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     

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.     

J estimation for shallow semi-elliptical surface cracks in wide plates under pure bending

Three-dimensional finite element models have been used to predict the J versus applied strain behaviour of shallow semi-elliptical surface cracks in wide steel plates. These models were loaded in pure bending up to fully plastic conditions that produced maximum strains remote from the crack of 10 times the yield strain. This study examined cracks with a crack depth to plate thickness ratio ( a / t ) in the range 0.05–0.15, and crack depth to crack half-length ratio ( a / c ) from 0.2 to 0.57, in elastic-hardening materials as well as those with a yield plateau before strain-hardening. The results were found to differ significantly from those obtained from plane strain analyses, and a J estimation scheme was developed from the three-dimensional results. Comparisons between the results obtained in this work and the J predictions provided by existing defect assessment methods show that the proposed equations are a significant improvement when large strains are likely.     

Tension of a finite-thickness plate with a pair of semi-elliptical surface cracks

This paper deals with the tension of a finite-thickness plate with a pair of semi-elliptical cracks on both of the free surfaces. The analysis is performed in a similar manner to the previous single crack problem, by using the body force method and the boundary conditions expressed in terms of resultant forces and displacements of the boundary elements. The stress intensity factor at the maximum depth of the crack front is calculated for various values of the parameters and these results are fitted by a reliable polynomial formula for convenience of engineering applications.     

Stress intensity factors for a wide range of long-deep semi-elliptical surface cracks, partly through-wall cracks and fully through-wall cracks in tubular members

To calculate the rate of fatigue crack growth in tubular members, one approach is to make use of the fracture mechanics based Paris law. Stress intensity factors (SIF) of the cracked tubular members are prerequisite for such calculations. In this paper, stress intensity factors for circumferential deep semi-elliptical surface crack (a/t > 0.8), semi-elliptical partly through-wall crack and fully through-wall crack cracks in tubular members subjected to axial tension are presented. The work has produced a comprehensive set of equations for stress intensity factors as a function of a/T, c/πR and R/T for deep surface cracks. For the partly through-wall cracks and fully through-wall cracks, two sets of bounding stress intensity factor equations were produced based on which all stress intensity factors within the range of parameters can be obtained by interpolation.     

Growth of three-dimensional cracks in finite-thickness plates

The stress field in a finite-thickness plate weakened by a three-dimensional crack and subjected to tension in a direction perpendicular to the crack plane is studied. The cases of an embedded elliptical, semi-elliptical and quarter-elliptical surface crack are considered. The stress analysis takes place by a finite element computer program which uses twenty-node isoparametric and fifteen-node enriched elements. The stress intensity factor variations along the periphery of the elliptical cracks are given for various plate thicknesses. The results of the stress analysis are used in conjunction with the strain energy density theory to study the growth characteristics of the cracks. The history of non-self-similar crack growth from initiation to final instability through the intermediate stage of stable growth is analyzed. The increments of crack growth from each point of its front are determined on the basis that the critical element in the direction of crack growth absorbs a critical amount of strain energy density. Crack growth becomes unstable when the last increment from the critical point of the crack front takes a limiting value. Results for the crack growth characteristics are given for the three types of cracks considered and various plate thicknesses.     

Approximate J estimates for tension-loaded plates with semi-elliptical surface cracks

This paper provides a simplified engineering J estimation method for semi-elliptical surface cracked plates in tension, based on the reference stress approach. Note that the essential element of the reference stress approach is the plastic limit load in the definition of the reference stress. However, for surface cracks, the definition of the limit load is ambiguous (“local” or “global” limit load), and thus the most relevant limit load (and thus reference stress) for the J estimation should be determined. In the present work, such limit load solution is found by comparing reference stress based J results with those from extensive 3-D finite element (FE) analyses. Based on the present FE results, the global limit load solution proposed by Goodall for surface cracked plates in combined bending and tension was modified, in the case of tension loading only, to account for a weak dependence on w/c and was defined as the reference normalizing load. Validation of the proposed equation against FE J results based on actual experimental tensile data of a 304 stainless steel shows excellent agreements not only for the J values at the deepest point but also for those at an arbitrary point along the crack front, including at the surface point. Thus the present results provide a good engineering tool for elastic-plastic fracture analyses of surface cracked plates in tension.     

