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History-dependent mixed variational problems in contact mechanics

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Abstract

We consider a new class of mixed variational problems arising in Contact Mechanics. The problems are formulated on the unbounded interval of time \([0,+\infty )\) and involve history-dependent operators. For such problems we prove existence, uniqueness and continuous dependence results. The proofs are based on results on generalized saddle point problems and various estimates, combined with a fixed point argument. Then, we apply the abstract results in the study of a mathematical model which describes the frictionless contact between a viscoplastic body and an obstacle, the so-called foundation. The process is quasistatic and the contact is modelled with normal compliance and unilateral constraint, in such a way that the stiffness coefficient depends on the history of the penetration. We prove the unique weak solvability of the contact problem, as well as the continuous dependence of its weak solution with respect to the viscoplastic constitutive function, the applied forces, the contact conditions and the initial data.

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Acknowledgments

This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, Project Number PN-II-ID-PCE-2011-3-0257.

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Authors and Affiliations

  1. Laboratoire de Mathématiques et Physique, Université de Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 , Perpignan, France

    Mircea Sofonea

  2. Department of Mathematics, University of Craiova, A.I. Cuza Street 13, 200585 , Craiova, Romania

    Andaluzia Matei

Authors
  1. Mircea Sofonea
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  2. Andaluzia Matei
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Correspondence to Mircea Sofonea.

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Sofonea, M., Matei, A. History-dependent mixed variational problems in contact mechanics. J Glob Optim 61, 591–614 (2015). https://doi.org/10.1007/s10898-014-0193-z

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  • DOI: https://doi.org/10.1007/s10898-014-0193-z

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