In-plane perturbation of the tunnel-crack under shear loading I: bifurcation and stability of the straight configuration of the front

One considers a planar tunnel-crack embedded in an infinite isotropic brittle solid and loaded in mode 2+3 through some uniform shear remote loading. The crack front is slightly perturbed within the crack plane, from its rectilinear configuration. Part I of this work investigates the two following questions: Is there a wavy “bifurcated” configuration of the front for which the energy release rate is uniform along it? Will any given perturbation decay or grow during propagation? To address these problems, the distribution of the stress intensity factors (SIF) and the energy release rate along the perturbed front is derived using Bueckner–Rice's weight function theory. A “critical” sinusoidal bifurcated configuration of the front is found; both its wavelength and the “phase difference” between the fore and rear parts of the front depend upon the ratio of the initial (prior to perturbation of the front) mode 2 and 3 SIF. Also, it is shown that the straight configuration of the front is stable versus perturbations with wavelength smaller than the critical one but unstable versus perturbations with wavelength larger than it. This conclusion is similar to those derived by Gao and Rice and the authors for analogous problems.     

In-plane perturbation of a system of two coplanar slit-cracks – I: Case of arbitrarily spaced crack fronts

In order to lay the grounds for a future study of the deformation of the fronts of coplanar cracks during their final coalescence, we consider the model problem of a system of two coplanar, parallel, identical slit-cracks loaded in mode I in some infinite body. The first, necessary task is to determine the distribution of the stress intensity factors along the crack fronts resulting from some small but otherwise arbitrary in-plane perturbation of these fronts. This is done here in the case where the distances between the various crack fronts are arbitrary and fixed.The first order expression of the local variation of the stress intensity factor is provided by a general formula of Rice (1989) in terms of some “fundamental kernel” tied to the mode I crack face weight function. In the specific case considered, this fundamental kernel reduces to six unknown functions; the problem is to determine them. This is done by using another formula of Rice (1989) which provides the variation of the fundamental kernel in a similar way. This second formula is applied to special perturbations of the crack fronts preserving the shape and relative dimensions of the cracks while modifying their absolute size and orientation. The output of this procedure consists of nonlinear integro-differential equations on the functions looked for, which are transformed into nonlinear ordinary differential equations through Fourier transform in the direction of the crack fronts, and then solved numerically.     

Evaluation of the Lazarus-Leblond constants in the asymptotic model of the interfacial wavy crack

The paper addresses the problem of a semi-infinite plane crack along the interface between two isotropic half-spaces. Two methods of solution have been considered in the past: Lazarus and Leblond [1998a. Three-dimensional crack-face weight functions for the semi-infinite interface crack-I: variation of the stress intensity factors due to some small perturbation of the crack front. J. Mech. Phys. Solids 46, 489-511, 1998b. Three-dimensional crack-face weight functions for the semi-infinite interface crack-II: integrodifferential equations on the weight functions and resolution J. Mech. Phys. Solids 46, 513-536] applied the “special” method by Bueckner [1987. Weight functions and fundamental fields for the penny-shaped and the half-plane crack in three space. Int. J. Solids Struct. 23, 57-93] and found the expression of the variation of the stress intensity factors for a wavy crack without solving the complete elasticity problem; their solution is expressed in terms of the physical variables, and it involves five constants whose analytical representation was unknown; on the other hand, the “general” solution to the problem has been recently addressed by Bercial-Velez et al. [2005. High-order asymptotics and perturbation problems for 3D interfacial cracks. J. Mech. Phys. Solids 53, 1128-1162], using a Wiener-Hopf analysis and singular asymptotics near the crack front.The main goal of the present paper is to complete the solution to the problem by providing the connection between the two methods. This is done by constructing an integral representation for Lazarus-Leblond's weight functions and by deriving the closed form representations of Lazarus-Leblond's constants.     

