Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



On the behavior of the solution of a viscoplastic contact problem

Authors: M. Barboteu, A. Matei and M. Sofonea
Journal: Quart. Appl. Math. 72 (2014), 625-647
MSC (2010): Primary 74M15, 74G25, 74G30; Secondary 74S05, 49J40
DOI: https://doi.org/10.1090/S0033-569X-2014-01345-4
Published electronically: September 25, 2014
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider 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 modeled with normal compliance and unilateral constraint. We provide a mixed variational formulation of the model which involves a dual Lagrange multiplier, and then we prove its unique weak solvability. We also prove an estimate which allows us to deduce the continuous dependence of the weak solution with respect to both the normal compliance function and the penetration bound. Finally, we provide a numerical validation of this convergence result.

References [Enhancements On Off] (What's this?)

  • [1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
  • [2] P. Alart and A. Curnier, A mixed formulation for frictional contact problems prone to Newton like solution methods, Comput. Methods Appl. Mech. Engrg. 92 (1991), no. 3, 353–375. MR 1141048, https://doi.org/10.1016/0045-7825(91)90022-X
  • [3] M. Barboteu, A. Matei, and M. Sofonea, Analysis of quasistatic viscoplastic contact problems with normal compliance, Quart. J. Mech. Appl. Math. 65 (2012), no. 4, 555–579. MR 2995754, https://doi.org/10.1093/qjmam/hbs016
  • [4] N. Cristescu and I. Suliciu, Viscoplasticity, Mechanics of Plastic Solids, vol. 5, Martinus Nijhoff Publishers, The Hague, 1982. Translated from the Romanian. MR 691135
  • [5] Dietrich Braess, Finite elements, Cambridge University Press, Cambridge, 1997. Theory, fast solvers, and applications in solid mechanics; Translated from the 1992 German original by Larry L. Schumaker. MR 1463151
  • [6] Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205
  • [7] Ivar Ekeland and Roger Temam, Convex analysis and variational problems, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. Translated from the French; Studies in Mathematics and its Applications, Vol. 1. MR 0463994
  • [8] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
  • [9] Weimin Han and Mircea Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity, AMS/IP Studies in Advanced Mathematics, vol. 30, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002. MR 1935666
  • [10] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes, Dunod, Paris, 1968.
  • [11] J. Haslinger, I. Hlaváček, and J. Nečas, Numerical methods for unilateral problems in solid mechanics, Handbook of numerical analysis, Vol. IV, Handb. Numer. Anal., IV, North-Holland, Amsterdam, 1996, pp. 313–485. MR 1422506
  • [12] Ioan R. Ionescu and Mircea Sofonea, Functional and numerical methods in viscoplasticity, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993. MR 1244578
  • [13] Jiří Jarušek and Mircea Sofonea, On the solvability of dynamic elastic-visco-plastic contact problems, ZAMM Z. Angew. Math. Mech. 88 (2008), no. 1, 3–22. MR 2376989, https://doi.org/10.1002/zamm.200710360
  • [14] H. B. Khenous, P. Laborde and Y. Renard, On the discretization of contact problems in elastodynamics, Lecture Notes in Applied Computational Mechanics 27 (2006), 31-38.
  • [15] Houari Boumediène Khenous, Julien Pommier, and Yves Renard, Hybrid discretization of the Signorini problem with Coulomb friction. Theoretical aspects and comparison of some numerical solvers, Appl. Numer. Math. 56 (2006), no. 2, 163–192. MR 2200937, https://doi.org/10.1016/j.apnum.2005.03.002
  • [16] Tod A. Laursen, Computational contact and impact mechanics, Springer-Verlag, Berlin, 2002. Fundamentals of modeling interfacial phenomena in nonlinear finite element analysis. MR 1902698
  • [17] A. Matei and R. Ciurcea, Contact problems for nonlinearly elastic materials: weak solvability involving dual Lagrange multipliers, ANZIAM J. 52 (2010), no. 2, 160–178. MR 2832610, https://doi.org/10.1017/S1446181111000629
  • [18] M. Shillor, M. Sofonea, and J. Telega, Models and Variational Analysis of Quasistatic Contact, Lecture Notes in Physics 655, Springer, Berlin Heidelberg, 2004.
  • [19] M. Sofonea and A. Matei, A mixed variational formulation for the Signorini frictionless problem in viscoplasticity, An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 12 (2004), no. 2, 157–170. MR 2209124
  • [20] M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series 398, Cambridge University Press, Cambridge, 2012.
  • [21] P. Wriggers, Computational Contact Mechanics, Wiley, Chichester, 2002.

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2010): 74M15, 74G25, 74G30, 74S05, 49J40

Retrieve articles in all journals with MSC (2010): 74M15, 74G25, 74G30, 74S05, 49J40

Additional Information

M. Barboteu
Affiliation: Laboratoire de Mathématiques et Physique, Université de Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan, France
Email: barboteu@univ-perp.fr

A. Matei
Affiliation: Department of Mathematics, University of Craiova, A.I. Cuza Street 13, 200585, Craiova, Romania
Email: andaluziamatei2000@yahoo.com

M. Sofonea
Affiliation: Laboratoire de Mathématiques et Physique, Université de Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan, France
Email: sofonea@univ-perp.fr

DOI: https://doi.org/10.1090/S0033-569X-2014-01345-4
Received by editor(s): August 30, 2012
Published electronically: September 25, 2014
Additional Notes: This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-RU-TE-2011-3-0223.
Article copyright: © Copyright 2014 Brown University

American Mathematical Society