横观各向同性压电空间中圆裂纹与合成点源的三维相互作用

研究了横观各向同性压电空间中的圆裂纹与包括力偶极子、电偶极子、点力矩、力张和力旋这些常见合成点源的相互作用．用初等函数的形式给出了在任意位置和方向的合成点源作用下,裂纹尖端3种应力强度因子和电位移强度因子的三维解．其中圆裂纹包括币形裂纹和外圆裂纹．这一系列的力电合成点源不仅其自身在工程中普遍存在,而且在某些情形下还可以模拟诸如微裂纹、空洞、夹杂和位错等缺陷．     

The axial displacement of a disc inclusion embedded in a penny-shaped crack

The present paper examines the axisymmetric problem of the axial translation of a rigid circular disc inclusion of finite thickness which is wedged in smooth contact in a penny-shaped crack. Results for the axial stiffness of the embedded inclusion and the stress intensity factor at the boundary of the penny shaped crack are evaluated in exact closed form.

---

Zusammenfassung Die vorliegende Arbeit untersucht das axisymmetrische Problem der axialen Verschiebung einer starren eingebetteten Kreisscheibe mit bestimmter Dicke, die in einer reibungslosen pfennigförmigen Spalte eingekeilt ist. Ergebnisse für die axiale Steifigkeit der eingebetteten Scheibe und der Druckintensitätsfaktor am Rande der pfennigförmigen Spalte sind als eine genaue geschlossene Lösung angegeben.

     

压电复合材料圆形夹杂中螺型位错和界面裂纹的干涉分析

研究了无穷远纵向剪切和面内电场共同作用下,压电复合材料圆形夹杂中螺型位错与界面裂纹的电弹耦合干涉作用．运用Riemann-Schwarz 对称原理,并结合复变函数奇性主部分析方法,获得了该问题的一般解答．作为典型算例,求出了界面含一条裂纹时基体和夹杂区域复势函数和电弹性场的封闭形式解．应用广义Peach-Koehler公式,导出了位错力的解析表达式．分析了裂纹几何参数和材料的电弹性常数对位错力的影响规律．结果表明,界面裂纹对位错力和位错平衡位置有很强的扰动效应,当界面裂纹长度达到临界值时,可以改变位错力的方向．该结果可以作为格林函数研究圆形夹杂内裂纹和界面裂纹的干涉效应．其公式的退化结果与已有文献完全一致．     

Topological sensitivity analysis of a multi‐scale constitutive model considering a cracked microstructure

This paper deals with the sensitivity analysis of the macroscopic elasticity tensor to topological microstructural changes of the underlying material. In particular, the microstucture is topologicaly perturbed by the nucleation of a small circular inclusion. The derivation of the proposed sensitivity relies on the concept of topological derivative, applied within a variational multi‐scale constitutive framework where the macroscopic strain and stress at each point of the macroscopic continuum are defined as volume averages of their microscopic counterparts over a representative volume element (RVE) of material associated with that point. We consider that the RVE can contain a number of voids, inclusions and/or cracks. It is assumed that non‐penetration conditions are imposed at the crack faces, which do not allow the opposite crack faces to penetrate each other. The derived sensitivity leads to a symmetric fourth‐order tensor field over the unperturbed RVE domain, which measures how the macroscopic elasticity parameters estimated within the multi‐scale framework changes when a small circular inclusion is introduced at the micro‐scale level. Copyright © 2009 John Wiley & Sons, Ltd.     

含裂纹的圆柱体的弯曲

含裂纹的圆柱体弯曲的研究有十分重要的意义。文献[1]～[4]研究过含径向裂纹或裂纹系的情形,[5]研究过有同心圆弧裂纹的圆柱体的弯曲。本文继续[6]对内部出现在任意位置的直线裂纹的圆柱体在力与裂纹垂直时的弯曲问题,用弹性理论复变函数方法进行了讨论;得到了位移、应力和应力强度因子用级数表示的表达式;对Ah小的这种弯曲问题的应力强度因子给出了好的近似式,分析了它们随裂纹中心的变化规律。最后对裂纹的一个尖端在原点的径向裂纹圆柱体的扭转率和弯曲中心进行了计算,其结果与[1]几乎完全相同。     

Steady-state thermoelastic stresses in an infinite transversely isotropic medium containing an external circular crack

The temperature and the normal components of stress and displacement around an external circular crack in an infinite transversely isotropic body have been calculated in the present paper. The stress intensity factor has been found and a comparison of the results with those for the isotropic case has been presented graphically.     

