Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Mircea Sofonea

Laboratoire de Mathématiques et Physique, Université de Perpignan, Perpignan, France

sofonea@univ-perp.fr

Keywords

  • Sweeping process
  • history-dependent operator
  • viscoelastic material
  • frictional contact problem
  • normal compliance

Mathematics Subject Classification

49J40, 47J20, 47J22, 34G25, 58E35, 74M10, 74M15, 74G25

References

  1. S. Adly: A Variational Approach to Nonsmooth Dynamics. Applications in Unilateral Mechanics and Electronics, SpringerBriefs in Mathematics, Springer, Cham (2017).https://doi.org/10.1007/978-3-319-68658-5Google Scholar
  2. S. Adly, T. Haddad: An implicit sweeping process approach to quasistatic evolution variational inequalities, SIAM J. Math. Analysis 50 (2018) 761–778.https://doi.org/10.1137/17M1120658Google Scholar
  3. S. Adly, M. Sofonea: Time-dependent inclusions and sweeping processes in contact mechanics, J. Appl. Math. Physics (ZAMP) 70 (2019) art. no. 39.https://doi.org/10.1007/s00033-019-1084-4Google Scholar
  4. H. Benabdellah: Existence of solutions to the nonconvex sweeping process, J. Diff. Equations 164 (2000) 286–295.https://doi.org/10.1006/jdeq.1999.3756Google Scholar
  5. F. Bernicot, J. Venel: Sweeping process by prox-regular sets in Riemannian Hilbert manifolds, J. Diff. Equations 259 (2015) 4086–4121.https://doi.org/10.1016/j.jde.2015.05.011Google Scholar
  6. M. Bounkhel, A.-Y. Rabab: First and second order convex sweeping processes in reflexive smooth Banach spaces, Set-Valued Var. Analysis 18 (2010) 151–182.https://doi.org/10.1007/s11228-010-0134-zGoogle Scholar
  7. T. H. Cao, B. S. Mordukhovich: Optimal control of a nonconvex perturbed sweeping process, J. Diff. Equations 266 (2019) 1003–1050.https://doi.org/10.1016/j.jde.2018.07.066Google Scholar
  8. C. Castaing: Version aléatoire de problème de rafle par un convexe variable, C. R. Acad. Sci. Paris, Sér. A 277 (1973) 1057–1059.Google Scholar
  9. C. Castaing, M. D. P. Monteiro Marques, P. R. de Fitte: A Skorokhod problem governed by a closed convex moving set, J. Convex Analysis 23 (2016) 387–423.https://doi.org/10.68381/jca23015Google Scholar
  10. N. Chemetov, M. D. P. Monteiro Marques: Non-convex quasi-variational differential inclusions, Set-Valued Analysis 15 (2007) 209–221.https://doi.org/10.1007/s11228-007-0045-9Google Scholar
  11. G. Colombo, V. V. Goncharov: The sweeping processes without convexity, Set-Valued Analysis 7 (1999) 357–374.https://doi.org/10.1023/A:1008774529556Google Scholar
  12. S. Di Marino, B. Maury, F. Santambrogio: Measure sweeping process, J. Convex Analysis 23 (2016) 567–601.https://doi.org/10.68381/jca23021Google Scholar
  13. J. F. Edmond, L. Thibault: BV solutions of nonconvex sweeping process differential inclusions with perturbation, J. Diff. Equations 226 (2006) 135–179.https://doi.org/10.1016/j.jde.2005.12.005Google Scholar
  14. W. Han, M. Sofonea: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics 30, Americal Mathematical Society, Providence (2002).https://doi.org/10.1090/amsip/030Google Scholar
  15. C. Henry: An existence theorem for a class of differential equations with multivalued right-hand side, J. Math. Analysis Appl. 41 (1973) 179–186.https://doi.org/10.1016/0022-247X(73)90192-3Google Scholar
