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A dynamic elastic-visco-plastic unilateral contact problem with normal damped response and Coulomb friction

Published online by Cambridge University Press:  07 April 2010

CHRISTOF ECK ,
JIŘÍ JARUŠEK  and
MIRCEA SOFONEA
CHRISTOF ECK
Affiliation:
Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, 70569 Stuttgart, Germany email: christof.eck@mathematik.uni-stuttgart.de
JIŘÍ JARUŠEK
Affiliation:
Mathematical Institute, Academy of Sciences of the Czech Republic, 115 67 Praha 1, Czech Republic
MIRCEA SOFONEA
Affiliation:
Laboratoire de Mathématiques, Physique et Systèmes, Université de Perpignan, 66860 Perpignan Cedex, France
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Abstract

We consider a dynamic frictional contact problem between an elastic-visco-plastic body and a foundation. The contact is modelled with a normal damped response condition of such a type that the normal velocity is restricted with unilateral constraint, associated with the Coulomb law in which the coefficient of friction may depend on the velocity. We derive a variational formulation of the problem which has the form of a system coupling an integro–differential equation for the stress field with an evolutionary variational inequality for the displacement field. This inequality is approximated by a variational equation using a smoothing of the friction and the penalty approximation of the unilateral condition. The existence of a weak solution to the variational equation is proved by the Galerkin method for an auxiliary problem with given viscoplastic part of the stress and a fixed point argument. The solvability of the original problem is proved by passing to the limit of the penalty parameter and the smoothing parameter. This convergence is based on a certain regularity of solutions which is verified with the use of a local rectification of the boundary and a translation method.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 21 , Issue 3 , June 2010 , pp. 229 - 251
Copyright
Copyright © Cambridge University Press 2010

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References

[1]Cristescu, N. & Suliciu, I. (1982) Viscoplasticity, Martinus Nijhoff, Editura Tehnica, Bucharest.Google Scholar
[2]Eck, C. (2002) Existence of solutions to a thermo-viscoelastic contact problem with Coulomb friction. Math. Mod. Methods Appl. Sci. 12, 1491–1511.Google Scholar
[3]Eck, C. & Jarušek, J. (2008) Solvability of static contact problems with Coulomb friction for orthotropic material. J. Elasticity 93, 93–104.CrossRefGoogle Scholar
[4]Eck, C., Jarušek, J. & Krbec, M. (2005) Unilateral Contact Problems: Variational Methods and Existence Theorems, Pure and Applied Mathematics 270, Chapman/CRC Press, New York.CrossRefGoogle Scholar
[5]Han, W. & Sofonea, M. (2002) Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics 30, American Mathematical Society, International Press.CrossRefGoogle Scholar
[6]Ionescu, I. R. & Sofonea, M. (1993) Functional and Numerical Methods in Viscoplasticity, Oxford University Press, Oxford.CrossRefGoogle Scholar
[7]Jarušek, J. (1990) Contact problem with given time-dependent friction force in linear viscoelasticity. Comment. Math. Univ. Carolinae 31, 257–262.Google Scholar
[8]Jarušek, J. (1996) Dynamic contact problems with given friction for viscoelastic bodies. Czech. Math. J. 46, 475–487.Google Scholar
[9]Jarušek, J. & Eck, C. (1999) Dynamic contact problems with small Coulomb friction for viscoelastic bodies. Existence of solutions. Math. Mod. Methods Appl. Sci. 9, 11–34.CrossRefGoogle Scholar
[10]Jarušek, J. & Sofonea, M. (2008) Dynamic contact problems with small Coulomb friction for viscoelastic bodies. ZAMM 88, 3–22.CrossRefGoogle Scholar
[11]Kuttler, K. L. (2007) Dynamic frictional contact for elastic viscoplastic material. Electron. J. Differ. Equ. 75, 11–20.Google Scholar
[12]Shillor, M., Sofonea, M. & Telega, J. J. (2004) Models and Analysis of Quasistatic Contact, Lecture Notes in Physics 655, Springer, Berlin.CrossRefGoogle Scholar
[13]Sofonea, M., Renon, N. & Shillor, M. (2004) Stress formulation for frictionless contact of an elastic-perfectly-plastic body. Appl. Anal. 83, 1157–1170.CrossRefGoogle Scholar