A differential variational inequality in the study of contact problems with wear |
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| Institution: | 1. College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, 610225, Sichuan Province, PR China;2. Jagiellonian University in Krakow, Faculty of Mathematics and Computer Science, Chair of Optimization and Control, ul. Lojasiewicza 6, 30348 Krakow, Poland;1. School of Science, Institute for Artificial Intelligence, State Key Laboratory of Oil and Gas Reservoir Geology and Exploration, Southwest Petroleum University, Chengdu, Sichuan 610500, PR China;2. Laboratoire de Mathématiques et Physique, University of Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan, France;3. Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, Guangxi, PR China;4. Jagiellonian University in Krakow, Faculty of Mathematics and Computer Science, ul. Lojasiewicza 6, 30348 Krakow, Poland;1. Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA;2. Faculty of Mathematics and Computer Science, Jagiellonian University in Krakow, Institute of Computer Science, ul. Łojasiewicza 6, 30348 Krakow, Poland;3. Laboratoire de Mathématiques et Physique, Université de Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan, France |
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| Abstract: | We start with a mathematical model which describes the sliding contact of a viscoelastic body with a moving foundation. The contact is frictional and the wear of the contact surfaces is taken into account. We prove that this model leads to a differential variational inequality in which the unknowns are the displacement field and the wear function. Then, inspired by this model, we consider a general differential variational inequality in reflexive Banach spaces, governed by four parameters. We prove the unique solvability of the inequality as well as the continuous dependence of its solution with respect to the parameters. The proofs are based on arguments of monotonicity, compactness, convex analysis and lower semicontinuity. Then, we apply these abstract results to the mathematical model of contact for which we deduce the existence of a unique solution as well as the existence of optimal control for an associate optimal control problem. We also present the corresponding mechanical interpretations. |
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| Keywords: | Frictional contact Wear Differential variational inequality Optimal control |
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