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On quasi-static contact problem with generalized Coulomb friction, normal compliance and damage

Published online by Cambridge University Press:  09 November 2015

LESZEK GASIŃSKI
Affiliation:
Faculty of Mathematics and Computer Science, Jagiellonian University, Lojasiewicza 6, 30348 Krakow, Poland email: leszek.gasinski@ii.uj.edu.pl, piotr.kalita@ii.uj.edu.pl
PIOTR KALITA
Affiliation:
Faculty of Mathematics and Computer Science, Jagiellonian University, Lojasiewicza 6, 30348 Krakow, Poland email: leszek.gasinski@ii.uj.edu.pl, piotr.kalita@ii.uj.edu.pl

Abstract

In this paper, we study a quasi-static frictional contact problem for a viscoelastic body with damage effect inside the body as well as normal compliance condition and multi-valued friction law on the contact boundary. The considered friction law generalizes Coulomb friction condition into multi-valued setting. The variational–hemi-variational formulation of the problem is derived and arguments of fixed point theory and surjectivity results for pseudo-monotone operators are applied, in order to prove the existence and uniqueness of solution.

Type
Papers
Copyright
Copyright © Cambridge University Press 2015 

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Footnotes

The research was supported by the Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme under Grant Agreement No. 295118, the International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under grant no. W111/7.PR/2012 and the National Science Center of Poland under Maestro Advanced Project no. DEC-2012/06/A/ST1/00262 and the project Polonium Mathematical and Numerical Analysis for Contact Problems with Friction 2014/15 between the Jagiellonian University and Université de Perpignan Via Domitia.

References

[1] Andrews, K. T. & Shillor, M. (2006) Thermomechanical behaviour of a damageable beam in contact with two stops. Appl. Anal. 85 (6–7), 845865.CrossRefGoogle Scholar
[2] Andrews, K. T., Anderson, S., Menike, R. S. R., Shillor, M., Swaminathan, R. & Yuzwalk, J. (2007) Vibrations of a damageable string, In Botelho, F., Hagen, T. & Jamison, J. (editors), Fluids and Waves – Recent Trends in Applied Analysis, Contemporary Mathematics, Vol. 440, AMS, Rhode Island, pp. 114.Google Scholar
[3] Barbu, V. (1984) Optimal Control of Variational Inequalities, Pitman, Boston.Google Scholar
[4] Bonetti, E. & Schimperna, G. (2004) Local existence for Frémond's model of damage in elastic materials. Continuum Mech. Therm. 16 (4), 319335.CrossRefGoogle Scholar
[5] Chipman, J. C., Roux, A., Shillor, M. & Sofonea, M. (2011) A damageable spring. Machine Dyn. Res. 35 (1), 8296.Google Scholar
[6] Clarke, F. H. (1983) Optimization and Nonsmooth Analysis, Wiley, New York.Google Scholar
[7] Denkowski, Z., Migórski, S. & Papageorgiou, N. S. (2003) An Introduction to Nonlinear Analysis: Theory, Kluwer Academic/Plenum Publishers, Boston.Google Scholar
[8] Denkowski, Z., Migórski, S. & Papageorgiou, N. S. (2003) An Introduction to Nonlinear Analysis: Applications, Kluwer Academic/Plenum Publishers, Boston.Google Scholar
[9] Frémond, M. (2002) Non-Smooth Thermomechanics, Springer, Berlin.CrossRefGoogle Scholar
[10] Frémond, M., Kuttler, K. L. & Shillor, M. (1999) Existence and uniqueness of solutions for a one-dimensional damage model. J. Math. Anal. Appl. 229 (1), 271294.CrossRefGoogle Scholar
[11] Gasiński, L. & Ochal, A. (2015) Dynamic thermoviscoelastic problem with friction and damage. Nonlinear Anal.-Real 21 (1), 6375.CrossRefGoogle Scholar
[12] Gasiński, L., Ochal, A. & Shillor, M. (2015) Variational-hemivariational approach to a quasistatic viscoelastic problem with normal compliance, friction and material damage. Z. Anal. Anwend. 34 (3), 251275.CrossRefGoogle Scholar
[13] Han, W., Shillor, M. & Sofonea, M. (2001) Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction and damage. J. Comput. Appl. Math. 137 (2), 377398.CrossRefGoogle Scholar
[14] Kuttler, K. L., Purcell, J. & Shillor, M. (2012) Analysis and simulations of a contact problem for a nonlinear dynamic beam with a crack. Q. J. Mech. Appl. Math. 65 (1), 125.CrossRefGoogle Scholar
[15] Kuttler, K. L. & Shillor, M. (2006) Quasistatic evolution of damage in an elastic body Nonlinear Anal. Real World Appl. 7 (4), 674699.CrossRefGoogle Scholar
[16] Li, Y. & Liu, Z. (2010) A quasistatic contact problem for viscoelastic materials with friction and damage. Nonlinear Anal. 73 (7), 22212229.CrossRefGoogle Scholar
[17] Migórski, S., Ochal, A. & Sofonea, M. (2013) Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, Vol. 26, Springer, New York, 288 pages.Google Scholar
[18] Naniewicz, Z. & Panagiotopoulos, P. D. (1995) Mathematical Theory of Hemivariational Inequalities and Applications, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 188, Marcel Dekker, New York, 267 pages.Google Scholar
[19] Shillor, M., Sofonea, M. & Telega, J. J. (2004) Models and Analysis of Quasistatic Contact, Lecture Notes in Physics, Vol. 655, Springer, Berlin.CrossRefGoogle Scholar
[20] Sofonea, M., Avramescu, C. & Matei, A. (2008) A fixed point result with applications in the study of viscoplastic frictionless contact problems. Commun. Pure Appl. Ana. 7 (3), 645658.CrossRefGoogle Scholar
[21] Sofonea, M., Han, W. & Shillor, M. (2006) Analysis and Approximations of Contact Problems with Adhesion or Damage, Pure and Applied Mathematics, Vol. 276, Chapman & Hall/CRC Press, Boca Raton, Florida, 220 pages.Google Scholar
[22] Szafraniec, P. Dynamic nonsmooth frictional contact problems with damage in thermoviscoelasticity. Math. Mech. Solids, to appear, doi: 10.1177/1081286514527860.Google Scholar
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