Elastic stress intensity factors and crack opening displacements for circumferential through-walled cracked elbows

This paper provides tabulated solutions of elastic stress intensity factors and crack opening displacements for circumferential through-wall cracked elbows under internal pressure and under in-plane bending, based on extensive three-dimensional elastic finite element analyses covering a wide range of crack lengths and elbow/pipe geometries. The effect of crack length and elbow/pipe geometry on the results is discussed, with particular emphasis on the crack closure behaviour under in-plane bending.     

Limit loads and approximate J estimates for axial through-wall cracked pipe bends

This paper presents plastic limit loads and approximate J estimates for axial through-wall cracked pipe bends under internal pressure and in-plane bending. These loads and estimates are based on small strain finite element limit analyses using elastic-perfectly plastic materials. Geometric variables associated with the crack and pipe bend are systematically varied, and three possible crack locations (intrados, crown and extrados) are considered. Effects of the bend and crack geometries on plastic limit loads are quantified, and closed-form limit load solutions are given. Based on the proposed limit load solutions, a reference stress based the J estimation scheme for axial through-wall cracked pipe bends under internal pressure and in-plane bending is proposed.     

Ratchetting limits for cracked bodies subjected to cyclic loads and temperatures

This paper describes a methodology capable of directly evaluating the ratchetting limits in cracked bodies, subjected to cyclic loads and temperatures, for an elastic-perfectly plastic material. The foundation behind the adopted numerical procedures, the Linear Matching Method (LMM), is described. This is then followed by the application of the method to the cracked Bree problem, under variations in cyclic loading histories and structural geometries. The results from the ensuing analyses revealed the method’s ability in coping with the effect of the elastic singularity at the crack tip. The applicability of such procedures is further examined, with additional solutions to an industrial problem considered in R5 the design and life assessment procedures used by British Energy in the UK.     

Structural integrity of pipelines: T-stress by line-spring

The elastic T‐stress is an important constraint parameter for characterizing elastic–plastic crack‐tip fields and in fracture assessment procedures. However, many of the methods reported in the literature for estimating T‐stress are not easily suited for surface‐cracked pipes because these are three‐dimensional in nature. Here, the line‐spring method is demonstrated to be an efficient and accurate tool for the constraint estimation in surface‐cracked pipes. Detailed three‐dimensional analyses are performed to verify the line‐spring results. Using the line‐spring method, the effects of different crack geometries and diameter‐to‐thickness ratio on stress‐intensity factor (SIF) and T‐stress in circumferentially surface‐cracked pipes are examined. Further, a compendium of normalised SIF and T‐stress values for surface‐cracked pipes in remote tension and bending, calculated from a total of 1000 analyses, is tabulated. Finally, the application of an ‘elastic–plastic’ T‐stress under large‐scale plasticity is explored.     

Determination of the Stress Intensity Factor of an elliptical breathing crack in a rotating shaft

The failures due to the propagation of fatigue cracks are one of the most frequent problems in rotating machines. Those failures sometimes are catastrophic and are sufficient to provoke the loss of the complete machine with high risks for people and other equipments. When a cracked shaft rotates, the breathing mechanism appears. The crack passes from an open state to a close state with a transition in which a partial opening or closing of the crack is produced. In this work, a new general expression that gives the Stress Intensity Factor (SIF) along the crack front of an elliptical crack in a rotating shaft in terms of the crack depth ratio, the crack aspect ratio, the relative position on the front and the angle of rotation has been developed for linear elastic materials. By the moment, no expressions of the SIF in term of these variables have been found in the literature. To this end, a quasi-static 3D numerical model of a cracked shaft with straight and elliptical cracks subjected to rotary bending using the Finite Element Method (FEM) has been made. To simulate the rotation of the shaft, different angular positions have been considered. The SIF in mode I along the crack front has been calculated for each angular position of the cracked shaft and for different crack geometries. The expression results have been compared with solutions obtained from the literature. It has been found that they are in good agreement. The model has been applied to other crack geometries with good results. The obtained SIF expression allows studying the dynamic behavior of cracked shafts and can be used to analyze the crack propagation.     

