A comparative method for improving the reliability of brittle components

Calculating the absolute reliability built in a product is often an extremely difficult task because of the complexity of the physical processes and physical mechanisms underlying the failure modes, the complex influence of the environment and the operational loads, the variability associated with reliability-critical design parameters and the non-robustness of the prediction models. Predicting the probability of failure of loaded components with complex shape for example is associated with uncertainty related to: the type of existing flaws initiating fracture, the size distributions of the flaws, the locations and the orientations of the flaws and the microstructure and its local properties. Capturing these types of uncertainty, necessary for a correct prediction of the reliability of components is a formidable task which does not need to be addressed if a comparative reliability method is employed, especially if the focus is on reliability improvement. The new comparative method for improving the resistance to failure initiated by flaws proposed here is based on an assumed failure criterion, an equation linking the probability that a flaw will be critical with the probability of failure associated with the component and a finite element solution for the distribution of the principal stresses in the loaded component. The probability that a flaw will be critical is determined directly, after a finite number of steps equal to the number of finite elements into which the component is divided. An advantage of the proposed comparative method for improving the resistance to failure initiated by flaws is that it does not rely on a Monte Carlo simulation and does not depend on knowledge of the size distribution of the flaws and the material properties. This essentially eliminates uncertainty associated with the material properties and the population of flaws.On the basis of a theoretical analysis we also show that, contrary to the common belief, in general, for non-interacting flaws randomly located in a stressed volume, the distribution of the minimum failure stress is not necessarily described by a Weibull distribution. For the simple case of a single group of flaws all of which become critical beyond a particular threshold value for example, the Weibull distribution fails to predict correctly the probability of failure. 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 these cases however, the probability of failure is correctly predicted by the suggested alternative equation. The suggested 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 any situation of locally initiated failure by non-interacting entities.