Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Solvability of a Dynamic Rational Contact with Limited Interpenetration for Viscoelastic Plates

  • 5 Accesses


Solvability of the rational contact with limited interpenetration of different kind of viscolastic plates is proved. The biharmonic plates, von Kármán plates, Reissner-Mindlin plates, and full von Kármán systems are treated. The viscoelasticity can have the classical (“short memory”) form or the form of a certain singular memory. For all models some convergence of the solutions to the solutions of the Signorini contact is proved provided the thickness of the interpenetration tends to zero.

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


  1. [1]

    I. Bock, J. Jarušek: Unilateral dynamic contact of viscoelastic von Kármán plates. Adv. Math. Sci. Appl. 16 (2006), 175–187.

  2. [2]

    I. Bock, J. Jarušek: Unilateral dynamic contact of von Kármán plates with singular memory. Appl. Math., Praha 52 (2007), 515–527.

  3. [3]

    I. Bock, J. Jarušek: Dynamic contact problem for a bridge modeled by a viscoelastic full von Kármán system. Z. Angew. Math. Phys. 61 (2010), 865–876.

  4. [4]

    I. Bock, J. Jarušek: Unilateral dynamic contact problem for viscoelastic Reissner-Mindlin plates. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74 (2011), 4192–4202.

  5. [5]

    J. M. Borwein, Q. J. Zhu: Techniques of Variational Analysis. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC 20, Springer, New York, 2005.

  6. [6]

    C. Eck, J. Jarušek, M. Krbec: Unilateral Contact Problems. Variational Methods and Existence Theorems. Pure and Applied Mathematics (Boca Raton) 270, Chapman & Hall/CRC, Boca Raton, 2005.

  7. [7]

    C. Eck, J. Jarušek, J. Stará: Normal compliance contact models with finite interpenetration. Arch. Ration. Mech. Anal. 208 (2013), 25–57.

  8. [8]

    J. Jarušek: Static semicoercive normal compliance contact problem with limited interpenetration. Z. Angew. Math. Phys. 66 (2015), 2161–2172.

  9. [9]

    J. Jarušek, J. Stará: Solvability of a rational contact model with limited interpenetration in viscoelastodynamics. Math. Mech. Solids 23 (2018), 1040–1048.

  10. [10]

    H. Koch, A. Stachel: Global existence of classical solutions to the dynamical von Kármán equations. Math. Methods Appl. Sci. 16 (1993), 581–586.

  11. [11]

    J. E. Lagnese: Boundary Stabilization of Thin Plates. SIAM Studies in Applied Mathematics 10, Society for Industrial and Applied Mathematics, Philadelphia, 1989.

  12. [12]

    A. Signorini: Sopra alcune questioni di statica dei sistemi continui. Ann. Sc. Norm. Super. Pisa, II. Ser. 2 (1933), 231–251. (In Italian.)

  13. [13]

    A. Signorini: Questioni di elasticità non linearizzata e semilinearizzata. Rend. Mat. Appl., V. Ser. 18 (1959), 95–139. (In Italian.)

Download references

Author information

Correspondence to Jiří Jarušek.

Additional information

This work was supported by the institutional research plan RVO 67985840.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jarušek, J. Solvability of a Dynamic Rational Contact with Limited Interpenetration for Viscoelastic Plates. Appl Math 65, 43–65 (2020). https://doi.org/10.21136/AM.2020.0216-19

Download citation


  • dynamic contact problem
  • limited interpenetration
  • viscoelastic plate
  • existence of solution

MSC 2010

  • 35Q74
  • 74D10
  • 74H20
  • 74K20
  • 74M15