The standard stress-based approach to fatigue is based on the use of S-N curves. They are obtained by applying cyclic loading of constant amplitude $S$ to identical and standardised specimens until they fail. The S-N curves actually depend on a reference probability $p$: for a given cycle amplitude $S$, they provide the number of cycles at which a proportion $p$ of specimens have failed. Based on the S-N curves, Miner's rule is next used to predict the number of cycles to failure of a specimen subjected to cyclic loading with variable amplitude. In this article, we present a probabilistic formulation of Miner's rule, which is based on the introduction of the notion of health of a specimen. We show the consistency of that new formulation with the standard approaches, thereby providing a precise probabilistic interpretation of these. 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.

翻译：标准的基于应力的疲劳分析方法依赖于S-N曲线，这些曲线通过对相同标准化试件施加恒定幅值$S$的循环载荷直至破坏获得。S-N曲线实际上取决于参考概率$p$：对于给定的循环幅值$S$，该曲线给出比例为$p$的试件发生破坏时的循环次数。基于S-N曲线，Miner准则用于预测承受变幅循环载荷试件的破坏循环次数。本文提出一种基于试件健康概念的概率论Miner准则表述，论证该新表述与标准方法的一致性，从而为后者提供精确的概率论解释。针对Weibull-Basquin模型推导出显式公式，进而转向完整机械结构情形：考虑尺寸效应并利用最弱链原理，建立结构生存概率公式。通过对I型钢梁进行数值模拟，计算其生存概率与破坏点密度，并展示如何利用拉普拉斯方法有效逼近这些量值。