STRESS INTENSITY FACTORS AND WEIGHT FUNCTIONS FOR SEMI-ELLIPTICAL SURFACE CRACKS IN FINITE-THICKNESS PLATES UNDER TWO-DIMENSIONAL STRESS DISTRIBUTION

Abstract— A Fourier series approach is proposed to calculate stress intensity factors using weight functions for semi-elliptical surface cracks in flat plates subjected to two-dimensional stress distributions. The weight functions were derived from reference stress intensity factors obtained by three-dimensional finite element analyses. The close form weight functions derived are suitable for the calculation of stress intensity factors for semi-elliptical surface cracks in flat plates under two-dimensional stress distributions with the crack aspect ratio in the range of 0.1 ≤ a/c ≤ 1 and relative depth in the range of 0 ≤ a/t ≤ 0.8. Solutions were verified using several two-dimensional non-linear stress distributions; the maximum difference being 6%.     

Stress intensity factors and weight functions for deep semi-elliptical surface cracks in finite-thickness plates

ABSTRACT Three-dimensional finite element analyses have been conducted to calculate the stress intensity factors for deep semi-elliptical cracks in flat plates. The stress intensity factors are presented for the deepest and surface points on semi-elliptic cracks with a/t -values of 0.9 and 0.95 and aspect ratios ( a/c ) from 0.05 to 2. Uniform, linear, parabolic or cubic stress distributions were applied to the crack face. The results for uniform and linear stress distributions were combined with corresponding results for surface cracks with a/t = 0.6 and 0.8 to derive weight functions over the range 0.05 ≤  a/c  ≤ 2.0 and 0.6 ≤  a/t  ≤ 0.95. The weight functions were then verified against finite element data for parabolic or cubic stress distributions. Excellent agreements are achieved for both the deepest and surface points. The present results complement stress intensity factors and weight functions for surface cracks in finite thickness plate developed previously.     

Calculation of stress intensity factors for a longitudinal semi-elliptical crack in a finite-length thick-walled cylinder

The purpose of this paper is to present the effect of finite boundary on the stress intensity factor of an internal semi-elliptical crack in a pressurized finite-length thick-walled cylinder  ( R _i/ t = 4)  . The three-dimensional finite element method, in conjunction with the weight function method, is used for computing the stress intensity factor at the deepest and surface points of an axial semi-elliptical crack in a cylinder. The transition aspect ratios, the aspect ratios in which the maximum stress intensity factor translates from the deepest to the surface points of the crack, are calculated for different relative depths and cylinder lengths. The results show that the stress intensity factor increases as the cylinder length decreases, especially at the corner point of the crack compared with the deepest point. The major advantage of this paper is that a closed-form expression is extracted for the stress intensity factor at the surface point of a semi-elliptical crack, which experiences higher changes due to the effect of the finite boundary of the cylinder.     

Stress intensity factor for semi-elliptical surface cracks loaded by stress gradients

The stress intensity factor at the deepest point of a semi-elliptical surface crack is calculated for stress gradients in direction of depth. The method is based on weight functions. The crack opening displacement for the reference problem is calculated with a method proposed by Petroski and Achenbach. The results are compared to finite element solutions given in the literature. As an example, the stress intensity factor is calculated for a crack in a thermally shocked pipe.     

Numerical and experimental evaluation of SIF for threaded connectors

This paper presents a methodology for fatigue crack growth analysis in tubular threaded connectors. A solution for stress intensity factor for semi-elliptical surface cracks emanating from a thread root in a screw connector is also discussed in the paper. The solution is based on a mixed approach incorporating weight function and finite element methods. The weight functions used are the universal functions for cracks in mode I and these are linked with a thread through-thickness stress distribution obtained from finite element analysis to produce a stress intensity factor for a crack at the critical tooth of a thread. The resulting crack growth data are then validated experimentally.     

Transient thermal stress intensity factors for an internal longitudinal semi-elliptical crack in a thick-walled cylinder

In this paper, the stress intensity factors are derived for an internal semi-elliptical crack in a thick-walled cylinder subjected to transient thermal stresses. First, the problem of transient thermal stresses in a thick-walled cylinder is solved analytically. Thermal and mechanical boundary conditions are assumed to act on the inner and outer surfaces of the cylinder. The quasi-static solution of the thermoelasticity problem is derived analytically using the finite Hankel transform and then, the stress intensity factors are extracted for the deepest point and the surface points of the semi-elliptical crack using the weight function method. The results show to be in accordance with those cited in the literature in the special case of steady-state problem. Using the closed-form relations extracted for the transient thermal stress intensity factors, some conclusive results are drawn.     