J and CTOD estimation procedure for circumferential surface cracks in pipes under bending

This work provides an estimation procedure to determine the J-integral and CTOD for pipes with circumferential surface cracks subjected to bending load for a wide range of crack geometries and material (hardening) based upon fully-plastic solutions. A summary of the methodology upon which J and CTOD are derived sets the necessary framework to determine nondimensional functions h₁ and h₂ applicable to a wide range of crack geometries and material properties characteristic of structural, pressure vessel and pipeline steels. The extensive nonlinear, 3-D numerical analyses provide a definite full set of solutions for J and CTOD which enters directly into fitness-for-service (FFS) analyses and defect assessment procedures of cracked pipes and cylinders subjected to bending load.     

Analysis of cracked piezoelectric solids by a mixed three-dimensional BE approach

This paper has two main objectives in relation to the analysis of three-dimensional crack problems in piezoelectric solids. The first one is to present the formulation, effective implementation and numerical treatment of a mixed boundary element technique for the study of this type of problems. The numerical procedure is based on the use of extended displacement and extended traction integral equations for external and crack boundaries, respectively. The boundary element formulation is presented with particular emphasis on numerical aspects related to singular kernels regularization and evaluation of boundary integrals. Quadratic boundary elements and quarter-point boundary elements are implemented in a computer code. By using these elements, electric and stress intensity factors are directly computed from nodal values at quarter-point elements. The second purpose is to study several realistic piezoelectric crack problems for the first time. Unbounded and bounded cracked piezoelectric three-dimensional (3D) solids with different geometries are studied. Results presented in this paper can be used as a reference for future research. Prior to the analysis of problems whose solution was previously unknown, the technique is validated by solving some simple problems with known analytical or numerical solution. Then, more realistic crack problems of engineering interest have been analysed for the first time. In all cases, results for the solid deformed shape, the crack opening displacements and the extended stress intensity factor components, are shown.     

Probabilistic fracture mechanics analysis of linear-elastic cracked structures by spline fictitious boundary element method

The inherent uncertainties in crack geometry, material properties and loadings have large influence on fracture response characteristics of cracked structures. This paper presents the probabilistic fracture mechanics analysis of linear-elastic cracked structures subjected to mixed-mode loading conditions using the spline fictitious boundary element method (SFBEM). The response surface method (RSM) is used to predict the fracture probability of the cracked structure. To determine the unknown coefficients of the response surface function, the SFBEM based on the Erdogan fundamental solutions for infinite cracked plates is adopted to perform deterministic analyses of stress intensity factors (SIFs) corresponding to different test points with given parameters. Numerical examples based on mode-I and mixed-mode crack problems are presented to illustrate the present method. The results show that the predicted failure probability obtained by the present approach is accurate in comparison with the Monte Carlo simulation (MCS) results. Since a much lesser number of numerical tests are required in RSM as compared with that needed in MCS, and since the SFBEM based on the Erdogan fundamental solutions has been used to conduct the numerical tests, reliability analysis of cracked structures can be performed efficiently using the present method.     

Probabilistic fracture mechanics analysis based on three-dimensional J-integral database

The development is described of a novel Probabilistic Fracture Mechanics (PFM) code based on the three-dimensional J-integral database, giving so-called fully plastic solutions. An efficient technique for the evaluation of leak and break probabilities is also utilized, based on the Stratified sampling Monte Carlo simulation (SMC).

The outline of the present PFM code is described, and the J-integral database and the numerical technique are presented. Nonlinear effects of materials on failure probabilities are discussed through the analysis of a surface cracked structure subjected to cyclic tension.     

