Evaluation of leak-before-break assessment methodology for pipes with a circumferential through-wall crack. Part II: J-integral estimation

Evaluation of the J-integral plays a central part in evaluation of the critical crack length for unstable fracture for piping systems. Simplified evaluation methods for the J-integral for a circumferential through-wall crack in pipes subjected to axial and bending loading or their combination is reviewed in this paper. Use of the LBB.ENG2 method and a similar approach based on the η-factor concept were found to result in significant underestimation of the J-integral for small and medium crack angles. On the other hand, the reference stress method based on the solutions for stress intensity factor and limit load recommended in the companion paper (Part I) provides solutions which agree well with the available non-linear finite-element solutions and can be utilized as a powerful tool for J-integral evaluation for arbitrary materials, not restricted to simple power-law hardening.     

Weight function method for transient thermomechanical fracture analysis of a functionally graded hollow cylinder possessing a circumferential crack

This article introduces a weight function method for fracture analysis of a circumferentially cracked functionally graded hollow cylinder subjected to transient thermomechanical loading. Analytical solutions for transient temperature and stress distributions in the uncracked cylinder are derived by applying finite Hankel transformation. These solutions are utilized to determine stress acting on the faces of the circumferential crack in the local perturbation problem. Thermomechanical material properties are assumed to be power functions of the radial coordinate in the derivations. Coefficients of the weight function are found using reference stress intensity factors computed through the finite element method. Domain form of the J-integral is used in the finite element calculations. Comparisons of the numerical results calculated by the proposed weight function method to those generated by finite element analysis demonstrate the high level of accuracy attained by the application of the developed procedures. Further parametric analyses are presented to illustrate the influences of dimensionless time, crack depth to thickness ratio, power law index, and convection coefficient upon transient mode I thermomechanical stress intensity factors.     

Reference stress method for evaluation of failure assessment curve of cracked pipes in nuclear power plants

In order to obtain a precise failure assessment curve (FAC) in the R6 defect assessment procedure, it is necessary to evaluate the J-value of cracked components. The reference stress method can be used for estimating J-values. However, the accuracy of estimation depends on the limit load used for evaluating the reference stress. In this study, the applicability of several limit load solutions was investigated through comparison with the results of elastic-plastic finite element analyses (FEA). A pipe containing a circumferential surface crack was analyzed under pure bending load. Six materials used in nuclear power plants were assumed. It was shown that the reference stress method is valid for FAC evaluation. The maximum non-conservativeness caused by using the reference stress method is less than 20% compared to the results obtained by FEA.     

Evaluation of leak-before-break assessment methodology for pipes with a circumferential through-wall crack. Part I: stress intensity factor and limit load solutions

The J-integral and the crack opening area are the main parameters required for a leak-before-break evaluation of a piping system. Stress intensity factor and limit load solutions have been widely used for evaluating these parameters in a simplified way. Solutions for the stress intensity factor and limit load for a pipe with a circumferential through-wall crack subjected to axial and bending loads are reviewed and compared in this study. Based on the comparisons, recommendations are then made on expressions for calculating these parameters.     

Assessment of a creep-fatigue crack initiation and crack propagation for a welded cylindrical shell

An assessment of the creep-fatigue crack bahaviour for a cylindrical structure with weldments has been carried out by using a structural test and an evaluation with an assessment procedure. The structural specimen with a diameter of 600 mm and thickness of 7 mm is a welded cylindrical shell made of 316L stainless steel (SS) for one half of the cylinder and 304 SS for the other half. Eight artificial defects were machined and the defect behaviours were examined. In the creep-fatigue test, the hold time was 1 h at 600 °C and the primary nominal stress was 45 MPa. The evaluation results for the creep-fatigue crack initiation and crack propagation were compared with those of the observation images from the structural test. The assessment results for the creep-fatigue crack behaviour by using the French A16 procedure showed that the A16 guide is reasonably conservative but overly conservative for the creep-fatigue crack propagation in the case of a short hold time. It was shown that the crack initiation and propagation were dominated by a creep.     

Experimental calibration of stress intensity factor for piping with circumferential through-wall crack

A survey of the literature shows that the existing stress intensity factor solutions for circumferential through-walled cracks in piping may be classified into three categories. One category is based on Sanders' analytical results for long pipe cracks, with various corrections in the short crack range and different curve-fitting formulae to give convenient closed form expressions. The second category consists of various independent finite element solutions. Each of these solutions is for a discrete pipe geometry and crack length and so is not practical to be used in fracture mechanics analysis. Lastly there is Kanninen & Zahoor's solution, derived independently of Sanders' results. Comparison showed that the results from the first two categories roughly agreed but were vastly different from Kanninen & Zahoor's results. Experimental calibration using the strain gauge method and the fatigue crack growth rate back-tracking method has been carried out. The experimental results agreed with the family of solutions derived from Sanders' work. Details about this experimental calibration are presented.     