An Equation for Stress Concentration Factor in Countersunk Holes

A detailed three-dimensional finite element stress analysis was conducted on straight-shank and countersunk rivet holes in a plate subjected to tension loading. The study included a wide range of plate width to radius, thickness to radius, countersunk depth to thickness ratios and countersunk angles(θ_c). The stress concentration is maximum at or near the countersunk edge. The stress concentration depends on countersunk depth, plate thickness and width and it is nearly independent of the countersunk angle for 80° ≤ θ_c ≤ 120°. Using the finite element results and limiting conditions, an equation for stress concentration factor is developed and verified.     

Stress-intensity factors for semi-elliptical surface cracks in welded joints

A method for calculating stress intensity factors for edge and surface cracks in weldments has been presented. The weight function method was applied and appropriate weight functions have been derived using the Petroski-Achenbach crack opening displacement expression. The derived weight functions account for both the global weldment geometry and the weld profile characterised by the weld angle.Finally analysis of several parameters such as the type of loading, crack aspect ratio, weld angle and weld toe radius have been carried out assessing their effect on the stress intensity factor.The calculated stress intensity factors were verified against available finite element data.Very close agreement was achieved between the finite element data and the weight function based calculations.

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Résumé On présente une méthode de calcul des facteurs d'intensité de contraintes dans le cas de fissures de bord et de surface dans des soudures. On applique la méthode des fonctions pondérales et, en utilisant l'expression du COD proposée par Petroski et Achenbach, on tire les fonctions pondérales appropriées, qui tiennent compte de la géométrie générale de la soudure et du profil du joint, caractérisé par l'angle à la racine. Enfin, on a procédé à l'analyse de divers paramètres tels que le type de sollicitation, l'aspect de la fissure, l'angle et le rayon du congé à la racine, en vue de déterminer leur effet sur le facteur d'intensité des contraintes.On vérifie les facteurs d'intensité de contraintes par rapport aux données par éléments finis qui sont disponibles. On obtient un accord très proche entre les données numériques venant des éléments finis et les calculs basés sur les fonctions pondérales.

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Notation a crack length for an edge crack or depth for a semi-elliptical surface crack - c half crack length for semi-elliptical surface crack - E modulus of elasticity - F, F(a/t) geometric stress intensity correction factor - F, F,(a/t) geometric correction factor for the reference stress intensity factor - F ^x _sf geometric stress intensity factor for an edge crack emanating from an angular corner in a semi-finite plate with a step - F ⁹⁰ _sf geometric stress intensity correction factor for a crack emanating from the right angle corner ( = 90°) in a semi-finite plate with a step - G(a/t) parameter of the crack opening displacement function - H generalised modulus of elasticityH = E — for plane stressH = E/(1 —v ²) — for plane strain - h weld leg length (or the step thickness) - I ₁ (a), I ₂ (a), I ₃ (a) parameters of the crack opening displacement function - K stress intensity factor - K _r reference stress intensity factor corresponding to the local reference stress _r(x) and nominal stressS _r - K ^p _e stress intensity factor for an edge crack in a flat plate - K ^w _e stress intensity factor for an edge crack in a weldment - K ^p _s stress intensity factor for a semi-elliptical surface crack in a flat plate - K ^w _s stress intensity factor for a semi-elliptical surface crack in a weldment - m(x, a) weight function - m(x, a, a) weight function for an edge crack emanating from an angular corner in a finite thickness plate with a step or weight function for an edge crack in a T-butt welded joint - m _B (x, a) Bueckner's weight function for an edge crack in a flat plate - m _s (x, a, c) weight function for a semi-elliptical surface crack in a flat plate - Q= /2 elliptical integral of second kind for a circular crack     

Calculation of Stress Intensity Factors for axial semi-elliptical surface cracks in a pipe with cladding loaded thermally

By means of the weight functions method stress intensity factors were calculated for axial semi-elliptical surface cracks in a pipe with cladding. The component is loaded by a thermoshock. Starting from a stress-free state the inner surface of the cladding is suddenly cooled down. The time-dependent temperature and hoop stress distributions of the uncracked component were calculated for the loading case considered. Numerical values of the stress intensity factors at the deepest point and at the surface points of the crack were evaluated at different time steps for a wide range of crack depths and crack lengths.     

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.     

自增强厚壁圆筒轴向内表面裂纹的应力强度因子

本文研究用有限元通用程序计算具有残余应力的自增强厚壁圆筒内半椭圆形表面裂纹的应力强度因子的方法。所考虑的应力强度因子被分为相应于工作内压及残余应力两部分，分别用三维有限元通用程序算得的裂纹前沿单元节点的垂直位移直接计算，对后者又运用了＂叠加原理＂。结果表明，残余应力的存在能有效地降低内裂纹的应力强度因子值，自增强度高者这一作用亦显著，残余应力引起的应力强度因子对裂织数目不敏感。     

Determination of stress intensity factor for embedded elliptical crack in turbine rotor

The stress intensity factor (SIF) for an embedded elliptical crack in a turbine rotor and the thermal shock stress intensity factor for a semi-elliptical surface crack in a finite plate are determined by means of Vainshtok's weight function method. The solution for the semi-elliptical surface crack is in good agreement with the previous one. The value of the SIF for the embedded elliptical crack in the turbine rotor under centrifugal and thermal loading is larger at the crack contour near the inner radius surface and almost constant at the opposite crack contour. The SIF decreases by increasing the crack ratio, and the distance between the inner radius surface and the crack center.     