In-plane perturbation of a system of two coplanar slit-cracks – II: Case of close inner crack fronts or distant outer ones

The study of the in-plane perturbation of a system of two coplanar slit-cracks carried out in Part I is specialized to the case where the distance between the inner crack fronts is small, or equivalently that between the outer fronts large. The limit process involved is complex because of appearance of a “boundary layer” in the limiting case considered; this boundary layer occurs near the origin in the Fourier space used to determine the unknown components of the fundamental kernel looked for. A technique of matched asymptotic expansions is used to tackle this difficulty.The problem is thus reduced to determining two unknown functions only, which characterize the “interactions” between the two inner fronts. These functions obey a system of nonlinear differential equations in Fourier’s space, which are solved analytically near the origin and numerically in general. The results evidence a very slow decrease of long-range interactions between distinct points on the same front or distinct ones. This represents a striking difference with respect to the cases considered earlier of a single semi-infinite crack and a single slit-crack.     

Crack paths in three-dimensional elastic solids. i: two-term expansion of the stress intensity factors—application to crack path stability in hydraulic fracturing

The aim of this work is to lay theoretical foundations for the prediction of crack paths within the theory of quasistatic LEFM under the most general hypotheses: arbitrary three-dimensional geometry, arbitrary loading. This objective requires to derive the expression of the stress intensity factors along the crack front after an arbitrary infinitesimal propagation. Only the first two terms of their expansion in powers of the crack extension length δ, proportional to δ⁰ = 1 and δfn1fn2, are considered in this paper. Fully general formulae for these terms are obtained by combining arguments of dimensional analysis (scale changes) and regularity properties (continuity, differentiability) of the stresses at a fixed, given point with respect to δ for δ = 0 derived from the Bueckner–Rice weight function theory. This notably allows to extend the Cotterell–Rice criterion for stable rectilinear propagation of a mode I crack under plane strain conditions to the three-dimensional case. As an application, a penny-shaped crack induced by hydraulic fracturing is considered. Conclusions concerning the influence of the orientation and depth of such a crack upon the stability of its coplanar propagation seem to be compatible with experimental evidence.     

Dielectric crack problem for a magnetoelectroelastic strip with functionally graded properties

The paper presents a fracture analysis for an electromagnetically dielectric crack in a functionally graded magnetoelectroelastic strip. It is considered that the material properties are varying exponentially along the width direction. Under the assumption of the in-plane magneto-electro-mechanical loadings, the dielectric crack is simulated by using the semi-permeable crack-face boundary conditions. The Fourier transform technique is applied to solve the boundary-value problem and four coupling singular integral equations are determined. A nonlinear system of algebraic equations is further derived and solved numerically to determine the electromagnetic field inside the crack. Then the field intensity factors of stress, electric displacement, and magnetic induction are given. Through the numerical computations, the effects of the material non-homogeneity and the permeability of crack interior on the electric displacement and the magnetic induction at the crack faces are studied. The variations of the intensity factors of stress, electric displacement, and magnetic induction versus the geometry of the crack, the strip width, and the material non-homogeneity are presented in graphics respectively.     

Stress intensity factors for a semi-infinite plane crack under a pair of point forces on the faces

Static three-dimensional stress intensity factors of a semi-infinite plane crack are investigated in this paper. The deformations are caused by a pair of normal and tangential point forces acting on the crack faces but located away from the crack front. Cases of symmetric and anti-symmetric loadings with respect to the crack plane are both considered. Analytic solutions are obtained by the application of Fourier transforms together with the Wiener-Hopf technique. The formulation departs significantly from the Papkovich-Neuber formulation used in previous works. This alternative formulation reduces the complexity of the calculations involved and has the same potential in regard to the elastodynamic problem. Several misprints in previous works are also noted.     

