International Journal of Fracture

, Volume 168, Issue 2, pp 133–143 | Cite as

A time-discrete model for dynamic fracture based on crack regularization

  • Blaise BourdinEmail author
  • Christopher J. Larsen
  • Casey L. Richardson
Original Paper


We propose a discrete time model for dynamic fracture based on crack regularization. The advantages of our approach are threefold: first, our regularization of the crack set has been rigorously shown to converge to the correct sharp-interface energy Ambrosio and Tortorelli (Comm. Pure Appl. Math., 43(8): 999–1036 (1990); Boll. Un. Mat. Ital. B (7), 6(1):105–123, 1992); second, our condition for crack growth, based on Griffith’s criterion, matches that of quasi-static settings Bourdin (Interfaces Free Bound 9(3): 411–430, 2007) where Griffith originally stated his criterion; third, solutions to our model converge, as the time-step tends to zero, to solutions of the correct continuous time model Larsen (Math Models Methods Appl Sci 20:1021–1048, 2010). Furthermore, in implementing this model, we naturally recover several features, such as the elastic wave speed as an upper bound on crack speed, and crack branching for sufficiently rapid boundary displacements. We conclude by comparing our approach to so-called “phase-field” ones. In particular, we explain why phase-field approaches are good for approximating free boundaries, but not the free discontinuity sets that model fracture.


Dynamic fracture Phase field Griffith’s criterion Crack regularization Variational fracture 


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Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Blaise Bourdin
    • 1
    Email author
  • Christopher J. Larsen
    • 2
  • Casey L. Richardson
    • 3
    • 4
  1. 1.Department of Mathematics and Center for Computation & TechnologyLouisiana State UniversityBaton RougeUSA
  2. 2.Department of Mathematical SciencesWorcester Polytechnic InstituteWorcesterUSA
  3. 3.Department of MathematicsWorcester Polytechnic InstituteWorcesterUSA
  4. 4.Center for Imaging Science, Clark 319BJohns Hopkins UniversityBaltimoreUSA

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