Elastic T-stress solutions for semi-elliptical surface cracks in finite thickness plates

Three-dimensional finite element analyses have been conducted to calculate the elastic T-stress for semi-elliptical surface cracks in finite thickness plates. Far-field tension and bending loads were considered. The analysis procedures and results were verified using both exact solutions and approximate solutions. The T-stress solutions are presented along the crack front for cracks with a/t values of 0.2, 0.4, 0.6 or 0.8 and a/c values of 0.2, 0.4, 0.6 or 1.0. Based on the present finite element calculations for T-stress, empirical equations for the T-stress at three locations: the deepest, the surface and the middle points of the crack front under tension or bending are presented. The numerical results are approximated by empirical formulae fitted with an accuracy of 1% or better. They are valid for 0.2?a/c?1 and 0?a/t?0.8. These T-stress results together with the corresponding K or J values for surface cracks are suitable for the analysis of constraint effects for surface cracked components.     

Stress intensity factors for semi-elliptical surface cracks in plates and cylindrical pressure vessels using the hybrid boundary element method

At first, a hybrid boundary element method used for three-dimensional linear elastic fracture analysis is established on the basis of the first and the second kind of boundary integral equations. Then the concerned basic theories and numerical approaches including the discretization of boundary integral equations, the divisions of different boundary elements, and the procedures for the calculations of singular and hypersingular integrals are presented in detail. Finally, the stress intensity factors of surface cracks in finite thickness plates and cylindrical pressure vessels are computed by the proposed method. The numerical results show that the hybrid boundary element method has very high accuracy for the analysis of surface crack.     

Weld magnification factors for semi-elliptical surface cracks in fillet welded T-butt joint models

The weld magnification factor method has been widely used in the determination of the stress intensity factor (SIF) for weld-toe cracks in welded structural components. Weld magnification factors M _kare normally derived from two-dimensional crack models with fillet weld profiles to take account of the effect of weld-notch stress concentration at the deepest point of the crack front. This paper presents a detailed three-dimensional analysis of weld-toe surface cracks in fillet welded T-butt joint models using the finite element method. Effects of the weld notch and the welded attachment stiffness on the SIFs of the weld-toe surface cracks have been studied quantitatively. Weld magnification factors applying to the whole surface crack fronts have been estimated. Numerical results show two contradictory effects; that the effect of weld notch increases SIF values throughout the shallow surface crack fronts which are in the region of notch stress concentration, while the effect of local structural constraint reduces the SIF values. The increase in the SIF values mainly depends upon the relative crack front depth and the decrease in the SIF values mainly depends upon the crack shape aspect ratio for a specific weld profile. Both effects on the weld magnification factors can be estimated separately. A simple approach for deriving the weld magnification factors for various weld-toe surface crack problems is proposed for engineering applications.     

Creep fracture parameter C* solutions for semi-elliptical surface cracks in plates under tensile and bending loads

In order to predict and assess creep life for plate structures with semi-elliptic surface cracks under high temperature condition, the accurate calculation of the creep fracture mechanics parameter C* is a critical step. In this paper, the effects of crack sizes, plate geometries, and material creep properties on the parameter C* have been investigated under tensile and bending loads by extensive finite element analyses. Based on the results, the creep influence functions Hc for calculating C* values were obtained and fitted into equations for surface cracks in plates under both loads. The equations have been verified by finite element calculations. The C* solutions were obtained through these equations which are suitable for wide ranges of crack sizes, plate geometries, and materials.     