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     

Transient response of a magnetoelectroelastic solid with two collinear dielectric cracks under impacts

Dynamic analysis of two collinear electro-magnetically dielectric cracks in a piezoelectromagnetic material is made under in-plane magneto-electro-mechanical impacts. Generalized semi-permeable crack-face boundary conditions are proposed to simulate realistic opening cracks with dielectric. Ideal boundary conditions of a combination of electrically permeable or impermeable and magnetically permeable or impermeable assumptions are several limiting cases of the semi-permeable dielectric crack. Utilizing the Laplace and Fourier transforms, the mixed initial-boundary-value problem is reduced to solving singular integral equations with Cauchy kernel. Dynamic intensity factors of stress, electric displacement, magnetic induction and crack opening displacement (COD) near the inner and outer crack tips are determined in the Laplace transform domain. Numerical results for a special magnetoelectroelastic solid are calculated to show the influences of the dielectric permittivity and magnetic permeability inside the cracks on the crack-face electric displacement and magnetic induction. By means of a numerical inversion of the Laplace transform, the variations of the normalized intensity factors of stress and COD are discussed against applied magnetoelectric impact loadings and the geometry of the cracks for fully impermeable, vacuum, fully permeable cracks and shown in graphics.     

Basic results for elastic fracture mechanics with frictionless contact between the crack lips

Some previous works have been devoted to problems of elastic fracture mechanics with frictionless unilateral contact between the crack lips, and especially to the asymptotic expression of the displacement and stress fields near the tip of a closed, ordinary or interface crack. However, not all energetic aspects of a general theory of such problems have been explored. Such features are developed here. The topics investigated include a thermodynamic analysis of a growing closed crack, leading in a natural way to some Griffith-like criterion, stiffness and compliance formulae for the energy release rate, with an appealing example, and another nice example of application of Rice's integral and Irwin's formula. It appears that in spite of the inherent non-linearity of the problems envisaged, most classical results of LEFM can be transposed to them, with some relatively minor adaptations. This is essentially due to the hypothesis of absence of friction.     

Two coplanar Griffith cracks in an infinitely long elastic strip

We consider the problem of determining the stress intensity factors and the crack energy in an infinitely long isotropic, homogeneous elastic strip containing two coplanar Griffith cracks. We assume that the cracks are opened by an internal pressure and the edges of the strip are rigidly fixed. By the use of Fourier transforms we reduce the problem to solving a set of triple integral equations with cosine kernel and a weight function. The triple integral equations are solved exactly. Exact analytical expressions are derived for the stress intensity factors, shape of the deformed crack and the crack energy. Solutions to some particular problems are derived as limiting cases.This work was supported by the National Research Council of Canada through NRC-Grant No. A4177.     

应用通用权函数法计算热机载荷作用下三维界面裂纹的应力强度因子

热机载荷共同作用下双材料、复合材料中的裂纹扩展往往发生在界面处,并且工程中实际遇到的裂纹大多数是三维裂纹,传统的求解热冲击及机械载荷共同作用下界面裂纹应力强度因子的数值方法如有限元、边界元法计算量大,计算效率低。由于通用权函数仅仅与裂纹体的几何形状有关,与载荷、时间无关,求解应力强度因子时避免了反复的应力分析,计算效率大大提高, 通用权函数法非常适合计算复杂冲击载荷下应力强度因子分布的过渡过程。根据Betti互易原理,本文推导出了三维界面裂纹问题通用权函数法的普遍表达式,并给出了热机载荷共同作用下三维界面I型、Ⅱ型和Ⅲ型裂纹问题通用权函数法的有限元格式. 通过实例计算比较,表明此方法得到的结果可以达到满意的工程应用精度。     

First order Oore–Burns integral for nearly circular cracks under uniform tensile loading

In this paper, by means of the Oore–Burns weight function, we have obtained a general explicit equation for the Stress Intensity Factors (SIFs) of a nearly circular internal crack subjected to remotely uniform tensile stress. We have expanded the crack border in Fourier series and have derived an analytic solution for the SIF at any point on the front crack, in terms of the homotopic transformation of a disk. More precisely, we have given the first order approximation of the SIF in closed form. From a theoretical point of view, the proposed equation can only be used for a small deviation from the circle, however some tests on elliptical cracks have shown that the equation also works well for slender ellipses (up to a ratio of 0.4 between the two semi-axes). Finally, as an example, we have given a suitable explicit formula for SIF derived from general equations for triangular cracks and square-like cracks.     

