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A Continuous Dependence Result for a Dynamic Debonding Model in Dimension One

  • Filippo RivaEmail author
Article
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Abstract

In this paper we address the problem of continuous dependence on initial and boundary data for a one-dimensional dynamic debonding model describing a thin film peeled away from a substrate. The system underlying the process couples the (weakly damped) wave equation with a Griffith’s criterion which rules the evolution of the debonded region. We show that under general convergence assumptions on the data the corresponding solutions converge to the limit one with respect to different natural topologies.

Mathematics Subject Classification (2010)

35B30 35L05 35Q74 35R35 70F40 74K35 

Keywords

Thin films dynamic debonding wave equation in time-dependent domains Griffith’s criterion continuous dependence 

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Notes

Acknowledgment

The author wishes to thank Prof. Gianni Dal Maso for many helpful discussions on the topic. The author is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

References

  1. 1.
    Bucur, D., Buttazzo, G., Lux, A.: Quasistatic evolution in debonding problems via capacitary methods. Arch. Rational Mech. Anal. 190, 281–306 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Burridge, R., Keller, J.B.: Peeling, slipping and cracking - some one-dimensional free boundary problems in mechanics. SIAM Review 20, 31–61 (1978)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Caponi, M.: Linear hyperbolic systems in domains with growing cracks. Milan J. of Mathematics 85, 149–185 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dal Maso, G., Lazzaroni, G., Nardini, L.: Existence and uniqueness of dynamic evolutions for a peeling test in dimension one. J. Differential Equations 261, 4897–4923 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dal Maso, G., Lucardesi, I.: The wave equation on domains with cracks growing on a prescribed path: existence, uniqueness, and continuous dependence on the data. Applied Mathematics Research eXpress 2017, 184–241 (2017)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Dumouchel, P.-E., Marigo, J.-J., Charlotte, M.: Rupture dynamique et fissuration quasi-static instable. Comptes Rendus Mécanique 335, 708–713 (2007)CrossRefGoogle Scholar
  7. 7.
    Dumouchel, P.-E., Marigo, J.-J., Charlotte, M.: Dynamic fracture: an example of convergence towards a discontinuous quasistatic solution. Contin. Mech. Thermodyn. 20, 1–19 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Freund, L.B.: Dynamic Fracture Mechanics. Cambridge University Press, Cambridge, Cambridge Monographs on Mechanics and Applied Mathematics (1990)CrossRefGoogle Scholar
  9. 9.
    Hellan, K.: Debond dynamics of an elastic strip-I. Timoshenko beam properties and steady motion, International Journal of Fracture 14, 91–100 (1978)Google Scholar
  10. 10.
    Hellan, K.: Debond dynamics of an elastic strip-II. Simple transient motion, International Journal of Fracture 14, 173–184 (1978)Google Scholar
  11. 11.
    Hellan, K.: Introduction to Fracture Mechanics. McGraw-Hill, New York (1984)Google Scholar
  12. 12.
    C.J. Larsen, Models for dynamic fracture based on Griffith's criterion, in: IUTAM Symp. on Variational Concepts with Applications to the Mechanics of Materials (K. Hackl, ed.), Springer, 2010, pp. 131–140Google Scholar
  13. 13.
    Lazzaroni, G., Bargellini, R., Dumouchel, P.-E., Marigo, J.-J.: On the role of kinetic energy during unstable propagation in a heterogeneous peeling test. Int. J. Fract. 175, 127–150 (2012)CrossRefGoogle Scholar
  14. 14.
    Lazzaroni, G., Nardini, L.: Analysis of a dynamic peeling test with speeddependent toughness. SIAM J. Appl. Math. 78, 1206–1227 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lazzaroni, G., Nardini, L.: On the quasistatic limit of dynamic evolutions for a peeling test in dimension one. J. Nonlinear Sci. 28, 269–304 (2018)MathSciNetCrossRefGoogle Scholar
  16. 16.
    G. Lazzaroni and L. Nardini, On the 1d wave equation in time-dependent domains and the problem of debond initiation, Preprint SISSA 56/2017/MATEGoogle Scholar
  17. 17.
    Maddalena, F., Percivale, D., Tomarelli, F.: Adhesive flexible material structures. Discr. Continuous Dynamic. Systems B 17(2), 553–574 (2012)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mielke, A., Roubíček, T.: Rate-Independent Systems: Theory and Application, Applied Mathematical Sciences, vol. 193. Springer, New York (2015)CrossRefGoogle Scholar
  19. 19.
    F. Riva and L. Nardini, Existence and uniqueness of dynamic evolutions for a onedimensional debonding model with damping, Preprint SISSA 28/2018/MATEGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.SISSATriesteItaly

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