Skip to main content
SpringerLink
Log in

Revisiting Energy Release Rates in Brittle Fracture

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

Abstract

We revisit in a 2d setting the notion of energy release rate, which plays a pivotal role in brittle fracture. Through a blow-up method, we extend that notion to crack patterns which are merely closed sets connected to the crack tip. As an application, we demonstrate that, modulo a simple meta-stability principle, a moving crack cannot generically kink while growing continuously in time. This last result potentially renders obsolete in our opinion a longstanding debate in fracture mechanics on the correct criterion for kinking.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Amestoy, M.: Propagation de fissures en élasticité plane. Thèse de Doctorat d’Etat, Université Pierre et Marie Curie, Paris (1987)

  • Amestoy, M., Leblond, J.-B.: Crack paths in plane situations. II. Detailed form of the expansion of the stress intensity factors. Int. J. Solids Struct. 29(4), 465–501 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  • Argatov, I.I., Nazarov, S.A.: Energy release in the kinking of a crack in a plane anisotropic body. Prikl. Mat. Mekh. 66(3), 502–514 (2002)

    MATH  MathSciNet  Google Scholar 

  • Ball, J.M.: Constitutive inequalities and existence theorems in nonlinear elastostatics. In: Nonlinear Analysis and Mechanics, vol. I, Heriot-Watt Symposium, Edinburgh, 1976. Res. Notes in Math., vol. 17, pp. 187–241. Pitman, London (1977)

    Google Scholar 

  • Bourdin, B., Francfort, G.A., Marigo, J.-J.: Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48, 797–826 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Bourdin, B., Francfort, G.A., Marigo, J.-J.: The Variational Approach to Fracture. Springer, Berlin (2008)

    Book  Google Scholar 

  • Bucur, D., Varchon, N.: Boundary variation for a Neumann problem. Ann. Scuola Norm. Super. Pisa Cl. Sci. (4) 29(4), 807–821 (2000)

    MathSciNet  Google Scholar 

  • Chambolle, A.: A density result in two-dimensional linearized elasticity, and applications. Arch. Ration. Mech. Anal. 167(3), 211–233 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Chambolle, A., Giacomini, A., Ponsiglione, M.: Crack initiation in brittle materials. Arch. Ration. Mech. Anal. 188(2), 309–349 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  • Chambolle, A., Francfort, G.A., Marigo, J.-J.: When and how do cracks propagate. J. Mech. Phys. Solids 56, 1614–1622 (2009)

    Article  MathSciNet  Google Scholar 

  • Costabel, M., Dauge, M., Duduchava, R.: Asymptotics without logarithmic terms for crack problems. Commun. Part. Differ. Equ. 28(5–6), 869–926 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Dauge, M.: Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions. Lecture Notes in Mathematics, vol. 1341. Springer, Berlin (1988)

    MATH  Google Scholar 

  • Destuynder, P., Djaoua, M.: Sur une interprétation mathématique de l’intégrale de Rice en théorie de la rupture fragile. Math. Methods Appl. Sci. 3(1), 70–87 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  • Falconer, K.J.: The Geometry of Fractal Sets. Cambridge University Press, Cambridge (1985)

    Book  MATH  Google Scholar 

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

    Google Scholar 

  • Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics, vol. 24. Pitman, Boston (1985). (Advanced Publishing Program)

    MATH  Google Scholar 

  • Grisvard, P.: Singularités en elasticité. Arch. Ration. Mech. Anal. 107(2), 157–180 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  • Irwin, G.R.: Fracture. In: Handbuch der Physik, herausgegeben von S. Flügge. Bd. 6. Elastizität und Plastizität, pp. 551–590. Springer, Berlin (1958)

    Google Scholar 

  • Knees, D., Mielke, A.: Energy release rate for cracks in finite-strain elasticity. Math. Methods Appl. Sci. 31(5), 501–528 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  • Larsen, C.J.: Epsilon-stable quasi-static brittle fracture evolution. Commun. Pure Appl. Math. 63, 630–654 (2010)

    MATH  Google Scholar 

  • Leblond, J.-B.: Crack paths in plane situation—I, General form of the expansion of the stress intensity factors. Int. J. Solids Struct. 25, 1311–1325 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  • Murat, F., Simon, J.: Quelques résultats sur le contrôle par un domaine géométrique. Technical report, Rapport du L.A. 189 No. 74003, Université Paris VI (1974)

  • Negri, M., Ortner, C.: Quasi-static crack propagation by Griffith’s criterion. Math. Models Methods Appl. Sci. 18(11), 1895–1925 (2008)

    Article  MathSciNet  Google Scholar 

  • Sneddon, I.N.: The distribution of stress in the neighbourhood of a crack in an elastic solid. Proc. R. Soc. Lond. 187, 229–260 (1946)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CMAP, Ecole Polytechnique, CNRS, 91128, Palaiseau, France

    Antonin Chambolle

  2. Institut Galilée, Université Paris 13, Avenue Jean-Baptiste Clément, 93430, Villetaneuse, France

    Gilles A. Francfort

  3. Institut Jean LeRond d’Alembert, Université Pierre et Marie Curie, 4, Place Jussieu, 75252, Paris Cedex 05, France

    Jean-Jacques Marigo

Authors
  1. Antonin Chambolle
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gilles A. Francfort
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jean-Jacques Marigo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gilles A. Francfort.

Additional information

Communicated by A. Mielke.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chambolle, A., Francfort, G.A. & Marigo, JJ. Revisiting Energy Release Rates in Brittle Fracture. J Nonlinear Sci 20, 395–424 (2010). https://doi.org/10.1007/s00332-010-9061-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00332-010-9061-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation