压电介质中受拉伸与弯曲联合作用的圆币形裂纹问题

以弹性位移分量和电势函数为基本未知量时，横观各向同性压电介质非轴对称三维问题的控制微分方程是四个二阶线性偏微分方程相联立的方程组。本文导出了用四个调和函数表示位移及电势的该方程组的势函数通解。作为通解的应用举例，文中求解了压电陶瓷材料中受拉伸与弯曲联合作用的圆币形裂纹问题，得到了裂纹尖端附近应力场及电位移场的解析表达式。结果表明裂尖场以及应力强度因子和电位移强度因子均表现出复杂的机－电耦合行为。     

Analysis of a crack in a half-plane piezoelectric solid with traction-induction free boundary

In this paper, the problem of a crack embedded in a half-plane piezoelectric solid with traction-induction free boundary is analyzed. A system of singular integral equations is formulated for the materials with general anisotropic piezoelectric properties and for the crack with arbitrary orientation. The kernel functions developed are in complex form for general anisotropic piezoelectric materials and are then specialized to the case of transversely isotropic piezoelectric materials which are in real form. The obtained coupled mechanical and electric real kernel functions may be reduced to those kernel functions for purely elastic problems when the electric effects disappear. The system of singular integral equations is solved numerically and the coupling effects of the mechanical and electric phenomena are presented by the generalized stress intensity factors for transversely isotropic piezoelectric materials.     

含币形裂纹的压电体在点力和点电荷作用下的裂纹张开位移

本文利用Chen和Shioya给出的在横观各向同性压电无限体内币形裂纹上下表面作用对称法向点力和点电荷情形下的解,结合压电材料之功的互等定,用初等函数的形式给出了在压电无限体中任意一点作用任意点力和点电荷情形下币形裂纹的张开位移,并对PZT－4压电陶瓷和非压电材料作了计算分析。     

An antisymmetric problem of a penny-shaped crack in a piezoelectric medium

Summary  An exact, three-dimensional analysis is developed for a penny-shaped crack in an infinite transversely isotropic piezoelectric medium. The crack is assumed to be parallel to the plane of isotropy, with its faces subjected to a couple of concentrated normal forces and a couple of point electric charges that are antisymmetric with respect to the crack plane. The fundamental solution of a concentrated force and a point charge acting on the surface of a piezoelectric half-space is employed to derive the integral equations for the general boundary value problem. For the above antisymmetric crack problem, complete expressions for the elastoelectric field are obtained. A numerical calculation is finally performed to show the piezoelectric effect in piezoelectric materials. It is noted here that the present analysis is an extension of Fabrikant's theory for elasticity. Received 30 August 1999; accepted for publication 1 March 2000     

Finite-part integral and boundary element method to solve three-dimensional crack problems in piezoelectric materials

Using the hypersingular integral equation method based on body force method, a planar crack in a three-dimensional transversely isotropic piezoelectric solid under mechanical and electrical loads is analyzed. This crack problem is reduced to solve a set of hypersingular integral equations. Compare with the crack problems in elastic isotropic materials, it is shown that for the impermeable crack, the intensity factors for piezoelectric materials can be obtained from those for elastic isotropic materials. Based on the exact analytical solution of the singular stresses and electrical displacements near the crack front, the numerical method of the hypersingular integral equation is proposed by the finite-part integral method and boundary element method, which the square root models of the displacement and electric potential discontinuities in elements near the crack front are applied. Finally, the numerical solutions of the stress and electric field intensity factors of some examples are given.     

横观各向同性材料的三维断裂力学问题

从三维横观各向同性材料弹性力学理论出发, 使用Hadamard有限部积分概念, 导出了三维状态下单位位移间断(位错)集度的基 本解. 在此基础上, 进一步运用极限理论, 将任意载荷作用下, 三维无限大横观各向 同性材料弹性体中, 含有一个位于弹性对称面内的任意形状的片状裂纹问题, 归结为求 解一组超奇异积分方程的问题. 通过二维超奇异积分的主部分析方法, 精确地求得了裂纹前沿光滑点附近的应力奇异指数和奇异应力场, 从而找到了以裂纹表面位移间断表示的应力强度因子表达式及裂纹局部扩展所提供 的能量释放率. 作为以上理论的实际应用，最后给出了一个圆形片状裂纹问题 的精确解例和一个正方形片状裂纹问题的数值解例. 对受轴对称法向均布载荷作用下圆形片状裂纹问题, 讨论了超奇异积分方程的精确求解方法, 并获得了位移间断和应力强度因子的封闭解, 此结果与现有理论解完全一致.     

