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

## Abstract

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.

## Authors

Flavius Patrulescu
(Tiberiu Popoviciu Institute of Numerical Analysis,

Mircea Sofonea
(Laboratoire de Mathématiques et Physique, Université de Perpignan)

## Keywords

viscoelastic material; frictional contact; normal compliance; rate-and-state friction; differential variational inequality; history-dependent operator; weak solution

## Cite this paper as

F. Patrulescu, M. Sofonea, Analysis of a rate-and-state friction problem with viscoelastic materials, Electron. J. Differential Equations, vol. 2017 (2017), no. 299, pp. 1-17.

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##### Publisher Name

Southwest Texas State University, Department of Mathematics, San Marcos, TX; North Texas State University, Department of Mathematics, Denton, TX

1072-6691

3748017

1386.74104