Application of the distributed dislocation technique for calculating cyclic crack tip plasticity effects

This paper describes a method for modelling cyclic crack tip plasticity effects based on the distributed dislocation technique (DDT). A strip‐yield model is utilised to allow for the determination of the crack opening displacement, size of the plastic zones and in the case of a fatigue crack, the wake of plasticity. The DDT can be easily implemented for a wide range of cracked geometries with reliable control over the accuracy and convergence. Thickness effects can also be incorporated through a recently obtained solution for an edge dislocation in an infinite plate of finite thickness. Results for finite length cracks that have had limited growth, such that there is no plastic wake, are presented for a range of applied loads and R‐ratios. Further results are provided for a steady‐state fatigue crack in a plate of finite thickness. The present results are compared with analytical solutions and they show an excellent agreement.     

有限宽中心裂纹板在裂纹面受一对偏心反平面集中力的弹塑性分析

有限宽裂纹板的弹塑性分析是弹塑性断裂力学中最困难的问题之一。本文对有限宽裂纹板在裂纹面任意点受一对反平面集中力的情形采用裂纹线场分析方法,将各场量在裂纹线附近展开,利用平衡方程和屈服准则进行弹塑性分析,这种分析不需要作小范围屈服的假定。通过裂纹线上的弹塑性应力场在弹塑性边界上进行匹配得出荷载与裂纹线上塑性区长度之间的关系,进而分析得出荷载的不同位置和板宽所对应的临界荷载。     

Fatigue crack growth analyses of offshore structural tubular joints

This study investigated various aspects of a fatigue crack growth analysis, ranging from the stress intensity factor solutions to the simulation of a fatigue crack coalescence process of a tubular joint weld toe surface flaw. Fracture mechanics fatigue crack growth analyses for offshore structural tubular joints are not simple, because of the difficulty to calculate the stress intensity factors due to their geometric complexity. The fully mixed-mode stress intensity factors of nine weld toe surface cracks of an X-shaped tubular joint under tension loading were calculated by detailed three-dimensional finite element analyses. Using these stress intensity factor solutions, a fatigue crack growth study was performed for the X-joint until (the crack surface length grew to two times the tube thickness. Through this study, the crack shape change during the fatigue crack propagation was investigated in detail. Fatigue life calculations were also performed for a range of crack geometries using the stress intensity factor solutions of the nine flaws. These calculations indicate that the natural fatigue crack growing path for a crack is its quickest growing path. The study demonstrated that detailed fracture mechanics fatigue analyses of tubular joints can be practical using the finite element method.     

Fully plastic analyses of plane strain single-edge-cracked specimens subject to combined tension and bending

Accurate yield surfaces of plane strain single-edge-cracked specimens having shallow as well as deep cracks are developed using finite element limit analyses and monotonic interpolation functions. Fully plastic shallow crack configurations are classified based on certain aspects of the yield surfaces. Relationships between incremental plastic crack tip and crack mouth opening displacements and incremental load point displacement/rotation are obtained for a wide range of relative crack depths and loading ratios. Fully plastic crack-tip fields for a sufficiently deep crack in a single-edge cracked specimen are examined to provide the stress triaxiality and the angular orientation of flow line at the crack tip in terms of the remotely applied tension-to-bending ratio. Evidence for fully plastic crack-tip stress fields consisting of an incomplete Prandtl fan and a crack plane constant state region is discussed.     

Plastic bending of an edge-cracked specimen

The plastic zones and the rotation of the cracked planes of an edge-cracked bend specimen are investigated by a numerical three-dimensional analysis. For an alloy-steel, for bending moments up to the first yielding moment of the uncracked specimen, it is shown that the crack tip is in small scale yielding.     

