Skip to main content

Existence of solutions to a phase–field model of dynamic fracture with a crack–dependent dissipation

Abstract

We propose a phase–field model of dynamic fracture based on the Ambrosio–Tortorelli’s approximation, which takes into account dissipative effects due to the speed of the crack tips. By adapting the time discretization scheme contained in Larsen et al. (Math Models Methods Appl Sci 20:1021–1048, 2010), we show the existence of a dynamic crack evolution satisfying an energy–dissipation balance, according to Griffith’s criterion. Finally, we analyze the dynamic phase–field model of Bourdin et al. (Int J Fract 168:133–143, 2011) and Larsen (in: Hackl (ed) IUTAM symposium on variational concepts with applications to the mechanics of materials, IUTAM Bookseries, vol 21. Springer, Dordrecht, 2010, pp 131–140) with no dissipative terms.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Adams, R.A.: Sobolev Spaces. Pure and Applied Mathematics, vol. 65. Academic Press, New York (1975)

    Google Scholar 

  2. 2.

    Almi, S., Belz, S., Negri, M.: Convergence of discrete and continuous unilateral flows for Ambrosio–Tortorelli energies and application to mechanics. ESAIM Math. Model. Numer. Anal. 53, 659–699 (2019)

    Google Scholar 

  3. 3.

    Ambrosio, L., Tortorelli, V.M.: Approximation of functionals depending on jumps by elliptic functionals via \(\Gamma \)-convergence. Commun. Pure Appl. Math. 43, 999–1036 (1990)

    Google Scholar 

  4. 4.

    Bourdin, B., Francfort, G.A., Marigo, J.J.: The variational approach to fracture. Reprinted from J. Elasticity 91 (2008), Springer, New York, 2008

  5. 5.

    Bourdin, B., Larsen, C.J., Richardson, C.L.: A time-discrete model for dynamic fracture based on crack regularization. Int. J. Fract. 168, 133–143 (2011)

    Google Scholar 

  6. 6.

    Dacorogna, B.: Direct Methods in the Calculus of Variations. Applied Mathematical Sciences, vol. 78. Springer, Berlin (1989)

    Google Scholar 

  7. 7.

    Dal Maso, G., Larsen, C.J.: Existence for wave equations on domains with arbitrary growing cracks. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 22, 387–408 (2011)

    Google Scholar 

  8. 8.

    Dal Maso, G., Scala, R.: Quasistatic evolution in perfect plasticity as limit of dynamic processes. J. Dyn. Differ. Equ. 26, 915–954 (2014)

    Google Scholar 

  9. 9.

    Dautray, R., Lions, J.L.: Analyse mathématique et calcul numérique pour les sciences et les techniques. Vol. 8. Évolution: semi-groupe, variationnel. Masson, Paris (1988)

  10. 10.

    Francfort, G.A., Marigo, J.J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998)

    Google Scholar 

  11. 11.

    Giacomini, A.: Ambrosio–Tortorelli approximation of quasi-static evolution of brittle fractures. Calc. Var. Partial Differ. Equ. 22, 129–172 (2005)

    Google Scholar 

  12. 12.

    Griffith, A.A.: The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. 221–A, 163–198 (1920)

    Google Scholar 

  13. 13.

    Ladyzenskaya, O.A.: On integral estimates, convergence, approximate methods, and solution in functionals for elliptic operators. Vestnik Leningrad. Univ. 13, 60–69 (1958)

    Google Scholar 

  14. 14.

    Larsen, C.J.: Models for dynamic fracture based on Griffith’s criterion. In: Hackl, K. (ed.) IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, IUTAM Bookseries, Vol. 21, pp. 131–140. Springer, Dordrecht (2010)

  15. 15.

    Larsen, C.J., Ortner, C., Süli, E.: Existence of solutions to a regularized model of dynamic fracture. Math. Models Methods Appl. Sci. 20, 1021–1048 (2010)

    Google Scholar 

  16. 16.

    Lazzaroni, G., Nardini, L.: Analysis of a dynamic peeling test with speed-dependent toughness. SIAM J. Appl. Math. 78, 1206–1227 (2018)

    Google Scholar 

  17. 17.

    Lazzaroni, G., Toader, R.: A model for crack propagation based on viscous approximation. Math. Models Methods Appl. Sci. 21, 2019–2047 (2011)

    Google Scholar 

  18. 18.

    Mott, N.F.: Brittle fracture in mild steel plates. Engineering 165, 16–18 (1948)

    Google Scholar 

  19. 19.

    Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577–684 (1989)

    Google Scholar 

  20. 20.

    Negri, M.: A unilateral \(L^2\)-gradient flow and its quasi-static limit in phase-field fracture by an alternate minimizing movement. Adv. Calc. Var. 12, 1–29 (2019)

    Google Scholar 

  21. 21.

    Oleinik, O.A., Shamaev, A.S., Yosifian, G.A.: Mathematical Problems in Elasticity and Homogenization. Studies in Mathematics and its Applications, vol. 26. North-Holland Publishing Co., Amsterdam (1992)

    Google Scholar 

  22. 22.

    Racca, S.: A viscosity-driven crack evolution. Adv. Calc. Var. 5, 433–483 (2012)

    Google Scholar 

  23. 23.

    Simon, J.: Compact sets in the space \(L^p(0, T;B)\). Ann. Mat. Pura Appl. 146, 65–96 (1987)

    Google Scholar 

  24. 24.

    Tasso, E.: Weak formulation of elastodynamics in domains with growing cracks. Online on Ann. Math. Pura Appl. (2019). https://doi.org/10.1007/s10231-019-00932-y

    Google Scholar 

Download references

Acknowledgements

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

Author information

Affiliations

Authors

Corresponding author

Correspondence to Maicol Caponi.

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

Verify currency and authenticity via CrossMark

Cite this article

Caponi, M. Existence of solutions to a phase–field model of dynamic fracture with a crack–dependent dissipation. Nonlinear Differ. Equ. Appl. 27, 14 (2020). https://doi.org/10.1007/s00030-020-0617-z

Download citation

Keywords

  • Dynamic fracture mechanics
  • Phase–field approximation
  • Elastodynamics
  • Griffith’s criterion
  • Energy balance
  • Crack path

Mathematics Subject Classification

  • 35L53
  • 35Q74
  • 49J40
  • 74R10