Controlling parameter of the stress intensity factors for a planar interfacial crack in three-dimensional bimaterials

Numerical solutions of singular integral equations are discussed in the analysis of a planar rectangular interfacial crack in three-dimensional bimaterials subjected to tension. The problem is formulated as a system of singular integral equations on the basis of the body force method. In the numerical analysis, unknown body force densities are approximated by the products of the fundamental density functions and power series, where the fundamental density functions are chosen to express singular behavior along the crack front of the interface crack exactly. The calculation shows that the present method gives smooth variations of stress intensity factors along the crack front for various aspect ratios. The present method gives rapidly converging numerical results and highly satisfied boundary conditions throughout the crack boundary. The stress intensity factors are given with varying the material combination and aspect ratio of the crack. It is found that the stress intensity factors K_I and K_II are determined by the bimaterial constant ε alone, independent of elastic modulus ratio and Poisson’s ratio.     

轴对称环形片状界面裂纹问题分析

讨论受拉伸载荷作用的轴对称环形片状界而裂纹问题．该问题归结为求解一组超奇异积分-微分方程．方程中的未知位移间断近似表示为基本密度函数与多项式之积,其中基本密度函数考虑到问题的对称性用二维界面裂纹精确解表示．在圆形片状裂纹的情况下,数值结果与现有理论解作比较的结果表明,数值结果与相应界面圆形片状裂纹和均质体圆形片状裂纹的精确解均吻合得很好．文中以图表形式给出应力强度因子与材料组合和几何条件之间的关系．     

Variation of the stress intensity factor along the front of a 3-D rectangular crack subjected to mixed-mode load

Summary  The singular integral equation method is applied to the calculation of the stress intensity factor at the front of a rectangular crack subjected to mixed-mode load. The stress field induced by a body force doublet is used as a fundamental solution. The problem is formulated as a system of integral equations with r ⁻³-singularities. In solving the integral equations, unknown functions of body-force densities are approximated by the product of polynomial and fundamental densities. The fundamental densities are chosen to express two-dimensional cracks in an infinite body for the limiting cases of the aspect ratio of the rectangle. The present method yields rapidly converging numerical results and satisfies boundary conditions all over the crack boundary. A smooth distribution of the stress intensity factor along the crack front is presented for various crack shapes and different Poisson's ratio. Received 5 March 2002; accepted for publication 2 July 2002     

Numerical solutions of singular integral equations for planar rectangular interfacial crack in three dimensional bimaterials

Stress intensity factors for a three dimensional rectangular interfacial crack were considered using the body force method. In the numerical calculations, unknown body force densities were approximated by the products of the fundamental densities and power series; here the fundamental densities are chosen to express singular stress fields due to an interface crack exactly. The calculation shows that the numerical results are satisfied. The stress intensity factors for a rectangular interface crack were indicated accurately with the varying aspect ratio, and bimaterial parameter.     

Numerical solution of singular integral equations in stress concentration problems under longitudinal shear loading

Summary The paper deals with numerical solutions of singular integral equations in stress concentration problems for longitudinal shear loading. The body force method is used to formulate the problem as a system of singular integral equations with Cauchy-type singularities, where unknown functions are densities of body forces distributed in the longitudinal direction of an infinite body. First, four kinds of fundamental density functions are introduced to satisfy completely the boundary conditions for an elliptical boundary in the range 0≤φ _k ≤2π. To explain the idea of the fundamental densities, four kinds of equivalent auxiliary body force densities are defined in the range 0≤φ _k ≤π/2, and necessary conditions that the densities must satisfy are described. Then, four kinds of fundamental density functions are explained as sample functions to satisfy the necessary conditions. Next, the unknown functions of the body force densities are approximated by a linear combination of the fundamental density functions and weight functions, which are unknown. Calculations are carried out for several arrangements of elliptical holes. It is found that the present method yields rapidly converging numerical results. The body force densities and stress distributions along the boundaries are shown in figures to demonstrate the accuracy of the present solutions. Received 26 May 1998; accepted for publication 27 November 1998     

Stress intensity factors of a rectangular crack meeting a bimaterial interface

Using the hypersingular integral equation method based on body force method, a planar crack meeting the interface in a three-dimensional dissimilar materials is analyzed. The singularity of the singular stress field around the crack front terminating at the interface is analyzed by the main-part analytical method of hypersingular integral equations. Then, the numerical method of the hypersingular integral equation for a rectangular crack subjected to normal load is proposed by the body force method, which the crack opening dislocation is approximated by the product of basic density functions and polynomials. Numerical solutions of the stress intensity factors of some examples are given.     