Analysis of crack arrest event in NESC-1 spinning cylinder experiment

In the NESC-1 spinning cylinder test, a large surface-breaking flaw in a thick steel cylinder component was subjected to high primary and secondary stresses produced by combined rotation and thermal shock loading. The crack was arrested after relatively small amounts of ductile tearing and cleavage crack extension. Finite element analyses have been carried out to obtain static elastic stress intensity factors for the initial and arrested crack under constant load and constant displacement boundary conditions. Applied static elastic stress intensity factors for the arrested crack have been compared with the plane strain crack arrest toughness values measured using small-scale compact crack arrest (CCA) specimens. The present analyses of the crack arrest event in the NESC-1 spinning cylinder test have concluded: (1) Applied static elastic stress intensity factors are reduced significantly for the lobe-shaped arrested crack which developed from the initial semi-elliptical surface crack as a result of the localised cleavage crack propagation. This reduction in crack driving force is likely to be the main reason for crack arrest. (2) The analysis carried out and comparison with the full-scale experiment confirm the prevailing approach to the assessment of crack arrest that brittle propagation will stop if the applied crack driving force falls below the crack arrest toughness. (3) The results justify the use of the static elastic stress intensity factor as the crack propagation driving force parameter and the static plane strain crack arrest toughness as the resistance parameter for crack arrest evaluation for small relative crack jump dimensions. (4) The small-scale CCA tests can be employed to evaluate crack arrest in a large cylinder of the same material.     

Experimental determination of the stress intensity factor for a part-through thickness crack

Based on the relationship between the stress intensity factor and the plastic zone size at the crack tip, an experimental method for the evaluation of the stress intensity factor for a part-through thickness crack is proposed. Approximating the geometrical shape of the crack to a semi-elliptical surface crack, an empirical K-expression for the surface crack in a plate under bend loading configuration is derived and verified experimentally where the new K-expression contains both crack depth and length as a crack tip singularity parameter. The experimental work makes use of newly developed etching techniques to reveal the plastic zone size in deformed A533B pressure vessel steel.     

Three-dimensional crack analysis applied to an LWR nozzle-cylinder intersection

Evaluations of the stress intensity factors around the profiles of four crack configurations at the inner crotch corner of a typical LWR nozzle-cylinder intersection in the pressure vessel are described. Three-dimensional finite element techniques are used, incorporating such recent developments to enhance the accuracy of the computed results as quadratic isoparametric elements with special singularity elements around the crack profile and a virtual crack extension method. Comparison with an alternative finite element set of results and suitable bounding approximate solutions are made.     

Proposal of reference stress for a surface flaw on a cylindrical component from a review-with-comparison of the local metal loss assessment rule between API 579-1 and the p-M diagram method

The Remaining Strength Factor (RSF) approach in Part 5 of API 579-1/ASME FFS-1 is an assessment method for a cylindrical component with a local metal loss based on surface correction factors. Also, reference stress solutions that are applied in the Failure Assessment Diagram (FAD) method for a cylindrical component with a crack-like flaw are provided in Annex D using surface correction factors. In the recently-developed p-M diagram method, the reference stress solution for local metal loss evaluation in a cylindrical component is derived using bulging factors, which are similar but not identical to the surface correction factors used in API 579-1/ASME FFS-1. This paper describes the results of a comparative study among the RSF approach, reference stress solutions for the FAD method, and the p-M diagram method, in terms of plastic collapse evaluation of a cylindrical component. These results were compared with the FEA and experimental results to confirm how these estimated stresses could be validated. This study also involves recommended reference stress solutions for a cylindrical component with a crack-like flaw or a local metal loss, which should be adopted as fitness-for-service rules, and a discussion on the influence of the design margin of the construction code on allowable flaw depth.     

Influence of reference stress formulae on creep and creep-fatigue crack initiation and growth prediction in plate components

Creep and creep-fatigue crack growth in pre-cracked plates of 316L(N) austenitic stainless steel, containing a semi-elliptical surface defect and tested at 650 °C under combined axial and bending loading, are investigated. The results have been interpreted in terms of the creep fracture mechanics parameter C^∗ and compared with data obtained on standard compact tension (CT) specimens of the same material and batch. In making the assessments, the reference stress method has been used to determine C^∗. Several formulae exist for calculating the reference stress depending on whether it is based on a ‘global’ or a ‘local’ collapse mechanism and the assessment procedure adopted. When using this approach, it has been found that the most satisfactory comparison of crack growth rates with standard CT specimen data is obtained when the ‘global’ reference stress solution is used in conjunction with mean uniaxial creep properties. It has been found that the main effect of changing the fatigue cycle range from 0.1 to −1.0 is to cause an acceleration in the early stage of cracking.     

