Comptes Rendus
Shakedown in frictional contact problems for the continuum
Comptes Rendus. Mécanique, Volume 336 (2008) no. 1-2, pp. 34-41.

Elastic systems with frictional interfaces subjected to periodic loading are often found to ‘shake down’ in the sense that frictional slip ceases after the first few loading cycles. The similarities in behaviour between such systems and monolithic bodies with elastic–plastic constitutive behaviour have prompted various authors to speculate that Melan's theorem might apply to them—i.e. that the existence of a state of residual stress sufficient to prevent further slip is a sufficient condition for the system to shake down.

In this article, we prove this result for ‘complete’ contact problems in the continuum formulation for systems with no coupling between relative tangential displacements at the interface and the corresponding normal contact tractions. This condition is satisfied for the contact of two half spaces, or of a rigid body with a half space if Dundurs' constant β=0. It is also satisfied for the contact of two symmetric bodies of similar materials at the plane of symmetry.

Les systèmes élastiques comportant des interfaces en contact frottant, soumis à des chargements périodiques, « s'adaptent » souvent dans le sens où le glissement cesse après les premiers cycles de chargement. Les similitudes entre le comportement de tels systèmes et celui de corps monolithiques à comportement constitutif élasto-plastique ont incité divers auteurs à penser que le théorème de Melan pourrait s'y appliquer—ce qui signifierait que l'existence d'un état d'efforts résiduels suffisants pour empêcher la poursuite du glissement est une condition suffisante pour que le système s'adapte. Dans cet article, nous prouvons ce résultat pour des problèmes « complets » de contact entre milieux continus, dans le cas de systèmes sans couplage entre les déplacements tangentiels relatifs à l'interface et les tractions normales de contact correspondantes. Cette condition est satisfaite pour le contact de deux demi-espaces, ou d'un corps rigide et d'un demi-espace, si la constante β de Dundurs est nulle. Elle est également satisfaite pour le contact de deux corps symétriques constitués de matériaux semblables.

Published online:
DOI: 10.1016/j.crme.2007.10.013
Keywords: Continuum mechanics, Contact problems, Fretting fatigue, Shakedown, Melan's theorem, Coulomb friction
Mot clés : Milieux continus, Problèmes de contact, Fatigue de fretting, Adaptation, Théorème de Melan, Frottement de Coulomb

James R. Barber 1; Anders Klarbring 2; Michele Ciavarella 3

1 Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2125, USA
2 Department of Mechanical Engineering, Linköping University, 58183 Linköping, Sweden
3 CEMEC–PoliBA – Centre of Excellence in Computational Mechanics, V.le Japigia 182, Politecnico di Bari, 70125 Bari, Italy
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James R. Barber; Anders Klarbring; Michele Ciavarella. Shakedown in frictional contact problems for the continuum. Comptes Rendus. Mécanique, Volume 336 (2008) no. 1-2, pp. 34-41. doi : 10.1016/j.crme.2007.10.013. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2007.10.013/

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