Problems on circular-arc cracks between dissimilar materials under antiplane concentrated forces

This paper deals with the problems of circular-arc crack between bonded dissimilar materials under antiplane concentrated forces. By applying the analytic continuation technique and the analyzing method of the Singularity in complex functions, we obtain universal solutions for the problems. For several typical cases the closed form solutions are obtained and the stress intensity factors are devived From these solutions, we provide a series of particular results, some of which coincide with those in Ref.(7).     

各向异性材料界面共线刚性线夹杂的反平面问题

研究两种各向异性材料焊接界面含共线刚性线夹杂的反平面问题,导出了一般问题的公式和几个典型问题的封闭形式解,求出了刚性线尖端的应力分布,从文中解答的特殊情况,直接导出各向性材料界面与均匀各向异性介质中相应问题的公式与结果,并与有关文献相一致。     

Problems on collinear cracks between bonded dissimilar materials under concentrated loads

Following Ref. [6], this paper deals with the problem on collinear cracks between bonded dissimilar materials under a concentrated force and moment at an arbitrary point. Several typical solutions of complex stress functions in closed form are formulated and the stress intensity factors are given. These solutions include a series of results of previous researchers, and redress some errors in the researches of problems containing semi-infinite cracks^[3],[4].     

两种各向异性材料界面共线裂纹的反平面问题

本文研究两种各向异性材料界面共线裂纹的反平面剪切问题。利用复变函数方法，提出了一般问题公式和某些实际重要问题的封闭形式解。考察了裂纹尖端附近的应力分布并给出了应力强度因子公式。从本文解签的特殊情形，可以直接导出两种各向同性材料界面裂纹，均匀各向异性材料共线裂纹以及均匀各向同性材料共线裂纹的相应问题公式，其中包括已有的经典结果。     

Problem on circular-arc cracks between bonded dissimilar materials under concentrated force and moment

The method is very efficient by applying extended Schwarz principle integrated with the analysis of the singularity of complex stress functions to solve some plane-elastic problems under concentrated loads, in Ref.[1], this method is used to deal with the elastic problems of homogeneous plane. In this paper, it is extended to the case of dissimilar materials with co-circular cracks under concentrated force and moment. For several typical cases the solutions of complex stress function in closed form are built up and the stress intensity factors are given. From these solutions, we provide a series of particular results, in which two of them coincide with those in Refs. [1] and [6].     

Antiplane Problems of Periodical Rigid Line Inclusions between Dissimilar Anisotropic Materials

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Interface crack problems for mode Ⅱ of double dissimilar orthotropic composite materials

The fracture problems near the interface crack tip for mode Ⅱ of double dissimilar orthotropic composite materials are studied. The mechanical models of interface crack for mode Ⅱ are given. By translating the governing equations into the generalized bi-harmonic equations,the stress functions containing two stress singularity exponents are derived with the help of a complex function method. Based on the boundary conditions,a system of non-homogeneous linear equations is found. Two real stress singularity exponents are determined be solving this system under appropriate conditions about himaterial engineering parameters. According to the uniqueness theorem of limit,both the formulae of stress intensity factors and theoretical solutions of stress field near the interface crack tip are derived. When the two orthotropic materials are the same,the stress singularity exponents,stress intensity factors and stresses for mode Ⅱ crack of the orthotropic single material are obtained.     

Interface crack problems for mode II of double dissimilar orthotropic composite materials

The fracture problems near the similar orthotropic composite materials are interface crack tip for mode Ⅱ of double disstudied. The mechanical models of interface crack for mode Ⅱ are given. By translating the governing equations into the generalized hi-harmonic equations, the stress functions containing two stress singularity exponents are derived with the help of a complex function method. Based on the boundary conditions, a system of non-homogeneous linear equations is found. Two real stress singularity exponents are determined be solving this system under appropriate conditions about bimaterial engineering parameters. According to the uniqueness theorem of limit, both the formulae of stress intensity factors and theoretical solutions of stress field near the interface crack tip are derived. When the two orthotropic materials are the same, the stress singularity exponents, stress intensity factors and stresses for mode II crack of the orthotropic single material are obtained.     

