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.     

一种镍基双晶材料疲劳裂纹扩展效应

对晶界平行裂纹和晶界垂直裂纹的双晶体进行三点弯曲疲劳实验，研究了双晶材料的疲劳裂纹扩展规律，测定了双晶的疲劳扩展速率，揭示了晶界对晶粒疲劳裂纹扩展的屏蔽效应：当裂纹距晶界某一特定长度时，裂纹扩展速率最快；而裂纹顶端交于晶界时，裂纹扩展速率最侵．进一步的晶体滑移有限元数值分析揭示了这种屏蔽效应的机理：晶界附近不协调的塑性变形，导致了裂纹尖端应力场的重新分布．     

Contour integrals for mixed-mode crack analysis: effect of nonsingular terms

An integral formulation for computing the nonsingular stresses (NSS) in a cracked body under mixed-mode static and dynamic loads is presented. The reciprocity theorems are applied to find the integral formula. The auxiliary fields are selected to eliminate the singular terms in the asymptotic expansion of the stresses near the crack tip. For elastodynamic crack problems, the integral representation of the NSS is presented in both the time and Laplace transform domain. Required variables along the integration path and region enclosed by the integration contour are obtained from the boundary element analysis. Influence of the NSS on predicting the crack growth direction is investigated for cracks under mixed-mode load conditions.     

载流薄板中裂纹形成瞬间尖端附近的应力场

利用电磁场的热效应对带有裂纹的载流导体进行裂纹止裂,是为了达到延长其工作寿命,提高安全性、可靠性的一种行之有效的方法。本文在文献[1]的基础上,以导电弹性体的麦克斯威尔方程为出发点,借助于边界条件和初始条件,推得了载流无限大薄板在形成裂纹的瞬间,裂纹尖端附近电流密度、温度和应力的具体表达式。通过算例分析证实了：在给定参数的情况下,通入适当强度的电流时,在电流所产生的焦耳热源的作用下,裂尖区域处的温度将瞬时升高,同时伴有压应力的产生,从而可达到阻止裂纹扩展的目的。     

异弹模界面裂缝的动态实验研究

本文利用电测法测量异弹模界面裂缝缝端的应力,进而推算出动态应力强度因子,并与特解边界元的计算结果作了比较,结果表明,两者的一致性较好.     

Boundary-layer analysis of propagating mode II cracks in porous elastic media

The steady advance of semi-infinite cracks in porous elastic media is examined in the limiting cases of very slow and very rapid propagation. The low-speed limit is the elastic solution representing fully-drained behavior, with no boundary layers present. In the high-speed limit, two boundary layers are identified: one along the crack face and another centered at the crack tip. The zero-order outer solution is the elastic solution representing fully-undrained behavior, with both boundary layers acting as zones of transition to the effectively drained conditions at the crack itself. Closedform expressions are derived for the stresses and pore pressure within the crack-tip boundary layer. The true stress intensity factor is found to be reduced by an amount related to the disparity between the drained and undrained Poisson's ratios of the medium.     

On the stress problem of large elliptical cutouts and cracks in circular cylindrical shells

Numerical solutions are presented for stresses around an elliptical hole in a long, thin, circular cylindrical shell subjected to axial tension for both the symmetric orientations of the hole with respect to the shell. The method of analysis involves obtaining a series solution to the governing shell equations in terms of Mathieu functions by the method of separation of variables and satisfying the boundary conditions numerically term by term in a Fourier series formulation. Results are presented in the form of charts from which stress concentration factors can be directly read over a wide range of the two parameters, namely, axis ratio of the ellipse and a curvature parameter defining the hole size with respect to dimensions of the shell.An interesting feature of the investigation is the analysis of limiting cases of circumferential and axial cracks for axial tension and internal pressure loadings respectively. The method developed involves determining the solution completely in elliptic coordinates and then determining the singular stresses by carrying out a transformation to polar coordinates with crack tip as the origin through a Taylor series expansion. Membrane and bending stress intensity factors are computed and plotted over a sufficiently wide range of the curvature parameter extending from small to large sized cracks. As an outcome of the analysis, a “hybrid” technique has been developed by which singularity conditions at the crack tip can be handled effectively in dealing with boundary conditions in crack problems.     

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.     

孔边裂纹对SH波的散射及其动应力强度因子

采用Ｇｒｅｅｎ函数法研究任意有限长度的孔边裂纹对ＳＨ波的散射和裂纹尖端场动应力强度因子的求解．取含有半圆形缺口的弹性半空间水平表面上任意一点承受时间谐和的出平面线源荷载作用时位移函数的基本解作为Ｇｒｅｅｎ函数,采用裂纹“切割”方法并根据连接条件建立起问题的定解积分方程,得到动应力强度因子的封闭解答．最后给出了孔边裂纹动应力强度因子的算例和结果,并讨论了圆孔的存在对动应力强度因子的影响     

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.     

