Crack shape developments for LBB applications

The flaw shape during sub-critical crack propagation (fatigue or stress corrosion) is important in order to estimate the crack opening area and leak rate at breakthrough and for the leaking crack. The ability to detect this leak rate is one of the cornerstones in the leak before break philosophy. In this study, the crack shape development is analysed and predicted using modern FEM techniques. The predictions are benchmarked against available fatigue experiments. Recommendations are given for crack shapes for both surface cracks and through-wall cracks in pipes and plates, and for sub-critical crack propagation as well as ductile tearing loading.     

J-integral analyses of plate and shell structures with through-wall cracks using thick shell elements

A new evaluation method using thick shell elements is proposed to calculate the distribution of the J-integral values along crack fronts of through-wall cracks in plate and shell structures. Dividing tentatively the thick shell elements into several layers in the thickness direction, integral paths are defined at each layer to obtain the thickness distributions of the J-integral values of through-wall cracks in plate and shell structures. The three-dimensional definition of the J-integral is used to calculate the J-integral value at each layer. The results of several stress intensity factor analyses are presented to show the effectiveness of the present evaluation method. The distribution of the J-integral value along a crack front is also compared between the methods using the thick shell elements and the three-dimensional solid elements.     

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.     

The T-stress solutions for through-wall circumferential cracks in cylinders subjected to general loading conditions

The elastic T-stress is a parameter used to define the level of constraint at a crack tip. It is important to provide T-stress solutions for practical geometries to apply the constraint-based fracture mechanics methodology. In the present work, T-stress solutions are provided for circumferential through-wall cracks in thin-walled cylinders. First, cylinders with a circumferential through-wall crack were analyzed using the finite element method. Three cylinder geometries were considered; defined by the mean radius of the cylinder (R) to wall thickness (t) ratios: R/t = 5, 10, and 20. The T-stress was obtained at eight crack lengths (θ/π = 0.0625, 0.1250, 0.1875, 0.2500, 0.3125, 0.3750, 0.4375, and 0.5000, θ is the crack half angle). Both crack face loading and remote loading conditions were considered including constant, linear, parabolic and cubic crack face pressures and remote tension and bending. The results for constant and linear crack face pressure were used to derive weight functions for T-stress for the corresponding cracked geometries. The weight functions were validated against several linear and non-linear stress distributions. The derived weight functions are suitable for T-stress calculations for circumferential cracks in cylinders under complex stress fields.     

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).     

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.     

Finite element analysis of fretting crack propagation

In this work, the finite elements method (FEM) is used to analyse the growth of fretting cracks. FEM can be favourably used to extract the stress intensity factors in mixed mode, a typical situation for cracks growing in the vicinity of a fretting contact. The present study is limited to straight cracks which is a simple system chosen to develop and validate the FEM analysis. The FEM model is tested and validated against popular weight functions for straight cracks perpendicular to the surface. The model is then used to study fretting crack growth and understand the effect of key parameters such as the crack angle and the friction between crack faces. Predictions achieved by this analysis match the essential features of former experimental fretting results, in particular the average crack arrest length can be predicted accurately.     

Calculation of approximate weight functions in fracture mechanics by FEM

A method is presented for the calculation of weight functions used in fracture mechanics to determine stress intensity factors of cracks loaded by stress gradients. The reference solution for the stress intensity factor and for the reference crack opening displacement field is computed numerically by use of finite elements. The accuracy of the method is checked by comparison with well-known solutions from the literature.     

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.     

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 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%.     

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.     

Calculation of weight functions for three-dimensional bodies with cracks

A new algorithm is proposed for the numerical calculation of weight functions used in the determination of stress intensity factors by the finite elements method. The algorithm is based on the method of equivalent volume integration. It is shown that weight functions can be obtained for cracks in three-dimensional bodies with the stress intensity factor averaged over a small section of the crack front. A numerical example demonstrating the usefulness of the algorithm is presented.Moscow. Moscow Physico-Engineering Institute. Translated from Problemy Prochnosti, No. 11, pp. 32–36, November, 1989.     

Approximate elastic-plastic J estimates of cylinders with off-centred circumferential through-wall cracks

This paper provides approximate J estimates for off-centred, circumferential through-wall cracks in cylinders under bending and under combined tension and bending. The proposed method is based on the reference stress approach, where the dependence of elastic and plastic influence functions of J on the cylinder/crack geometry, the off-centred angle and strain hardening is minimised through the use of a proper normalising load. Based on published limited FE results for off-centred, circumferential through-wall cracks under bending, such normalising load is found, based on which the reference stress based J estimates are proposed for more general cases, such as for a different cylinder geometry and for combined loading. Comparison of the estimated J with extensive FE J results shows overall good agreements for different crack/cylinder geometries and for combined tension and bending, which provides sufficient confidence in the use of the proposed method for fracture mechanics analyses of off-centred circumferential cracks. Furthermore, the proposed method is simple to use, giving significant merits in practice.     

