Is Weibull distribution the correct model for predicting probability of failure initiated by non-interacting flaws?

Abstract

The utility of the Weibull distribution has been traditionally justified with the belief that it is the mathematical expression of the weakest-link concept in the case of flaws locally initiating failure in a stressed volume. This paper challenges the Weibull distribution as a mathematical formulation of the weakest-link concept and its suitability for predicting probability of failure locally initiated by flaws. The paper shows that the Weibull distribution predicts correctly the probability of failure locally initiated by flaws if and only if the probability that a flaw will be critical is a power law or can be approximated by a power law of the applied stress. Contrary to the common belief, on the basis of a theoretical analysis and Monte Carlo simulations we show that in general, for non-interacting flaws randomly located in a stressed volume, the distribution of the minimum failure stress is not necessarily a Weibull distribution. For the simple cases of a single group of identical flaws or two flaw size groups each of which contains identical flaws, for example, the Weibull distribution fails to predict correctly the probability of failure. Furthermore, if in a particular load range, no new critical flaws are created by increasing the applied stress, the Weibull distribution also fails to predict correctly the probability of failure of the component. In all these cases however, the probability of failure is correctly predicted by the suggested alternative equation. This equation is the correct mathematical formulation of the weakest-link concept related to random flaws in a stressed volume. The equation does not require any assumption concerning the physical nature of the flaws and the physical mechanism of failure and can be applied in cases of locally initiated failure by non-interacting entities. © 2008 Elsevier Ltd. All rights reserved.