A B S T R A C T Quantification of the enhancement in cleavage fracture toughness of ferritic steels following warm pre-stressing has received great interest in light of its significance in the integrity assessment of such structures as pressure vessels. A Beremin type probability distribution model, i.e., a local stress-based approach to cleavage fracture, has been developed and used for estimating cleavage fracture following prior loading (or warm pre-stressing, WPS) in two ferritic steels with different geometry configurations. Firstly, the Weibull parameters required to match the experimental scatter in lower shelf toughness of the candidate steels are identified. These parameters are then used in two-and three-dimensional finite element simulations of prior loading on the upper shelf followed by unloading and cooling to lower shelf temperatures (WPS) to determine the probability of failure. Using both isotropic hardening and kinematic hardening material models, the effect of hardening response on the predictions obtained from the suggested approach has been examined. The predictions are consistent with experimental scatter in toughness following WPS and provide a means of determining the importance of the crack tip residual stresses. We demonstrate that for our steels the crack tip residual stress is the pivotal feature in improving the fracture toughness following WPS. Predictions are compared with the available experimental data. The paper finally discusses the results in the context of the non-uniqueness of the Weibull parameters and investigates the sensitivity of predictions to the Weibull exponent, m, and the relevance of m to the stress triaxiality factor as suggested in the literature. a = crack length (mm) W = ligament (mm) a/W = crack/ligament ratio B = specimen thickness (mm) B 0 = reference thickness (mm) B/B 0 = thickness correction ratio i = order number (ascending) of a specific specimen tested in a group (i = 1, . . . , N ) β = 'shape parameter' exponent for toughness-based distribution m = Weibull exponent N = total number of specimens tested under the same conditions (sample size) K 0f = reference fracture toughness (MPa √ m) K f = fracture toughness after WPS (MPa √ m) K min f = minimum (threshold) fracture toughness (MPa √ m)

A series of experiments were undertaken using a multiple bar assembly to measure elastic follow-up and relaxation of an initial residual stress. A test rig was designed to permit different levels of elastic follow-up to occur. The general features of the experimental results confirmed predictions provided by simple models. The most reliable measure of elastic followup was obtained by measuring the relaxation of the initial residual stress. The rate of relaxation of the residual stress is found to be proportional to the elastic follow-up factor.

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This paper reviews the concepts and definitions related to elastic follow-up, Z, together with its potential use in stress classification. Based on the principles governing benchmark multiple bar structures elastic follow-up (EFU) is quantified. Local nonlinearities arising within a structure influence elastic follow-up. These include variations in the geometry of structure, its material properties, effects of plasticity and creep, structural discontinuities and boundary conditions. Elastic follow-up is shown to be simple to evaluate, is physically meaningful (as it relates strain accumulation in the structure to its cause) and is useful in design practice. In this generalised definition Z = 1 indicates no follow-up and represents a fully displacement controlled situation. In contrast Z = ∞ represents the extreme case of fully load controlled situation. Presence of mixed boundary conditions is interpreted as 1 < Z < ∞. A methodology that overcomes the singularity problem of cracked structure to determine Z is then proposed. The distinctive characteristic of the proposed approach is that it takes account of situations where the structure contains defects.

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