Linear and non-linear analyses for semi-elliptical surface cracks in pipes under bending

In this paper, the J integral was calculated for semi-elliptical surface cracks in pipes under bending using three-dimensional finite element analysis. The computations were performed for elastic and elastic-plastic behaviours. For the elastic case, the numerical results allowed the extrapolation of shape functions for analytical determination of the J integral. The results are in a good agreement with those in the literature if the ratio between the radius and the thickness of the pipe (R/t) is from 1 to 10. The analysis was extended to values of the ratio R/t higher than 10. For the elastic-plastic, the numerical results are in good agreement with the analytical solution found in the literature for thick pipes (R/t ≥ 10). The effect of the ratio R/t becomes sensible when the ratio of the applied moment to the moment of reference (M/Mor) exceeds 0.9.     

Post-yield fracture mechanics theory and its application to pressure vessels

A post-yield fracture mechanics theory is presented based on Bilby, Cottrell and Swinden-Dugdale model solutions for cracks in bodies of finite width and subject to stress gradients. The model solutions are in good agreement with computed elastic-plastic values of the path independent integral J up to loads of 0·8 of the collapse load in the case of a cracked plate subject to bending. As an example, the model is applied to a thermal transient in a pressure vessel. The possibility of cracking and failure of a nozzle was considered. It is shown that a semi-circular crack will grow into an extended defect. Therefore, in the main analyses, axial symmetry and infinitely long longitudinal and fully circumferential cracks were considered. The post-yield fracture mechanics solutions are presented in the form of elastic-plastic stress intensity factors. Within the geometric approximations it is shown that longitudinal cracks are more dangerous than circumferential ones and can result in general yielding across, and full penetration of, the pressure vessel wall. In addition, linear elastic fracture mechanics under-estimates the danger of deep cracks and in some circumstances can considerably overestimate the critical size of small cracks. It is shown that the addition of a residual pressure stress considerably reduces the predicted critical defect sizes.     

Relevance of plastic limit loads to reference stress approach for surface cracked cylinder problems

To investigate the relevance of the definition of the reference stress to estimate J and C* for surface crack problems, this paper compares finite element (FE) J and C* results for surface cracked pipes with those estimated according to the reference stress approach using various definitions of the reference stress. Pipes with part circumferential inner surface cracks and finite internal axial cracks are considered, subject to internal pressure and global bending. The crack depth and aspect ratio are systematically varied. The reference stress is defined in four different ways using (i) a local limit load, (ii) a global limit load, (iii) a global limit load determined from the FE limit analysis, and (iv) the optimised reference load. It is found that the reference stress based on a local limit load gives overall excessively conservative estimates of J and C*. Use of a global limit load clearly reduces the conservatism, compared to that of a local limit load, although it can sometimes provide non-conservative estimates of J and C*. The use of the FE global limit load gives overall non-conservative estimates of J and C*. The reference stress based on the optimised reference load gives overall accurate estimates of J and C*, compared to other definitions of the reference stress. Based on the present findings, general guidance on the choice of the reference stress for surface crack problems is given.     

Net-section limit moments and approximate J estimates for circumferential cracks at the interface between elbows and pipes

This paper firstly presents net-section limit moments for circumferential through-wall and part-through surface cracks at the interface between elbows and attached straight pipes under in-plane bending. Closed-form solutions are proposed based on fitting results from small strain FE limit analyses using elastic–perfectly plastic materials. Net-section limit moments for circumferential cracks at the interface between elbows and attached straight pipes are found to be close to those for cracks in the centre of elbows, implying that the location of the circumferential crack within an elbow has a minimal effect on the net-section limit moment. Accordingly it is also found that the assumption that the crack locates in a straight pipe could significantly overestimate the net-section limit load (and thus maximum load-carrying capacity) of the cracked component. Based on the proposed net-section limit moment, a method to estimate elastic–plastic J based on the reference stress approach is proposed for circumferential cracks at the interface between elbows and attached straight pipes under in-plane bending.     

