁的振动,记录两路压电传感器采集的振动信号,进行小波包分解并计算损伤指标.将这些损伤指标进行组合,作为小波神经网络的输入特征参数,进行裂缝深度即损伤程度的识别.实验结果表明:能量比相对变化量的H2范数对柔性梁的裂缝损伤敏感,对测试噪声不敏感;采用的小波神经网络可以精确识别柔性梁的裂缝深度.