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Journal of Nonlinear Science

, Volume 29, Issue 3, pp 1041–1094 | Cite as

Fatigue Effects in Elastic Materials with Variational Damage Models: A Vanishing Viscosity Approach

  • Roberto Alessi
  • Vito CrismaleEmail author
  • Gianluca Orlando
Article

Abstract

We study the existence of quasistatic evolutions for a family of gradient damage models which take into account fatigue, that is the process of weakening in a material due to repeated applied loads. The main feature of these models is the fact that damage is favoured in regions where the cumulation of the elastic strain (or other relevant variables, depending on the model) is higher. To prove the existence of a quasistatic evolution, we follow a vanishing viscosity approach based on two steps: we first let the time step \(\tau \) of the time discretisation and later the viscosity parameter \(\varepsilon \) go to zero. As \(\tau \rightarrow 0\), we find \(\varepsilon \)-approximate viscous evolutions; then, as \(\varepsilon \rightarrow 0\), we find a rescaled approximate evolution satisfying an energy-dissipation balance.

Keywords

Fatigue Gradient damage models Variational methods Vanishing viscosity approach 

Mathematics Subject Classification

74C05 74A45 74R20 35Q74 49J45 

Notes

Acknowledgements

The authors wish to thank Adriana Garroni for several interesting discussions and fruitful advices. Roberto Alessi has been supported by the MATHTECH-CNR-INdAM project and the MIUR-DAAD Joint Mobility Program: “Variational approach to fatigue phenomena with phase-field models: modeling, numerics and experiments”. Vito Crismale has been supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, and acknowledges the financial support from the Laboratory Ypatia and the CMAP. He is currently funded by the Marie Skłodowska-Curie Standard European Fellowship No. 793018. Gianluca Orlando has been supported by the Alexander von Humboldt Foundation. Vito Crismale and Gianluca Orlando acknowledge the kind hospitality of the Department of Mathematics of Sapienza University of Rome, where part of this research was developed.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Civil and Industrial EngineeringUniversità di PisaPisaItaly
  2. 2.CMAP, École Polytechnique, UMR CNRS 7641Palaiseau CedexFrance
  3. 3.Zentrum MathematikTechnische Universität MünchenGarching bei MünchenGermany

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