Thermal Shock loading of a surface crack in a hollow sphere

Temperature and stress distributions in a hollow sphere are calculated, caused by a sudden cooling (thermal shock) of the inner surface of a hollow sphere. The thermal stresses are acting as load of surface cracks of approximate semi-elliptic shape. By means of the weight functions method stress intensity factors are estimated at the deepest point of the cracks using the well-known Newman-Raju solution for semi-elliptical surface cracks in a plate as reference solution.     

Fatigue analysis of non-load-carrying fillet welded cruciform joints

The goal of this investigation was to study the effect of local geometrical variations of the weld on the fatigue strength. Therefore the fatigue behaviour of non-load-carrying cruciform fillet welded joint under tensile loading has been studied parametrically. Several two-dimensional (2D) finite element models of the joint were analysed using plane strain linear elastic fracture mechanics (LEFM) calculations in order to get the magnification function M_k. A maximum tangential stress criterion was used to predict the crack growth direction under mixed mode K_I-K_II conditions. The derived M_k solution was then applied both for continuous weld toe cracks and also for semi-elliptical toe cracks at the deepest point of the crack front. An experimental crack aspect ratio development curve was used for propagating semi-elliptical cracks. The as-welded condition was assumed with the result that no crack initiation period was considered and stress ranges were fully effective. The Paris crack growth law was used to predict the growth rate. The effects of weld toe radius, flank angle and weld size on the fatigue strength were systematically studied. Finally, predicted fatigue strength values corresponding to different assumed crack sizes were compared with the available test results.     

Spannungsintensitätsfaktoren für halbelliptische Umfangsoberflächenrisse in einem Rohr bei Innendruck- und Biegebelastung

Stress Intensity Factors for Semi-Elliptical Circumferential Surface Cracks in a Pipe Loaded by Internal Pressure and Bending Pipes are often loaded by superposed tensile and bending stresses. Flaws in circumferential direction, for example at welded joints, may be caused by these stresses to grow. In this paper, semi-elliptical circumferential surface cracks in a pipe are studied. By means of the weight function method stress intensity factors at the deepest point and at the surface points of the cracks are evaluated in dependence on crack length and crack depth. The application of the weight function method in the form used here requires that the half crack length measured by the angle of circumference is not greater than 15 degrees. Longer cracks should be studied by the finite element method.     

Stress intensity factor solutions for estimation of fatigue lives of laser welds in lap-shear specimens

Fatigue behavior of laser welds in lap-shear specimens of high strength low alloy (HSLA) steel is investigated based on experimental observations and two fatigue life estimation models. Fatigue experiments of laser welded lap-shear specimens are first reviewed. Analytical stress intensity factor solutions for laser welded lap-shear specimens based on the beam bending theory are derived and compared with the analytical solutions for two semi-infinite solids with connection. Finite element analyses of laser welded lap-shear specimens with different weld widths were also conducted to obtain the stress intensity factor solutions. Approximate closed-form stress intensity factor solutions based on the results of the finite element analyses in combination with the analytical solutions based on the beam bending theory and Westergaard stress function for a full range of the normalized weld widths are developed for future engineering applications. Next, finite element analyses for laser welded lap-shear specimens with three weld widths were conducted to obtain the local stress intensity factor solutions for kinked cracks as functions of the kink length. The computational results indicate that the kinked cracks are under dominant mode I loading conditions and the normalized local stress intensity factor solutions can be used in combination with the global stress intensity factor solutions to estimate fatigue lives of laser welds with the weld width as small as the sheet thickness. The global stress intensity factor solutions and the local stress intensity factor solutions for vanishing and finite kinked cracks are then adopted in a fatigue crack growth model to estimate the fatigue lives of the laser welds. Also, a structural stress model based on the beam bending theory is adopted to estimate the fatigue lives of the welds. The fatigue life estimations based on the kinked fatigue crack growth model agree well with the experimental results whereas the fatigue life estimations based on the structural stress model agree with the experimental results under larger load ranges but are higher than the experimental results under smaller load ranges.     