Interaction between elliptical and ellipsoidal inclusions under bending stress fields

Summary  This paper deals with interaction problems of elliptical and ellipsoidal inclusions under bending, using singular integral equations of the body force method. The problems are formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where unknown functions are densities of body forces distributed in the x,y and r,θ,z directions in infinite bodies having the same elastic constants as those of the matrix and inclusions. In order to satisfy the boundary conditions along the elliptical and the ellipsoidal boundaries, the unknown functions are approximated by a linear combination of fundamental density functions and polynomials. The present method is found to yield the exact solutions for a single elliptical or spherical inclusion under a bending stress field. It yields rapidly converging numerical results for interface stresses in the interaction of inclusions. Received 9 September 1999; accepted for publication 15 January 2000     

三维体内含垂直于双材料界面混合型裂纹的超奇异积分解法

利用双材料位移基本解和Somigliana公式,将三维体内含垂直于双材料界面混合型裂纹问题归结为求解一组超奇异积分方程。使用主部分析法,通过对裂纹前沿应力奇性的分析,得到用裂纹面位移间断表示的应力强度因子的计算公式,进而利用超奇异积分方程未知解的理论分析结果和有限部积分理论,给出了超奇异积分方程的数值求解方法。最后,对典型算例的应力强度因子做了计算,并讨论了应力强度因子数值结果的收敛性及其随各参数变化的规律。     

Singular integral equations and boundary element method of cracks in thermally stressed planar solids

Ｉｔｉｓｋｎｏｗｎｔｈａｔｍｏｓｔａｇｒｉｃｕｌｔｕｒａｌｐｒｏｄｕｃｔｓａｎｄｆｏｏｄｓａｒｅｐｒｏｃｅｓｓｅｄａｎｄｔｒａｎｓｐｏｒｔｅｄｕｎｄｅｒｃｅｒｔａｉｎｔｅｍｐｅｒａｔｕｒｅｃｏｎｄｉｔｉｏｎｓ,ａｎｄｔｈｅｓｔｒｕｃｔｕｒａｌｃｏｍｐｏｎｅｎｔｓａｌｓｏｗｏｒｋｕｎｄｅｒａｔｈｅｒｍａｌｅｎｖｉｒｏｎｍｅｎｔ.Ｔｅｍｐｅｒａｔｕｒｅｉｎｄｕｃｅｄｓｔｒｅｓｓｅｓｕｓｕａｌｌｙｌｅａｄｔｏｄａｍａｇｅｏｆｆｌａｗｅｄｓｏｌｉｄｓ.Ｔｈｕｓ,ｔｈｅｉｎｖｅｓｔｉｇａｔｉｏｎｏｆｔｈｅｃｒ…     

垂直磁压电材料界面三维裂纹的超奇异积分法

应用有限部积分概念和广义位移基本解，垂直于磁压电双材料界面三维复合型裂纹问题被转 化为求解一组以裂纹表面广义位移间断为未知函数的超奇异积分方程问题. 进而，通过主部 分析法精确地求得裂纹尖端光滑点附近的奇性应力场解析表达式. 然后，通过将裂纹表面 位移间断未知函数表达为位移间断基本密度函数与多项式之积，使用有限部积分法对超奇异 积分方程组建立了数值方法. 最后，通过典型算例计算，讨论了广义应力强度因子的变化规 律.     

Asymmetric problem of a row of revolutional ellipsoidal cavities using singular integral equations

This paper deals with numerical solution of singular integral equations of the body force method in an interaction problem of revolutional ellipsoidal cavities under asymmetric uniaxial tension. The problem is solved on the superposition of two auxiliary loads; (i) biaxial tension and (ii) plane state of pure shear. These problems are formulated as a system of singular integral equations with Cauchy-type singularities, where the unknowns are densities of body forces distributed in the r, θ, z directions. In order to satisfy the boundary conditions along the ellipsoidal boundaries, eight kinds of fundamental density functions proposed in our previous papers are applied. In the analysis, the number, shape, and spacing of cavities are varied systematically; then the magnitude and position of the maximum stress are examined. For any fixed shape and size of cavities, the maximum stress is shown to be linear with the reciprocal of squared number of cavities. The present method is found to yield rapidly converging numerical results for various geometrical conditions of cavities.     

Elastic wave scattering from a penny-shaped interface crack in coated materials

This paper is concerned with the elastic wave scattering induced by a penny-shaped interface crack in coated materials. Using the integral transform, the problem of wave scattering is reduced to a set of singular integral equations in matrix form. The singular integral equations are solved by the asymptotic analysis and contour integral technique, and the expressions for the stress and displacement as well as the dynamic stress intensity factors (SIFs) are obtained. Using numerical analysis, this approach is verified by the finite element method (FEM), and the numerical results agree well with the theoretical results. For various crack sizes and material combinations, the relations between the SIFs and the incident frequency are analyzed, and the amplitudes of the crack opening displacements (CODs) are plotted versus incident wavenumber. The investigation provides a theoretical basis for the dynamic failure analysis and nondestructive evaluation of coated materials.     

