Journal of Convex Analysis 29 (2022), No. 1, 077--100
Copyright Heldermann Verlag 2022
History-Dependent Sweeping Processes in Contact Mechanics
Florent Nacry
Laboratoire de Mathématiques et Physique, Université de Perpignan, Perpignan, France
florent.nacry@univ-perp.fr
Mircea Sofonea
Laboratoire de Mathématiques et Physique, Université de Perpignan, Perpignan, France
sofonea@univ-perp.fr
We consider a special type of sweeping process in real Hilbert spaces, governed by two (possibly history-dependent) operators. We associate to this problem an auxiliary time-dependent inclusion for which we establish an existence and uniqueness result. The proof is based on arguments of convex analysis and fixed point theory. From the unique solvability of the intermediate inclusion, we derive the existence of a unique solution to the considered sweeping processes. Our theoretical results find various applications in contact mechanics. As an example, we consider a frictional contact problem for viscoelastic materials. We list the assumptions on the data and provide a variational formulation of the problem, in a form of a sweeping process for the strain field. Then, we prove the unique solvability of the sweeping process and use it to obtain the existence of a unique weak solution to the viscoelastic contact problem.
Keywords: Sweeping process, history-dependent operator, viscoelastic material, frictional contact problem, normal compliance.
MSC: 49J40, 47J20, 47J22, 34G25, 58E35, 74M10, 74M15, 74G25.
[ Fulltext-pdf (178 KB)] for subscribers only.