SpringerLink
Log in

A Class of Nonlinear Inclusions and Swee** Processes in Solid Mechanics

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

We consider a new class of inclusions in Hilbert spaces for which we provide an existence and uniqueness result. The proof is based on arguments of monotonicity, convexity and fixed point. We use this result to establish the unique solvability of an associated class of Moreau’s swee** processes. Next, we give two applications in Solid Mechanics. The first one concerns the study of a time-dependent constitutive law with unilateral constraints and memory term. The second one is related to a frictional contact problem for viscoelastic materials. For both problems we describe the physical setting, list the assumptions on the data and provide existence and uniqueness results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Germany)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adly, S., Haddad, T.: An implicit swee** process approach to quasistatic evolution variational inequalities. SIAM J. Math. Anal. 50, 761–778 (2018)

    Article  MathSciNet  Google Scholar 

  2. Adly, S., Sofonea, M.: Time-dependent inclusions and swee** processes in contact mechanics. J. Appl. Math. Phys. (ZAMP) 70, 39 (2019). https://doi.org/10.1007/s00033-019-1084-4

    Article  MathSciNet  MATH  Google Scholar 

  3. Benabdellah, H.: Existence of solutions to the nonconvex swee** process. J. Differ. Equ. 164, 286–295 (2000)

    Article  MathSciNet  Google Scholar 

  4. Bernicot, F., Venel, J.: Swee** process by prox-regular sets in Riemannian Hilbert manifolds. J. Differ. Equ. 259, 4086–4121 (2015)

    Article  MathSciNet  Google Scholar 

  5. Bounkhel, M., Rabab, A.-Y.: First and second order convex swee** processes in reflexive smooth Banach spaces. Set-Valued Var. Anal. 18, 151–182 (2010)

    Article  MathSciNet  Google Scholar 

  6. Cao, T.H., Mordukhovich, B.S.: Optimal control of a nonconvex perturbed swee** process. J. Differ. Equ. 266, 1003–1050 (2019)

    Article  MathSciNet  Google Scholar 

  7. Castaing, C.: Version aléatoire de problème de rafle par un convexe variable. C. R. Acad. Sci. Paris, Sér. A 277, 1057–1059 (1973)

    MathSciNet  MATH  Google Scholar 

  8. Colombo, G., Goncharov, V.V.: The swee** processes without convexity. Set-Valued Anal. 7, 357–374 (1999)

    Article  MathSciNet  Google Scholar 

  9. Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Theory. Kluwer Academic/Plenum, Dordrecht/New York (2003)

    Book  Google Scholar 

  10. Drozdov, A.D.: Finite Elasticity and Viscoelasticity–a Course in the Nonlinear Mechanics of Solids. World Scientific, Singapore (1996)

    Book  Google Scholar 

  11. Edmond, J.F., Thibault, L.: BV solutions of nonconvex swee** process differential inclusions with perturbation. J. Differ. Equ. 226, 135–179 (2006)

    Article  MathSciNet  Google Scholar 

  12. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)

    MATH  Google Scholar 

  13. Haddad, T.: Nonconvex differential variational inequality and state-dependent swee** process. J. Optim. Theory Appl. 159, 386–398 (2013)

    Article  MathSciNet  Google Scholar 

  14. Haddad, T., Noel, J., Thibault, L.: Perturbed swee** process with a subsmooth set depending on the state. Linear Nonlinear Anal. 2, 155–174 (2016)

    MathSciNet  MATH  Google Scholar 

  15. Jourani, A., Vilches, E.: Positively \(\alpha \)-far sets and existence results for generalized perturbed swee** processes. J. Convex Anal. 23, 775–821 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Kunze, M., Monteiro Marques, M.D.P.: On parabolic quasi-variational inequalities and state-dependent swee** processes. Topol. Methods Nonlinear Anal. 12, 179–191 (1998)

    Article  MathSciNet  Google Scholar 

  17. Kurdila, A.J., Zabarankin, M.: Convex Functional Analysis. Birkhäuser, Basel (2005)

    MATH  Google Scholar 

  18. Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems. Advances in Mechanics and Mathematics, vol. 26. Springer, New York (2013)

