Crack opening analysis of complex cracked pipes

Based on the reference stress approach, two sets of the crack opening displacement (COD) estimation equations are proposed for a complex cracked pipe. One set of equations can be used for the case when full stress-strain data are known, and the other for the case when only yield and tensile strengths are available. To define the reference stress, a simple plastic limit analysis for the complex cracked pipe subject to combined bending and tension is performed, considering the crack closure effect in the compressive-stressed region. Comparison with ten published test data and the results from the existing method shows that the present method not only reduces non-conservatism associated with the existing method, but also provides consistent and overall satisfactory results. These results provide sufficient confidence in the use of the present method to estimate the COD (and thus the leak rate) for the Leak-before-Break (LBB) analysis of complex cracked pipes. Finally, the J-estimation equations are also provided for complex cracked pipes, for the LBB analysis of complex cracked pipes.     

Elastic–plastic J and COD estimation schemes for 90° elbow with throughwall circumferential crack at intrados under in-plane opening moment

Leak-before-break (LBB) assessment of primary heat transport piping of nuclear reactors involves detailed fracture assessment of pipes and elbows with postulated throughwall cracks. Fracture assessment requires the calculation of elastic–plastic J-integral and crack opening displacement (COD)¹ for these piping components. Analytical estimation schemes to evaluate elastic–plastic J-integral and COD simplify the calculations. These types of estimation schemes are available for pipes with various crack configurations subjected to different types of loading. However, such schemes for elbow (or pipe bend), which is one of the important components for LBB analyses, is very meager. Recently, elastic–plastic J and COD estimation scheme has been developed for throughwall circumferentially cracked elbow subjected to closing bending moment. However, it is well known that the elbow deformation characteristics are distinctly different for closing and opening bending modes because the ovalisation patterns of elbow cross section are different under these two modes. Development of elastic–plastic J and COD estimation scheme for an elbow with throughwall circumferential crack at intrados subjected to opening bending moment forms the objective of the present paper. Experimental validation of proposed J-estimation scheme has been provided by comparing the crack initiation, unstable ductile tearing loads and crack extension at instability with the test data. The COD estimation scheme has been validated by comparing the COD of test data with the predictions of the proposed scheme.     

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.     

Quantification of pressure-induced hoop stress effect on fracture analysis of circumferential through-wall cracked pipes

This paper provides engineering J and crack opening displacement (COD) estimation equations for through-wall cracked (TWC) pipes under internal pressure and under combined internal pressure and bending. Based on selected 3-D FE calculations for the TWC pipe under internal pressure using power law materials, elastic and plastic influence functions for fully plastic J and COD solutions are tabulated as a function of the normalised crack length and the mean radius-to-thickness ratio. These developed GE/EPRI-type solutions are then re-formulated based on the reference stress concept. Such re-formulation not only provides simpler equations for J and COD estimation, but also can be easily extended to combined internal pressure and bending. The proposed reference stress based J and COD estimation equations are compared with elastic-plastic 3-D FE results using actual stress-strain data for Type 316 stainless steels. The FE results for both internal pressure cases and combined internal pressure and bending cases compare very well with the proposed J and COD estimates.     

Estimations of creep fracture mechanics parameters for through-thickness cracked cylinders and finite element validation

This paper compares engineering estimation schemes of C* and creep crack opening displacement (COD) for cylinders with circumferential and axial through‐thickness cracks at elevated temperatures with detailed 3D elastic‐creep finite element results. Engineering estimation schemes include the GE/EPRI method; the reference stress (RS) method where the reference stress is defined based on the plastic limit load; and the enhanced reference stress (ERS) method where the reference stress is defined based on the optimised reference load, recently proposed by the authors. Systematic investigations are made not only on the effect of creep‐deformation behaviour on C* and creep COD, but also on effects of the crack location, the cylinder geometry, the crack length and the loading mode. Comparison of the finite element (FE) results with engineering estimations provides that for idealised power law creep, estimated C* and COD rate results from the GE/EPRI method agree best with FE results, suggesting that published plastic influence functions for plastic J and COD for through‐thickness cracked cylinders are reliable. For general creep‐deformation laws where either primary or tertiary creep is important and thus the GE/EPRI method is hard to apply, on the other hand, the ERS method provides more accurate and robust estimations for C* and COD rate than the reference stress method. As these two methods differ only in the definition of the reference stress, the ERS method maintains benefits of the reference stress method in terms of simplicity, but improves accuracy of the estimated J, C* and COD results.     