  16. M. Kunze, M. D. P. Monteiro Marques: On parabolic quasi-variational inequalities and state-dependent sweeping processes, Topol. Methods Nonlinear Analysis 12 (1998) 179–191.https://doi.org/10.12775/TMNA.1998.036Google Scholar
  17. B. Maury, J. Venel: A discrete contact model for crowd motion, ESAIM Math. Model. Numer. Analysis 45 (2011) 145–168.https://doi.org/10.1051/m2an/2010035Google Scholar
  18. S. Migórski, M. Sofonea, S. Zeng: Well-posedness of history-dependent sweeping processes, SIAM J. Math. Analysis 51 (2019) 1082–1107.https://doi.org/10.1137/18M1201561Google Scholar
  19. M. D. P. Monteiro Marques: Differential Inclusions in Nonsmooth Mechanical Problems. Shocks and Dry Friction, Progress in Nonlinear Differential Equations and their Applications 9, Birkhäuser Verlag, Basel (1993).Google Scholar
  20. J. J. Moreau: Sur l’évolution d’un système élastoplastique, C. R. Acad. Sci. Paris, Sér A-Bn 273 (1971) A118–A121.Google Scholar
  21. J. J. Moreau: Rafle par un convexe variable I, Travaux Sém. Analysis Convexe, Montpellier (1971), Exposé 15.Google Scholar
  22. J. J. Moreau: Rafle par un convexe variable II, Travaux Sém. Analysis Convexe, Montpellier (1972), Exposé 3.Google Scholar
  23. J. J. Moreau: On unilateral constraints, friction and plasticity, in: New Variational Techniques in Mathemaical Physics, G. Capriz and G. Stampacchia (eds.), C.I.M.E. II, Ciclo 1973, Edizione Cremonese, Roma (1974) 173–322.Google Scholar
  24. J. J. Moreau: Evolution problem associated with a moving convex in a Hilbert space, J. Diff. Equations 26 (1977) 347–374.https://doi.org/10.1016/0022-0396(77)90085-7Google Scholar
  25. F. Nacry: Truncated nonconvex state-dependent sweeping process: implicit and semi-implicit adapted Moreau’s catching-up algorithms, J. Fixed Point Theory Appl. 20 (2018), art. no. 121.https://doi.org/10.1007/s11784-018-0601-yGoogle Scholar
  26. F. Nacry, J. Noel, L. Thibault: On first and second order state-dependent prox-regular sweeping processes, Pure Appl. Functional Analysis 6/6 (2021) 1453–1493.Google Scholar
  27. F. Nacry, M. Sofonea: A class of nonlinear inclusions and sweeping processes in solid mechanics, Acta Appl. Math. 176 (2021), art. no. 16.https://doi.org/10.1007/s10440-020-00380-4Google Scholar
  28. F. Nacry, L. Thibault: Regularization of sweeping process: old and new, Pure Appl. Functional Analysis 4 (2019) 59–117.Google Scholar
  29. M. Sofonea, A. Matei: Mathematical Models in Contact Mechanics, London Math. Soc. Lecture Notes Series 398, Cambridge University Press, Cambridge (2012).https://doi.org/10.1017/CBO9781139104166Google Scholar
  30. M. Sofonea, S. Migórski: Variational-Hemivariational Inequalities with Applications, Pure and Applied Mathematics, Chapman & Hall/CRC Press, Boca Raton (2018).https://doi.org/10.1201/9781315153261Google Scholar
  31. L. Thibault: Sweeping process with regular and nonregular sets, J. Diff. Equations 193 (2003) 1–26.https://doi.org/10.1016/S0022-0396(03)00129-3Google Scholar

Suggested citation

F. Nacry, M. Sofonea. “History-Dependent Sweeping Processes in Contact Mechanics.” Journal of Convex Analysis 29 (2022), No. 1, 77–100. https://doi.org/10.68381/jca29004

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2022. Rights now held by Banach Press LLC.