Transferability of fracture parameters from specimens to component level

The structural integrity of components is usually performed using the specimen fracture resistance curve. However, the specimen fracture resistance curve significantly differs from the component fracture resistance. This is the most serious limitation of classical fracture mechanics. To address this issue, several tests have been carried out on fractured specimens and piping components under an Indo-German bilateral project. Two approaches, namely, two-parameter fracture mechanics and micro-mechanical models are considered to investigate the feasibility of transferability. For the two-parameter fracture mechanics approach, the J-integral has been used as the crack driving force and q is used as a measure of stress triaxiality. The triaxiality quotient q is proportional to the ratio of the hydrostatic stress and the von Mises effective stress and is an additional parameter to make a decision about the initiation value of the J-integral for the failure behaviour of a component. It is shown that if the triaxial conditions match for any two arbitrary geometries, it is feasible to transfer the fracture parameters. The difficulty in transferability is largely overcome by damage mechanics, which models the drop in load carrying capacity of a material with increase in plastic strain. Such modeling is done considering nucleation, growth and coalescence of voids in a material following large-scale plasticity. The Gurson–Tvergaard–Needleman and Rousselier models are used. Some of the results obtained by these models and comparisons with experimental results are presented in this paper to demonstrate the usefulness of damage mechanics in analyzing components with flaws.     

Limit load and J estimates of a centre cracked plate with an asymmetric crack in a mismatched weld

The limit load and J estimates of a centre cracked plate with an asymmetric crack in the tensile properties mismatched weld were investigated. A limit load expression was derived on the basis of a simplified slip-line field. A good agreement between the predictions of the expression and finite element (FE) results was found for ratios of half-weld width to the crack ligament, H/l, of less than 0.5. The equivalent stress–strain relationship method (ESSRM) was used to predict elastic–plastic J values. Results from FE analyses show that the ESSRM is accurate for the crack with asymmetry in the mismatched weld provided an accurate theoretical or numerical value of the limit load of the same specimen is available. Defect assessment methods are discussed, and it is found that the failure assessment diagram (FAD) of an asymmetrically cracked mismatched weld can be constructed from the equivalent stress–strain relationship for the same mismatched geometry with a symmetric crack. The effect of an asymmetric crack on the FAD may then be covered by the limit load solution for the asymmetrically cracked mismatched weld.     

Fracture analysis of hydrogen storage composite cylinders with liner crack accounting for autofrettage effect

An understanding of fracture behavior is crucial to the safe installation and operation of high-pressure composite cylinders for hydrogen storage. This work has developed a comprehensive finite element model to investigate axial surface flaws in cylinder liners using the fracture mechanics and a global–local finite element technique. Since the autofrettage process has a strong influence on cylinder fracture behavior, it is also considered in this analysis. The simulation process is broken down into three steps in order to precisely extract fracture parameters and incorporate the autofrettage effect. In the first step, the global model performs the autofrettage simulation to study the residual stress with consideration of both material hardening and the Bauschinger effect. In the second step, the global model uses residual stress to compute displacement for the local model. Finally, in the third step, the local model extracts the values of stress intensity factor and J-integral. Comparison is conducted on the fracture parameters with various autofrettage levels and crack shapes. The vicinity of the crack front is also studied by the size and shape of the plastic zone, and the validity of stress intensity factor and J-integral dominances is examined.     

Allowable sizes of axial flaws in pressurized pipes made of moderate toughness materials

Failure stresses for axially part-through flawed pipes made of moderately tough materials are predicted by several fracture mechanics. However, allowable flaw sizes using these fracture mechanics cannot be simply described because there are many effective parameters such as pipe diameter, wall thickness, material properties, etc. To establish codes and standards to evaluate flaws for piping of light water reactors, we determine unified allowable sizes for axial flaws in pipes subjected to internal pressure from J-integral based fracture mechanics. The allowable sizes are simply tabulated using a single parameter which consists of pipe geometry and material properties.     

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.     

