D. W. Beardsmore

JEDI: A Code for the Calculation of J for Cracks Inserted in Initial Strain Fields and the Role of J and Q in the Prediction of Crack Extension and Fracture

When crack tip constraint is high, the crack tip contour integral J characterises the asymptotic stress, strain and displacement fields of a stationary crack in an elastic-plastic material. In other cases, the crack tip fields can be related to J and a second parameter Q which governs the crack tip constraint. These observations are the basis of J-Q fracture mechanics assessments. In the most usual procedure J is compared to an effective, constraint-corrected fracture toughness Jc which is derived from Q and the fracture toughness of the material. The difference Jc – J is a measure of the margin of safety. The assessment procedure assumes there are no initial inelastic strains in the component or the fracture toughness specimen prior to introducing the crack and subsequent loading. However, plant components may contain inelastic strains prior to cracking arising from welding and other manufacturing or fit-up processes. This initial strain field can be established by a finite element analysis that simulates the welding and/or manufacture sequence. Weld residual stresses develop due to the accumulation of incompatible, inelastic strains, including thermal, plastic and transformation strains in the material. If a crack is inserted into an initial strain field, a procedure is required to calculate J by analysis of the resulting crack tip fields. Moreover, for the fracture assessment method to remain valid, it must be demonstrated that the values of J and Q continue to govern the onset of crack extension or fracture so that a meaningful comparison of J with Jc can be made. This paper describes a domain integral for calculating J when inelastic strains exist prior to cracking, and its implementation in the JEDI computer code. The code is used to determine J for a crack inserted into a three-point bend specimen containing an initial inelastic strain field representative of one that might develop during welding. The extent to which the crack tip stress field is characterised by J and Q is examined by comparing it to the field for high constraint, small-scale yielding conditions.Copyright

Allowance for Residual Stresses and Material Interfaces When Calculating J in and Close to Welded Joints

The J-integral is used widely as a measure of crack driving force within structural integrity assessments of engineering structures. The calculation of J in and close to welds using the Equivalent Domain Integral (EDI) found in some finite element (FE) codes can give unreliable, domain-dependent, results. This arises because FE codes typically use simplified, “default” implementations of the EDI which do not allow for the non-uniform strain field, material interface and/or material inhomogeneity which are present as a result of the welding process. This paper provides guidance on the corrections that must be made to restore the domain-independence of the EDI, allowing J to be determined from the far stress field for cracks in and close to welds. Example calculations are presented to illustrate the effect of the modifications. The results are compared with those obtained using the default J post-processor built into a commercial FE code. Calculations are presented for cases where there is a material interface, where there are initial non-uniform strains, and where both of these phenomena occur together. It is shown that the default post-processor results are highly domain dependent, but that independence is restored using the corrected EDI formulation.© 2003 ASME

An Analysis of the Roles of J and Q in the Assessment of Fracture for Quasi-Statically Extending Cracks in Residual Stress Fields

It is well known that the crack tip stress and strain fields for a crack in an elastic-plastic body depend on the crack tip contour integral J, the Q-stress, and the elastic-plastic properties of the material. This dependence is the fundamental basis of conventional two-parameter J-Q fracture mechanics assessments. It is normally assumed that the crack is created in an unstressed body, or else is inserted concurrently into an existing non-zero stress and strain field such that the crack tip fields build up monotonically and dominate at the crack tip. In such cases, the crack may be regarded as stationary and the J-Q procedure is valid provided that care is taken to calculate J and Q properly when initial stress and/or strains exist. When a crack is introduced progressively and quasi-statically into a component, the location of the crack tip will move along a distinct path. If the component contains residual stress and this is of a significant size along the crack tip path, a re-distribution of the residual stress will occur as the crack tip moves. Specifically, the stress field ahead of the crack tip will unload as the crack tip advances so that non-proportional loading will occur behind the advancing crack tip. In elastic-plastic materials, a wake of plasticity will usually be deposited in the material behind the moving or growing crack tip. Similar effects will also occur when a stationary crack extends due to critical or sub-critical processes. The presence of a plastic wake alters the stress and strain fields at the crack tip so that they do not generally match the fields of a stationary crack. Moreover, J and Q may not describe the stress and strain fields, invalidating the use of the fracture mechanics procedure for such cases. In this paper, a Finite Element analysis of J and Q is carried out for a quasi-statically extending crack inserted in a strip of elastic-plastic material containing an initial residual stress field. Care is taken to model the crack tip conditions appropriately as the crack extends and J is determined using the JEDI post-processing program which can allow for the effects of initial plastic strains and non-proportional loading. An assessment is made of the crack tip field and the likelihood of further extension or fracture is made using local approach models. The analysis considers both cleavage and ductile fracture. The extent of the relationship between J and Q and the crack tip fields is established and the validity of the J-Q procedure to such cases is discussed. The paper considers whether the procedure is conservative when J and Q are determined from an analysis of a stationary crack of the same size inserted into the same initial field.Copyright

