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January  2014, 13(1): 371-387. doi: 10.3934/cpaa.2014.13.371

## A viscoplastic contact problem with a normal compliance with limited penetration condition and history-dependent stiffness coefficient

 1 Laboratoire de Mathématiques et Physique pour les Systèmes, Université de Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan 2 Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309

Received  April 2013 Revised  June 2013 Published  July 2013

We consider a mathematical model that describes frictionless contact between a viscoplastic body and a deformable obstacle or foundation. The process is quasistatic and contact is modeled with the normal compliance with limited penetration condition, which has been introduced recently. Moreover, the contact stiffness coefficient is allowed to depend on the history of the contact process. We derive a variational formulation of the problem, which is in the form of a strongly nonlinear system coupling an integral equation and a time-dependent variational inequality. Then, we provide the analysis of the problem, which includes its unique weak solvability and the continuous dependence of the solution on the problem data. The proofs are based on results from the theory of history-dependent variational inequalities, on monotonicity and a fixed point argument.
Citation: Mircea Sofonea, Meir Shillor. A viscoplastic contact problem with a normal compliance with limited penetration condition and history-dependent stiffness coefficient. Communications on Pure and Applied Analysis, 2014, 13 (1) : 371-387. doi: 10.3934/cpaa.2014.13.371
##### References:
 [1] M. Barboteu, A. Matei and M. Sofonea, Analysis of quasistatic viscoplastic contact problems with normal compliance, Quarterly Journal of Mechanics and Applied Mathematics, 65 (2012), 555-579. [2] M. Barboteu, A. Matei and M. Sofonea, On the behaviours of the solution of a viscoplastic contact problem,, Quarterly of Applied Mathematics, (). [3] N. Cristescu and I. Suliciu, "Viscoplasticity," Martinus Nijhoff Publishers, Editura Tehnica, Bucharest, 1982. [4] G. Duvaut and J.-L. Lions, "Inequalities in Mechanics and Physics," Springer-Verlag, Berlin, 1976. [5] C. Eck, J. Jarušek and M. Krbeč, "Unilateral Contact Problems: Variational Methods and Existence Theorems," Pure and Applied Mathematics 270, Chapman/CRC Press, New York, 2005. [6] J. Jarušek and M. Sofonea, On the solvability of dynamic elastic-visco-plastic contact problems, Zeitschrift für Angewandte Matematik und Mechanik (ZAMM), 88 (2008), 3-22. [7] A. Klarbring, A. Mikelic and M. Shillor, Frictional contact problems with normal compliance, Int. J. Engng. Sci., 26 (1988), 811-832. [8] A. Klarbring, A. Mikelic and M. Shillor, On friction problems with normal compliance, Nonlinear Analysis, 13 (1989), 935-955. [9] T. Laursen, "Computational Contact and Impact Mechanics," Springer, Berlin, 2002. [10] J. A. C. Martins and J. T. Oden, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlinear Analysis TMA, 11 (1987), 407-428. [11] J. T. Oden and J. A. C. Martins, Models and computational methods for dynamic friction phenomena, Computer Methods in Applied Mechanics and Engineering, 52 (1985), 527-634. [12] J. Piotrowski, Smoothing dry friction damping by dither generated in rolling contact of wheel and rail and its influence on ride dynamics of freight wagons, Vehicle System Dynamics, 48 (2010), 675-703. doi: 10.1080/00423110903126478. [13] M. Shillor, M. Sofonea and J. Telega, "Models and Variational Analysis of Quasistatic Contact," Lecture Notes in Physics 655, Springer, 2004. doi: 10.1007/b99799. [14] M. Sofonea and A. Matei, A mixed variational formulation for the Signorini frictionless problem in viscoplasticity, Annals Univ. Ovidius Constanta, 12 (2004), 157-170. [15] M. Sofonea and A. Matei, "Mathematical Models in Contact Mechanics," London Mathematical Society Lecture Note Series 398, Cambridge University Press, Cambridge, 2012. [PMid:23005564] doi: 10.1017/CBO9781139104166. [16] P. Wriggers, "Computational Contact Mechanics," Wiley, Chichester, 2002. doi: PMCid:PMC123642.

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##### References:
 [1] M. Barboteu, A. Matei and M. Sofonea, Analysis of quasistatic viscoplastic contact problems with normal compliance, Quarterly Journal of Mechanics and Applied Mathematics, 65 (2012), 555-579. [2] M. Barboteu, A. Matei and M. Sofonea, On the behaviours of the solution of a viscoplastic contact problem,, Quarterly of Applied Mathematics, (). [3] N. Cristescu and I. Suliciu, "Viscoplasticity," Martinus Nijhoff Publishers, Editura Tehnica, Bucharest, 1982. [4] G. Duvaut and J.-L. Lions, "Inequalities in Mechanics and Physics," Springer-Verlag, Berlin, 1976. [5] C. Eck, J. Jarušek and M. Krbeč, "Unilateral Contact Problems: Variational Methods and Existence Theorems," Pure and Applied Mathematics 270, Chapman/CRC Press, New York, 2005. [6] J. Jarušek and M. Sofonea, On the solvability of dynamic elastic-visco-plastic contact problems, Zeitschrift für Angewandte Matematik und Mechanik (ZAMM), 88 (2008), 3-22. [7] A. Klarbring, A. Mikelic and M. Shillor, Frictional contact problems with normal compliance, Int. J. Engng. Sci., 26 (1988), 811-832. [8] A. Klarbring, A. Mikelic and M. Shillor, On friction problems with normal compliance, Nonlinear Analysis, 13 (1989), 935-955. [9] T. Laursen, "Computational Contact and Impact Mechanics," Springer, Berlin, 2002. [10] J. A. C. Martins and J. T. Oden, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlinear Analysis TMA, 11 (1987), 407-428. [11] J. T. Oden and J. A. C. Martins, Models and computational methods for dynamic friction phenomena, Computer Methods in Applied Mechanics and Engineering, 52 (1985), 527-634. [12] J. Piotrowski, Smoothing dry friction damping by dither generated in rolling contact of wheel and rail and its influence on ride dynamics of freight wagons, Vehicle System Dynamics, 48 (2010), 675-703. doi: 10.1080/00423110903126478. [13] M. Shillor, M. Sofonea and J. Telega, "Models and Variational Analysis of Quasistatic Contact," Lecture Notes in Physics 655, Springer, 2004. doi: 10.1007/b99799. [14] M. Sofonea and A. Matei, A mixed variational formulation for the Signorini frictionless problem in viscoplasticity, Annals Univ. Ovidius Constanta, 12 (2004), 157-170. [15] M. Sofonea and A. Matei, "Mathematical Models in Contact Mechanics," London Mathematical Society Lecture Note Series 398, Cambridge University Press, Cambridge, 2012. [PMid:23005564] doi: 10.1017/CBO9781139104166. [16] P. Wriggers, "Computational Contact Mechanics," Wiley, Chichester, 2002. doi: PMCid:PMC123642.
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