Optimum material distributions for prescribed apparent fracture toughness in thick-walled FGM circular pipes

This study treats the inverse problem of evaluating optimum material distributions intending to realize prescribed apparent fracture toughness in thick-walled functionally graded material (FGM) circular pipes. The incompatible eigenstrain induced in the pipes after cooling from the sintering temperature due to the nonhomogeneous coefficient of thermal expansion is taken into consideration. An approximation method of finding stress intensity factors for a crack in the FGM pipes is introduced in which the nonhomogeneous material properties are simulated by a distribution of equivalent eigenstrain. A radial edge crack emanating from the inner surface of the homogenized pipes is considered for the case of a uniform internal pressure applied to the surfaces of the pipes and the crack. The stress intensity factors determined for the crack in the homogenized pipes represent the approximate values of the stress intensity factors for the same crack in the FGM pipes, and are used in the inverse problem of evaluating optimum material distributions intending to realize prescribed apparent fracture toughness in the FGM pipes. Numerical results obtained for a thick-walled TiC/Al₂O₃ FGM pipe reveal that the apparent fracture toughness significantly depends on the material distributions, and can be controlled within possible limits by choosing an optimum material distribution profile.     

The effect of mechanical heterogeneity and limit load of a weld joint with longitudinal weld crack on the J-integral and failure assessment curve

According to the CEGB R6 (Rev.3) approach, the influence of strength mis-matching and the limit load of a weld joint with a longitudinal weld crack on the J-integral and failure assessment curve can be studied by using an elastic–plastic finite element method for Center-Cracked Panel (CCP) specimens. The results indicate that the values of the J-integral and the shapes and positions of the failure assessment curves are greatly affected by the strength mis-matching factor M, a ratio of weld metal yield strength to that of base metal. If the limit load of the base metal is adopted to normalize the applied load, then the greater the value of M, the larger the safe area is in the failure assessment curve (FAC). However, if the limit load of the weld metal is adopted to normalize the applied load, then the greater the value of M, the smaller the safe area is. Therefore, for the undermatched and evenmatched joints, it is safer to choose the limit load of the base metal as the normalized load, and for the overmatched joints, it is safer to choose the limit load of the weld metal as the normalized load. Moreover, when M is less than 0.8 for the former situation, the option 1 curve of the R6 is not a conservative assessment curve. Considering that there is no simple theoretical formula which is suitable for calculating the limit load of a mechanical heterogeneous weld joint under plane stress and a variety of crack geometries, it is recommended that no matter what the strength of the overmatched or undermatched weld joint is, it is safer to use the limit load of that metal which has the higher strength grade of base metal and weld metal as the normalized load.     

Fracture mechanics analysis of stable crack growth under sustained load and instability of circumferential cracks in straight water-steam pipes

In order to determine the component behaviour of pipes made from X 20 CrMoV 121 steel for the water-steam circulation system of a helium-cooled high-temperature reactor component, tests were carried out under load conditions similar to the case of emergency cooling. The aim of the tests was to determine the stable and unstable crack growth in the less tough weld filler metal of girt-welded pipes of 18·5 mm wall thickness and 400 mm inside diameter. The tests were carried out with weld filler metals of two different toughnessesSuch stable crack growth could be of particular importance in the case of residual heat removal. For this reason two tests were carried out on pipes with a circumferential crack under sustained load.In the second part of the test programme pipes with defects were loaded under sustained internal pressure, and additionally an increasing bending moment was applied. The aim of this investigation was to determine the boundary conditions for failure occurring by leak or by fracture with regard to the different toughness of the weld filler metal.The fracture mechanics analysis was carried out using a modified version of the flow-stress criterion for circumferential cracks (Batelle approach) and using also the so-called ‘effective fracture toughness’ $\text{K}{}_{\mathrm{<mtext>eff</mtext>}}$.The investigation has revealed that the component toughness is adequately high and component failure by fracture is not expected. The experimental results can be described theoretically using either the Battelle approach or the formulae of linear-elastic fracture mechanics if the data obtained in the test characterising component behaviour (flow stress or effective toughness) are used as a basis. The results show that an evaluation of component behaviour using $\text{K}{}_{\mathrm{<mtext>Jo</mtext>}}$ values determined by CT specimens according to the $\text{J}$ integral procedure is too conservative.     

Use of local and global limit load solutions for plates with surface cracks under tension

Some available experimental results for the ductile failure of plates with surface cracks under tension are reviewed. The response of crack driving force, J, and the ligament strain near the local and global limit loads are investigated by performing elastic-perfectly plastic finite element (FE) analysis of a plate with a semi-elliptical crack under tension. The results show that a ligament may survive until the global collapse load is reached when the average ligament strain at the global collapse load, which depends on the uniaxial strain corresponding to the flow stress of the material and the crack geometry, is less than the true fracture strain of the material obtained from uniaxial tension tests. The FE analysis shows that ligament yielding corresponding to the local limit load has little effect on J and the average ligament strain, whereas approach to global collapse corresponds to a sharp increase in both J and the average ligament strain. The prediction of the FE value of J using the reference stress method shows that the global limit load is more relevant to J-estimation than the local one.     

