A local and probabilistic model for low-cycle fatigue

Fatigue describes crack formation in material under cyclic loading which can often result in failure of engineering components. Activation and deactivation operations of technical units are frequently occurring examples in engineering, where fatigue and, in particular, low-cycle fatigue (LCF) play an important role. For many materials, fatigue life is marked by significant scatter. Furthermore, size effects have a non-negligible influence on the fatigue life of components. The deterministic safe-life approach in fatigue design employs results of standardized specimen tests to estimate the design life of a component. Thereby, factors of safety are applied to consider the inherent scatter in fatigue life, size effects, and uncertainties such as those in the loading and temperature conditions. In this thesis, a local and probabilistic model for LCF in the context of polycrystalline metal is presented. The model can be derived from reliability statistics or from the concept of point processes. It quantifies risks for LCF crack initiation and takes size effects and inhomogeneous strain and temperature fields into account. The model has been calibrated and validated based on fatigue tests with different specimen geometries. The uncertainty of parameter estimation is considered by means of bootstrapping. Moreover, the probabilistic fatigue life of engineering components, which are already in service, is analyzed. By varying the geometry of a component to minimize the probability for LCF crack initiation, new designs can be developed according to the local and probabilistic model. This leads to shape optimization, where cost functionals are optimized under certain constraints and with respect to sets of admissible shapes. Since the probabilistic cost functional is not as regular as the compliance functional which itself cannot be used for fatigue design, an existence analysis based on sufficiently smooth admissible shapes and state solutions of the mixed problem of linear elasticity is conducted within this thesis. Thereby, Schauder estimates, the Arzela-Ascoli theorem and Lebesgue’s dominated convergence theorem are employed. Finally, the prospect of conducting sensitivity analysis in the probabilistic framework of fatigue design, which provides a more regular cost functional than the deterministic safe-life approach, is discussed.