A study on three-dimensional crack tip fields

In this paper several kinds of crack problems are numerically analyzed by the three-dimensional finite element method. They are (1) CT specimens with different thicknesses; (2) stable crack growth in CT specimens: (3) surface cracked plate subjected to bending: (4) pipe with surface crack subjected to bending: (5) CT and surface cracked specimens made of welded plate. The crack tip singular fields are compared with HRR solutions and the Q-factor is evaluated. The effects of the J-integral and Q-factor on the crack growth behavious are discussed with the comparison of the experimental results. It is concluded that the J-integral does not uniquely control the stable crack growth behaviors in many three-dimensional crack problems. It is also found that both J-integral and Q-factor play important roles in the stable crack growth especially in the welded plate.     

Modeling the fatigue crack growth and propagation life of a joint of two elastic materials using interface elements

In this study, a new approach for the simulation of fatigue crack growth in two dissimilar elastic materials has been developed and specifically, the concept has been applied to a butt-welded joint. The phenomena of crack propagation and interface debonding can be regarded as the formation of new surface. Thus, it is possible to model these problems by introducing the mechanism of surface formation. In the proposed method, the formation of new surface is represented by an interface element based on the interface potential energy. The properties of this interface element represent the bonding strength of the material. As the cyclic load continues, the bonding strength decreases between the interacting surfaces and the crack propagates slowly. Based on this concept, an ANSYS code has been written for the simulation of crack propagation. Using this code, the fatigue crack growth rate and fatigue crack propagation life of a 2D Finite Element Method (FEM) model of a butt-welded joint for different stress ratios have been analyzed and presented in this paper. The variation of the crack opening displacement (COD) over crack lengths and crack tip stress and strain over crack tip stress have also been presented. The trend of fatigue crack growth rate is similar to results found in the literature. For validation, the predictions are compared with experimental results and reasonably good comparisons are found. The method is relatively simple compared to other conventional FEM methods and it eliminates the limitation of crack propagation of one element length per cycle and saves computer time significantly.     

Analysis of an interface crack of arbitrary shape in a three-dimensional transversely isotropic magnetoelectrothermoelastic bimaterial—part 2: Numerical method

The extended displacement discontinuity (EDD) boundary integral equation and boundary-element method are extended and developed to analyze an arbitrarily shaped, planar interface crack in a three-dimensional, transversely isotropic, magnetoelectrothermoelastic bimaterial under combined, thermoelectromagnetomechanical loadings. The fundamental solutions for uniformly distributed EDDs applied over a constant triangular element are obtained through integrating the fundamental solutions for the unit-point EDDs given by Part 1 over the triangular area. To eliminate the oscillatory singularity near the crack front, the Dirac delta function in the integral–differential equations is approximated by the Gaussian distribution function, and accordingly, the Heaviside step function is replaced by the Error function. The extended stress intensity factors without oscillatory singularities, the energy release rate, and the local J-integral in terms of intensity factors are all obtained. To validate the solution, the EDD boundary-element method is proposed. As an application, an elliptical interface crack is numerically simulated. The influences of the applied combined loadings and material-mismatch as well as the ellipticity ratio on the multiphysical response are studied.     

Fundamental solutions in piezoelectricity. Penny-shaped crack solution

The problem of electroelasticity for piezoelectric materials is considered. For axially symmetric states three potentials are introduced, which determine the displacements, the electric potentials, the stresses, the components of the electric field vector and the electric displacements in a piezoelectric body. These fundamental solutions are utilized to solve the penny-shaped crack problem. Two cases of boundary—value problems are considered, namely the permeable and impermeable crack boundary conditions. Exact solutions are obtained for elastic and electric fields. The main results are the stress intensity factor for singular stress and the electric displacement intensity factor. The numerical results are presented graphically to show the influence of applied mechanical and electrical loading on the analyzed quantities and to clarify the effect of anisotropy of piezoelectric materials. It is show that the influence of anisotropy of the materials on these fields is significant.     