A boundary element application for mixed modeloading idealized sawtooth fracture surface

In this paper, a 2-D elastic-plastic BEM formulation predicting the reduced mode IIand the enhanced mode I stress intensity factors are presented. The dilatant boundary conditions (DBC) are assumed to be idealized uniform sawtooth crack surfaces and an effective Coulombsliding law. Three types of crack face boundary conditions, i.e. (1) BEM sawtooth model-elasticcenter crack tip; (2) BEM sawtooth model-plastic center crack tip; and (3) BEM sawtoothmodel-edge crack with asperity wear are presented. The model is developed to attempt todescribe experimentally observed non-monotonic, non-linear dependence of shear crack behavioron applied shear stress, superimposed tensile stress, and crack length. The asperity sliding isgoverned by Coulombs law of friction applied on the inclined asperity surface which hascoefficient of friction μ. The traction and displacement Greens functions which derive fromNaviers equations are obtained as well as the governing boundary integral equations for an infiniteelastic medium. Accuracy test is performed by comparison stress intensity factors of the BEMmodel with analytical solutions of the elastic center crack tip. The numerical results show thepotential application of the BEM model to investigate the effect of mixed mode loading problemswith various boundary conditions and physical interactions.     

On the crack face boundary conditions in electromechanical fracture and an experimental protocol for determining energy release rates

The fracture mechanics of electromechanical materials has been investigated for well over a decade, yet there still exists controversy over the appropriate crack face boundary conditions for non-conducting cracks. In this paper an experimental protocol for measuring the energy release rate in a non-linear reversible electromechanical body is proposed and summarized. The potential results from the proposed experimental approach are capable of shedding light on the true physical nature of the conditions prevailing at the crack surface and in the space within the crack. The experimental procedure is simulated numerically for a linear piezoelectric specimen in a four point bending configuration subjected to electrical loading perpendicular to the crack. The focus of these investigations is on a comparison between the commonly used exact crack face boundary condition and the recently proposed energetically consistent boundary conditions. To perform the numerical calculation with a wide range of electrical and mechanical loadings, two efficient finite element formulations are presented for the general analysis of crack problems with non-linear crack face boundary conditions. Methods for the numerical determination of the crack tip energy release rate and the simulation of the experimental method for obtaining the total energy release rate are developed. Numerical results for the crack tip and total energy release rate are given for both the exact and energetically consistent boundary conditions. It is shown that the crack tip energy release rate calculated under energetically consistent boundary conditions is equal to the total energy release rate generated from the simulated experimental method. When the exact boundary conditions are used, there is no such agreement.     

Boundary-layer corrections for stress and strain fields in randomly heterogeneous materials

Boundary-layer effects on the effective response of fibre-reinforced media are analysed. The distribution of the fibres is assumed random. A methodology is presented for obtaining non-local effective constitutive operators in the vicinity of a boundary. These relate ensemble averaged stress to ensemble averaged strain. Operators are also developed which re-construct the local fields from their ensemble averages. These require information on the local configuration of the medium. Complete information is likely not to be available, but averages of these operators conditional upon any given local information generate corresponding conditional averages of the fields. Explicit implementation is performed within the framework of an approximation of Hashin-Shtrikman type. Two types of geometry are considered in examples: a half-space and a crack in an infinite heterogeneous medium. These are representative, asymptotically, of the field in the vicinity of any smooth boundary, and in the vicinity of a crack tip, respectively. Results have been obtained for the case of anti-plane deformation, realized by the imposition of either Dirichlet or Neumann conditions on the boundary; those for the Neumann condition are presented and discussed explicitly. The stresses in both fibre and matrix adjacent to a crack tip are shown to differ substantially from the values that would be predicted by ordinary homogenization.     

A small-scale magnetic-yielding model for an infinite magnetostrictive plane with a crack-like flaw

The effect of an external magnetic field on the fracture toughness of magnetostrictive materials has been investigated by determining the local stress fields around the tip of a very slender elliptical flaw embedded in an infinite magnetostrictive plane subjected to magnetic loading, based on the assumption of linear magnetization. In this paper, the above-mentioned analytical approach is extended to develop a small-scale magnetic-yielding model. The magnetic saturation zone is constructed and the distributions of magnetic field and magnetization are obtained around the tip of a slender elliptical crack. Based on the complex potential theory, the stress field is obtained in the vicinity of the tip of the slender elliptical crack by implementing the continuity conditions of displacement and resultant force at the interface between the magnetic saturation and magnetoelastic zones. The stress fields near the tip of the slender elliptical crack are obtained for two kinds of soft ferromagnetic materials each with a small induction magnetostrictive modulus. The theoretical results obtained show that the stresses in the neighborhood of a crack-tip are finite even when the elliptical crack reduces to a sharp crack, and are much smaller than the yield stress or the nominal fracture stress of the material. This suggests that, generally, the magnetic field has no obvious effects on the apparent fracture toughness of soft ferromagnetic materials, which is in agreement with the existing experimental results published in the existing literature. In addition, the theoretical analysis illustrates that no crack is magnetically impermeable, and the corresponding boundary conditions are inappropriate for fracture analysis of soft ferromagnetic materials.     

On the Finite Motions Generated by a Mode I Propagating Crack

The motion field surrounding a rapidly propagating crack, loaded symmetrically about the plane of the crack, is investigated. The problem is formulated within the framework of finite elastodynamics for thin slabs composed of compressible hyperelastic material. Writing the motion equations, the initial and the internal boundary conditions, with respect to a coordinate system that translates with the moving crack tip, we perform an asymptotic local analysis for a traction-free straight crack that suddenly grows at constant velocity. Moreover, the asymptotic Piola–Kirchhoff and Cauchy stress fields are computed, and we discuss the order of singularity of the dynamic stresses. This revised version was published online in August 2006 with corrections to the Cover Date.     

Special rotational deformation as a toughening mechanism in nanocrystalline solids

A theoretical model is suggested which describes the effect of special rotational deformation on crack growth in deformed nanocrystalline ceramics and metals. Within the model, the special rotational deformation (driven by the external stress concentrated near the tip of a mode I crack) occurs in a nanograin through formation of immobile disclinations whose strengths gradually increase during the formation process conducted by grain boundary sliding and diffusion. The special rotational deformation releases, in part, local stresses near the crack tip, thus serving as a toughening mechanism in nanocrystalline materials. The effects of the special rotational deformation on the growth of pre-existent, comparatively large cracks in nanocrystalline metals and ceramics are estimated.     

Kinked crack analysis by a hybridized boundary element/boundary collocation method

In this research a two dimensional displacement discontinuity method (which is a kind of indirect boundary element method) using higher order elements (i.e. a source element with a cubic variation of displacement discontinuities having four sub-elements) is used to obtain the displacement discontinuities along each boundary element. In this paper, three kinds of the higher order boundary elements are used: the ordinary elements, the kink elements and the special crack tip elements.The boundary collocation technique is used for the calculation of the displacement discontinuities at the center of each sub-elements. Again a special boundary collocation technique is used to treat the kinked source elements occur in the crack analysis. Considering the two source elements (each having four sub-elements) joined at a corner (kink point). The collocation points in the cubic element model which are outside of the kink point are moved to the crack kink then the displacement discontinuities on the left and right sides of the kink are calculated. The displacement discontinuities of the kink point are obtained by averaging the corresponding values of its left and right sides. The special crack tip elements are also treated by the boundary displacement collocation technique considering the singularity variation of the displacements and stresses near the crack tip. Some simple example problems are solved numerically by the proposed method. The numerical results are compared with the corresponding results obtained by the previous methods cited in the literature. This comparison shows a very good agreement between the results and verify the accuracy and validity of the proposed method.     

A rigid triangular inclusion partially bonded in an elastic matrix

The two-dimensional problem of a rigid rounded-off angle triangular inclusion partially bonded in an infinite elastic plate is studied. The unbonded part of the inclusion boundary forms an interfacial crack. Based on the complex variable method for curvilinear boundaries, the problem is reduced to a non-homogeneous Hilbert problem and the stress and displacement fields in the plate are obtained in closed form. Special attention is paid in the investigation of the stress field in the vicinity of the crack tip. It is found that the stresses present an oscillatory singularity and the general equations for the local stresses are derived. The singular stress field is coupled with the maximum circumferential stress and the minimum strain energy density criteria to study the fracture characteristics of the composite plate. Results are given for the complex stress intensity factors, the local stresses, the crack extension angles and the critical applied loads for unstable crack growth from its more vulnerable tip or two types of interfacial cracks along the inclusion boundary.     

THE STRESS FIELD AT MODE I CRACK TIP ON STEEL PIPE UNDER AXIAL TENSION

采用弹性断裂力学Westergaard 的方法,分析在轴向拉伸载荷作用下钢管的Ⅰ型裂纹尖端处的应力场. 基于Ⅰ型裂纹钢管应力场的特征,设定尖端处的应力艾雷函数,给出其应力场模型边界条件和应力场的解析函数,并利用裂纹尖端处的切平面研究裂纹尖端局部应力场,建立了钢管裂纹尖端应力场模型. 通过钢管与平板Ⅰ型裂纹应力场模型的对比,结果表明二者明显不同,钢管裂纹尖端处应力峰值影响范围仅与裂纹长度、拉应力相关.     

Pre-kinking of a moving crack in a magnetoelectroelastic material under in-plane loading

A constant moving crack in a magnetoelectroelastic material under in-plane mechanical, electric and magnetic loading is studied for impermeable crack surface boundary conditions. Fourier transform is employed to reduce the mixed boundary value problem of the crack to dual integral equations, which are solved exactly. Steady-state asymptotic fields near the crack tip are obtained in closed form and the corresponding field intensity factors are expressed explicitly. The crack speed influences the singular field distribution around the crack tip and the effects of electric and magnetic loading on the crack tip fields are discussed. The crack kinking phenomena is investigated using the maximum hoop stress intensity factor criterion. The magnitude of the maximum hoop stress intensity factor tends to increase as the crack speed increases.