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Bifurcations in piecewise-smooth steady-state problems: abstract study and application to plane contact problems with friction

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Abstract

The paper presents a local study of bifurcations in a class of piecewise-smooth steady-state problems for which the regions of smooth behaviour permit analytical expressions. A system of piecewise-linear equations capturing the essential features of branching scenarios around points of non-smoothness is derived under the assumptions that (i) the points lie in the intersection of the boundaries of the regions where the gradients of the respective smooth selections have the full rank, (ii) there is no solution branch whose tangential direction is tangent to the boundary of any of the regions. The simplest cases of this system are studied in detail and the most probable branching scenarios are described. A criterion for detecting bifurcation points is proposed and a procedure for its realisation in the course of numerical continuation of solution curves is designed for large problems. Application of the general frame to discretised plane contact problems with Coulomb friction is explained. Simple as well as more realistic model examples of bifurcations are shown.

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Acknowledgments

The basis of this paper was laid down at a postdoctoral stay of T. Ligurský at INSA-Lyon funded by Manufacture Française des Pneumatiques Michelin. During further developments, T. Ligurský received support from the project “Support for building excellent research teams and inter-sectoral mobility at Palacký University in Olomouc II” (CZ.1.07/2.3.00/30.0041). Besides, both authors would like to thank anonymous referees for careful reading of the manuscript and their stimulating comments that have helped to improve the exposition.

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Authors and Affiliations

  1. Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, 771 46, Olomouc, Czech Republic

    T. Ligurský

  2. Université de Lyon, CNRS, INSA-Lyon, ICJ UMR5208, 69621, Villeurbanne, France

    Y. Renard

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  1. T. Ligurský
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  2. Y. Renard
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Correspondence to T. Ligurský.

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Ligurský, T., Renard, Y. Bifurcations in piecewise-smooth steady-state problems: abstract study and application to plane contact problems with friction. Comput Mech 56, 39–62 (2015). https://doi.org/10.1007/s00466-015-1154-1

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  • DOI: https://doi.org/10.1007/s00466-015-1154-1

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