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     

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。     

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.     

Variation of the stress intensity factor along the crack front of interacting semi-elliptical surface cracks

Summary  In this study, the interaction between two semi-elliptical co-planar surface cracks is considered when Poisson's ratio ν = 0.3. The problem is formulated as a system of singular integral equations, based on the idea of the body force method. In the numerical calculation, the unknown density of body force density is approximated by the product of a fundamental density function and a polynomial. The results show that the present method yields smooth variations of stress intensity factors along the crack front very accurately, for various geometrical conditions. When the size of crack 1 is larger than the size of crack 2, the maximum stress intensity factor appears at a certain point, β₁=177^∘, of crack 1. Along the outside of crack 1, that is at β₁=0∼90^∘, the interaction can be negligible even if the two cracks are very close. The interaction can be negligible when the two cracks are spaced in such a manner that their two closest points are separated by a distance exceeding the small crack's major diameter. The variations of stress intensity factor of a semi-elliptical crack are tabulated and charted. Received 30 August 1999; accepted for publication 22 February 2000     

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.     

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.     

Analysis of stress intensity factors of a ring-shaped interface crack

In this paper, numerical solutions of singular integral equations are discussed in the analysis of axi-symmetric interface cracks under torsion and tension. The problems of a ring-shaped interface crack are formulated in terms of 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 densities are chosen to express a two-dimensional interface crack exactly. The accuracy of the present analysis is verified by comparing the present results with the results obtained by other researchers for the limiting cases of the geometries. The calculation shows that the present method gives rapidly converging numerical results for those problems as well as for ordinary crack problems in homogeneous material. The stress intensity factors of a ring-shaped interface crack are shown in tables and charts with varying the material combinations and also geometrical conditions.     

Arbitrarily oriented crack near interface in piezoelectric bimaterials

Arbitrarily oriented crack near interface in piezoelectric bimaterials is considered. After deriving the fundamental solution for an edge dislocation near the interface, the present problem can be expressed as a system of singular integral equations by modeling the crack as continuously distributed edge dislocations. In the paper, the dislocations are described by a density function defined on the crack line. By solving the singular integral equations numerically, the dislocation density function is determined. Then, the stress intensity factors (SIFs) and the electric displacement intensity factor (EDIF) at the crack tips are evaluated. Subsequently, the influences of the interface on crack tip SIFs, EDIF, and the mechanical strain energy release rate (MSERR) are investigated. The J-integral analysis in piezoelectric bimaterals is also performed. It is found that the path-independent of J₁-integral and the path-dependent of J₂-integral found in no-piezoelectric bimaterials are still valid in piezoelectric bimaterials.     

Hypersingular integral equation method for a three-dimensional crack in anisotropic electro-magneto-elastic bimaterials

Using Green’s functions, the extended general displacement solutions of a three-dimensional crack problem in anisotropic electro-magneto-elastic (EME) bimaterials under extended loads are analyzed by the boundary element method. Then, the crack problem is reduced to solving a set of hypersingular integral equations (HIE) coupled with boundary integral equations. The singularity of the extended displacement discontinuities around the crack front terminating at the interface is analyzed by the main-part analysis method of HIE, and the exact analytical solutions of the extended singular stresses and extended stress intensity factors (SIFs) near the crack front in anisotropic EME bimaterials are given. Also, the numerical method of the HIE for a rectangular crack subjected to extended loads is put forward with the extended crack opening dislocation approximated by the product of basic density functions and polynomials. At last, numerical solutions of the extended SIFs of some examples are obtained.     

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.     

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.     

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．     

Cracking of an orthotropic substrate reinforced by an orthotropic plate

A solution method is derived to determine the stress intensity factors for both an internal crack and an edge crack in an orthotropic substrate that is reinforced on its boundary by a finite-length orthotropic plate. The method utilizes the Green’s functions for a pair of dislocations and a concentrated force on the boundary while invoking the concept of superposition. Enforcing the traction-free boundary condition along the crack surfaces and the continuity of displacement gradients along the plate/substrate interface results in a coupled system of singular integral equations. An asymptotic analysis of the kernels in these equations for the region of the junction point between the plate corner and the substrate boundary reveals the strength of the singularity in the case of an edge crack. The numerical solution of the integral equations provides results for the stress intensity factors for both an internal crack and an edge crack perpendicular to the substrate boundary and aligned with one of the corners of the plate. The present results have been validated against previously published stress intensity factors for an internal crack and an edge crack in an isotropic substrate.     