A semi-empirical approach and uncertainty analysis to pipes under hydrogen embrittlement degradation

The Hydrogen Embrittlement (HE) appearance is the main reason for the desire to evaluate the burst pressure of pipes. Several existing models predict burst pressure of hydrogen-induced pre-cracking pipes with the fracture mechanics criterion (fracture toughness, J_cr). Still, none of them has taken into account the model uncertainty. In this paper, we propose a surrogated model associated with molecular models. The discrete values of the dimensionless influence functions presented for elastic and plastic components of J are connected by machine learning (Random Forest Regression, RFR) and third-order polynomial functions with optimized factors to avoid the inconvenience of using the lookup table. The molecular empirical models are obtained by the up-to-date Balancing Composite Motion Optimization (BCMO) algorithm. Because samples of elastic and plastic databases are limited and the final output, P_b, is not directly derived from molecular models, all databases are used for training without the conventional data splitting to train and test sets.Consequently, the proposed approaches (RFR model and Empirical model) are validated based on the experiments collected globally. The final models are accounted for the residual distribution as the unavoidable component. The efficiency of the Random Forest and empirical models is validated by experiments or simulations from the literature when the evaluation metrics (i.e., R-square, Mean Absolute Error) are (0.9666, 0.8695 MPa) and (0.9701, 0.7996 MPa), respectively. The drawbacks of the proposed models, heavily dependent on databases, are also illustrated and discussed for further development. Strict boundaries of input and output, especially the fracture toughness, which is commonly degraded due to the effect of HE, from the test set combined with the uncertainty of the models based on the analysis of model residuals are also proposed for validation.     

Influences of some materials on J-estimation methods for pipes with circumferential surface cracks

The approximate calculation methods (SC.ENG1 and SC.ENG2) for the J-integral for pipes with circumferential surface cracks are discussed and three-dimensional elastic–plastic finite element models for circumferentially surface-cracked pipe are conducted to evaluate the accuracy of these methods for different pipe materials used in China. The numerical studies verify that the SC.ENG2 method provides more accurate estimates of J than SC.ENG1. Based on three-dimensional elastic–plastic fracture analysis, the distribution of the local J-integral along the front of a circumferential constant-depth internal surface crack is investigated and the influences of different pipe materials with different yield plateaux on J-integral values are discussed. The validity of SC.ENG1 and SC.ENG2 J-integral estimation methods for pipe steel materials with different yield plateaux used in China are examined in detail and the SC.ENG2 method is found to provide reasonable estimates of J for materials with yield plateaux.     

Leak-before-break qualification of primary heat transport piping of 500 MWe Tarapur atomic power plant

The advent of Leak-Before-Break (LBB) concept has now replaced the traditional design basis event of the Double-Ended-Guillotine-Break (DEGB) to design the Primary Heat Transport (PHT) system piping of the Pressurised Heavy Water Reactor (PHWR) and Pressurised Water Reactor (PWR). This approach is being adopted to design the PHT system piping of 500 MWe Indian PHWR to be built at Tarapur (Tarapur Atomic Power Plant 3 and 4). The LBB concept basically demonstrates through fracture mechanics analysis that there is negligible chance of any catastrophic break of PHT pipes without prior indication of leakage. There are several steps in this work of LBB qualification, namely, evaluation of loads on the piping components, generation of tensile and fracture properties of PHT pipe base and weld material, determination of leakage size crack (LSC) and the elastic–plastic fracture mechanics (EPFM) and limit load analysis of the piping components with postulated LSC to evaluate the critical load at unstable ductile tearing and the limit load, respectively. The paper deals with the fracture analysis of the straight pipes and elbows of three pipe lines in the PHT system of TAPP 3 and 4. Three crack configurations are considered in the analysis. These are throughwall circumferential crack at the weld location of straight pipe and extrados of the elbow and throughwall axial crack at the elbow crown. In all the cases, necessary factor of safety with respect to the anticipated safe shutdown earthquake (SSE) load and LSC are shown to be more than the minimum required values for LBB qualification.     

