Optimal Reliability in Design for Fatigue Life

Abstract

The failure of a component often is the result of a degradation process that originates with the formation of a crack. Fatigue describes the crack formation in the 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, optimization of reliability is of vital interest in engineering, where with respect to fatigue the cost functionals are motivated by the predicted probability for the integrity of the component after a certain number of load cycles. The natural mathematical language to model failure, here understood as crack initiation, is the language of spatio-temporal point processes and their first failure times. The local crack formation intensities thereby need to be modeled as a function of local stress states and thus as a function of the derivatives of the displacement field $u$ obtained as the solution to the PDE of linear elasticity. This translates the problem of optimal reliability in the framework of shape optimization. The cost functionals derived in this way for realistic optimal reliability problems are too singular to be $H^1$-lower semicontinuous as many damage mechanisms, like LCF, lead to crack initiation as a function of the stress at the component's surface. Realistic crack formation models therefore impose a new challenge to the theory of shape optimization. In this work, we have to modify the existence proof of optimal shapes, for the case of sufficiently smooth shapes using elliptic regularity, uniform Schauder estimates, and compactness of strong solutions via the Arzela--Ascoli theorem. This result applies to a variety of crack initiation models and in particular applies to a recent probabilistic model for LCF.