Interfacial crack-tip constraints and <Emphasis Type="Italic">J</Emphasis>-integral for bi-materials with plastic hardening mismatch

This paper investigates interfacial crack tip stress fields and the J-integral for bi-materials with plastic hardening mismatch via detailed elastic-plastic finite element analyses. For small scale yielding, the modified boundary layer formulation with the elastic T-stress is employed. For fully plastic yielding, plane strain single-edge- cracked specimens under pure bending are considered. Interfacial crack tip stress fields are explained by modified Prandtl slip-line fields. It is found that, for bi-materials consisting of two elastic-plastic materials, increasing plastic hardening mismatch increases both crack-tip stress constraint in the lower hardening material and the J-contribution there. The implication of asymmetric J-integral in bi-materials is also discussed.     

Approximate J estimates for axial part-through surface-cracked pipes

This paper provides net‐section limit pressures and a reference stress based J estimation method for pipes with constant depth, internal axial surface cracks under internal pressure. Based on systematic small strain finite element (FE) limit analyses using elastic perfectly plastic materials, net‐section limit pressures are firstly determined, and based on FE results, a closed‐form limit pressure solution is proposed. Furthermore, based on the proposed limit pressure solution, a method to estimate elastic–plastic J is proposed based on the reference stress approach. When the reference stress is defined by the proposed (global) limit pressure, estimated J values based on the reference stress approach are overall slightly lower than FE results, implying that the method is non‐conservative. By re‐defining the reference using optimised reference loads, resulting J estimates agree well with FE results.     

Elastic-plastic fracture mechanics analyses of circumferential through-wall cracks between elbows and pipes

The present work proposes a method for elastic-plastic fracture mechanics analysis of the circumferential through-wall crack in weldment joining elbows and attached straight pipes, subject to in-plane bending. Heterogeneous nature of weldment is not explicitly considered and thus, the proposed method assumes cracks in homogeneous materials. Based on small strain finite element limit analyses using elastic-perfectly plastic materials, closed-form limit loads for circumferential through-wall cracks between elbows and straight pipes under bending are given. Then applicability of the reference stress-based method to approximately estimate J and crack opening displacement (COD) is evaluated. It was found that the limit moments for circumferential cracks between elbows and attached straight pipes can be much lower than those for cracks in straight pipes, particularly for a crack length of less than 30% of the circumference; this result is of great interest in practical cases. This result implies that, if one assumes that the crack locates in the straight pipe, limit moments could be overestimated significantly, and accordingly, reference stress-based J and COD could be significantly overestimated. For the leak-before-break analysis, accurate J and COD estimation equations based on the reference stress approach are proposed.     

Non-linear fracture mechanics analyses of part circumferential surface cracked pipes

This paper provides engineering estimates of non-linear fracture mechanics parameters for pipes with part circumferential inner surface cracks, subject to internal pressure and global bending. Solutions are given in the form of two different approaches, the GE/EPRI approach and the reference stress approach. For the GE/EPRI approach, the plastic influence functions for fully plastic J solutions are tabulated based on extensive 3-D FE calculations using deformation plasticity, covering a wide range of pipe and crack geometries. The developed GE/EPRI-type fully plastic J estimation equations are then re-formulated using the concept of the reference stress approach for wider applications. The proposed reference stress based estimates are validated against detailed 3-D elastic-plastic and elastic-creep FE results. For a total of 26 cases considered in this paper, agreement between the proposed reference stress based J and C ^* estimates and the FE results is excellent. An important aspect of the proposed estimates is that they not only are simple and accurate but also can be used to estimate J and C ^* at an arbitrary point along the crack front.     

On the transfer of specimen J–R curve to piping components with throughwall circumferential flaw

Transferability of the specimen J–R/J–T curve to the component level is an important issue in the field of fracture mechanics. Towards this goal, fracture experiments have been carried out on single‐edge bend (SE(B)) and compact tension (CT) specimens and throughwall circumferentially cracked straight pipes/elbows of 200 mm nominal bore (NB) diameter. The pipe material is SA 333 Gr 6 steel (low strength and high toughness material) and specimens are machined from the pipes. Subsequently, elastic–plastic finite‐element analyses have been carried out on these cracked components/specimens in order to evaluate the stress triaxiality levels. It is found that the triaxial levels for these cracked components are similar. Hence, similar fracture behaviour is expected for these components. Consequently, one of the pipe J–R curves is used as a reference J–R curve to consider the crack growth in the analysis and the load deformation behaviour of other pipes/elbows is predicted. The load deformation behaviour of the piping components is also predicted using an extrapolated J–R curve from a specimen that exhibits the similar triaxiality level to that of the cracked components. The predicted results are in good agreement with the experiments.     

