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The standard stress-based approach to fatigue is based on the use of S-Ncurves. They are obtained by applying cyclic loading of constant amplitude $S$to identical and standardised specimens until they fail. The S-N curvesactually depend on a reference probability $p$: for a given cycle amplitude$S$, they provide the number of cycles at which a proportion $p$ of specimenshave failed. Based on the S-N curves, Miner's rule is next used to predict thenumber of cycles to failure of a specimen subjected to cyclic loading withvariable amplitude. In this article, we present a probabilistic formulation ofMiner's rule, which is based on the introduction of the notion of health of aspecimen. We show the consistency of that new formulation with the standardapproaches, thereby providing a precise probabilistic interpretation of these.Explicit formulas are derived in the case of the Weibull--Basquin model. Wenext turn to the case of a complete mechanical structure: taking into accountsize effects, and using the weakest link principle, we establish formulas forthe survival probability of the structure. We illustrate our results bynumerical simulations on a I-steel beam, for which we compute survivalprobabilities and density of failure point. We also show how to efficientlyapproximate these quantities using the Laplace method.

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