Time-dependent inclusions and sweeping processes in contact mechanics

  • Samir Adly
  • Mircea SofoneaEmail author


We consider a class of time-dependent inclusions in Hilbert spaces for which we state and prove an existence and uniqueness result. The proof is based on arguments of variational inequalities, convex analysis and fixed point theory. Then, we use this result to prove the unique weak solvability of a new class of Moreau’s sweeping processes with constraints in velocity. Our results are useful in the study of mathematical models which describe the quasistatic evolution of deformable bodies in contact with an obstacle. To provide some examples, we consider three viscoelastic contact problems which lead to time-dependent inclusions and sweeping processes in which the unknowns are the displacement and the velocity fields, respectively. Then we apply our abstract results in order to prove the unique weak solvability of the corresponding contact problems.


Nonlinear inclusion Sweeping process Contact problem Unilateral constraint Weak solution 

Mathematics Subject Classification

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



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.


  1. 1.
    Adly, S., Haddad, T.: An implicit sweeping process approach to quasistatic evolution variational inequalities. SIAM J. Math. Anal. 50(1), 761–778 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Adly, S., Haddad, T., Thibault, L.: Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities. Math. Program. Ser. B 148, 5–47 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Capatina, A.: Variational Inequalities Frictional Contact Problems, Advances in Mechanics and Mathematics, vol. 31. Springer, New York (2014)zbMATHGoogle Scholar
  4. 4.
    Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Theory. Kluwer Academic, Boston (2003)CrossRefGoogle Scholar
  5. 5.
    Duvaut, G., Lions, J.-L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)CrossRefGoogle Scholar
  6. 6.
    Eck, C., Jarušek, J., Krbeč, M.: Unilateral Contact Problems: Variational Methods and Existence Theorems, Pure and Applied Mathematics, vol. 270. CRC Press, New York (2005)zbMATHGoogle Scholar
  7. 7.
    Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)zbMATHGoogle Scholar
  8. 8.
    Han, W., Migórski, S., Sofonea, M. (eds.): Advances in Variational and Hemivariational Inequalities, Advances in Mechanics and Mathematics 33. Springer, New York (2015)Google Scholar
  9. 9.
    Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics, vol. 30. Americal Mathematical Society, Providence (2002)CrossRefGoogle Scholar
  10. 10.
    Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988)CrossRefGoogle Scholar
  11. 11.
    Kunze, M., Monteiro Marques, M.D.P.: On discretization of degenerate osweeping process. Portigalliae Mathematica 55, 219–232 (1998)zbMATHGoogle Scholar
  12. 12.
    Kurdila, A.J., Zabarankin, M.: Convex Functional Analysis. Birkhäuser, Basel (2005)zbMATHGoogle Scholar
  13. 13.
    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)zbMATHGoogle Scholar
  14. 14.
    Moreau, J.J.: Sur l’évolution d’un système élastoplastique. C. R. Acad. Sci. Paris, Sér A-Bn 273, A118–A121 (1971)Google Scholar
  15. 15.
    Moreau, J.J.: On unilateral constraints, friction and plasticity. In: Capriz, G., Stampacchia, G. (eds.) New Variational Techniques in Mathemaical Physics, C.I.M.E. II, Ciclo 1973, Edizione Cremonese, Roma, p. 173–322 (1974)Google Scholar
  16. 16.
    Moreau, J.J.: Intersection of moving convex sets in a normed space. Math. Scand. 36, 159–173 (1975)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Moreau, J.J.: Evolution problem associated with a moving convex in a Hilbert space. J. Differ. Eqs. 26, 347–374 (1977)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker Inc, New York (1995)zbMATHGoogle Scholar
  19. 19.
    Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Birkhäuser, Boston (1985)CrossRefGoogle Scholar
  20. 20.
    Panagiotopoulos, P.D.: Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer, Berlin (1993)CrossRefGoogle Scholar
  21. 21.
    Sofonea, M., Matei, A.: Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series, vol. 398. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  22. 22.
    Sofonea, M., Migórski, S.: Variational–Hemivariational Inequalities with Applications, Pure and Applied Mathematics. Chapman & Hall, Boca Raton (2018)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Laboratoire XLIMUniversity of LimogesLimogesFrance
  2. 2.Laboratoire de Mathématiques et PhysiqueUniversity of Perpignan Via DomitiaPerpignanFrance

Personalised recommendations