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

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

Elastic-plastic J and COD estimation schemes for throughwall circumferentially cracked elbow under in-plane closing moment

Leak-before-break (LBB) assessment of primary heat transport piping of nuclear reactors involves detailed fracture assessment of pipes and elbows with postulated throughwall cracks. Fracture assessment requires the calculation of elastic-plastic J-integral and crack opening displacement (COD)¹ for these piping components. Analytical estimation schemes to evaluate elastic-plastic J-integral and COD simplify the calculations. These types of estimation schemes are available for pipes with various crack configurations subjected to different types of loading. However, no such schemes are available for throughwall circumferentially cracked elbow (or pipe bend), an important component for LBB analysis. In this paper, simple J and COD estimation schemes are proposed for throughwall circumferentially cracked elbow subjected to closing bending moment. The ovalisation of elbow cross-section has a significant bearing on its fracture behavior. Therefore, unlike conventional deformation theory plasticity analysis, incremental flow theory is adopted considering both material and geometric non-linearities in the development of the proposed estimation schemes. Although it violates Ilyushin’s theorem, it has been shown that the resulting estimation schemes is still reasonably accurate for engineering purposes. Finally, experimental/numerical validation has been provided by comparing the J-integral and COD between numerical/test data and predictions of the proposed estimation schemes.     

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.     

Automatic consistency integrations for the stress update of J2 elastoplastic rate constitutive equations with combined hardening

A novel method is introduced to study numerical integrations of J₂ elastoplastic rate constitutive equations with general combined hardening. The basic idea is to transform the usual time rate constitutive equations into those with reference to the equivalent plastic strain. By virtue of tensorial matrix operations, we show that these transformed equations may be converted to a linear differential system governing the shifted stress and the plastic multiplier. From this system, we derive explicit integrations for the shifted stress and then for the back stress and the Cauchy stress. We demonstrate that these results are accurate up to within a third order term of the equivalent plastic strain increment. In particular, for pure kinematic hardening, we show that the integrations obtained can achieve automatic enforcement of both the plastic consistency condition and the loading condition, thus bypassing the numerical treatment of the latter two. Furthermore, we explain that, with the new algorithm for the stress update, the continuum tangent moduli may be used to ensure a quadratic rate of convergency in Newton's iteration scheme for the balance equation. Numerical examples suggest that the new algorithm may be more accurate and efficient than the widely used return algorithm. Copyright © 2014 John Wiley & Sons, Ltd.     

Advances in J‐integral estimation of circumferentially surface cracked pipes

This paper describes enhanced J‐integral estimation schemes for pipes with circumferential semi‐elliptical cracks subjected to tensile loading, global bending and internal pressure. These schemes are given in two different forms to cover the wide ranges of geometries and material parameters; the modified GE/EPRI method and the modified reference stress method. In the former method, new plastic influence functions for fully plastic J‐integral estimation are developed based on extensive three‐dimensional finite element calculations. In the latter method, new optimized reference loads are suggested and utilized to predict the J values. To verify the feasibility of these two schemes, J‐integral values obtained from further detailed FE analyses are compared to those from the proposed schemes. Because the estimated J‐integrals agree fairly well with the detailed FE analysis results, the new solutions can be applied for accurate structural integrity assessment of different size pipes with a circumferential surface crack.     

Computations of energy release rate under monotonic and cyclic loading conditions

In case of an elastic–plastic fracture mechanics analysis, the determination of the energy release rate distribution is a crucial point. In the present paper, three numerical techniques: the virtual crack closure technique (VCCT), J-integral and energy derivative technique (EDT), are used to compute the energy release rate in a middle-crack tension specimen with the combined isotropic/kinematic hardening model. The results obtained by these methods are compared with each other under monotonic and cyclic loading conditions. Finally, it comes out that the difference of the VCCT method to the J-integral is rather insensitive to load increasing, especially when the traction >40% of yield stress, however, the deviation of the VCCT and J-integral results are within 10%, suggesting that one may use the VCCT for plastic cracked specimen analysis. The computations show that the EDT provides the same values for the monotonic as the J-integral if the plastic deformations are not large, but for high plastic loading the EDT overestimates the fracture energy. For cyclic loading case, VCCT method offers closer results as the elastic analytical results, also suggesting that the whole plastic dissipated energy in the loading process should be integrated. While EDT method gives the smaller results than the J-integral because of the energy dissipated in the unloading phase is considered in the loading process.     

