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     

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.     

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     

Analysis of 2D infinite anisotropic elastic strips and semi-strips

Summary  Using Stroh's formalism and the theory of analytic functions, simple and explicit solutions for a line dislocation in an infinite anisotropic elastic strip are obtained. The two boundaries of the strip are free of traction. The problem of a dislocation in an anisotropic elastic semi-infinite strip with traction-free boundaries is also studied. A set of singular integral equations governing the unknown functions is derived. When the medium is orthogonal anisotropic and the coordinate axes x ₁ x ₂ x ₃ are coincident with the material principal axes, all the eigenvalues of the material coefficient matrix are pure imaginary. Explicit expressions of the unknown functions are given for this case. The results obtained are valid not only for plane and anti-plane problems but also for coupled problems between in-plane and out-of-plane deformations. Received 30 October 2000; accepted for publication 28 March 2001     

Stress concentration of an ellipsoidal inclusion of revolution in a semi-infinite body under biaxial tension

Summary This paper deals with the stress concentration problem of an ellipsoidal inclusion of revolution in a semi-infinite body under biaxial tension. The problem is formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where unknowns are densities of body forces distributed in the r- and z-directions in semi-infinite bodies having the same elastic constants as the ones of the matrix and inclusion. In order to satisfy the boundary conditions along the ellipsoidal boundary, four fundamental density functions proposed in [24, 25] are used. The body-force densities are approximated by a linear combination of fundamental density functions and polynomials. The present method is found to yield rapidly converging numerical results for stress distribution along the boundaries even when the inclusion is very close to the free boundary. The effect of the free surface on the stress concentration factor is discussed with varying the distance from the surface, the shape ratio and the elastic modulus ratio. The present results are compared with the ones of an ellipsoidal cavity in a semi-infinite body.accepted for publication 11 November 2003     

Generalized stress intensity factors in the interaction within a rectangular array of rectangular inclusions

Summary To evaluate the mechanical strength of fiber-reinforced composites, it is necessary to consider singular stresses at the end of fibers because they cause crack initiation, propagation, and final failure. A square array of rectangular inclusions under longitudinal tension is considered in this paper. The body-force method is applied to a unit cell region. Then, the problem is formulated as a system of singular integral equations, where the unknown functions are the densities of body forces distributed in infinite plates having the same elastic constants as those of the matrix and inclusions. The unknown functions are expressed as piecewise-smooth functions using power series and two types of fundamental densities which express singular stresses. Generalized stress intensity factors at the corners of inclusions are systematically calculated with varying the shape and spacing of a square array of square and rectangular inclusions.     

压电材料中椭圆片状裂纹问题精确解的一种方法

在线性压电陶瓷本构关系和裂纹边界绝缘的框架下,用超奇异积分方程的方法对椭圆类片状裂纹问题进行了重新研究．超奇异积分方程中的未知位移间断和电势间断近似地表示为基本密度函数与多项式之积,其中基本密度函数反映了椭圆片状裂纹前沿电弹性场的奇异性,而多项式在均布载荷作用下可用一个常数来表达．引入椭球坐标系后,得到了均布载荷作用下未知位移间断和电势间断的解析解．使用这些解析解和电弹性场强度的定义,得到了裂纹前沿Ⅰ型、Ⅱ型和Ⅲ型应力强度因子以及电位移强度因子的精确表达式．法向均布载荷作用下的结果与现有精确解完全一致,切向均布载荷作用下的结果则尚未见有其它报道．     

Interaction of plane elastic nonstationary waves with an elastic inclusion under complete adhesion

We solve the problem on the interaction of plane elastic nonstationary waves with a thin elastic strip-shaped inclusion. The inclusion is contained in an unbounded body (matrix) which in under conditions of plane strain. It is assumed that the condition of perfect adhesion between the inclusion and the matrix is satisfied. Because of the small thickness of the inclusion we assume that the bending and shear displacements at any inclusion point coincide with the displacements of the corresponding points of its midplane. The displacements on the midplane itself are found from the corresponding equations of the theory of plates. The statement of the boundary conditions for these equations takes into account the forces and moments acting on the inclusion edges from the matrix. The solution method is based on representing the displacements in the space of Laplace transforms as a discontinuous solution of the Lame’ equations for the plane strain with subsequent determining the transforms of the unknown jumps from integral equations. The passage to the original functions is performed numerically by methods based on replacement of the Mellin integral by the Fourier series. As a result, we obtain approximate formulas for calculating the stress intensity factors for the inclusion. These formulas are used to study the time dependence of the stress intensity factors and the influence of the inclusion rigidity on their values. We also study the possibility of treating inclusions of high rigidity as absolutely rigid inclusions.     

