International Journal of Fracture

, Volume 204, Issue 1, pp 79–100 | Cite as

Dynamic crack propagation with a variational phase-field model: limiting speed, crack branching and velocity-toughening mechanisms

  • Jérémy BleyerEmail author
  • Clément Roux-Langlois
  • Jean-François Molinari
Original Paper


We address the simulation of dynamic crack propagation in brittle materials using a regularized phase-field description, which can also be interpreted as a damage-gradient model. Benefiting from a variational framework, the dynamic evolution of the mechanical fields are obtained as a succession of energy minimizations. We investigate the capacity of such a simple model to reproduce specific experimental features of dynamic in-plane fracture. These include the crack branching phenomenon as well as the existence of a limiting crack velocity below the Rayleigh wave speed for mode I propagation. Numerical results show that, when a crack accelerates, the damaged band tends to widen in a direction perpendicular to the propagation direction, before forming two distinct macroscopic branches. This transition from a single crack propagation to a branched configuration is described by a well-defined master-curve of the apparent fracture energy \(\varGamma \) as an increasing function of the crack velocity. This \(\varGamma (v)\) relationship can be associated, from a macroscopic point of view, with the well-known velocity-toughening mechanism. These results also support the existence of a critical value of the energy release rate associated with branching: a critical value of approximately 2\(G_c\) is observed i.e. the fracture energy contribution of two crack tips. Finally, our work demonstrates the efficiency of the phase-field approach to simulate crack propagation dynamics interacting with heterogeneities, revealing the complex interplay between heterogeneity patterns and branching mechanisms.


Dynamic fracture Crack branching Brittle materials Phase-field model Damage-gradient model 



The authors would like to acknowledge Corrado Maurini and Li Tianyi for sharing FEniCS-based implementation of damage-gradient models.

Supplementary material

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Supplementary material 2 (mov 348 KB)

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Supplementary material 5 (mov 368 KB)


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Civil Engineering, Department of Materials ScienceEcole Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.Mechanics and Glasses Department, Institute of physics of RennesUMR UR1-CNRS 6251, University of Rennes 1Rennes CedexFrance

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