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Mechanical components that are exposed to cyclic mechanical loading fail atloads that are well below the ultimate tensile strength. This process is knownas fatigue. The failure time, that is the time when a first crack forms, ishighly random. In this work we review some recent developments in the modellingof probabilistic failure times, understood as the time to the formation of afatigue crack.  We also discuss the how probabilistic models can be used in shape design withthe design intent of optimizing the component's reliability. We give review arecent existence result for optimal shapes and we discuss continuous anddiscrete shape derivatives. Another application is optimal service scheduling.The mathematical fields involved range from reliability statistics overstochastic point processes, multiscale modeling, PDEs on variable geometries,shape optimization and numerical analysis to operations research.

Modeling, Minimizing and Managing the Risk of Fatigue for Mechanical Components