三维裂纹分析的主值型面力边界积分方程法

给出了一组只包含Ｃａｕｃｈｙ主值积分、不含有强奇异积分的三维静动力边界积分方程及其应用于裂纹问题的具体列式，并给出了几何轴对称问题的相应半解析边界元求解方法，将三维问题降阶为一维数值问题．文中分析了无限、半无限介质中圆裂纹、平行圆裂纹系、球面裂纹等在静载及应力波作用下的静力或瞬态动力响应问题，求得了相应的应力强度因子．     

Interaction of plane elastic harmonic waves with a perfectly bonded elastic inclusion

The interaction of plane harmonic waves with a thin elastic inclusion in the form of a strip in an infinite body (matrix) under plane strain conditions is studied. It is assumed that the bending and shear displacements of the inclusion coincide with the displacements of its midplane. The displacements in the midplane are found from the theory of plates. The priblem-solving method represents the displacements as discontinuous solutions of the Lamé equations and finds the unknown discontinuities solving singular integral equations by the numerical collocation method. Approximate formulas for the stress intensity factors at the ends of the inclusion are derived     

Intensity of singular stress fields causing interfacial debonding at the end of a fiber under pullout force and transverse tension

In this study, singular stress fields at the ends of fibers are discussed by the use of models of rectangular and cylindrical inclusions in a semi-infinite body under pullout force. Those singular stresses have not been discussed yet in the previous studies for pullout problems although they are important for causing interfacial initial debonding. The body force method is used to formulate those problems as a system of singular integral equations where unknowns are densities of the body forces distributed in a semi-infinite body having the same elastic constants as those of the matrix and inclusions. In order to compare the results with the previous solutions, tension problems of a fiber in a semi-infinite body are also considered. Then, generalized stress intensity factors at the corner of rectangular and cylindrical inclusions are systematically calculated for various geometrical conditions with varying the elastic ratio, length, and spacing of the location from edge to inner of the body. The effects of elastic modulus ratio and aspect ratio of inclusion upon the stress intensity factors are discussed for pullout problems.     

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     

Elastic Fields for the Ellipsoidal Cavity Problem

Lur’e (Three-dimensional Problem of the Theory of Elasticity. Interscience, New York, 1964, §6.9) presented an approach to solve the problem of an ellipsoidal cavity in a linear, elastic and isotropic medium loaded by uniform principal stresses at infinity. In this paper we show that the approach by Lur’e may have no solution. Derivation mistakes are first pointed out in his (6.9.22), (6.9.23), (6.9.30) and (6.9.31). With the correct expressions, we then prove the coefficient matrix in his (6.9.32) to be singular. Therefore constants A,A ₄,A ₅ may have no solution. The problem lies in the harmonic functions chosen by Lur’e for the Papkovich-Neuber solution. From the solutions obtained by the Eshelby equivalent inclusion method, the present paper derives new Papkovich-Neuber harmonic functions for the ellipsoidal cavity problem.     

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.     

Solving the Problem of Bending of Multiply Connected Plates with Elastic Inclusions

This paper describes a method for determining the strain state of a thin anisotropic plate with elastic arbitrarily arranged elliptical inclusions. Complex potentials are used to reduce the problem to determining functions of generalized complex variables, which, in turn, comes down to an overdetermined system of linear algebraic equations, solved by singular expansions. This paper presents the results of numerical calculations that helped establish the influence of rigidity of elastic inclusions, distances between inclusions, and their geometric characteristics on the bending moments occurring in the plate. It is found that the specific properties of distribution of moments near the apexes of linear elastic inclusions, characterized by moment intensity coefficients, occur only in the case of sufficiently rigid and elastic inclusions.     

