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Zbl 1231.74065
Migórski, Stanisław; Ochal, Anna; Sofonea, Mircea
History-dependent subdifferential inclusions and hemivariational inequalities in contact mechanics. (English)
[J] Nonlinear Anal., Real World Appl. 12, No. 6, 3384-3396 (2011). ISSN 1468-1218

Summary: We consider a class of subdifferential inclusions involving a history-dependent term for which we provide an existence and uniqueness result. The proof is based on arguments on pseudomonotone operators and fixed point. Then we specialize this result in the study of a class of history-dependent hemivariational inequalities. Such kind of problems arises in a large number of mathematical models which describe quasistatic processes of contact between a deformable body and an obstacle, the so-called foundation. To provide an example we consider a viscoelastic problem in which the frictional contact is modeled with subdifferential boundary conditions. We prove that this problem leads to a history-dependent hemivariational inequality in which the unknown is the velocity field. Then we apply our abstract result in order to prove the unique weak solvability of the corresponding contact problem.

MSC 2000:: *74D10 Nonlinear constitutive equations
74M15 Contact
49J40 Variational methods including variational inequalities
35J87
47J22

Keywords: nonlinear inclusion; clarke subdifferential; history-dependent term; hemivariational inequality; viscoelastic material; frictional contact; existence and uniqueness; weak solution

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