Views of TAGSI on effects of neutron irradiation on ductile tearing in ferritic steels

The paper reviews information pertaining to effects of neutron irradiation on “upper-shelf” Charpy impact behaviour and on elastic/plastic fracture mechanics characterising parameters, again for “upper shelf” conditions, in which the initiation and early growth of a crack involve ductile tearing. The hardening and associated reduction in strain-hardening capacity induced by irradiation gives rise to a decrease in Charpy upper shelf energy. Effects on J-based parameters are more complicated. The material resistance parameters tend to increase for low dose, but decrease at high dose, when the decrease in crack-tip ductility outweighs the effect of hardening. High doses can produce “fast shear” fracture, which propagates rapidly and is therefore more likely to induce brittle cleavage fracture. The situation is exacerbated if the irradiation also promotes inter-granular segregation and fracture, hence reducing the local brittle fracture stress. For the levels of irradiation experienced by the types of UK civil reactors in operation, no fracture instability is expected to arise as a result of ductile fracture mechanisms alone.     

Biaxial stress effects on estimating J under combined mechanical and thermal stresses

In this paper, the applicability to bi-axial stress states of a simple approximation proposed in our previous work for quantifying the elastic–plastic J under combined primary and secondary stresses is investigated.To produce the bi-axial stress states, circumferential cracked pipes under combined pressure, axial tension (or compression) and thermal stresses are considered. The results suggest that the proposed approximation can be applied to bi-axial stress states, and is slightly more conservative for bi-axial mechanical stresses than for uni-axial ones. The degree of conservatism decreases with increasing strain hardening exponent and with increasing relative magnitude of secondary stress.     

Limit loads of circumferentially flawed pipes and cylindrical vessels under internal pressure

Published limit load formulae for circumferential defects overestimate the burst pressure for penetrating defects in pipes by the factor two in the short crack limit, because they only consider axial stress. Therefore, a class of limit load solution is discussed which takes the triaxial state of stress into account. The solutions for pressure loaded crack faces are improved analytically. Primal–dual limit analysis with the finite element method is used to adjust all solutions to numerical results. Limit loads are obtained for circumferential cracks of all sizes in thick-walled cylinders.     

Principles of the German approach to the analysis of flawed structures

The analysis of flawed structures in the linear elastic regime (LEFM) has been treated very extensively, especially in the framework of the 4th German nuclear programme. Investigations with large specimens have demonstrated the applicability of the LEFM for a variety of materials ranging from low to high toughness levels. In the elastic-plastic (EPFM) regime, however, especially the J-integral method has been used for quantifying the safety margin, because with a required value of upper shelf Charpy energy alone this quantification is not possible.

A correlation between the crack initiation value J_i and Charpy upper shelf energy was evaluated experimentally, justifying the generally upgraded upper shelf energy level adopted in the Code. Furthermore, this correlation can be employed when fracture toughness values are not available as is mostly the case for irradiated material from surveillance programmes.

To cope with the problems resulting from repeated transients in a complex component, incipient crack intitiation and propagation under cyclic thermal load have been investigated experimentally and theoretically on a RPV nozzle corner in the HDR plant and on a thick-walled (200 mm) hollow cylinder subjected to pressurized thermal shock (PTS). Since the OCA code covers only the linear elastic range, for this loading case elastic-plastic fracture mechanics calculations have been carried out by means of the Finite Element Method. The first test performed with high toughness material has shown good agreement with the J-integral approach. Additional validation tests are under way to demonstrate the lowest tolerable toughness level to withstand PTS without catastrophic failure.

With respect to critical flaw sizes in degraded piping and vessels, the ‘leak before break’ limit curve for different loading conditions has been established and experimentally validated using piping and model vessels of different sizes, crack configurations and toughness levels.

The still existing uncertainties in the detection, sizing and interpretation of signals from nondestructive examination are the background of NDE validation programmes for both medium size and full size reactor pressure vessels. Acoustic emission trials as well as extensive ultrasonic (US) examinations will be pursued in cooperation programmes, last but not least in the framework of PISC III.