Stress-strain state of an anisotropic plate with an elliptic hole and thin rigid inclusions

The stress-strain state of an anisotropic plate containing an elliptic hole and thin, absolutely rigid, curvilinear inclusions is studied. General integral representations of the solution of the problem are constructed that satisfy automatically the boundary conditions on the elliptic-hole contour and at infinity. The unknown density functions appearing in the potential representations of the solution are determined from the boundary conditions at the rigid inclusion contours. The problem is reduced to a system of singular integral equations which is solved by a numerical method. The effects of the material anisotropy, the degree of ellipticity of the elliptic hole, and the geometry of the rigid inclusions on the stress concentration in the plate are studied. The numerical results obtained are compared with existing analytical solutions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 173–180, July–August, 2007.     

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.     

Interaction of harmonic axisymmetric waves with a thin circular absolutely rigid separated inclusion

We study stress concentration near a circular rigid inclusion in an unbounded elastic body (matrix). In the matrix, there are wave motions symmetric with respect to the axis passing through the inclusion center and perpendicular to the inclusion. It is assumed that one of the inclusion sides is completely fixed to the matrix, while the other side is separated and the conditions of smooth contact are realized on that side. The solution method is based on the fact that the displacements caused by waves reflected from the inclusion are represented as a discontinuous solution of the Lamé equations. This permits reducing the original problem to a system of singular integral equations for functions related to the stress and displacement jumps on the inclusion. Its solution is constructed approximately by the collocation method with the use of special quadrature formulas for singular integrals. The approximate solution thus obtained permits numerically studying the stress state in the matrix near the inclusion. Technological defects or constructive elements in the form of thin rigid inclusions contained in machine parts and engineering structure members are stress concentration sources, which may result in structural failure. It is shown that the largest stress concentration is observed near separated inclusions. Static problems for elastic bodies with such inclusions have been studied rather comprehensively [1, 2]. The stress concentration near separated inclusions under dynamic actions on the bodies has been significantly less studied even in the case of harmonic vibrations. The results of these studies can be found in [3, 4], where bodies with a thin separated inclusion were considered, and in [5], where the problem about torsional vibrations of a body with a thin circular separated inclusion was studied. The aim of the present paper is to study stress concentration near such an inclusion in the case of interaction with harmonic waves under axial symmetry conditions.     

Lyapunov and Sacker–Sell Spectral Intervals

In this work, we show that for linear upper triangular systems of differential equations, we can use the diagonal entries to obtain the Sacker and Sell, or Exponential Dichotomy, and also –under some restrictions– the Lyapunov spectral intervals. Since any bounded and continuous coefficient matrix function can be smoothly transformed to an upper triangular matrix function, our results imply that these spectral intervals may be found from scalar homogeneous problems. In line with our previous work [Dieci and Van Vleck (2003), SIAM J. Numer. Anal. 40, 516–542], we emphasize the role of integral separation. Relationships between different spectra are shown, and examples are used to illustrate the results and define types of coefficient matrix functions that lead to continuous Sacker–Sell spectrum and/or continuous Lyapunov spectrum.      

A class of nonlinear boundary value problems for the second-order<Emphasis Type="Italic">E</Emphasis><Subscript>2</Subscript> class elliptic systems in general form

A class of nonlinear boundary value problems (BVP) for the second-order E₂ class elliptic systems in general form is discussed. By introducing a kind of transformation, this kind of BVP is reduced to a class of generalized nonlinear Riemann-Hilbert BVP. And then some singular integral operators are introduced to establish the equivalent nonlinear singular integral equations. The solvability is proved under some suitable hypotheses by means of the properties of singular integral operators and the function theoretic methods. Foundation items: the National Natural Science Foundation of China (19671056); Shanghai Municipal Natural Scientific Foundation (99ZA14030, 01ZA14023); Jiangxi Provincial Natural Scientific Foundation (981102, 0211014) Biographies: LI Ming-zhong (1935−); XU Ding-hua (1967−)