THREE-DIMENSIONAL INTERACTIONS OF CIRCULAR CRACK IN TRANSVERSELY ISOTROPIC PIEZOELECTRIC SPACE WITH RESULTANT SOURCES

Exact solutions in form of elementary functions were derived for the stress and electric displacement intensity factors of a circular crack in a transversely isotropic piezoelectric space interacting with various stress and charge sources: force dipoles, electric dipoles, moments, force dilatation and rotation. The circular crack includes penny-shaped crack and external circular crack and the locations and orientations of these resultant sources with respect to the crack are arbitrary. Such stress and charge sources may model defects like vacancies, foreign particles, and dislocations. Numerical results are presented at last.     

THREE-DIMENSION INTERACTIONS OF CIRCULAR CRACK IN TRANSVERSELY ISOTROPIC PIEZOELECTRIC SPACE WITH RESULTANT SOURCES

Exact solutions in form of elementary functions were derived for the stress and electric displacement intensity factors of a circular crack in a transversely isotropic piezoelectric space interacting with various stress and charge sources: force dipoles, electric dipoles, moments, force dilatation and rotation. The circular crack includes penny-shaped crack and external circular crack and the locations and orientations of these resultant sources with respect to the crack are arbitrary. Such stress and charge sources may model defects like vacancies, foreign particles, and dislocations. Numerical results are presented at last.     

Analysis of multiple axisymmetric annular cracks in a piezoelectric medium

An axisymmetric annular electric dislocation is defined. The solution of axisymmetric electric and Volterra climb and glide dislocations in an infinite transversely isotropic piezoelectric domain is obtained by means of Hankel transforms. The distributed dislocation technique is used to construct integral equations for a system of co-axial annular cracks with so-called permeable and impermeable electric boundary conditions on the crack faces where the domain is under axisymmetric electromechanical loading. These equations are solved numerically to obtain dislocation densities on the crack surfaces. The dislocation densities are employed to determine field intensity factors for a system of interacting annular and/or penny-shaped cracks.     

Penny-shaped and half-plane cracks in a transversely isotropic piezoelectric solid under arbitrary loading

Summary The problem of a penny-shaped crack in a transversely isotropic piezoelectric material loaded by both normal and tangential tractions and by electric charges is analyzed. Closed-form solutions are obtained for the full electroelastic fields as well as for the stress and electric displacement intensity factors. Solutions are also obtained for the (non-trivial) limiting case of a half-plane crack. The results are illustrated on the example of piezoceramics PZT-6B. Received 12 July 1999; accepted for publication 20 July 1999     

EXACT SOLUTION TO THE PROBLEM OF A HALF-PLANE CRACK IN A TRANSVERSELY ISOTROPIC PIEZOELECTRIC BODY SUBJECTED TO ANTISYMMETRIC TANGENTIAL POINT FORCES

An exact and complete solution of the problem of a half-plane crack in an infinite transversely isotropic piezoelectric body is presented. The upper and lower crack faces are assumed to be loaded antisymmetrically by a couple of tangential point forces in opposite directions. The solution is derived through a limiting procedure from that of a penny-shaped crack. The expressions for the electroelastic field are given in terms of elementary functions. Finally, the numerical results of the second and third mode stress intensity factorsk ₂ andk ₃ of piezoelectric materials and elastic materials are compared in figures. Project supported by the National Natural Science Foundation of China (No. 19872060 and 69982009) and the Postdoctoral Foundation of China.     

Circular crack in a transversely isotropic piezoelectric space under point forces and point charges

In this paper, two kinds of circular crack including external circular crack and penny-shaped crack in a transversely isotropic piezoelectric space are considered. Firstly, we obtain the solution to the problem of an external circular crack in a transversely isotropic piezoelectric space subjected to antisymmetric normal point forces and point charges. Based on this, the solution of one-sided loading of an external circular crack is constructed. Secondly, the real shape of an external circular crack and the opening displacement of a penny-shaped crack under an arbitrary point force and point charge are further obtained. At last, the results are presented in a graphical form. The project supported by the National Natural Science Foundation of China (19872060 and 69982009) and the Postdoctoral Foundation of China     

横观各向同性压电材料中裂纹问题的边界元法

采用Somigiliana公式给出了三维横观各向同性压电材料中的非渗漏裂纹问题的一般解和超奇异积分方程,其中未知函数为裂纹面上的位移间断和电势间断．在此基础上,使用有限部积分和边界元结合的方法,建立了超奇异积分方程的数值求解方法,并给出了一些典型数值算例的应力强度因子和电位移强度因子的数值结果,结果令人满意．     

Subinterface crack in an anisotropic piezoelectric bimaterial

In this paper, the problem of a subinterface crack in an anisotropic piezoelectric bimaterial is analyzed. A system of singular integral equations is formulated for general anisotropic piezoelectric bimaterial with kernel functions expressed in complex form. For commonly used transversely isotropic piezoelectric materials, the kernel functions are given in real forms. By considering special properties of one of the bimaterial, various real kernel functions for half-plane problems with mechanical traction-free or displacement-fixed boundary conditions combined with different electric boundary conditions are obtained. Investigations of half-plane piezoelectric solids show that, particularly for the mechanical traction-free problem, the evaluations of the mechanical stress intensity factors (electric displacement intensity factor) under mechanical loadings (electric displacement loading) for coupled mechanical and electric problems may be evaluated directly by considering the corresponding decoupled elastic (electric) problem irrespective of what electric boundary condition is applied on the boundary. However, for the piezoelectric bimaterial problem, purely elastic bimaterial analysis or purely electric bimaterial analysis is inadequate for the determination of the generalized stress intensity factors. Instead, both elastic and electric properties of the bimaterial’s constants should be simultaneously taken into account for better accuracy of the generalized stress intensity factors.     

