Optimal Reliability in Design for Fatigue Life

Fatigue describes the damage or failure of material under cyclic loading. Activation and deactivation operations of technical units are important examples in engineering where fatigue and especially low-cycle fatigue (LCF) play an essential role. A significant scatter in fatigue life for many materials results in the necessity of advanced probabilistic models for fatigue. Moreover, structural shape optimization is of increasing interest in engineering, where with respect to fatigue the cost functionals are motivated by their predictability for the integrity of the component after a certain number of load cycles. But mathematical properties such as the existence of the shape derivatives are desirable, too. Deterministic design philosophies that derive a predicted component life from the average life of the most loaded point on the component plus a safety factor accounting for the scatter band do not have this favorable property, as taking maxima is not a differentiable operation. In this work we present a new local probabilistic model for LCF. This model constitutes a new link between reliability statistics, shape optimization and structural analysis which considers the perspective of fatigue but also fits into the mathematical setting of shape optimization. The cost functionals derived in this way are too singular to beH1 lower semi-continuous. We therefore have to modify the existence proof of optimal shapes [20] for the case of sufficiently smooth shapes using elliptic regularity, uniform Schauder estimates and compactness of certain subsets in Ck(Ωext,R) via the ArzelaAscoli theorem, where Ωext is some shape containing all admissible shapes. Here, we analyze a class of cost functionals which consists of volume and surface integrals whose integrands include derivatives of the displacement field up to second order. Moreover, we extend our existence results to high-cycle fatigue (HCF) and deterministic models of fatigue.