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。     

压电材料中的Bueckner功共轭积分及和J积分M积分的关系

研究横观各向同性压电材料中裂纹问题,提出了Bueckner功共轭积分在这类材料中的表达式：并通过引出两类辅助的应力-位移-电位移-电势场,证明功共轭积分和这类材料中的J积分和M积分仍然存在简单的两倍关系由此,各类在脆性材料断裂问题中已广泛应用的权函数方法可顺理成章地推广到压电材料的研究中来．这对独立地确定电位移强度因子和经典的I、II型应力强度因子提供了有力的数学上的工具．进而通过计算机械应变能释放率对压电材料中裂纹的稳定做出判断．     

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.     

A new approach to the pseudo-orthogonal properties of eigenfunction expansion form of the crack-tip complex potential function in anisotropic and piezoelectric fracture mechanics

Over the past twenty years, the well-known weight function theory based on the Bueckner work conjugate integral has been widely used to calculate crack tip fracture dominant parameter such as the stress intensity factor, the energy release rate (or the J-integral) and the T-stress in various kinds of cracked materials (e.g. isotropic materials, anisotropic materials and piezoelectric materials). Meanwhile, the pseudo-orthogonal property of the eigenfunction expansion form of the crack tip stress complex potential function has been proved to play a very important role in the theory. In this paper, we provide a new approach to establish the pseudo-orthogonal properties for crack problems in anisotropic and/or piezoelectric materials. In the latter case associated with mechanical-electric coupling, the electrical boundary conditions under both impermeable and permeable crack models are considered. The approach developed is much simpler than the classical complex variable separation technique proposed by previous researchers and hence the cumbersome and lengthy manipulations are avoided. Moreover, it is shown that, unlike previous works, the orthogonal properties of the material characteristic matrices A and B induced by the Stroh theory are no longer necessary in establishing the pseudo-orthogonal properties of eigenfunction expansion form in cracked piezoelectric materials. The approach can be easily extended to treat many other different crack problems concerning the Bueckner integral involving several complex arguments.     

求解三维裂纹前缘SIF分布时间历程的通用权函数法和有限变分法

将三维热权函数法扩展为适用于表面力、体积力和温度载荷的通用权函数法(UWF).推导出以变分型积分方程表达的UWF法基本方程,从变分的角度,将求解三维热权函数法基本方程的多虚拟裂纹扩展法(MVCE)改造为可以适用于一般的变分型积分方程的一类新型数值方法--有限变分法(FVM).在FVM中可以引入无穷多种线性无关的局部变分模式,可以根据计算要求在求解域中插入任意多个计算节点,单一型裂纹问题FVM所得到的最终方程组的系数矩阵总是一个对称的窄带矩阵,而且对角元总是大数,具有良好的数值计算性能.FVM对于SIF沿裂纹前缘急剧变化的复杂情况具有较好的数值模拟能力和较高的计算精度,利用自身一致性,可以求得三维裂纹前缘SIF的高精度解.     

Geometrical disorder of the fronts of a tunnel-crack propagating in shear in some heterogeneous medium

A statistical analysis of the deformation of the fronts of a tensile tunnel-crack propagating in fatigue in some medium with spatially varying Paris constant was recently performed, with special emphasis on the evolution of the power spectra and correlation functions of the fluctuations of the fronts around reference straight lines. This study is extended here to coplanar propagation (along a weak plane) of a tunnel-crack loaded in mode 2+3. The results are rather similar to those previously obtained for mode 1. In particular, just like for tensile loadings, there is an effect of gradual selection in time of Fourier components of the fluctuations of the fronts of large wavelength. One novelty, however, is that for shear loadings, the fronts no longer tend to become symmetrical in time, so that correlations between crack front fluctuations at two points are higher for points located on the same front than for points located on distinct ones.     

The Problem of an External Circular Crack Under Asymmetric Loadings

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