Elliptical crack and punch problems solved by integral equation method for piezoelectric media

The problem of an elliptical crack embedded in an unbounded transversely isotropic piezoelectric media with the crack-plane parallel to the plane of isotropy of the media and subjected to remote normal mechanical as well as electric loading is considered first. The problem has been successfully reduced to a pair of coupled integral equations that are suitable for the application of an integral equation method developed earlier for three-dimensional problems of LEFM. Solution to the mechanical displacement and electric potentials are obtained for prescribed uniform loadings and expressions for corresponding intensity factors and crack opening displacement are deduced. The above method has further been applied to solve the problem of a rigid flat-ended elliptical punch indenting a transversely isotropic piezoelectric half-space surface with the plane of isotropy parallel to the surface. Solutions to mechanical stress and electric displacement are obtained for prescribed constant normal displacement and constant electric potential interior to the elliptical region and expression for the total force required to maintain a prescribed indentation is deduced.     

Effects of electric field on crack growth for a penny-shaped dielectric crack in a piezoelectric layer

The assumptions of impermeable and permeable cracks give rise to significant errors in determining electro-elastic behavior of a cracked piezoelectric material. The former simply imposes that the permittivity or electric displacement of the crack interior vanishes, and the latter neglects also the effects of the dielectric of an opening crack interior. Considering the presence of the dielectric of an opening crack interior and the permeability of the crack surfaces for electric field, this paper analyzes electro-elastic behavior induced by a penny-shaped dielectric crack in a piezoelectric ceramic layer. In the cases of prescribed displacement or prescribed stress at the layer surfaces, the Hankel transform technique is employed to reduce the problem to Fredholm integral equations with a parameter dependent nonlinearly on the unknown functions. For an infinite piezoelectric space, a closed-form solution can be derived explicitly, while for a piezoelectric layer, an iterative technique is suggested to solve the resulting nonlinear equations. Field intensity factors are obtained in terms of the solution of the equations. Numerical results of the crack opening displacement intensity factors are presented for a cracked PZT-5H layer and the effect of applied electric field on crack growth are examined for both cases. The results indicate that the fracture toughness of a piezoelectric ceramic is affected by the direction of applied electric fields, dependent on the elastic boundary conditions.     

Axisymmetric displacement boundary value problem for a penny-shaped crack

Summary A solution is derived from equations of equilibrium in an infinite isotropic elastic solid containing a penny-shaped crack where displacements are given. Abel transforms of the second kind stress and displacement components at an arbitrary point of the solid are known in the literature in terms of jumps of stress and displacement components at a crack plane. Limiting values of these expressions at the crack plane together with the boundary conditions lead to Abel-type integral equations, which admit a closed form solution. Explicit expressions for stress and displacement components on the crack plane are obtained in terms of prescribed face displacements of crack surfaces. Some special cases of the crack surface shape functions have been given in the paper.     

Isolated crack in three-dimensional piezoelectric solid. Part II: Stress intensity factors for circular crack

This is part II of the work concerned with finding the stress intensity factors for a circular crack in a solid with piezoelectric behavior. The method of solution involves reducing the problem to a system of hypersingular integral equations by application of the unit concentrated displacement discontinuity and the unit concentrated electric potential discontinuity derived in part I [1]. The near crack border elastic displacement, electric potential, stress and electric displacement are obtained. Stress and electric displacement intensity factors can be expressed in terms of the displacement and the potential discontinuity on the crack surface. Analogy is established between the boundary integral equations for arbitrary shaped cracks in a piezoelectric and elastic medium such that once the stress intensity factors in the piezoelectric medium can be determined directly from that of the elastic medium. Results for the penny-shaped crack are obtained as an example.     

A general solution and the application of space axisymmetric problem in piezoelectric material

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横观各向同性三维热弹性力学通解及其势理论法

通过引入两个位移函数，对用位移表达的运动平衡方程作了简化．利用算子理论，严格地导出了横观各向同性非耦合热弹性动力学问题的通解．对于静力学问题，通解的形式可进一步简化成用4个准调和函数来表示．具体考察了横观各向同性体内平面裂纹上下表面有对称分布温度作用的问题，推广了势理论方法，导出了一个积分方程和一个微分-积分方程．针对币状裂纹表面受均布温度作用情形，给出了具体的解。