Stress intensity factors for semi-elliptical surface cracks in a plate under tension based on the Isida's solution

The stress intensity factor solution given by Isida et al. for semi-elliptical surface cracks in plates is improved and extended. The data tabulated for the ranges 0a/t0.6 and 0.25a/c1.0 were checked using the energy relation and found to be superior to solutions available in the literature.     

Subcritical Crack Growth of semi-elliptical surface cracks in a turbine rotor

The subcritial growth of semi-elliptical surface cracks in the front face of a cylindrical disk was calculated. The disk represents a part of a turbine rotor. The stress intensity factors, which are the basis for the calculations presented here, were given in a former paper [1]. The loads steps considered consist of an inhomogeneous hoop stress distribution, which occurs at a distinct time-step of the start – up procedure of the turbine, and of the stress free state, respectively. By use of a Paris-law the growth of different initial cracks was evaluated under the assumption that the crack geometry remains semi-elliptic.     

Calculation of Stress Intensity Factors for semi-elliptical surface cracks in a turbine rotor

Stress intensity factors based on linear elastic behaviour were calculated for semi-elliptical surface cracks in the front face of a cylindrical disk. The small semi-axis of the cracks coincides with the axis of rotation of the disk. The disk represents a part of a turbine rotor and is used to simplify the calculations. The uncracked rotor is loaded by a radial varying hoop stress distribution, caused by rotation, thermal gradients and the influence of the turbine blades. Following a procedure proposed by Mattheck at al. [1] the stress points of the cracks were calculated by means of the weight functions method.     

Vibrodiagnostic parameters of fatigue damage in rectangular plates. Part 3. Through-the-thickness and surface semi-elliptical cracks

The paper addresses the analytical determination of vibrodiagnostic parameters that describe the presence of normal-rupture flat through-the-thickness and surface semi-elliptical central cracks in a rectangular homogeneous plate of constant thickness for various plate fixing conditions and vibration modes. It is shown that the most sensitive vibrodiagnostic damage parameter of a plate is the variation of the logarithmic decrement in the case of a through-the thickness crack and the second-harmonic ratio of the vibration process in superharmonic resonance in the case of a surface crack. __________ Translated from Problemy Prochnosti, No. 5, pp. 27–47, September–October, 2006.     

Stress intensity factors for semi-elliptical surface cracks in pressurised cylinders using the boundary integral equation method

Three dimensional linear elastic fracture mechanics analyses of the problem of an internally pressurised thick-walled cylinder with a semi-elliptical surface crack are carried out using the Boundary Integral Equation method. The cases treated are for ratios of external to internal cylinder radii of 2 and 3, with maximum crack depths ranging from 20% to 80% of the wall thickness. For a typical crack, the predicted value of stress intensity factor decreases slightly along the crack front moving away from the point of deepest penetration, reaching a minimum before increasing rapidly as the free internal surface of the cylinder is approached. The results presented will be of considerable use in the prediction of residual or safe life of, for example, chemical reactor tubes known to be cracked to a certain depth.

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Résumé Une analyse tri-dimensionnelle élastique et linéaire en mécanique de rupture du problème d'un cylindre à paroi épaisse soumis à pression interne et comportant une fissure de surface semi-elliptique a été effectuée en utilisant la méthode d'équation intégrale aux limites. Les cas envisagés dans l'étude sont relatifs à des rapports des rayons de cylindre externe et interne de 2 et de 3 avec des profondeurs maximum de fissure s'étalant de 20% à 80% de l'épaisseur de paroi. Dans le cas d'une fissure typique, la valeur prédite pour le facteur d'intensité des contraintes décroit légèrement le long du front de la fissure lorsque l'on se meut du point de pénétration le plus profond, et passe par un minimum pour ensuite s'accroître rapidement lorsqu'on approche de la surface libre interne du cylindre. Les résultats présentés s'avèreront d'une aide considérable pour prédire la vie résiduelle ou la durée de service fiable, par exemple, de tubes de réacteur chimique dont on sait qu'ils sont fissurés sur une certaine profondeur.