Discrete and Continuous Dynamical Systems - Series S (DCDS-S)

Long-time behaviour of a thermomechanical model for adhesive contact

Pages: 273 - 309, Volume 4, Issue 2, April 2011      doi:10.3934/dcdss.2011.4.273

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Elena Bonetti - Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy (email)
Giovanna Bonfanti - Dipartimento di Matematica Università di Brescia, Via Valotti 9, I–25133 Brescia, Italy (email)
Riccarda Rossi - Dipartimento di Matematica, Università di Brescia, Via Valotti 9, I–25133 Brescia, Italy (email)

Abstract: This paper deals with the large-time analysis of a PDE system modelling contact with adhesion, in the case when thermal effects are taken into account. The phenomenon of adhesive contact is described in terms of phase transitions for a surface damage model proposed by M. Frémond. Thermal effects are governed by entropy balance laws. The resulting system is highly nonlinear, mainly due to the presence of internal constraints on the physical variables and the coupling of equations written in a domain and on a contact surface. We prove existence of solutions on the whole time interval $(0,+\infty)$ by a double approximation procedure. Hence, we are able to show that solution trajectories admit cluster points which fulfil the stationary problem associated with the evolutionary system, and that in the large-time limit dissipation vanishes.

Keywords:  Nonlinear PDE system, contact with adhesion, long-time behaviour.
Mathematics Subject Classification:  Primary: 35K55, 74A15; Secondary: 74M15.

Received: April 2009;      Available Online: November 2010.