Enhanced reference stress-based J and crack opening displacement estimation method for leak-before-break analysis and comparison with GE.EPRI method

Three-dimensional, elastic–plastic finite element analyses for circumferential through-wall cracked pipes are performed using actual tensile data of stainless steels, for two purposes. The first one is to validate the recently proposed enhanced reference stress (ERS) method to estimate the J -integral and rack-opening displacement (COD) for circumferential through-wall cracked pipes. The second one is to compare the J and COD results from the ERS method with those based on the GE/EPRI method, together with the finite element results. It is found that the GE/EPRI method generally provides poor estimations of J and COD, and such poor estimation results from the Ramberg–Osgood (R–O) fit. Moreover, the GE/EPRI results can be very sensitive to how the stress–strain data are fitted using the R–O relation. The most appropriate fitting range for the R–O fit depends on the material, and thus no specific guideline can be given. On the contrary, the J and COD estimations based on the ERS method give more accurate and robust results than the GE/EPRI estimation. The present results, together with the experimental validation presented in the previous paper, provide sufficient confidence in using the proposed method in the leak-before-break analysis.     

Reference stress based elastic-plastic fracture analysis for circumferential through-wall cracked pipes under combined tension and bending

This paper proposes an engineering method to estimate the J-integral and the crack opening displacement (COD) for circumferential through-wall cracked pipes under combined tension and bending. The proposed method to estimate the COD is validated against three published pipe test data, generated from a monotonically increasing bending load with a constant internal pressure, which shows excellent agreements. Further validation is performed against three-dimensional, elastic-plastic finite element results using actual tensile data of a TP316 stainless steel at the temperature of 288°C. The FE results of the J-integral and the COD, resulting from six cases of proportional and non-proportional combined tension and bending, compare very well with those estimated from the proposed method.Excellent agreements of the proposed method with experimental data and the detailed FE results firstly provide sufficient confidence in the use of the proposed method to the LBB analysis of through-wall cracked pipes under combined tension and bending. More importantly, these validations suggest that the proposed method can be used to any combination of the bending-to-tension ratio, not only for proportional loading but also for non-proportional loading. Finally the proposed method is simple to use, which gives significant merits in practice, and thus is easy to be generalised to more complex situations.     

Theoretical investigation of elastoplastic notch fields under triaxial stress constraint

In this paper, an exact elastic-plastic solution has been obtained based on the J ₂-deformation theory of plasticity for a plate having a circular hole under biaxial tension and triaxial stress constraint in linear elastic strain-hardening materials. The theoretical solution shows that a linear elastic solution of the equivalent strain can be used to linear elastic-power hardening plastic situation just by a simple variable replacement. Then a strain equivalent rule (SER) is proposed to predict the elastoplastic notch fields by use of the elastic solution. Validations against theoretical analyses and finite element calculation for various combinations of material properties, triaxial stress constraints, load levels show that the SER can be used to predict stress-strain distributions in the whole plastic zone effectively and conveniently.     

Bending modified J–Q theory and crack-tip constraint quantification

It is well known that the J–Q theory can characterize the crack-tip fields and quantify constraint levels for various geometry and loading configurations in elastic–plastic materials, but it fails at bending-dominant large deformation. This drawback seriously restricts its applications to fracture constraint analysis. A modification of J–Q theory is developed as a three-term solution with an additional term to address the global bending stress to offset this restriction. The nonlinear bending stress is approximately linearized in the region of interest under large-scale yielding (LSY), with the linearization factor determined using a two-point matching method at each loading for a specific cracked geometry in bending. To validate the proposed solution, detailed elastic–plastic finite element analysis (FEA) is conducted under plane strain conditions for three conventional bending specimens with different crack lengths for X80 pipeline steel. These include single edge notched bend (SENB), single edge notched tension (SENT) and compact tension (CT) specimens from small-scale yielding (SSY) to LSY. Results show that the bending modified J–Q solution can well match FEA results of crack-tip stress fields for all bending specimens at all deformation levels from SSY to LSY, with the modified Q being a load- and distance-independent constraint parameter under LSY. Therefore, the modified parameter Q can be effectively used to quantify crack-tip constraint for bending geometries. Its application to fracture constraint analysis is demonstrated by determining constraint corrected J–R curves.