非对称载荷作用的外部圆形裂纹问题

使用边界积分方程方法,研究了三维无限弹性体中受非对称载荷作用的外部圆形裂纹问题。通过使用Fourier级数和超几何函数,将问题的二维边界奇异积分方程简化为Abel型方程,获得了一般非对称载荷作用的外部圆形裂纹问题的应力强度因子精确解,比用Hankel变换法得到的结果更为一般。结果表明:边界积分方程法在解析分析方面还有很大的潜力。     

External circular crack under arbitrary shear loading

An exact closed form solution in terms of elementary functions has been obtained to the governing integral equation of an external circular crack in a transversely isotropic elastic body. The crack is subjected to arbitrary tangential loading applied antisymmetrically to its faces. The recently discovered method of continuity solutions was used here. The solution to the governing integral equation gives the direct relationship between the tangential displacements of the crack faces and the applied loading. Now a complete solution to the problem, with formulae for the field of all stresses and displacements, is possible.     

A contact problem for a smooth rigid disc inclusion in a penny-shaped crack

The present paper examines the problem of the complete indentation of the surface of a penny-shaped crack by a smooth rigid disc inclusion. The integral equation governing the problem is solved numerically to evaluate the axial stiffness of the rigid inclusion and the stress intensity factors at the tip of the penny-shaped crack.     

有一条裂纹的圆形焊接问题

本文讨论了在带圆孔的无限平面中焊接一个不同材料的带裂纹的近似圆板的问题.该问题化为求解解析函数边值问题然后又转化为求解沿裂纹的奇异积分方程.后者的数字解法也已给出.文末并对Ⅰ型、Ⅱ型情况得出了应力强度因子的公式以及数字结果.     

Topological gradient for fourth-order PDE and application to the detection of fine structures in 2D images

In this paper we describe a new approach for the detection of fine structures in an image. This approach is based on the computation of the topological gradient associated with a cost function defined from a regularization of the data (possibly noisy). We get this approximation by solving a fourth-order PDE. The study of the topological sensitivity is made in the cases of both a circular inclusion and a crack. We illustrate our approach by giving two experimental results.     

On elastic bodies with thin rigid inclusions and cracks

This paper is concerned with the analysis of equilibrium problems for two‐dimensional elastic bodies with thin rigid inclusions and cracks. Inequality‐type boundary conditions are imposed at the crack faces providing a mutual non‐penetration between the crack faces. A rigid inclusion may have a delamination, thus forming a crack with non‐penetration between the opposite faces. We analyze variational and differential problem formulations. Different geometrical situations are considered, in particular, a crack may be parallel to the inclusion as well as the crack may cross the inclusion, and also a deviation of the crack from the rigid inclusion is considered. We obtain a formula for the derivative of the energy functional with respect to the crack length for considering this derivative as a cost functional. An optimal control problem is analyzed to control the crack growth. Copyright © 2010 John Wiley & Sons, Ltd.     

任意形状孔口单边裂纹问题的边界配位解法

本文提出了一组复应力函数,采用边界配位方法对不同形状孔口(包括圆、椭圆、矩形及菱形孔口)的单边裂纹平板的应力强度因子进行了计算.计算结果表明,对长度和宽度远大于孔口和裂纹几何尺寸的试件,配位法与用其他方法所得的无限大板含圆或椭圆孔边裂纹问题的解符合得很好.同时,对其他孔口问题,特别是有限大板情形,本文给出了一系列计算结果.本文所提出的函数及计算过程可以应用于任意形状孔口单边裂纹平板的计算.     