Statistical Analysis of Residual Stress Profiles Using a Heuristic Method

In this paper we present a recently developed heuristic method for statistical analysis of residual stress that is based on a combination of the weighted least-squares method and the application of expert judgement. The least-squares method allows a model of the best residual stress profile to be determined as a linear combination of basis functions; the expert knowledge gives the flexibility of applying expert judgement to determine the weights from the observed scatter in the residual stress data. The heuristic method has been applied to a set of measurement data of a Welded Bead-on-Plate specimen. The results show that with the heuristic method, it is possible to obtain less conservative residual stress profile to a known confidence level.Copyright

Further Studies of Multiple Co-Planar Surface Breaking Flaws for Cleavage Fracture

In assessing the integrity of structures, complex multiple flaws located in close proximity to each other are generally characterised as one, larger, single flaw. Guidance for the characterisation of multiple flaws is provided in procedures such as R6 and BS 7910, which are routinely used in the UK and elsewhere in the structural integrity assessment of structures and components. For this approach to be valid, the characterisation process must be conservative. That is to say, the probability of failure must be higher for the characterised flaw than for the system of multiple flaws. However, previous studies showed that the current characterisation rules may be non-conservative under some circumstances, in particular under cleavage fracture conditions. A combined experimental and analytical programme of work has been undertaken within the UK in order to further investigate this potential non-conservatism for situations where the possibility of cleavage failure may have to be taken into account when assessing structures or components containing multiple flaws. Details of early stages of the analytical programme were reported at the 2006 and 2007 ASME PVP Conferences and comprised a number of finite element analyses to evaluate cleavage failure probability, via a Master Curve-based approach, for interacting twin flaws and the corresponding characterised single flaw, under applied tensile and bending loads, at low temperatures. These analyses considered surface-breaking semi-elliptical flaws all having the same depth, but with four different aspect ratios. For each aspect ratio the separation of the twin flaws was varied. It was found that non-conservatism of the characterisation rules was indicated for flaws of high depth to length aspect ratio (a/c) in contact. This paper describes further work that has been undertaken to extend the results previously reported. The further work described has been centred on: • for the most onerous aspect ratio (a/c = 1.0) extending the experimental results to higher temperatures in the cleavage transition regime. • performing finite element analyses to complement these experiments. • revisiting the methods for calculation of cleavage failure probability to obtain improved agreement with the experimental results. • examining the rules governing the characterisation process, to determine if modification is necessary.Copyright

Advanced Probabilistic Fracture Mechanics Using the R6 Procedure

Deterministic Fracture Mechanics (DFM) assessments of structural components (e.g. pressure vessels and piping used in the nuclear industry) containing defects can usually be carried out using the R6 procedure. The aim of such an assessment is to demonstrate that there are sufficient safety margins on the applied loads, defect size and fracture toughness for the safe continual operation of the component. To ensure a conservative assessment is made, a lower-bound fracture toughness, and upper-bound defect sizes and applied loads are used. In some cases, this approach will be too conservative and will provide insufficient safety margins. Probabilistic Fracture Mechanics (PFM) allow a way forward in such cases by allowing for the inherent scatter in material properties, defect size and applied loads explicitly. Basic Monte Carlo Methods (MCM) allow an estimate of the probability of failure to be calculated by carrying out a large number of fracture mechanics assessments, each using a random sample of the different random variables (loads, defect size, fracture toughness etc). The probability of failure is obtained by counting the proportion of simulations which lead to assessment points that lie outside the R6 failure assessment curve. This approach can give good results for probabilities greater than 10−5 . However, for smaller probabilities, the calculation may be inefficient and a very large number of assessments may be necessary to obtain an accurate result, which may be prohibitive. Engineering Reliability Methods (ERM), such as the First Order Reliability method (FORM) and the Second Order Reliability Method (SORM), can be used to estimate the probability of failure in such cases, but these methods can be difficult to implement, do not always give the correct result, and are not always robust enough for general use. Advanced Monte Carlo Methods (AMCM) combine the two approaches to provide an accurate and efficient calculation of probability of failure in all cases. These methods aim to carry out Importance Sampling so that only assessment points that lie close to or outside the failure assessment curve are calculated. Two methods are described in this paper: (1) orthogonal sampling, and (2) spherical sampling. The power behind these methods is demonstrated by carrying out calculations of probability of failure for semi-elliptical, surface breaking, circumferential cracks in the inside of a pressure vessel. The results are compared with the results of Basic Monte Carlo and Engineering Reliability calculations. The calculations use the R6 assessment procedure.Copyright

Determination of Residual Stress Profiles of Pipe Girth Weld Using a Unified Parametric Function Form

