Acta Applicandae Mathematicae

, Volume 153, Issue 1, pp 125–146 | Cite as

A Mixed Variational Formulation of a Contact Problem with Wear

  • Mircea Sofonea
  • Flavius Pătrulescu
  • Ahmad Ramadan


We consider a mathematical model which describes the sliding frictional contact between a viscoplastic body and an obstacle, the so-called foundation. The process is quasistatic, the material’s behavior is described with a viscoplastic constitutive law with internal state variable and the contact is modelled with normal compliance and unilateral constraint. The wear of the contact surfaces is taken into account, and is modelled with a version of Archard’s law. We derive a mixed variational formulation of the problem which involve implicit history-dependent operators. Then, we prove the unique weak solvability of the contact model. The proof is based on a fixed point argument proved in Sofonea et al. (Commun. Pure Appl. Anal. 7:645–658, 2008), combined with a recent abstract existence and uniqueness result for mixed variational problems, obtained in Sofonea and Matei (J. Glob. Optim. 61:591–614, 2014).


viscoplastic material Frictional contact Normal compliance Unilateral constraint Wear Mixed variational formulation History-dependent operator Weak solution 

Mathematics Subject Classification

74M15 74G25 74G30 49J40 



The work of the authors has been partially supported by project LEA Math Mode 2014/2015. The work of the second author has been partially supported by project POSDRU/159/1.5/S/132400: Young successful researchers-professional development in an international and interdisciplinary environment at Babeş-Bolyai University.


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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  • Mircea Sofonea
    • 1
  • Flavius Pătrulescu
    • 2
  • Ahmad Ramadan
    • 1
  1. 1.Laboratoire de Mathématiques et PhysiqueUniversité de PerpignanPerpignanFrance
  2. 2.Tiberiu Popoviciu Institute of Numerical AnalysisCluj-NapocaRomania

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