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:按照标准的应力为基础的疲劳方法,需要使用S-N曲线。它们是通过对相同且标准化的试样施加恒定振幅 $S$ 的循环加载,直到试样破裂的方法获得的。 S-N曲线实际上取决于参考概率 $p$:对于给定的循环振幅 $S$,它们提供了在多少循环次数下 $p$ 的试样已经失效。然后运用矿工准则基于S-N曲线来预测受变振幅循环载荷的试样的失效次数。在本文中,我们根据引入试样 “健康程度” 的概念,提出了矿工准则的概率形式。我们证明了这种新形式与标准方法的一致性,从而提供了对这些方法的精确概率解释。在Weibull-Basquin模型的情况下,得到了明确的公式。我们接下来转向完整机械结构的情况:考虑尺寸影响,并使用最薄弱环节原则,建立了结构生存概率的公式。我们通过在I型钢梁上进行数值模拟来说明我们的结果,计算生存概率和失败点密度。我们还展示了如何使用Laplace方法有效地逼近这些量。