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.     

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.     

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.     

Engineering C*‐integral and COD estimates for nonidealized circumferential through‐wall cracked pipes at elevated temperature

In this study, creep fracture mechanics parameters, C^*‐integral and crack opening displacement (COD) rate, are estimated for a nonidealized circumferential through‐wall crack (TWC) in pipes. The GE/EPRI and enhanced reference stress (ERS) methods are employed. As for creep condition, the Norton and RCC‐MRx creep models are considered for secondary and primary‐secondary creep strain, respectively. The bending moment, axial tension, and internal pressure are applied to a pipe with a nonidealized circumferential TWC, as individual loads. Three‐dimensional elastic‐creep finite element (FE) analyses are performed, and the predictions from the GE/EPRI and ERS methods are compared with FE results. For the Norton creep model, both methods show good agreement with the FE results. For the RCC‐MRx creep model, only the ERS method can be used, and it provides accurate predictions comparing with FE results. Based on the comparison results, the use of the present engineering C^*‐integral and COD estimation methods for nonidealized circumferential TWC is validated.     

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.     

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.     

Reference Stress Approximations for J and COD of Circumferential Through-Wall Cracked Pipes

Reference stress approximations for the J integral and crack tip opening displacement (COD) for circumferential through-wall cracked pipes under tension and under bending are reported. The proposed J estimation equation is fully compatible with the existing reference stress based J estimation, currently embedded in the R6 assessment procedure, but involves a slightly different definition of the reference stress, using an optimised reference load instead of the limit load. This modification enhances the accuracy of the J estimation for circumferentially cracked pipes. Confidence in the proposed equation is gained from the significantly reduced hardening dependence of the plastic influence functions in the GE/EPRI method. The proposed COD estimation equation includes two further modifications. One is the use of a power-law fit to the plastic portion of the stress strain data, instead of the use of the actual stress strain data. In this context, a robust estimation equation for the strain hardening index is given. The other modification is to the plasticity correction term in contained yielding. A lower bound COD estimation equation is also given, similar to the R6 option 1 Jestimation curve, which is suitable when only limited tensile properties are available. The resulting estimation equations are simple to use. Comparisons with experimental pipe test data show that the proposed COD estimation equations provide overall good agreement, which gives confidence in applying them to Leak-before-Break (LBB) analyses.     

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.     

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.     

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.     

Yield locus for circumferential part-through surface cracked pipes under combined pressure and bending

This paper provides analytical plastic limit load solutions of pipes with circumferential part-through surface cracks under combined pressure and bending. A key issue is to postulate discontinuous hoop stress distributions in the net-section. General equations for arbitrary-shaped cracks are derived and closed-form solutions are given for two idealized cracks; a constant depth crack and circular crack. Validity of the proposed limit load solutions are checked against the results from three-dimensional (3D) finite element (FE) limit analyses using elastic-perfectly plastic material behaviour.     

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.     

Effect of internal pressure on plastic loads of 90° elbows with circumferential part-through surface cracks under in-plane bending

This paper quantifies the effect of internal pressure on plastic loads for elbows with circumferential through-wall and constant-depth part-through surface cracks under in-plane bending. Results are based on FE limit analyses using elastic-perfectly plastic materials and the non-linear geometry option. It is found that, for larger values of r/t, the pressure effect on plastic loads is more pronounced and plastic loads for combined loading cases could be much higher than those for pure bending cases. Full FE data of plastic loads are tabulated, and effects of crack geometries on interaction curves are discussed.     

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.     

Closed-form plastic collapse loads of pipe bends under combined pressure and in-plane bending

Based on three-dimensional (3-D) FE limit analyses, this paper provides plastic limit, collapse and instability load solutions for pipe bends under combined pressure and in-plane bending. The plastic limit loads are determined from FE limit analyses based on elastic-perfectly-plastic materials using the small geometry change option, and the FE limit analyses using the large geometry change option provide plastic collapse loads (using the twice-elastic-slope method) and instability loads. For the bending mode, both closing bending and opening bending are considered, and a wide range of parameters related to the bend geometry is considered. Based on the FE results, closed-form approximations of plastic limit and collapse load solutions for pipe bends under combined pressure and bending are proposed.     

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.     

Stress intensity factors for slanted through-wall cracks based on elastic finite element analyses

Based on detailed 3‐dimensional (3‐D) elastic finite element (FE) analyses, the present paper provides stress intensity factors (SIFs) for plates with slanted through‐wall crack (TWC) and cylinders with slanted circumferential TWC. Regarding loading conditions, axial tension was considered for the plates, whereas axial tension, global bending and internal pressure were considered for the cylinders. To cover a practical range, the geometric variables affecting the SIF were systematically varied. Based on FE results, SIFs along the crack front, including the inner and outer surface points, were provided. The present results can be used to evaluate the fatigue crack growth or stress corrosion cracking behaviour of a slanted TWC and furthermore to perform detailed Leak‐Before‐Break analysis considering a more realistic crack shape.     

Stress intensity factors for cracks in thin plates

This paper presents stress intensity factor solutions for several crack configurations in plates. The loadings considered include internal pressure, and also combined bending and tension. The dual boundary element method is used to model the plate and mixed mode stress intensity factors are evaluated by a crack surface displacement extrapolation technique and the J-integral technique. Several cases including centre crack, edge crack and cracks emanating from a hole in finite width plates are presented.     

Plastic loads of pipe bends under combined pressure and out-of-plane bending

This paper provides plastic limit and TES (twice-elastic-slope) plastic load solutions for 90° pipe bends under combined pressure and out-of-plane bending, via three-dimensional non-linear FE analyses using elastic-perfectly plastic materials. Without internal pressure, a closed-form approximation is given. For combined pressure and out-of-plane bending, tabulated data are given, from which TES plastic loads can be interpolated. It is found that TES plastic loads for pipe bends under out-of-plane bending are lower than those under in-plane opening bending, but are higher than those under in-plane closing bending. It suggests that the in-plane closing bending mode is the most critical loading mode for 90° pipe bends, which is fully consistent to existing findings.