Probabilistic formulation of Miner's rule and application to structural fatigue

The standard stress-based approach to fatigue relies on the use of S-N curves, which are obtained by applying cyclic loading of constant amplitude S to identical and standardised specimens until they fail. For some reference probability p, the S-N curve indicates the number of cycles N at which a proportion p of specimens have failed. Based on these curves, Miner's rule is a widely employed method which yields a predicted number of cycles to failure of a specimen subjected to cyclic loading with variable amplitude. The first main contribution of this article is to introduce a probabilistic model for the number of cycles to failure and to show that, under mild assumptions, the deterministic number returned by Miner's rule is the quantile of order p of this random number of cycles to failure, which demonstrates the consistency of our formulation with standard approaches. Our formulation is based on the introduction of the notion of health of a specimen. Explicit formulas are derived in the case of the Weibull-Basquin model. We next turn to the case of a complete mechanical structure: taking into account size effects, and using the weakest link principle, we establish formulas for the survival probability of the structure. We illustrate our results by numerical simulations on a I-steel beam, for which we compute survival probabilities and density of failure point. We also show how to efficiently approximate these quantities using the Laplace method.