Methods for Fitting a 3-Parameter Weibull Distribution on Fracture Toughness Data

Abstract

Probabilistic fracture mechanics is a major element to evaluate the risk of failure of the reactor pressure vessel when it is subjected to pressurized-thermal-shock transient. One of the key issues concerning this probabilistic evaluation is the validity of the statistical model for the fracture toughness of the vessel steel. The Master Curve methodology has been developed by Wallin [1] to analyse the scatter in the results of steel toughness measures. This scatter is treated using the weakest-link theory applied to a three-parameter Weibull distribution as follows: where P f if the failure probability, K IC the fracture toughness, K min the location parameter, K 0 — K min the scale parameter and α the shape parameter. Wallin has determined that and α = 4 for reactor vessel steels. Thus only K 0 has to be determined for a specific material using a maximum likelihood estimator. However due to the heterogeneity of the existing K IC databases, with different materials, different sizes of specimen and different expressions of toughness, the characterisation of the scatter in the toughness databases by a Weibull model with a unique slope α= 4 does not seem sufficient. In addition, the toughness is very dependent on material temperature and for the relative high temperatures, the assumption on the location parameter, seems too conservative. For these reasons, it was decided to develop a purely statistical model without physical considerations except the use of a Weibull distribution. In this paper we present three estimation techniques for fitting a 3-parameter Weibull distribution on a toughness database. We have also tested various sampling techniques in the field of the temperatures. For this purpose, the software WOLF3 (Weibull Or Lognormal distribution Fitting with 3-parameters) has been developed under the MATLAB data-processing system.