Abstract
We consider a new class of variational-hemivariational inequalities which arise in the study of quasistatic models of contact. The novelty lies in the special structure of these inequalities, since each inequality of the class involve unilateral constraints, a history-dependent operator and two nondifferentiable functionals, of which at least one is convex. We prove an existence and uniqueness result of the solution. The proof is based on arguments on elliptic variational-hemivariational inequalities obtained in our previous work [23], combined with a fixed point result obtained in [30]. Then, we prove a convergence result which shows the continuous dependence of the solution with respect to the data. Finally, we present a quasistatic frictionless problem for viscoelastic materials in which the contact is modeled with normal compliance and finite penetration and the elasticity operator is associated to a history-dependent Von Mises convex. We prove that the variational formulation of the problem cast in the abstract setting of history-dependent quasivariational inequalities, with a convenient choice of spaces and operators. Then we apply our general results in order to prove the unique weak solvability of the contact problem and its continuous dependence on the data.
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Research supported by the Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme under Grant Agreement No. 295118, the National Science Center of Poland under the Maestro Advanced Project No. DEC-2012/06/A/ST1/00262, and the project Polonium “Mathematical and Numerical Analysis for Contact Problems with Friction” 2014/15 between the Jagiellonian University and Universit´e de Perpignan Via Domitia. The second author is also partially supported by the International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under Grant No. W111/7.PR/2012.
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Sofonea, M., Migórski, S. A class of history-dependent variational-hemivariational inequalities. Nonlinear Differ. Equ. Appl. 23, 38 (2016). https://doi.org/10.1007/s00030-016-0391-0
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DOI: https://doi.org/10.1007/s00030-016-0391-0