Analysis of two quasistatic history-dependent contact models

Xiaoliang Cheng; Stanisław Migórski; Anna Ochal; Mircea Sofonea

doi:10.3934/dcdsb.2014.19.2425

Article Contents

2014, Volume 19, Issue 8: 2425-2445. Doi: 10.3934/dcdsb.2014.19.2425

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Analysis of two quasistatic history-dependent contact models

1.
Department of Mathematics, Zhejiang University, Hangzhou 310027
2.
Jagiellonian University, Faculty of Mathematics and Computer Science, Institute of Computer Science, ul. Łojasiewicza 6, 30348 Krakow
3.
Laboratoire de Mathématiques et Physique pour les Systèmes, Université de Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan

Received: September 30, 2013

Revised: December 31, 2013

Published: September 2014

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Abstract

We consider two mathematical models which describe the evolution of a viscoelastic and viscoplastic body, respectively, in contact with a piston or a device, the so-called obstacle or foundation. In both models the contact process is assumed to be quasistatic and the friction is described with a Clarke subdifferential boundary condition. The novelty of the models consists in the constitutive laws as well as in the contact conditions we use, which involve a memory term. We derive a variational formulation of the problems which is in the form of a system coupling a nonlinear integral equation with a history--dependent hemivariational inequality. Then, we prove the existence of a weak solution and, under additional assumptions, its uniqueness. The proof is based on a result on history--dependent hemivariational inequalities obtained in [18].

Keywords:

Viscoelastic material,
viscoplastic material,
normal compliance,
memory term,
frictional contact,
Clarke subdifferential,
history-dependent hemivariational inequality,
weak solution.

Mathematics Subject Classification: Primary: 74M15, 74M10, 74G25, 74G30, 74C10, 49J40; Secondary: 47J20, 35Q74, 35J85.

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