Journal of Elasticity

, Volume 98, Issue 2, pp 159–187 | Cite as

From Rate-Dependent to Rate-Independent Brittle Crack Propagation



On the base of many experimental results, e.g., Ravi-Chandar and Knauss (Int. J. Fract. 26:65–80, 1984), Sharon et al. (Phys. Rev. Lett. 76(12):2117–2120, 1996), Hauch and Marder (Int. J. Fract. 90:133–151, 1998), the object of our analysis is a rate-dependent model for the propagation of a crack in brittle materials. Restricting ourselves to the quasi-static framework, our goal is a mathematical study of the evolution equation in the geometries of the ‘Single Edge Notch Tension’ and of the ‘Compact Tension’. Besides existence and uniqueness, emphasis is placed on the regularity of the evolution making reference also to the ‘velocity gap’. The transition to the rate-independent model of Griffith is obtained by time rescaling, proving convergence of the rescaled evolutions and of their energies. Further, the discontinuities of the rate-independent evolution are characterized in terms of unstable points of the free energy. Results are illustrated by a couple of numerical examples in the above mentioned geometries.

Brittle fracture Rate-dependent evolutions Rate-independent evolutions 


62.20.Mm 81.40.Np 

Mathematics Subject Classification (2000)

74R10 74R15 


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  1. 1.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, New York (2000) MATHGoogle Scholar
  2. 2.
    Anderson, T.: Fracture Mechanics: Fundamentals and Applications. CRC Press, Boca Raton (1995) MATHGoogle Scholar
  3. 3.
    Aubin, J.P., Cellina, A.: Differential Inclusions. Springer, Berlin (1984) MATHGoogle Scholar
  4. 4.
    Bourdin, B., Francfort, G., Marigo, J.J.: The variational approach to fracture. J. Elast. 91, 5–148 (2008) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cherepanov, G.P., Germanovich, L.N.: An employment of the catastrophe theory in fracture mechanics as applied to brittle strength criteria. J. Mech. Phys. Solids 41(10), 1637–1649 (1993) MATHCrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Ciarlet, P.: Mathematical Elasticity. Three-Dimensional Elasticity. North-Holland, Amsterdam (1988) Google Scholar
  7. 7.
    Dal Maso, G., Rampazzo, F.: On systems of ordinary differential equations with measures as controls. Differ. Integral Equ. 4(4), 739–765 (1991) MATHMathSciNetGoogle Scholar
  8. 8.
    Dal Maso, G., DeSimone, A., Mora, M., Morini, M.: A vanishing viscosity approach to quasistatic evolution in plasticity with softening. Arch. Ration. Mech. Anal. 189(3), 469–544 (2008) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    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) MATHCrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992) MATHGoogle Scholar
  11. 11.
    Francfort, G., Marigo, J.J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46(8), 1319–1342 (1998) MATHCrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Freund, L.: Dynamic Fracture Mechanics. Cambridge University Press, Cambridge (1990) MATHCrossRefGoogle Scholar
  13. 13.
    Griffith, A.: The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. 18, 163–198 (1920) Google Scholar
  14. 14.
    Hauch, J., Marder, M.: Energy balance in dynamic fracture, investigated by a potential drop technique. Int. J. Fract. 90, 133–151 (1998) CrossRefGoogle Scholar
  15. 15.
    Knees, D., Mielke, A., Zanini, C.: On the inviscid limit of a model for crack propagation. Math. Models Methods Appl. Sci. 18(9), 1529–1569 (2008) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Larsen, C., Ortiz, M., Richardson, C.: Fracture paths from front kinetics: relaxation and rate-independence. Arch. Ration. Mech. Anal. 193, 539–583 (2009) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Negri, M.: A comparative analysis on variational models for quasi-static brittle crack propagation. Adv. Calc. Var. (submitted) Google Scholar
  18. 18.
    Negri, M., Ortner, C.: Quasi-static propagation of brittle fracture by Griffith’s criterion. Math. Models Methods Appl. Sci. 18(11), 1895–1925 (2008) CrossRefMathSciNetGoogle Scholar
  19. 19.
    Puglisi, G., Truskinovsky, L.: Thermodynamics of rate-independent plasticity. J. Mech. Phys. Solids 53(3), 655–679 (2005) MATHCrossRefMathSciNetADSGoogle Scholar
  20. 20.
    Ravi-Chandar, K., Knauss, W.: An experimental investigation into dynamic fracture: II. Microstructural aspects. Int. J. Fract. 26, 65–80 (1984) CrossRefGoogle Scholar
  21. 21.
    Rice, J.: Mathematical analysis in the mechanics of fracture. In: Liebowitz, H. (ed.) Fracture: An Advanced Treatise, pp. 192–308. Academic Press, San Diego (1968) Google Scholar
  22. 22.
    Sharon, E., Gross, S., Fineberg, J.: Energy dissipation in dynamic fracture. Phys. Rev. Lett. 76(12), 2117–2120 (1996) CrossRefADSGoogle Scholar
  23. 23.
    Toader, R., Zanini, C.: An artificial viscosity approach to quasi-static crack growth. Boll. Unione Mat. Ital. 2(1), 1–35 (2009) MATHMathSciNetGoogle Scholar
  24. 24.
    Zanini, C.: Singular perturbations of finite dimensional gradient flows. Discrete Contin. Dyn. Syst. 18(4), 657–675 (2007) MATHMathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PaviaPaviaItaly

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