Stress intensity factors for an interface crack along an elliptical inclusion

The problem of a crack along the interface of an elliptical elastic inclusion embedded in an infinite plate subjected to uniform stresses at infinity is analyzed by the body force method. The crack tip stress intensity factors are calculated for various inclusion geometries and material combinations. Based on numerical results, the effect of the inclusion geometry on the stress intensity factors is investigated. It is found that for small interface cracks the stress intensity factors are mainly determined by the stresses, occurring at the crack center point before the crack initiation, and interface curvature radius alone.     

Weight Function Method for Stress Intensity Factor of Internal Surface Semi-elliptical Crack in Elliptical Holes

The hatches for inspecting are usually designed with elliptical holes in airplane structures, so computation of the stress intensity factor of three dimensional crack at elliptical holes is pivotal for damage tolerance analysis of these structures. In this paper, weight function is derived for a two dimensional through cracks at elliptical holes by applying a compounding method. Stress intensity factor formulas for an internal surface semi-elliptical crack in elliptical holes are obtained wing the three dimensional weight function method. Stress intensity factors for an internal surface semi-elliptical crack in elliptical holes under remote tension are computed. At the same time, research on how radius of curvature for elliptical holes affect stress intensity factors was conducted. Stress intensity factors decrease when radius of curvature increases. Some results and conclusions which are of practical value are given.     

Plane strain stress intensity factors for branched cracks

A method of calculating stress intensity factors for branched and bent cracks embedded in an infinite body has been developed. The branches are always assumed to be sharp cracks and are modelled by dislocation distributions. The original crack may be either sharp or of elliptical cross-section with finite root radius. Hence, the method which has a precision ±2%, is also applicable to the study of crack branches emanating from elliptical holes and, approximately, also from notches. The following detailed calculations have been made assuming mode I loading: branched sharp crack with branches of equal and different length, bent sharp crack, and one and two crack branches emanating from the crack with a finite root radius. Bending of a sharp crack under mixed mode loading has also been studied. The criteria of maximum tensile stress and maximum energy release rate used in the study of direction of crack propagation are discussed.     

The interacting stress field around a composite crack with a misfitting inclusion

The problem of a composite crack interacting with a circular misfitting inclusion in an infinite elastic medium is investigated. The close-formed solutions of the stress fields in the inclusion and the matrix are obtained by using the dislocation model of cracks and the point force method as well as the complex function technique. The stress intensity factors at two tips of the crack are calculated from the stress components.     

含界面裂纹圆形夹杂的十二次对称二维准晶热应力分析

针对点热源作用下,无限大十二次对称二维准晶基体和圆形弹性夹杂界面之间含多条裂纹的问题进行了研究。基于复变函数分区全纯理论、留数定理、广义 Liouville 定理、Riemann-Schwarz 解析延拓定理及复应力函数奇性主部分析方法,获得了集中热源作用于准晶基体内任意一点时,准晶基体和圆形弹性夹杂内外温度场、声子场热应力的一般复势解。由此获得了含一条界面裂纹和两条界面裂纹时温度场以及声子场热应力的封闭形式解答,将所得结果与已有结果进行了对比,验证了该方法的有效性。最后通过数值算例分析了夹杂半径、点热源强度及裂纹角度对热应力和裂纹尖端热应力强度因子的影响规律。结果表明：随着热源强度的增大,裂纹尖端的声子场热应力也逐渐增大;随着裂纹角度的增大,裂纹尖端的声子场热应力强度因子变大;随着半径的增大,热应力强度因子的变化趋势越来越明显,并且取得的峰值越高,即裂纹角度和夹杂半径的增加,促进了裂纹的扩展。这些结论为准晶材料的结构设计和使用提供了科学依据。     

Variations of stress intensity factor of a semi-elliptical surface crack subjected to mixed mode loading

Maximum stress intensity factors of a surface crack usually appear at the deepest point of the crack, or a certain point along crack front near the free surface depending on the aspect ratio of the crack. However, generally it has been difficult to obtain smooth distributions of stress intensity factors along the crack front accurately due to the effect of corner point singularity. It is known that the stress singularity at a corner point where the front of 3 D cracks intersect free surface is depend on Poisson's ratio and different from the one of ordinary crack. In this paper, a singular integral equation method is applied to calculate the stress intensity factor along crack front of a 3-D semi-elliptical surface crack in a semi-infinite body under mixed mode loading. The body force method is used to formulate the problem as a system of singular integral equations with singularities of the form r ⁻³ using the stress field induced by a force doublet in a semi-infinite body as fundamental solution. In the numerical calculation, unknown body force densities are approximated by using fundamental density functions and polynomials. The results show that the present method yields smooth variations of mixed modes stress intensity factors along the crack front accurately. Distributions of stress intensity factors are indicated in tables and figures with varying the elliptical shape and Poisson's ratio.     

Determination of approximate point load weight functions for embedded elliptical cracks

This paper presents the application of weight function method for the calculation of stress intensity factors in embedded elliptical cracks under complex two-dimensional loading conditions. A new general mathematical form of point load weight function is proposed based on the properties of weight functions and the available weight functions for two-dimensional cracks. The existence of this general weight function form has simplified the determination of point load weight functions significantly. For an embedded elliptical crack of any aspect ratio, the unknown parameters in the general form can be determined from one reference stress intensity factor solution. This method was used to derive the weight functions for embedded elliptical cracks in an infinite body and in a semi-infinite body. The derived weight functions are then validated against available stress intensity factor solutions for several linear and non-linear stress distributions. The derived weight functions are particularly useful for the fatigue crack growth analysis of planer embedded cracks subjected to fluctuating non-linear stress fields resulting from surface treatment (shot peening), stress concentration or welding (residual stress).     

Stress intensity factors of an inclined elliptical crack near a bimaterial interface

In this paper the stress intensity factors are discussed for an inclined elliptical crack near a bimaterial interface. The solution utilizes the body force method and requires Green’s functions for perfectly bonded semi-infinite bodies. The formulation leads to a system of hypersingular integral equation whose unknowns are three modes of crack opening displacements. In the numerical calculation, unknown body force densities are approximated by using fundamental density functions and polynomials. The results show that the present method yields smooth variations of stress intensity factors along the crack front accurately. Distributions of stress intensity factors are presented in tables and figures with varying the shape of crack, distance from the interface, and elastic modulus ratio. It is found that the inclined crack can be evaluated by the models of vertical and parallel cracks within the error of 24% even for the cracks very close to the interface.     

Analysis of elliptical cracks perpendicular to the interface of two joined transversely isotropic solids

This paper presents a boundary element analysis of elliptical cracks in two joined transversely isotropic solids. The boundary element method is developed by incorporating the fundamental singular solution for a concentrated point load in a transversely isotropic bi-material solid of infinite space into the conventional displacement boundary integral equations. The multi-region method is used to analyze the crack problems. The traction-singular elements are employed to capture the singularity around the crack front. The values of stress intensity factors (SIFs) are obtained by using crack opening displacements. The results of the proposed method compare well with the existing exact solutions for an elliptical crack parallel to the isotropic plane of a transversely isotropic solid of infinite extent. Elliptical cracks perpendicular to the interface of transversely isotropic bi-material solids of either infinite extent or occupying a cubic region are further examined in detail. The crack surfaces are subject to the uniform normal tractions. The stress intensity factor values of the elliptical cracks of the two types are analyzed and compared. Numerical results have shown that the stress intensity factors are strongly affected by the anisotropy and the combination of the two joined solids.     

Interacting dissimilar semi-elliptical surface flaws under tension and bending

Stress intensity factors for two dissimilar interacting semi-elliptical coplanar surface flaws (cracks) in a semi-infinite elastic body are obtained under overall tension and bending. First the basic equations for a general planar crack normal to the free surface are established, using the method of equivalent eigen- or transformation strains (the body force method). Then the results are specialized for application to elliptical cracks. Numerical values are obtained for various configurations and crack shapes. Results are compared with those of two-dimensional collinear cracks. Finally, an approximate procedure for estimating the stress intensity factors for a general three-dimensional crack is suggested.     

Weight functions and Stress Intensity Magnification factors for elliptical and semi-elliptical cracks under variable normal stress – Part II

Surface cracks under peak stresses are investigated. The calculational procedure is based on the general form of the weight function for an elliptical crack embedded in an infinite solid. Two points on the contour of the ellipse are investigated. A new correction procedure for transfer from the embedded crack to surface crack configurations is presented, which is valid for all a/t-values. Weight functions for both points have been found with the crack aspect ratio a/c as parameter. For the point at the end of the minor axis all weight functions for embedded cracks are describable by one equation only (using Heuman's lambda function). For various a/c-ratios of the surface crack under different stress distributions the stress intensity magnification factors are given.     

半空间中椭圆夹杂与裂纹对SH波的散射

采用Green函数、复变函数和多极坐标方法求解弹性半空间中椭圆形弹性夹杂与任意方位的裂纹对SH波的散射问题。利用“保角映射”技术求解椭圆夹杂对SH波的散射位移场,并构造适合本问题的Green函数,即含椭圆形弹性夹杂的弹性半空间内任意一点承受时间谐和的反平面荷载作用时的位移基本解,结合裂纹“切割”法构造裂纹,进而得到椭圆夹杂和裂纹同时存在条件下的位移场与应力场。最后,通过具体算例,讨论了不同参数对地表位移、弹性夹杂周边动应力集中系数和裂纹尖端动应力强度因子的影响规律。     

Weight functions and Stress Intensity Magnification factors for elliptical and semi-elliptical cracks under variable normal stress – part I

Surface cracks under peak stresses are investigated. The calculational procedure is based on the general form of the weight function for an elliptical crack embedded in an infinite solid. Two points on the contour of the ellipse are investigated. The superposition method is used for transfer from the embedded crack to surface crack configurations. Weight functions for both points have been found with the crack aspect ratio a/c as parameter. For the point at the end of the minor axis all weight functions are describable by one equation only (Heuman's lambda function). For various a/c ratios of the surface crack under different stress distributions the stress intensity magnification factors are given.     

Boundary element analysis of cracks at a fastener hole in an anisotropic sheet

Stress intensity factors for cracks emanating from a fastener hole are obtained, using the principle of superposition and a Green's function for a point force applied in an anisotropic sheet with an elliptical hole. Various loading cases, such as a point load, a uniform pressure applied on an arc, and a cosine distribution of pressure acting over half the circumference of the hole are considered to simulate the bolted-joint load. Both cases of a single crack and of two cracks of unequal lengths are studied. The accuracy and simplicity of the technique are demonstrated by comparing numerical results for various loading conditions and crack lengths with their known isotropic counterparts.     

Developments in the analysis of interacting cracks

This paper presents an overview of the finite element alternating technique for the analysis of interacting cracks. To illustrate the ease and accuracy of this method the technique is used to analyse several problems associated with both widespread fatigue and multi-site damage, a problem which is attracting worldwide attention. Whilst this paper presents an overview of the technique for both two- and three-dimensional problems attention is focused on three-dimensional problems. In particular, the interaction effects between two fully embedded elliptical flaws and between two semi-elliptical surface flaws, and the effects of crack proximity and crack aspect ratio on the stress intensity factors are presented. For semi-elliptical surface flaws these results indicate that as the cracks approach each other the position of the point on the crack front with the highest stress intensity factor shifts. This subsequently suggests that surface cracks will tend to grow preferentially towards each other. The same trend is evidenced for fully embedded cracks. However, in this case there is no shift in the position of the maximum stress intensity factor. A discussion of the results in terms of stress intensity magnification factors is also presented.     

Axisymmetric finite cylinder with one end clamped and the other under uniform tension containing a penny-shaped crack

This study considers the axisymmetric analysis of a finite cylinder containing a penny-shaped transverse crack. Material of the cylinder is assumed to be linearly elastic and isotropic. One end of the cylinder is bonded to a fixed support while the other end is subjected to uniform axial tension. Solution is obtained by superposing the solutions for an infinite cylinder loaded at infinity and an infinite cylinder containing four cracks and a rigid inclusion loaded along the cracks and the inclusion. When the radius of the inclusion approaches the radius of the cylinder, its mid-plane becomes fixed and when the radius of the distant cracks approach the radius of the cylinder, the ends become cut and subject to uniform tensile loads. General expressions for the perturbation problem are obtained by solving Navier equations with Fourier and Hankel transforms. Formulation of the problem is reduced to a system of five singular integral equations. By using Gauss-Lobatto and Gauss-Jacobi integration formulas, these five singular integral equations are converted to a system of linear algebraic equations which is solved numerically. Stress distributions along the rigid support, stress intensity factors at the edges of the rigid support and the crack are calculated.     

Analysis of stress intensity factors for three-dimensional cruciform cracks

The stress intensity factors for three-dimensional cruciform surface cracks in a semi-infinite body are numerically calculated by the body force method. Mindlin's point force solution is used for the derivation of basic equations to express the influence coefficient of triangular elements, into which the crack is divided. The interactions between crossed crack planes as well as contact between crack surfaces are considered in the iterative manner. Stress intensity factors for a cruciform median crack and a cruciform semicircular crack under a point force on the surface of a semi-infinite solid are analyzed. The possibility of growth of a median crack toward the free surface of the semi-infinite solid is discussed. A cruciform semicircular surface crack under remote uniaxial tension, or under combined tension and compression is also analyzed. The effect of contact of crack surfaces on stress intensity factors is discussed.     

Stress intensity factors at tips of branched cracks under various loadings

Several papers have been published on branched cracks by using various analytical methods, but most of them are concerned with special crack geometries or special loading conditions, and often give unreliable values for cracks with short branches or with small branching angles. The purpose of this paper is to give reliable formulae and new results of the stress intensity factors of various branched cracks in a wide plate. The analysis is based on the body force method combined with a perturbation procedure, and the stress intensity factors at the tips of all the branches and the main crack are given by power series formulae. Numerical results for typical branched cracks are discussed.     

Crack propagation from a pre-existing flaw at a notch root. I. Introduction and general form of the stress intensity factors at the initial crack tip

This paper and its companion are devoted to the study of crack kinking from some small pre-existing crack originating from a notch root (the notch root radius being zero). Both the notch boundaries and the initial crack are allowed to be curved; also, the geometry of the body and the loading are totally arbitrary. The ingredients required are knowledge of the stress intensity factors at the initial crack tip and use of a suitable mixed mode propagation criterion. This paper is devoted to the first point, and more specifically to establishing the general (that is, not yet fully explicit) form of the formulae giving these stress intensity factors. The method used is based on changes of scale (homogeneity properties of the equations of elasticity) on the one hand, and on continuity of the displacement and stresses at a given, fixed point with respect to the crack length on the other hand. The formulae derived for the stress intensity factors at the tip of the small crack are of universal value: they apply to any situation, whatever the geometry of the body, the notch and the crack and whatever the loading, the stress intensity factors depending always only upon the `stress intensity factor of the notch' (the multiplicative coefficient of the singular stress field near the notch root in the absence of the crack), the length of the crack, the aperture angle of the notch and the angle between its bisecting line and the direction of the crack.     

Approximate weight function method for elliptical arc through cracks in mechanical joints

Mechanical joints such as bolted, riveted or pinned joints are widely used to join the constituent parts of structural components. Reliable stress intensity factor analysis of arbitrary cracks in mechanical joints is required for the safety evaluation or fracture mechanics design. It has been reported that cracks in mechanical joints usually nucleate as the corner crack and grow as the elliptical arc through crack. The weight function method is a useful technique to calculate the stress intensity factor using the appropriate weight function for a cracked body and the stress field of an uncracked body. In this paper, the weight function method for the two surface points of elliptical arc through cracks in mechanical joints is developed to analyze the mixed-mode stress intensity factors. Unknown coefficients included in the weight function are determined using the reference stress intensity factors obtained from finite element analysis.