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Signorini-Type Problems over Non-Convex Sets for Composite Bodies Contacting by Sharp Edges of Rigid Inclusions

Author

Listed:
  • Nyurgun P. Lazarev

    (Department of Mathematics, North-Eastern Federal University, Belinsky Str., 58, 677891 Yakutsk, Russia
    These authors contributed equally to this work.)

  • Victor A. Kovtunenko

    (Institute for Mathematics and Scientific Computing, Karl-Franzens University of Graz, NAWI Graz, Heinrichstr. 36, 8010 Graz, Austria
    Lavrentyev Institute of Hydrodynamics, Siberian Division of the Russian Academy of Sciences, 630090 Novosibirsk, Russia
    These authors contributed equally to this work.)

Abstract

A new type of non-classical 2D contact problem formulated over non-convex admissible sets is proposed. Specifically, we suppose that a composite body in its undeformed state touches a wedge-shaped rigid obstacle at a single contact point. Composite bodies under investigation consist of an elastic matrix and a rigid inclusion. In this case, the displacements on the set, corresponding to a rigid inclusion, have a predetermined structure that describes possible parallel shifts and rotations of the inclusion. The rigid inclusion is located on the external boundary and has the form of a wedge. The presence of the rigid inclusion imposes a new type of non-penetration condition for certain geometrical configurations of the obstacle and the body near the contact point. The sharp-shaped edges of the obstacle effect such sets of admissible displacements that may be non-convex. For the case of a thin rigid inclusion, which is described by a curve and a volume (bulk) rigid inclusion specified in a subdomain, the energy minimization problems are formulated. The solvability of the corresponding boundary value problems is proved, based on analysis of auxiliary minimization problems formulated over convex sets. Qualitative properties of the auxiliary variational problems are revealed; in particular, we have found their equivalent differential formulations. As the most important result of this study, we provide justification for a new type of mathematical model for 2D contact problems for reinforced composite bodies.

Suggested Citation

  • Nyurgun P. Lazarev & Victor A. Kovtunenko, 2022. "Signorini-Type Problems over Non-Convex Sets for Composite Bodies Contacting by Sharp Edges of Rigid Inclusions," Mathematics, MDPI, vol. 10(2), pages 1-11, January.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:2:p:250-:d:724691
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    References listed on IDEAS

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    1. Andreas Rademacher & Korinna Rosin, 2018. "Adaptive optimal control of Signorini’s problem," Computational Optimization and Applications, Springer, vol. 70(2), pages 531-569, June.
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    Cited by:

    1. Evgeny Rudoy & Sergey Sazhenkov, 2023. "Variational Approach to Modeling of Curvilinear Thin Inclusions with Rough Boundaries in Elastic Bodies: Case of a Rod-Type Inclusion," Mathematics, MDPI, vol. 11(16), pages 1-14, August.

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