An axisymmetric steady-state thermoelastic problem of an external circular crack in an isotropic thick plate

A steady state thermoelastic mixed boundary value problem for an isotropic thick plate is considered in this paper. The faces of an external circular crack situated in the mid-plane of the plate are opened up by the application of temperature while the bounding surface of the plate are maintained at a constant zero temperature. Solution valid for large values of the ratio of the plate thickness to the diameter of the crack has been obtained. Expressions for various quantities of physical interest are derived by finding iterative solutions of the equations and the results are shown graphically.     

矩形弹性夹杂与裂纹相互干扰的边界元分析

使用边界元法研究了无限弹性体中矩形弹性夹杂对曲折裂纹的影响,导出了新的复边界积分方程．通过引入与界面位移密度和面力有关的未知复函数H(t),并使用分部积分技巧,使得夹杂和基体界面处的面力连续性条件自动满足,而边界积分方程减少为2个,且只具有1/r阶奇异性．为了检验该边界元法的正确性和有效性,对典型问题进行了数值计算．所得结果表明:裂纹的应力强度因子随着夹杂弹性模量的增大而减小,软夹杂有利于裂纹的扩展,而刚性较大的夹杂对裂纹有抑制作用．     

State of stress in a plat circular ring with a crack : PMM vol. 39, n 6, 1975, pp. 1118–1128

The stress distribution in a circular isotropie ring with a crack on part of the concentric circle is investigated. A system of functional equations governing the coefficients of the complex Fourier series expansion of the stresses acting on the circle on which the crack is located is obtained. The solution of the mentioned system of equations is obtained by using a factorization method, which permitted reduction of the initial system of equations to two coupled infinite systems of algebraic equations. The possibility of using the method of truncation to solve these systems is proved. The singularity originating in the neighborhood of ends of the crack in the formulas governing the stresses is isolated. Stress intensity coefficients for the effect of a uniform load on the external contour are presented.     

On the solution of boundary value problems on an inhomogeneous plane with a crack and a barrier connected in series

We consider boundary value problems for an equation in divergence form on a plane divided into two inhomogeneous half-planes by a film inclusion in the form of a strongly permeable crack and a weakly permeable barrier connected in series; this models a contact of heterogeneous media under inhomogeneous external conditions. The desired potentials have prescribed singular points (sources, drains, etc.). The coefficients of the equation are nonconstant and may increase or decrease when moving away from the film inclusion along a family of parabolas. We obtain representations of solutions of the considered problems via harmonic functions with the corresponding singular points on the plane.     

三角形弹性夹杂对裂纹的影响

研究了三角形弹性夹杂和裂纹之间的相互影响问题。应用Chau和Wang导出的面力边值问题的边界积分方程为基本方程,用夹杂和基体交界面上的面力和位移的连续性条件为补充方程,从而得到了一组能够解决夹杂和裂纹相互影响问题的方程,最后的方程组用一种新的边界单元法求解。计算了各种不同的夹杂和基体的材料常数以及夹杂和基体之间不同距离情况下裂纹尖端的应力强度因子。文中结果对研究新型复合材料有一定的应用价值。     

Joint deformation of a circular inclusion and a matrix

An elastic infinite plane containing a circular inclusion with given jumps of tractions and displacements along the interface and nonzero conditions at infinity is considered. Explicit expressions are derived for the Goursat-Kolosov complex potentials of this problem. The solution constructed can be used to examine various circular interfacial defects, including interfacial cracks and rigid parts of the interface. The problem under consideration is fundamental for the superposition method, which solves many problems in which a circular region is an element of a polyphase elastic medium. In such cases, the well-posedness of the problem, which depends on the interrelation between the jumps of tractions and displacements, follows from the very superposition method. The application techniques of this method are demonstrated for singular problems on the action of a point force and an edge dislocation located inside an inclusion or in the matrix. Computational results for the tractions arising at the interface under the action of a point force concentrated in the inclusion are given.     

Differentiation of the energy functional in the equilibrium problem for a Timoshenko plate with a crack on the boundary of an elastic inclusion

Under consideration is the equilibrium of a composite plate containing a through vertical crack of variable length at the interface between thematrix and the elastic inclusion. The deformation of the matrix is described by the Timoshenko model, and the deformation of the elastic inclusion, by the Kirchhoff–Love model. Some formula is obtained for the derivative of the energy functional with respect to the crack length.