Validation of surface crack stress intensity factors of a tubular K-joint

Tubular K-joints are encountered widely in offshore structures, and the prediction of damaged joints depends very much on the accuracy of stress intensity factor solutions (SIFs). No parametric equations and very few results have been proposed and published in the literature for estimating the SIFs of any K-joints subjected to complex loading conditions. In this paper, a mesh generation method proposed previously for the Y-joint and T-joint has been extended to the K-joint. This method is realized by dividing the K-joint into several sub-zones with each zone consisting of different types of elements and mesh densities. This method has a distinct advantage of controlling the mesh quality, and most importantly the aspect ratio of the elements along the crack front. When the mesh of all the sub-zones has been generated automatically and completely, they are merged to form the complete model. The two most commonly used methods, namely the J-integral and displacement extrapolation, are used to evaluate the SIF values along the crack front of a typical K-joint. To validate the accuracy of these computed SIFs, a full-scale K-joint specimen was tested to failure under fatigue loading conditions. The standard alternating current potential drop (ACPD) technique was used to monitor the rate of crack propagation of the surface crack located at the hot spot stress region. Using the given material parameters C and m, the experimental SIFs were deduced, and they are found to be in good agreement with the computed SIFs obtained from the generated models. Hence, the proposed finite element models are both efficient and reliable.     

The transient thermo-electro-elastic fields in a piezoelectric plate with a crack

This paper is concerned with a thin plate made by a piezoelectric ceramic material and containing a crack perpendicular to its surfaces. It is assumed that the transient thermal stress is set up by the application of a heat flux as a function of the time and position along the crack edge and the heat flow by convection from the plate surfaces. The plate is also subjected to mechanical and electric loadings. The exact analytical formulae are obtained for transient thermo-electro-elastic fields in the plate. The exact analytical solutions for the stress and electric displacement intensity factors and crack-opening displacement are obtained. Numerical examples show, among others, a dependence of the stress and electric displacement intensity factors on the thermal and elastic, piezoelectric and dielectric constants of the piezoelectric materials.     

Plasticity around an axial surface crack in a cylindrical shell

This paper presents a plasticity model for deep axial surface cracks in pressurised pipes. The model is used in an investigation of the relative merits of fracture criteria based on COD and plastic instability.Recent investigations have shown that the inconsistency of the singular bending stress field in an axially cracked cylindrical shell arising from use of classical eighth order shallow shell theory is removed when use is made of a tenth order shell theory which accounts for transverse shear deformations. Although the membrane stresses are only moderately affected, the influence on the bending stresses is considerable. In the case of surface cracks moments are induced due to the eccentricity of the crack and transverse shear effects should therefore be included.A plasticity model for a rectangular axial surface crack is developed. Like a previous surface crack model by Erdogen and Ratwani,^3–5 it generalises Dugdale's assumption of a concentrated yield zone in the plane of the crack but, contrary to that model, transverse shear effects are included and a continuous stress distribution is assumed in the yield zone. The inherent difficulties arising from the use of shell theory to model a three-dimensional problem can be overcome when the crack is sufficiently deep and the material is so ductile that full yield of the section around the crack develops before failure. In that case the calculations confirm the initial assumption of separation of the crack surfaces and the sides of the yield zone.The model is used to analyse published test data on surface cracked pressurised pipes. The analysis consists in COD evaluation and estimate of failure as a consequence of plastic instability. A method is proposed which deals with the problem by simultaneous analysis of a number of cracks with increasing depth. The method avoids iterations and enables, for any load and crack length, calculation of the smallest crack depth which would cause instability.     

The shape of a surface crack in a plate based on a given stress intensity factor distribution

A new method is developed for the evaluation of a crack shape based on a given stress intensity factor (SIF) distribution for a surface crack under Mode-I loading conditions. The SIF distribution along the crack front is investigated using a direct simulation technique, in which the effect of crack closure on fatigue crack growth is considered. Then a SIF distribution function is chosen based on the numerical results. Crack shape (and the SIF) is achieved based on the given SIF distribution function using a numerical iterative procedure. Empirical SIF equations for surface cracks in plates subjected to tension and pure bending fatigue load are determined by systematic curve fitting of the numerical results. The depth ratio and the aspect ratio are considered in the ranges of 0.1–0.9 and 0.2–1.2, respectively. The aspect-ratio variation of surface cracks under fatigue loading is predicted. The application of the new method to predict the shape of a surface crack in plates subjected to tension and pure bending and comparisons of the results obtained with the predictions of the empirical equations proposed by Newman and Raju are presented.     

A phenomenological theory of subcritical creep crack growth in a cylindrical vessel weakened by an axial part-through crack

A phenomenological theory of subcritical creep crack growth is formulated for materials with large creep exponent. The subcritical crack growth is shown to be mainly controlled by the average net section stress and the plastic deformations in the vicinity of the crack tip. Plastic zone size is evaluated by considering the effects of relaxation on the plastic stress singularity at the crack tip for a power hardening material. The theory has been applied to a cylindrical vessel weakened by an axial part-through crack. The predicted rupture behaviour compares favourably with published creep rupture tests on 9Cr 1Mo steel tubing pressurised at 550°C. The concept of flaw size stress is introduced and is used in failure mode prediction. Failure will be by breakage or leakage, depending on the relative values of flaw size stress and material yield stress.