Abstract
Mechanical components that are exposed to cyclic mechanical loading fail at loads that are well below the ultimate tensile strength. This process is known as fatigue . The failure time , that is the time when a first crack forms, is highly random. In this work we review some recent developments in the modelling of probabilistic failure times , understood as the time to the formation of a fatigue crack . We also discuss how probabilistic models can be used in shape design with the intent of optimizing the components’ reliability . We review a recent existence result for optimal shapes and we discuss continuous and discrete shape derivatives. Another application is optimal service scheduling . The mathematical fields involved range from reliability statistics over stochastic point processes , multiscale modeling , PDEs on variable geometries , shape optimization and numerical analysis to operations research .
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Notes
- 1.
www.calculix.de.
- 2.
Structure of several unit cells having the same orientation.
- 3.
U is distributed according to the Haar measure.
- 4.
Or von Mises stress.
- 5.
According to the von Mises shape modification hypothesis hydrostatic stress conditions with similar principal stress in all directions lead to a value of zero.
- 6.
Confer for example [Evans 2010, 5.5].
- 7.
In (66), the Hessian \(\nabla ^2u\) is a three-dimensional matrix with one index regarding the components of u which contracts with the index of the partial derivatives of \(\nabla \mathcal {F}_{\text {sur}}\), and the other two indices with respect to the partial derivatives which contract with the remaining index of \(\nabla \mathcal {F}_{\text {sur}}\) and with \(\nu \).
- 8.
Here, \(\nabla [M]_{\text {vM}}\) denotes the gradient of the von Mises stress at the value of a matrix \(M\in \mathbb {R}^{3\times 3}\).
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Acknowledgements
Hanno Gottschalk would like to thank Sergio Albeverio, Ana Bela Cruzeiro and Darryl Holm for their kind invitation to the CIB and the staff of CIB for their hospitality. Nadine Moch and Mohamed Saadi have been supported by AG Turbo Project 4.1.2 and 4.1.13 co financed by BMWi and Siemens Energy. We also thank T. Beck (TU Kaiserslautern), B. Beckmann, H. Harders, G. Rollmann and A. Sohail (Siemens Energy) for interesting discussion.
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Bittner, L., Gottschalk, H., Gröger, M., Moch, N., Saadi, M., Schmitz, S. (2017). Modeling, Minimizing and Managing the Risk of Fatigue for Mechanical Components. In: Albeverio, S., Cruzeiro, A., Holm, D. (eds) Stochastic Geometric Mechanics . CIB-SGM 2015. Springer Proceedings in Mathematics & Statistics, vol 202. Springer, Cham. https://doi.org/10.1007/978-3-319-63453-1_10
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