圆形界面裂纹反平面问题的基本奇异解

本文研究反平面集中力作用下,不同弹性材料的圆形界面上有多条裂纹的问题。运用复变函数的解析延拓技术与奇性主部分析方法,首次获得该问题的一般解答,求出了几种典型情况的封闭解;算出了应力强度因子,并由此导出一系列特殊结果,其中几个与已有文献完全吻合。     

多重应力奇异性及其强度系数的数值分析方法

以具有两个应力奇异性次数的平面问题为例,提出了一种利用普通的数值分析结果确定奇异点附近多重应力奇异性的各阶次数以及相应的应力强度系数的数值分析方法,计算实例表明,本方法可以精确地求得各阶应力奇异性的次数,并且可以很方便地应用外插法确定出对应的应力强度系数。     

Mode-III interface edge crack between two bonded quarter-planes of dissimilar piezoelectric materials

Summary  The problem of an interface edge crack between two bonded quarter-planes of dissimilar piezoelectric materials is considered under the conditions of anti-plane shear and in-plane electric loading. The crack surfaces are assumed to be impermeable to the electric field. An integral transform technique is employed to reduce the problem under consideration to dual integral equations. By solving the resulting dual integral equations, the intensity factors of the stress and the electric displacement and the energy release rate as well as the crack sliding displacement and the electric voltage across the crack surfaces are obtained in explicit form for the case of concentrated forces and free charges at the crack surfaces and at the boundary. The derived results can be taken as fundamental solutions which can be superposed to model more realistic problems. Received 10 November 2000; accepted for publication 28 March 2001     

Paradox solution on elastic wedge dissimilar materials

According to the Hellinger-Reissner variational principle and introducing proper transformation of variables , the problem on elastic wedge dissimilar materials can be led to Hamiltonian system, so the solution of the problem can be got by employing the separation of variables method and symplectic eigenfunction expansion under symplectic space, which consists of original variables and their dual variables . The eigenvalue - 1 is a special one of all symplectic eigenvalue for Hamiltonian system in polar coordinate . In general, the eigenvalue - 1 is a single eigenvalue, and the classical solution of an elastic wedge dissimilar materials subjected to a unit concentrated couple at the vertex is got directly by solving the eigenfunction vector for eigenvalue - 1. But the eigenvalue - 1 becomes a double eigenvalue when the vertex angles and modulus of the materials satisfy certain definite relationships and the classical solution for the stress distribution becomes infinite at this moment, that is, the para     

Antiplane crack at the interface of two bonded dissimilar graded piezoelectric materials

The problem of an antiplane crack situated in the interface of two bonded dissimilar graded piezoelectric half-spaces is considered under the permeable crack assumption. The mechanical and electrical properties of the half-spaces are considered for a class of functional forms for which the equilibrium equation has analytic solutions. By using an integral transform technique, the problem is reduced to dual integral equations which are transformed into a Fredholm integral equation by introducing an auxiliary function. The stress intensity factors are obtained in explicit form in terms of auxiliary functions. By solving the Fredholm integral equation numerically, the numerical results for stress intensity factors are obtained which have been displayed graphically to show the influence of the graded piezoelectric materials.     

The analysis of thermal residual stresses near the apex in bonded dissimilar materials

By employing the complex variable method and constructing the particular solution sequences in the form of complex functions, all the cases of the thermal residual stress field near the apex in dissimilar materials bonded with two arbitrary angles are researched theoretically, and the corresponding classical solutions are obtained. Moreover, the primary paradox, the secondary paradox and even the triple paradox are discovered in the classical solutions and also resolved here, thereby it is confirmed that thermal residual stresses near the apex in bonded dissimilar materials probably possess the singularities of lnr (when the primary paradox occurs) , ln²r (when the secondary paradox occurs) and even ln³r (when the triple paradox occurs) . In addition, the systematic method to solve multiple paradox problems is put forward. © 1999 Elsevier Science Ltd. All rights reserved.     

