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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.

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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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