双材料基本解弹塑性边界元法

提出并研究采用双材料基本解的弹塑性边界元法,得到了内点应力公式中有关奇点塑性应变自由项的完整表达式,并利用非连续边界单元和非连续区域单元解决了当奇点位于界面上时该自由项难于确定,以及计算区域Cauchy主值积分的常塑性应变场法在与界面相连的奇异区域单元上无法实施的困难．采用双材料基本解的弹塑性边界无法针对双材料的结构特点,特别适于分析有关弹塑性双材料界面及界面裂纹问题．     

Interaction between an interface crack and subinterface microcracks in metal/piezoelectric bimaterials

The J-integral analysis is presented for the interaction problem between a semi-infinite interface crack and subinterface matrix microcracks in dissimilar anisotropic materials. After deriving the fundamental solutions for an interface crack subjected to different loads and the fundamental solutions for an edge dislocation beneath the interface, the interaction problem is deduced to a system of singular integral equations with the aid of a superimposing technique. The integral equations are then solved numerically and a conservation law among three values of the J-integral is presented, which are induced from the interface crack tip, the microcracks and the remote field, respectively. The conservation law not only provides a necessary condition to confirm the numerical results derived, but also reveals that the microcrack shielding effect in such materials could be considered as a redistribution of the remote J-integral. It is this redistribution that does lead to the phenomenological shielding effect.     

三维间断位移法及强奇异和超奇异积分的处理方法

从积分方程Somigliana等式出发,导出三维状态下单位位错集度的基本解.在此基础上,建立了边界积分方程,并给出了其离散形式.对强奇异和超奇异积分,采用了Hadamard定义的有限部分积分来处理.最后,给出了计算裂纹应力强度因子的算例,并与解析解进行了比较,证实了该方法的有效性.     

与界面垂直相触的三维裂纹的超奇异积分方程和奇性指数

本文使用有限部积分原理和两相材料空间弹性力学问题的点力基本解导出了与界面垂直相触的三维平片解纹的超奇异积分方程组;     

半平面多边缘裂纹反平面问题的奇异积分方程

利用复变函数和奇异积分方程方法,求解弹性范围内半平面多边缘裂纹的反平面问题．提出了满足半平面边界自由的由分布位错密度表示的单边缘裂纹的基本解,此基本解由主要部分和辅助部分组成．将半平面多边缘裂纹问题看作是许多单边缘裂纹问题的叠加,建立了一组Cauchy型奇异积分方程．然后,利用半开型积分法则求解该奇异积分方程,得到了裂纹端处的应力强度因子．最后,给出了几个数值算例．     

Thermal response of a partially insulated interface crack in a graded coating-substrate structure under thermo-mechanical disturbance: Energy release and density

Consider the thermal fracture problem of a functionally graded coating-substrate structure of finite thickness with a partially insulated interface crack subjected to thermal-mechanical supply. A new model is proposed that the heat conduction through the crack region occurs and the temperature drop across the crack surfaces is the result of the thermal resistance. For the first time, real fundamental solutions are derived for the fracture analysis of functionally graded materials. The complicated mixed boundary problems of equations of heat conduction and elasticity are converted analytically into singular integral equations, which are solved numerically. The asymptotic expressions with higher order terms for the singular integral kernels are considered to improve the accuracy and efficiency of the numerical integration. Explicit expressions of various failure modes including stress intensity factors, energy release rate and strain energy density, are provided. Numerical results are presented to illustrate the effects of non-homogeneity parameters and the dimensionless thermal resistance on the temperature distribution along the crack surfaces and extended crack line, the thermal stress intensity factors and minimum strain energy density.     

The scattering of general SH plane wave by interface crack between two dissimilar viscoelastic bodies

The scattering of general SH plane wave by an interface crack between two dissimilar viscoelastic bodies is studied and the dynamic stress intensity factor at the crack-tip is computed. The scattering problem can be decomposed into two problems: one is the reflection and refraction problem of general SH plane waves at perfect interface (with no crack); another is the scattering problem due to the existence of crack. For the first problem, the viscoelastic wave equation, displacement and stress continuity conditions across the interface are used to obtain the shear stress distribution at the interface. For the second problem, the integral transformation method is used to reduce the scattering problem into dual integral equations. Then, the dual integral equations are transformed into the Cauchy singular integral equation of first kind by introduction of the crack dislocation density function. Finally, the singular integral equation is solved by Kurtz's piecewise continuous function method. As a consequence, the crack opening displacement and dynamic stress intensity factor are obtained. At the end of the paper, a numerical example is given. The effects of incident angle, incident frequency and viscoelastic material parameters are analyzed. It is found that there is a frequency region for viscoelastic material within which the viscoelastic effects cannot be ignored. This work was supported by the National Natural Science Foundation of China (No.19772064) and by the project of CAS KJ 951-1-20     

层体边界元法在二维层体断裂分析中的应用

在刚度矩阵法的基础上建立了用于进行二维多层体结构断裂分析的边界单元法（ＢＥＭＬＭ）由于ＢＥＭＬＭ的基本方程中已经包含了层体表面和裂纹缝面的边界条件，因而不需要对这些边界进行单元离散，从而其断裂分析可望有较好的精度通过与柯西积分方程法进行结合，算例表明ＢＥ ＭＬＭ是可靠并有效的