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On Abstract Variational Inequalities in Viscoplasticity with Frictional Contact

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Abstract

In this paper, we study quasistatic abstract variational inequalities with time-dependent constraints. We prove existence results and present an approximation method valid for nonsmooth constraints. Then, we apply our results to the approximation of the quasistatic evolution of an elastic body in bilateral contact with a rigid foundation. The contact involves viscous friction of the Tresca or Coulomb type. We prove existence results for approximate problems and give a full asymptotic analysis, proving strong or weak convergence results. Our work is motivated by the numerical study in the paper [Delost, M.: Quasistatic Problem with Frictional Contact: Comparison between Numerical Methods and Asymptotic Analysis Related to Semi Discrete and Fully Discrete Approximations. University of Nice, Nice (2007, to appear)] and explains the choice of the approximation made in it.

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References

  1. Cocu, M., Pratt, E., Raous, M.: Formulation and approximation of quasistatic frictional contact. Int. J. Eng. Sci. 34, 783–798 (1996)

    Article  MATH  Google Scholar 

  2. Amassad, A., Fabre, C.: Existence for viscoplastic contact with Coulomb friction law problems. Int. J. Math. Math. Sci. 32, 411–437 (2002)

    Article  MATH  Google Scholar 

  3. Shillor, M., Sofonea, M.: A quasistatic viscoelastic contact problem with friction. Int. J. Eng. Sci. 38(14), 1517–1533 (2000)

    Article  Google Scholar 

  4. Amassad, A., Fabre, C.: On the analysis of viscoelastic unilateral contact problem involving Coulomb friction law. J. Optim. Theory Appl. 116, 465–481 (2003)

    Article  MATH  Google Scholar 

  5. Brezis, H.: Problèmes unilatéraux. J. Math. Pures Appl. 51, 1–168 (1972)

    Google Scholar 

  6. Duvaut, G., Lions, J.-L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)

    MATH  Google Scholar 

  7. Glowinski, R., Lions, J.-L., Tremolieres, R.: Analyse Numérique des Inéquations Variationnelles. Collection des Méthodes Mathématiques de l’Informatique (1981)

  8. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and their Applications. Academic, New York (1980)

    MATH  Google Scholar 

  9. Panagiotopoulos, P.-D.: Inequalities Problems in Mechanics and Applications. Birkhauser, Basel (1985)

    Google Scholar 

  10. Delost, M., Fabre, C.: On abstract variational inequalities in viscoplasticity with frictional contact. Rapport interne 2006-10 du laboratoire de mathématique de l’université Paris Sud (2006)

  11. Amassad, A., Sofonea, M.: Analysis of a quasistatic viscoplastic problem involving tresca friction law. Discret. Contin. Dyn. Syst. 4, 55–72 (1998)

    MATH  Google Scholar 

  12. Delost, M.: Quasistatic Problem with Frictional Contact: Comparison between Numerical Methods and Asymptotic Analysis Related to Semi Discrete and Fully Discrete Approximations. University of Nice, Nice (2007, to appear)

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Authors and Affiliations

  1. Observatoire Ocanologique, UMR 6526, UR 082 BP 48, 06235, Villefranche sur Mer, France

    M. Delost

  2. Université Paris Sud, CNRS UMR 8628, Bat 425, 91405, Orsay, France

    C. Fabre

Authors
  1. M. Delost
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  2. C. Fabre
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Correspondence to C. Fabre.

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Communicated by R. Glowinski.

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Delost, M., Fabre, C. On Abstract Variational Inequalities in Viscoplasticity with Frictional Contact. J Optim Theory Appl 133, 131–150 (2007). https://doi.org/10.1007/s10957-007-9176-2

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  • DOI: https://doi.org/10.1007/s10957-007-9176-2

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