Electron. J. Differential Equations, Vol. 2017 (2017), No. 299, pp. 1-17.

Analysis of a rate-and-state friction problem with viscoelastic materials

Flavius Patrulescu, Mircea Sofonea

We consider a mathematical model which describes the frictional contact between a viscoelastic body and a foundation. The contact is modelled with normal compliance associated to a rate-and-state version of Coulomb's law of dry friction. We start by presenting a description of the friction law, together with some examples used in geophysics. Then we state the classical formulation of the problem, list the assumptions on the data and derive a variational formulation of the model. It is in a form of a differential variational inequality in which the unknowns are the displacement field and the surface state variable. Next, we prove the unique weak solvability of the problem. The proof is based on arguments of history-dependent variational inequalities and nonlinear implicit integral equations in Banach spaces.

Submitted September 21, 2017. Published December 5, 2017.
Math Subject Classifications: 74M15, 74M10, 74G25, 74G30, 49J40.
Key Words: Viscoelastic material; frictional contact; normal compliance; rate-and-state friction; differential variational inequality; history-dependent operator; weak solution.

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Flavius Patrulescu
Tiberiu Popoviciu Institute of Numerical Analysis
Romanian Academy, P.O. Box 68-1
400110 Cluj-Napoca, Romania
email: fpatrulescu@ictp.acad.ro
Mircea T. Sofonea
Laboratoire de Mathématiques et Physique
Université de Perpignan Via Domitia
52 Avenue de Paul Alduy
66 860 Perpignan, France
email: sofonea@univ-perp.fr

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