J Smart

A numerical study on the application of the Weibull theory to brittle materials

Abstract In this paper, the effects of assuming that a brittle material can be modelled by the two-parameter Weibull distribution, if the true representation of the material behaviour is actually a three-parameter distribution, are examined. It is shown that if data from a material characterisation test is used to predict the failure of a component and the stressed surface areas are very different, then this can lead to major discrepancies in the predicted failure stress. It is also shown that if the three-parameter model is a true representation of the material behaviour, then the Weibull modulus for the two-parameter approximation will vary with surface area.

The mechanical testing of nuclear graphite

Two billets of nuclear grade medium-grained semi-isotropic graphite were machined into rectangular 4-point bend and L-shaped specimens and tested to failure. The material was not irradiated. During the testing, as well as determining the failure load, the failure was monitored by a high speed camera. The results showed that: there was a difference in the failure loads both along a billet and between the billets, in the L-shaped specimens the cracks did not fail instantaneously but needed further movement of the testing machine’s crosshead before total failure, and the speed of the crack varied in the different specimens. The data were analysed and it was found that the Weibull theory does not predict the failure well but fracture mechanics does provide a way of correlating the data, particularly the crack propagation.

The relationship between irradiation induced dimensional change and the coefficient of thermal expansion: A modified Simmons relationship

Abstract In the 1960s, a theoretical relationship between the dimensional changes and the coefficient of thermal expansion of irradiated graphite was derived by J.H.W. Simmons. The theory was shown to be comparable with experimental observations at low irradiation doses, but shown to diverge at higher irradiation doses. However, various modified versions of this theory have been used as the foundation of design and life prediction calculations for graphite-moderated reactors. This paper re-examines the Simmons relationship, summarising its derivation and assumptions. The relationship was then modified to incorporate the high dose, high strain changes that were assumed to be represented in the changes in Young’s modulus with irradiation dose. By scrutinising the behaviour of finite element analyses, it was possible to use a modified Simmons relationship to predict the dimensional changes of an isotropic and anisotropic graphite to high irradiation doses. These issues are important to present high-temperature reactors (HTRs) as the life of HTR graphite components is dependent upon their dimensional change behaviour. A greater understanding of this behaviour will help in the selection and development of graphite materials.

The effect of the threshold stress on the determination of the Weibull parameters in probabilistic failure analysis

In the analysis of brittle materials and components the probability of failure is commonly modelled using a two-parameter Weibull distribution. Occasionally, a three-parameter model is used when the material shows significant threshold behaviour. In this paper two methods for determining the three-parameter constants are discussed. Two theoretical two- and three-parameter distributions are then analysed to examine the number of samples needed to determine the parameters accurately. The two-parameter models are the best fits of the three-parameter models and their failure distributions are very similar to the three-parameter distributions. It is concluded that far more specimens need to be tested than is usually the case to be confident that the correct distribution has been found.

Discriminating between fracture criteria in the prediction of failure probability of brittle materials

Abstract Many different fracture criteria have been proposed for the failure prediction of brittle materials using Batdorfs or Evans weakest link formulation. There are also many reports where efforts have been made to determine which is the most suitable criterion. However, the difference between the predictions is usually small, making it difficult to decide which criterion best describes a given material. A shear-sensitive and a shear-insensitive criterion are critically examined in this paper under various stress states and it is proposed that to distinguish between the criteria the basic material data should be obtained from biaxial tensile tests, e.g. ring-on-ring tests, and that to obtain the greatest difference between the predictions of the different criteria, a torsion test should be performed. Whether predictions are conservative or non-conservative is also discussed. Finally, comparisons are made with a simpler weakest link model which uses the principle of independent actions.

Discriminating between volume and surface integral theories for brittle materials

Abstract Many theories have been proposed for the failure of brittle materials. Most of these are based on the Weibull weakest link theory and require an evaluation of an integral of the stress state either over the volume or surface of the stressed component. In this paper, the predictions using volume and surface integrals are compared and it is shown that to distinguish between them it is best to obtain material data from a test state which is essentially bending and predict tests in which the stress is relatively constant throughout the volume. Using these tests the failure theories based on volume and surface integrals can be separated, as the spread is considerably greater than the probable experimental error. The relative size of the specimens is also examined and it is found that the “bending specimen” should be thin and the “constant stress specimen” thick.