In this paper we present an improved analysis of residual stress data of a pipe girth weld by applying the developed heuristic method to one set of high-quality residual stress measurement data. The through-thickness residual stress is expressed as a parametric function form which is a combination of three stress components: membrane, bending and self-equilibrating. This parametric function form provides not only a clear physical basis for the residual stress profile, but is also closely related to two important governing parameters, i.e. the pipe geometry and the welding heat input. The residual stress profiles obtained are also compared with results predicted by the Bayesian method as well as the profiles from the UK R6 procedure and the US API 579 code.Copyright

Load History Effects on Crack Driving Force for Cracks in Residual Stress Fields

Fracture mechanics assessments of engineering components and structures containing defects are made by comparing an estimate of the crack driving force KJ with an effective fracture toughness KJc . The assessments must account for the combined effect of primary loads, such as internal pressure in pressurised components, and secondary stresses arising from welding and/or thermal loading. Elastic-plastic finite element analysis, or simplified methods set out in standard assessment procedures, can be used to estimate the crack driving force KJ as a function of the applied primary load on the component. The effective fracture toughness KJc should take account of the material fracture toughness and the crack tip constraint. For the assessment of defects in weld residual stress fields, it is usually assumed that the defect is inserted into the as-welded stress distribution in such a way that traction free crack surfaces are created simultaneously at all positions on the crack faces. However, it may be beneficial to take account of any relaxation in the residual stress field that might arise during proof-testing or in-service cyclic loading, and to consider a more gradual, progressive introduction of the defects. These benefits could, in principle, result in a reduction in the crack driving force. This paper describes work that has been undertaken to provide estimates of the crack driving force KJ for a fully-circumferential defect in a circumferential repair weld in a cylindrical pipe. Calculations have been carried out to establish KJ for a number of cases where different pressure overloads are applied to the uncracked pipe and different methods of crack insertion are applied. Estimates of the margin of safety on fracture toughness and pressure loading were calculated. At the outset, it was assumed that the fracture toughness of relevance for the defects is the material fracture toughness KJc * derived from strain free, high constraint fracture toughness specimens. No allowance was made for constraint effects associated with the finite geometry or initial strains in the pipe. The values of KJ were derived from values of J calculated using the JEDI post-processing code; this allows for initial inelastic strains present in the model prior to the start of the crack insertion process.© 2008 ASME

Effects of Neutron Irradiation on the Fracture Toughness of RPV Materials: Prediction of Material Property Changes for Irradiated Euro Reference Material ‘A’ and Other RPV Materials

A micromechanistic model is used to estimate the irradiation-induced change in the Master Curve reference temperature for cleavage fracture as a function of the associated change in material yield stress relative to the yield stress in the unirradiated condition. The model is shown to predict well the behaviour of Euro Reference Material A (quenched and tempered 22NiMoCr37 ring forging) irradiated at temperatures of T = 285°C and T = 150°C with neutron fluences of 4.3E+19 n/cm2 (En > 1MeV) and 3.1E+19 n/cm2 (En > 1MeV) respectively. Further validation of the model is provided with reference to published data for a range of irradiated RPV plate, forging and weld materials, where measured and predicted values of the change in the Master Curve reference temperature are successfully compared. For the LWR materials and irradiation conditions considered, the fitted parameters of the model are consistent with the view that the primary effect of neutron irradiation is to increase the friction stress for plastic flow of crack-tip material, whereby the irradiation-induced change in yield stress may be associated with a change in a non-hardening, athermal term e.g. as described in the Zerelli-Armstrong constitutive equation. The model predictions compare well with trend curves due to Sokolov and Nanstad, and Wallin and Laukkanen. A particular advantage of the model, compared with these more general formulations, is that it is potentially better suited to a more detailed analysis and interpolation of RPV material datasets covering a range of irradiation conditions where flow properties are reasonably well characterised.Copyright

The Effects of Warm Pre-Stressing of a Pre-Cracked Pressurised Thermal Shock Disk Specimen Under a LUCF Loading Cycle

A finite element analysis has been performed to investigate the effects of warm prestressing of a pre-cracked PTS-D (Pressurized Thermal Shock Disk) specimen, for comparison with the experimental work conducted by the Belgium SCK-CEN organisation under the European NESC VII project. The specimen was loaded to a maximum loading at −50 °C, unloaded at the same temperature, cooled down to −150 °C, and then re-loaded to fracture at −150 °C. This is a loading cycle known as a LUCF cycle. The temperature-dependant tensile stress-strain data was used in the model and the finite element software ABAQUS was used in the analysis. The finite element results were used to derive the apparent fracture toughness by three different methods: (1) Chell’s displacement superposition method; (2) the local stress matching method; and (3) Wallin’s empirical formula. The apparent fracture toughness values were derived at the deepest point of the semi-elliptical crack for a 5% un-prestressed fracture toughness of 43.96 MPam1/2 at −150 °C. The detailed results were presented in the paper.Copyright