Effects of local wall thinning on plastic limit loads of elbows using geometrically linear FE limit analyses

This paper provides closed-form plastic limit load solutions for elbows under in-plane bending and internal pressure, via three-dimensional (3D), geometrically linear FE limit analyses using elastic-perfectly plastic materials. Wide ranges of elbow and thinning geometries are considered. To investigate the effect of the axial thinning length on limit loads systematically, two limiting cases are considered; a sufficiently long thinning, and the circumferential part-through surface crack. Closed-form plastic limit load solutions for wall thinning with intermediate longitudinal extents are then obtained from these two limiting cases. The effect of the axial extent of wall thinning on plastic limit loads for elbows is highlighted by comparing that for straight pipes. Although the proposed solutions are developed for the case when wall thinning exists in the center of elbows, it is also shown that they can be applied to the case when thinning exists anywhere within the elbow.     

A probabilistic fracture mechanics model including 3D ductile tearing of bi-axially loaded pipes with surface cracks

This paper presents a probabilistic fracture mechanics model established from three-dimensional FEM analyses of surface cracked pipes subjected to tension load in combination with internal pressure. The models are particularly interesting for offshore pipelines under operational conditions or during laying, where inelastic deformations may occur. In the numerical models, the plastic deformations, including ductile tearing effects, are accounted for by use of the Gurson-Tvergaard-Needleman model. This model is calibrated to represent a typical X65 pipeline steel behaviour under ductile crack growth and collapse. Several parameters are taken into account, such as crack depth, crack length and material hardening. Another important topic is the examination of the influence of bi-axial loading due to internal pressure on capacity. From the results of the deterministic analyses a probabilistic fracture mechanics model is established using the response surface methodology. Two failure criteria are examined to represent the structural capacity. Based on the established model, we illustrate the methodology by examples employing the two different failure criteria solved with first and second order reliability methods.     

Characterization of crack tip stress fields in test specimens using mode mixity parameters

The aim of this study is to represent the combined effect of mode mixity, specimen geometry and relative crack length on the $T$ -stress, elastic–plastic stress fields, integration constant $I_{n}$ , angle of initial crack extension, and the plastic stress intensity factor. The analytical and numerical results are obtained for the complete range of mixed modes of loading between mode I and mode II. For comparison purposes, the reference fields for plane mixed-mode problems governing the asymptotic behavior of the stresses and strains at the crack tip are developed in a power law elastic–plastic material. For the common experimental fracture mechanics specimen geometries considered, the numerical constant of the plastic stress field $I_{n}$ and the $T$ -stress distributions are obtained as a function of the dimensionless crack length and mode mixity. A method is also suggested for calculating the plastic stress intensity factor for any mixed-mode I/II loading based on the $T$ -stress and power law solutions. It is further demonstrated that in both plane stress and the plane strain, the plastic stress intensity factor can be used to characterize the crack tip stress fields for a variety of specimen geometries and different mixed-mode loading. The applicability of the plastic stress intensity factor to analysis of the in-plane and out-of-plane constraint effect is also discussed.     

Probabilistic fracture mechanics by Galerkin meshless methods – part I: rates of stress intensity factors

 This is the first in a series of two papers generated from a study on probabilistic meshless analysis of cracks. In this paper (Part I), a Galerkin-based meshless method is presented for predicting first-order derivatives of stress-intensity factors with respect to the crack size in a linear-elastic structure containing a single crack. The method involves meshless discretization of cracked structure, domain integral representation of the fracture integral parameter, and sensitivity analysis in conjunction with a virtual crack extension technique. Unlike existing finite-element methods, the proposed method does not require any second-order variation of the stiffness matrix to predict first-order sensitivities, and is, consequently, simpler than existing methods. The method developed herein can also be extended to obtain higher-order derivatives if desired. Several numerical examples related to mode-I and mixed-mode problems are presented to illustrate the proposed method. The results show that first-order derivatives of stress-intensity factors using the proposed method agree very well with reference solutions obtained from either analytical (mode I) or finite-difference (mixed mode) methods for the structural and crack geometries considered in this study. For mixed-mode problems, the maximum difference between the results of proposed method and finite-difference method is less than 7. Since the rates of stress-intensity factors are calculated analytically, the subsequent fracture reliability analysis can be performed efficiently and accurately. Received 20 February 2001 / Accepted 19 December 2001