High-pH inclined stress corrosion cracking in Australian and Canadian gas pipeline X65 steels

High-pH stress corrosion cracking is a form of environmental degradation of gas pipeline steels. The crack path is intergranular by nature and typically perpendicular to the maximum applied (hoop) stress (i.e. perpendicular to the pipe outer surface). Some unusual instances of cracks have been observed in Canadian and Australian X65 pipes, where cracks grow away from the perpendicular for considerable distances. This paper presents a comparative study in terms of crack morphology, mechanical properties and crystallographic texture for these Australian and Canadian pipe steels. It is shown that the crack morphologies are quite similar, the main difference being the angle at which the cracks propagate into the material. This difference could be explained by the different through-wall texture and grain aspect ratio measured in the two materials. The interdependency of crack tip plasticity, crack tip electrochemistry and anisotropy in microstructural texture seems to heavily affect the resulting inclined crack path.     

Determination of failure pressure for tubes with two non-aligned axial through-wall cracks

The 40% of wall thickness criterion which has been used as a plugging rule is applicable only to a single cracked steam generator tubes. In the previous studies performed by authors, several failure prediction models were introduced to estimate the plastic collapse pressures of steam generator tubes containing two adjacent collinear or parallel axial through-wall cracks. The objective of this study is to examine the failure prediction models and propose optimum ones for two non-aligned axial through-wall cracks in steam generator tubes. In order to determine the optimum ones, a series of plastic collapse tests and finite element analyses were carried out for steam generator tubes with two machined non-aligned axial through-wall cracks. Thereby, either the plastic zone contact model or COD based model was selected as the optimum one according to axial distance between two cracks. Finally, the optimum failure prediction model was used to demonstrate the conservatism of flaw characterization rules for multiple cracks having equal lengths according to ASME code.     

Effect of porosity and stress concentration associated with pores on elastic creep by crack growth in brittle solids

The solution for the creep rate of a porous solid as a function of pore volume fraction, pore size and the radial or annular flaw size, is presented. The analysis is based on the concept of crack opening displacement which assumes that the displacement of a solid under stress is solely due to the opening of all radial or annular cracks emanating from the surface of the cavities. It is shown that the strain rate by crack growth is a strong function of pore size and preexisting flaw size. A linear relationship was found to exist between the strain rate by crack growth and the porosity volume fraction.     

Stress intensity factors and crack opening displacements for slanted axial through-wall cracks in pressurized pipes

This paper proposes elastic stress intensity factors and crack opening displacements (CODs) for a slanted axial through-wall cracked cylinder under an internal pressure based on detailed three-dimensional (3D) elastic finite element (FE) analyses. The FE model and analysis procedure were validated against existing solutions for both elastic stress intensity factor and COD of an idealized axial through-wall cracked cylinder. To cover a practical range, four different values of the ratio of the mean radius of cylinder to the thickness ( R _m/ t ) were selected. Furthermore, four different values of the normalized crack length and five different values of the ratio of the crack length at the inner surface to the crack length at the outer surface representing the slant angle were selected. Based on the elastic FE results, the stress intensity factors along the crack front and CODs through the thickness at the centre of the crack were provided. These values were also tabulated for three selected points, that is, the inner and outer surfaces and at the mid-thickness. The present results can be used to evaluate the crack growth rate and leak rate of a slanted axial through-wall crack due to stress corrosion cracking and fatigue. Moreover, the present results can be used to perform a detailed leak-before-break analysis considering more realistic crack shape development.     

Universal features of weight functions for cracks in mode I

An analysis of known analytical and numerical weight functions for cracks in mode I has revealed that they all have a similar singular term and that it is possible to approximate them with one universal expression with three unknown parameters. The unknown parameters can be determined directly from reference stress intensity factor expressions without using the crack opening displacement function. The universal weight function expression, with suitable reference stress intensity factors, was used to derive the weight functions for internal and external radial cracks in a thick cylinder. These weight functions were then further used to calculate the stress intensity factors for radial cracks in a cylinder subjected to several nonlinear stress fields and were compared to available numerical data.     

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.