Application of hypersingular integral equation method to three-dimensional crack in electromagnetothermoelastic multiphase composites

A three-dimensional crack problem in electromagnetothermoelastic multiphase composites (EMTE-MCs) under extended loads is investigated in this paper. Using Green’s functions, the extended general displacement solutions are obtained by the boundary element method. This crack problem is reduced to solving a set of hypersingular integral equations coupled with boundary integral equations, in which the unknown functions are the extended displacement discontinuities. Then, the behavior of the extended displacement discontinuities around the crack front terminating at the interface is analyzed by the main-part analysis method of hypersingular integral equations. Analytical solutions for the extended singular stresses, the extended stress intensity factors (SIFs) and the extended energy release rate near the crack front in EMTE-MCs are provided. Also, a numerical method of the hypersingular integral equations for a rectangular crack subjected to extended loads is put forward with the extended displacement discontinuities approximated by the product of basic density functions and polynomials. In addition, distributions of extended SIFs varying with the shape of the crack are presented. The results show that the present method accurately yields smooth variations of extended SIFs along the crack front.     

Mixed-mode stress intensity factors of 3D interface crack in fully coupled electromagnetothermoelastic multiphase composites

This contribution presents an extended hypersingular intergro-differential equation (E-HIDE) method for modeling the 3D interface crack problem in fully coupled electromagnetothermoelastic anisotropic multiphase composites under extended electro-magneto-thermo-elastic coupled loads through theoretical analysis and numerical simulations. First, based on the extended boundary element method, the 3D interface crack problem is reduced to solving a set of E-HIDEs coupled with extended boundary integral equations, in which the unknown functions are the extended displacement discontinuities. Then, the behavior of the extended singular stress indices around the interface crack front terminating at the interface is analyzed by the extended main-part analysis. The extended stress intensity factors near the crack front are defined. In addition, a numerical method for a 3D interface crack problem subjected to extended loads is proposed, in which the extended displacement discontinuities are approximated by the product of basic density functions and polynomials. Finally, the radiation distribution of extended stress intensity factors at the interface crack surface are calculated, and the results are presented toward demonstrating the applicability of the proposed method.     

同震断层三维裂纹扩展波动时域超奇异积分法

应用波动时域超奇异积分法将P波、S波和磁电热弹多场耦合作用下同震断层任意形状三维裂纹扩展问题转化为求解以广义位移间断率为未知函数的超奇异积分方程组问题;定义了广义应力强度因子,得到裂纹前沿广义奇异应力增量解析表达式;应用波动时域有限部积分概念及体积力法,为超奇异积分方程组建立了数值求解方法,编制了FORTRAN程序,以三维矩形裂纹扩展问题为例,通过典型算例,研究了广义应力强度因子随裂纹位置变化规律;分析了同震断层裂纹扩展中力、磁、电场辐射规律.      

Anti-plane shear of kinked interface crack in bonded dissimilar anisotropic solid

Complex potentials are derived to describe the anti-plane singular shear stress fields around a kinked crack, the main portion of which is embedded along the interface of two dissimilar anisotropic elastic media. This is accomplished by formulating the problem as singular integral equations with generalized Cauchy kernels. The shear stress singularity at the kink differs from the familiar inverse square root of the local distance; it is found to influence the magnitude of the Mode III crack tip stress intensity factor, K₃. Numerical results of K₃ are obtained and displayed in graphical forms for different degree of material anisotropy and crack dimensions.     

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     

SINGULAR SOLUTIONS OF ANISOTROPIC PLATE WITH AN ELLIPTICAL HOLE OR A CRACK

In the present paper, closed form singular solutions for an infinite anisotropic plate with an elliptic hole or crack are derived based on the Stroh-type formalism for the general anisotropic plate. With the solutions, the hoop stresses and hoop moments around the elliptic hole as well as the stress intensity factors at the crack tip under concentrated in-plane stresses and bending moments are obtained. The singular solutions can be used for approximate analysis of an anisotropic plate weakened by a hole or a crack under concentrated forces and moments.They can also be used as fundamental solutions of boundary integral equations in BEM analysis for anisotropic plates with holes or cracks under general force and boundary conditions.     

三维界面裂纹的奇性应力场和应力强度因子分析

使用有限部积分概念和极限方法得到了三维平片界面裂纹的超奇异积分－微分方程组后,进一步利用二维超奇异积分主部分析方法,对裂纹前沿的应力场作了理论分析,并获得了其奇性应力场和裂纹面位移间断表示复位应力强度因子的精确表达式,为三维平片界面裂纹的超奇异积分－微分方程组的求解建立了数值方法,并分析了界面椭圆平片裂纹问题,和现有解比较,所得数值结果令人满意。