振荡应力奇异性及其强度系数的数值分析方法

本文以具有振荡应力奇异性的平面问题为例,提出了一种利用普通的数值分析结果（由有限地或边界元程序计算得到的应力分量或位移分量）,来确定奇异性次数以及相应的复应力强度系数的数值分析方法。为了验证该方法的有效性,应用平面应变情况下的边界元计算结果,对界面端模型进行了分析。计算结果表明,本方法可以精确地求得振动应力奇异性次数,并且与奇异性对应的复应力强度系数也可以很方便地应用外插法得到。     

Analysis of bi-material interface cracks with complex weighting functions and non-standard quadrature

A boundary element formulation is developed to determine the complex stress intensity factors associated with cracks on the interface between dissimilar materials. This represents an extension of the methodology developed previously by the authors for determination of free-edge generalized stress intensity factors on bi-material interfaces, which employs displacements and weighted tractions as primary variables. However, in the present work, the characteristic oscillating stress singularity is addressed through the introduction of complex weighting functions for both displacements and tractions, along with corresponding non-standard numerical quadrature formulas. As a result, this boundary-only approach provides extremely accurate mesh-insensitive solutions for a range of two-dimensional interface crack problems. A number of computational examples are considered to assess the performance of the method in comparison with analytical solutions and previous work on the subject. As a final application, the method is applied to study the scaling behavior of epoxy–metal butt joints.     

Uniformly moving screw dislocation interacting with interface cracks in anisotropic bimaterials

This paper attempts to investigate the problem for the interaction between a uniformly subsonic moving screw dislocation and interface cracks in two dissimilar anisotropic materials. Using Riemann–Schwarz’s symmetry principle integrated with the analysis singularity of complex functions, we present the general elastic solutions of this problem and the closed form solutions for interface containing one and two cracks. The expressions of stress intensity factors at the crack tips and image force acting on moving dislocation are derived explicitly. The results show that the stress intensity factors at the crack tips decrease with increasing velocity of dislocation, and larger dislocation velocity leads to the equilibrium position of dislocation leaving from crack tips. The presented solutions contain previously known results as the special cases.     

Stress intensity factor for a kinked interfacial crack in dissimilar anisotropic materials under antiplane deformation

The asymptotic problem of a kinked interfacial crack in dissimilar anisotropic materials under antiplane deformation is investigated. The linear transformation method for the problem of the anisotropic bimaterial with a straight interface is proposed. The stress intensity factor for the kinked interfacial crack in the anisotropic composite is obtained from the solution of the transformed problem of the kinked interfacial crack in the isotropic bimaterial based on the linear transformation method. The effects of the material parameters as well as the kink angle on the stress intensity factor are discussed from numerical results of the stress intensity factor. The finite element analysis is carried out to verify the stress intensity factor obtained by using the linear transformation. The influence of the material orientations on the stress intensity factor is investigated for the kinked crack in the bimaterial consisting of dissimilar inclined orthotropic materials.     

Solution of the plane stress problems of strain-hardening materials described by power-law using the complex pseudo-stress function

In the present paper,the compatibility equation for the plane stress problems of power-law materials is transformed into a biharmonic equation by introducing the so-calledcomplex pseudo-stress function,which makes it possible to solve the elastic-plastic planestress problems of strain hardening materials described by power-law using the complexvariable function method like that in the linear elasticity theory.By using this generalmethod,the close-formed analytical solutions for the stress,strain and displacementcomponents of the plane stress problems’of power-law materials is deduced in the paper,which can also be used to solve the elasto-plastic plane stress problems of strain-hardeningmaterials other than that described by power-law.As an example,the problem of a power-law material infinite plate containing a circular hole under uniaxial tension is solved byusing this method,the results of which are compared with those of a known asymptoticanalytical solution obtained by the perturbation method.     

与界面相交的裂纹尖端的应力奇异性分析

为了确定与结合材料的界面相交的裂纹尖端附近的应力奇异性次数，提出了一种基于最小势能原理的一维特殊有限元法，以奇异点为原点半径r0的扇形奇异区域，可以简化为一维线性领域，即一条以代表结合材料的两个自由表面为端点的线段。对该一维线性领域作网格划分，采用三节点一维等参数二次单元。数值计算结果与已有理论解的比较表明，该方法具有很高的精度和效率，最后，利用文中给出的方法，得到了各向异性结合材料中与界面以任意角相交的裂纹尖端的奇异性次数随裂纹的变化规律。