Slip-dependent friction in dynamic elasticity - Sylvie Wolf

Since w is a solution of the wave equation w(t; z) − [email protected] ..... [19] G. Perrin, J.R. Rice, G. Zheng, Self-healing slip pulse on a frictional surface, J. Mech. Phys.
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Nonlinear Analysis 53 (2003) 375 – 390

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Slip-dependent friction in dynamic elasticity Ioan R. Ionescua;∗ , Quoc-Lan Nguyenb , Sylvie Wolf a; c a Laboratoire

de Math ematiques, Universit e de Savoie, Campus Scientique, 73376 Le Bourget-du-Lac Cedex, France b National Polytechnic Institute, Ho-Shi-Minh, 143/2D Xo Viet nghe Tinh (he m), P25, Q.BT, Ho-Shi-Minh, Viet Nam c Laboratoire de G eophysique Interne, Universit e Joseph Fourier, BP 53X, 38041 Grenoble Cedex, France Received 1 August 2001; accepted 15 October 2001

Abstract The dynamic evolution with frictional contact of an elastic body is considered. In modeling the contact the Tresca model and a slip-dependent friction law are used. The existence of a solution is proved in the two-dimensional case. The uniqueness is proved for the one-dimensional shearing problem. The convergence, for a vanishing viscosity, of the unique solution of the viscoelastic problem to a solution of the elastic problem is obtained. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Slip-dependent friction; Dynamic elasticity; Tresca contact; Existence; Uniqueness; Viscoelasticity; Vanishing viscosity

1. Introduction A lot of applications in solid mechanics and geophysics involve frictional contact problems between elastic structures. The 5rst existence and uniqueness results for contact problems with friction in elastodynamics were obtained by Duvaut and Lions [5]. Martins and Oden [16] relaxed, a few years later, the non-penetrability of mass by considering the normal compliance model of contact with friction. In the viscoelastic case they obtained in [16] existence and uniqueness results (see also [14]). In the study of some frictional processes, such as stick–slip motions and earthquakes modeling, it is important to consider that the friction coe