A. Brückner-Foit

Estimation procedure for the Weibull parameters used in the local approach

The local approach was recently proposed by Beremin and Mudry for evaluating the statistical behaviour of toughness results of materials. This approach introduces a stress parameter σw, termed the Weibull stress, as a measure of the fracture resistance of materials instead of the conventional toughness parameters such as Kc, δc and Jcl (critical stress intensity factor, CTOD and J-integral, respectively). The Weibull stress σw obeys the Weibull distribution with the two parameters m and σu (the shape and the scale parameter, respectively). The first parameter m is normally estimated to be 22 irrespective of the kind of material. In this paper a procedure for the determination of the Weibull parameters m and σu is developed. This procedure consists of the determination of the plastic zone ahead of the crack tip, from which cleavage fracture originates, and of the maximum likelihood estimation of the parameters m and σu based on the stress distribution in the plastic zone. Calculations using this procedure confirm that the distribution of the Weibull stress σw is a material property independent of specimen thickness, and in particular that the shape parameter m depends on the material, e.g. m≃12 for a German reactor pressure vessel steel (20 Mn Mo Ni 5 5). Using these parameters for the distribution of the Weibull stress the size effect in fracture toughness values is predicted and an improved agreement between theory and experiments is obtained compared to the Weakest Link model.

Calculation of the second fracture parameter for finite cracked bodies using a three-term elastic-plastic asymptotic expansion

A three-term asymptotic expansion which is controlled by two amplitude parameters is used to describe the stress field in the vicinity of the crack tip in a power-hardening material. The first parameter is the well-known J-integral. The second parameter (amplitude A) characterizes the following terms. A least squares procedure is developed for the determination of the amplitude parameter A by fitting of finite element data. The convergence of computed A values is investigated for a small scale yielding modified boundary layer problem. It is shown that the three-term expansion has certain advantages over the Q-stress approach. Values of the amplitude parameter A are determined for an edge cracked plate, center cracked plate, three-point bend specimen and compact tension specimen.

A stochastic simulation model for microcrack initiation in a martensitic steel

Specimens of the martensitic steel F82H-mod were fatigued under strain control. During the tests, crack initiation and growth were monitored on the specimen surface using an optical microscope. The experiments revealed microcrack initiation at the martensitc laths in the grains. These microcracks grew transgranularly until they were stopped at the grain boundaries. Empirical relations for crack initiation could be inferred from a statistical analysis of the microcrack patterns observed on the specimen surface. The probability that a microcrack of a specific length and orientation is initiated in a grain was determined and inserted in a stochastic simulation model based on a random cell structure. The simulation results correlated well with the experimental findings.

Discrimination of multiaxiality criteria with the Brazilian disc test

A fixture was developed which allows us to test circular discs of high strength ceramic materials under diametral compression. This so-called Brazilian disc test can be used to determine the failure behaviour of natural flaws contained in ceramic materials under multiaxial loading, as a stress state with both negative and positive principal stresses is induced in a disc during the test. The results of the test series performed with low and high strength ceramics are analysed using the multiaxial Weibull theory. Comparison of the Weibull distributions of the fracture stresses obtained with the Brazilian disc test and the outcome of the other test series clearly shows that a shear-insensitive failure criterion is more suitable to determine the critical flaw size under mixed mode loading than a shear-sensitive criterion.

WELFEP: a round robin for weakest-link finite element postprocessors

Abstract A numerical round robin for finite element postprocessors for weakest-link failure probability predictions has been analysed. Three selected problems were analysed by the participants using a number of different postprocessors. In general, good agreement was obtained for the predicted mean nominal failure stress. The predicted values for the failure probability show much more scatter due to the sensitivity of this quantity to integration errors.

Fracture toughness transition curve estimation from a notched round bar specimen using the local approach method

The local approach method for the brittle or transition region describes the fracture probability of specimens or structures using Weibull statistics. Many papers have discussed the characteristics of the Weibull parameter using notched tensile specimens and the applicability to fracture toughness scatter evaluation using CT specimens. However few papers have made clear whether the Weibull parameter of the Weibull stress is a material property or not. In this paper the distribution of Weibull stress in the brittle fracture region using notched round bar specimens and CT specimens were investigated and it was confirmed that both distributions agreed well. Furthermore the estimation method for the fracture toughness transition curve including its scatter from notched round bar tensile tests was proposed based on the relation between the Weibull stress and the Wallins fracture toughness transition curve. As a result, the estimated fracture toughness curve in the brittle and lower transition region from the notched round bar specimens coincided with the measured fracture toughness curve from CT specimens. This method will be applicable to fracture toughness curve estimation under plane strain conditions even if there is no possibility of obtaining thick enough CT specimens from a structure because of geometry or some other restrictions.

Discrimination of multiaxiality criteria using brittle fracture loci

The statistical distribution of brittle fracture loci was measured in the Brazilian disc test. A logical circuit was used to identify the fracture event in addition to fractographic examination. The empirical distribution of brittle fracture loci is compared with the predictions of the multiaxial Weibull theory. The results indicate that a shear-insensitive criterion is more suitable than a shear-sensitive criterion to describe the fracture behaviour of natural flaws in ceramic materials.

On criteria for accepting a response surface model

Abstract One of the key problems in response surface analysis is the judgement whether a response surface model is a sufficiently accurate representation of the true response function based on a finite set of experiments. The use of response surface techniques in the context of structural reliability led to the definition of a special lack of fit measure and the formulation of a criterion for accepting response surface models using this measure. The purpose of this paper is to derive and justify two alternative criteria based on this lack of fit measure. Some examples are given to demonstrate the validity of the derived criteria.

On the contribution of notches to the failure probability of ceramic components

The failure probability of tensile bars containing circular notches is calculated using the multiaxial Weibull theory. The influence exerted by the stress concentration factor, the stress gradient in the notch root, and the Weibull exponent are analysed. A local risk of rupture is defined, and it is shown how characteristic sub-volumes of the notched tensile bar contribute to the failure probability. An analytical expression is derived which is based on a linear approximation of the stress field in the vicinity of the notch root, and which yields the dependence of the leading term of the failure probability on characteristic quantities such as the stress concentration factor and the Weibull modulus.

Probabilistic fracture mechanics applied to high temperature reliability

Abstract An example is used to demonstrate the applicability of Probabilistic Fracture Mechanics (PFM) methods in high temperature reliability assessment. The failure probability of a pipe under pure bending at a temperature of 973 K is calculated using both Monte Carlo simulation and the First Order Reliability Method. The advantages and the accuracy of approximative methods for calculating failure probabilities are demonstrated. Additionally, probabilistic and deterministic methods for reliability assessment are compared with each other. It is shown that a deterministic reliability assessment becomes inadequate in cases where the failure probability is determined by equally significant contributions of several random variables.