    MATH  Google Scholar 

  19. Migórski, S., Sofonea, M., Zeng, S.: Well-posedness of history-dependent swee** processes. SIAM J. Math. Anal. 51, 1082–1107 (2019)

    Article  MathSciNet  Google Scholar 

  20. Monteiro Marques, M.D.P.: Differential Inclusions in Nonsmooth Mechanical Problems. Shocks and Dry Friction. Progress in Nonlinear Differential Equations and Their Applications, vol. 9. Birkhäuser, Basel (1993)

    Book  Google Scholar 

  21. Moreau, J.J.: Rafle par un convexe variable I. Travaux Sém. Anal. Convexe Montpellier (1971). Exposé 15

  22. Moreau, J.J.: Sur l’évolution d’un système élastoplastique. C. R. Acad. Sci. Paris, Sér A-Bn 273, A118–A121 (1971)

    Google Scholar 

  23. Moreau, J.J.: On unilateral constraints, friction and plasticity. In: Capriz, G., Stampacchia, G. (eds.) New Variational Techniques in Mathematical Physics. C.I.M.E. II, Ciclo 1973, pp. 173–322. Edizione Cremonese, Roma (1974)

    Google Scholar 

  24. Moreau, J.J.: Evolution problem associated with a moving convex in a Hilbert space. J. Differ. Equ. 26, 347–374 (1977)

    Article  MathSciNet  Google Scholar 

  25. Nacry, F.: Truncated nonconvex state-dependent swee** process: implicit and semi-implicit adapted Moreau’s catching-up algorithms. J. Fixed Point Theory Appl. 20, 121 (2018)

    Article  MathSciNet  Google Scholar 

  26. Nacry, F., Thibault, L.: Regularization of swee** process: old and new. Pure Appl. Funct. Anal. 4, 59–117 (2019)

    MathSciNet  Google Scholar 

  27. Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Dekker, New York (1995)

    MATH  Google Scholar 

  28. Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Birkhäuser, Boston (1985)

    Book  Google Scholar 

  29. Panagiotopoulos, P.D.: Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer, Berlin (1993)

    Book  Google Scholar 

  30. Sofonea, M.: History-dependent inequalities for contact problems with locking materials. J. Elast. 134, 127–148 (2019)

    Article  MathSciNet  Google Scholar 

  31. Sofonea, M., Matei, A.: Mathematical Models in Contact Mechanics. London Mathematical Society Lecture Note Series, vol. 398. Cambridge University Press, Cambridge (2012)

    Book  Google Scholar 

  32. Sofonea, M., Migórski, S.: Variational-Hemivariational Inequalities with Applications. Pure and Applied Mathematics. Chapman & Hall/CRC Press, London/Boca Raton (2018)

    MATH  Google Scholar 

  33. Thibault, L.: Swee** process with regular and nonregular sets. J. Differ. Equ. 193, 1–26 (2003)

    Article  MathSciNet  Google Scholar 

  34. Tolstonogov, A.A.: Swee** process with unbounded nonconvex perturbation. Nonlinear Anal. 108, 291–301 (2014)

    Article  MathSciNet  Google Scholar 

  35. Tsukada, M.: Convergence of best approximations in a smooth Banach space. J. Approx. Theory 40, 301–309 (1984)

    Article  MathSciNet  Google Scholar 

  36. Valadier, M.: Quelques problèmes d’entraînement unilatéral en dimension finie. Travaux Sém. Anal. Convexe Montpellier (1988). Exposé 8

Download references

Acknowledgement

This project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No 823731 CONMECH.

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques et Physique, Université de Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860, Perpignan, France

    Florent Nacry & Mircea Sofonea

Authors
  1. Florent Nacry
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mircea Sofonea
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mircea Sofonea.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nacry, F., Sofonea, M. A Class of Nonlinear Inclusions and Swee** Processes in Solid Mechanics. Acta Appl Math 171, 16 (2021). https://doi.org/10.1007/s10440-020-00380-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10440-020-00380-4

Mathematics Subject Classification

Keywords

Search

Navigation