Weight functions for the determination of stress intensity factor and T‐stress for semi‐elliptical cracks in finite thickness plate

This paper presents the application of weight function method for the calculation of stress intensity factors (K) and T‐stress for surface semi‐elliptical crack in finite thickness plates subjected to arbitrary two‐dimensional stress fields. New general mathematical forms of point load weight functions for K and T have been formulated by taking advantage of the knowledge of a few specific weight functions for two‐dimensional planar cracks available in the literature and certain properties of weight function in general. The existence of the generalised forms of the weight functions simplifies the determination of specific weight functions for specific crack configurations. The determination of a specific weight function is reduced to the determination of the parameters of the generalised weight function expression. These unknown parameters can be determined from reference stress intensity factor and T‐stress solutions. This method is used to derive the weight functions for both K and T for semi‐elliptical surface cracks in finite thickness plates, covering a wide range of crack aspect ratio (a/c) and relative depth (a/t) at any point along the crack front. The derived weight functions are then validated against stress intensity factor and T‐stress solutions for several linear and nonlinear two‐dimensional stress distributions. These derived weight functions are particularly useful for the development of two‐parameter fracture and fatigue models for surface cracks subjected to fluctuating nonlinear stress fields, such as these resulting from surface treatment (shot peening), stress concentration or welding (residual stress).     

Stress intensity factors for three‐dimensional weld toe cracks using weld toe magnification factors

In welded components, particularly those with complex geometrical shapes, evaluating stress intensity factors is a difficult task. To effectively calculate the stress intensity factors, a weld toe magnification factor is introduced that can be derived from data obtained in a parametric study performed by finite element method (FEM). Although solutions for the weld toe magnification factor have been presented, these are applicable only to non‐load‐carrying cruciform or T‐butt joints, due possibly to the requirement of very complicated calculations. In the majority of cases for various welded joints, the currently used weld toe magnification factors do not adequately describe the behaviour of weld toe cracks. In this study, the weld toe magnification factor solutions for the three types of welded joints such as cruciform, cover plate and longitudinal stiffener joints were provided through a parametric study using three‐dimensional finite elements. The solutions were formed with exponents and fractions that have polynomial functions in terms of a/c and a/t – that is, crack depths normalised by corresponding half crack lengths and specimen thickness. The proposed weld toe magnification factors were applied to evaluate the fatigue crack propagation life considering the propagation mechanisms of multiple‐surface cracks for all welded joints. It showed good agreement within a deviation factor of two between the experimental and calculated results for the fatigue crack propagation life.     

Effects of weld geometry and sheet thickness on stress intensity factor solutions for spot and spot friction welds in lap-shear specimens of similar and dissimilar materials

In this paper, three-dimensional finite element analyses for spot welds with ideal geometry in lap-shear specimens of different materials and thicknesses were first conducted. The computational results indicate that the stress intensity factor and J integral solutions based on the finite element analyses agree well with the analytical solutions and that the analytical solutions can be used with a reasonable accuracy. Three-dimensional finite element analyses based on the micrographs of an aluminum 6111 resistance spot weld, an aluminum 5754 spot friction weld, and a dissimilar Al/Fe spot friction weld were also conducted. The computational results indicate that the stress intensity factor and J integral solutions based on the finite element analyses for the aluminum 6111 resistance spot weld and aluminum 5754 spot friction weld with complex geometry are in good agreement with the analytical solutions for the equivalent spot welds with ideal geometry. However, the stress intensity factor and J integral solutions based on the finite element analysis for the Al/Fe spot friction weld with complex geometry are completely different from the analytical solutions for the equivalent spot weld with ideal geometry. Different three-dimensional finite element analyses based on the meshes that represent different features of the complex geometry of the Al/Fe spot friction weld were then conducted. The computational results indicate that the stress intensity factor and J integral solutions for the Al/Fe spot friction weld based on the finite element analysis agree reasonably well with the analytical solutions for the equivalent spot weld with consideration of gap and bend. The computational and analytical results suggest that the stress intensity factor and J integral solutions based on the finite element analysis and the analytical solutions with consideration of gap and bend may be used to correlate with the fatigue crack growth patterns of Al/Fe spot friction welds observed in experiments.     

Stress intensity factor calculation of steel-lined hoop-wrapped cylinders with internal semi-elliptical circumferential crack

The purpose of this paper is to present mode I stress intensity factor for a circumferential semi-elliptical crack on the inner surface of a hoop-wrapped steel-lined CNG cylinder. The stress intensity factors along the crack front are directly computed by 3D finite element method for a wide range of variations of the crack geometry. Also influence of many parameters such as cylinder internal pressure, composite layer thickness, composite material properties and undertaking Auto-Frettage pressure are studied on the stress intensity factor of the crack and some conclusive results are drawn. For the sake of validation of the results and because of lack of the results for a circumferential semi-elliptical crack in the literature, a semi-elliptical axial crack in a composite hoop-wrapped cylinder has been modeled and the results have been compared with those in the literature showing a good agreement.