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


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.

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    with sometimes a non-essential rescaling of the internal length \(l_0\).


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

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Correspondence to Jérémy Bleyer.

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Supplementary material 4 (mov 349 KB)

Supplementary material 5 (mov 368 KB)

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Bleyer, J., Roux-Langlois, C. & Molinari, J. Dynamic crack propagation with a variational phase-field model: limiting speed, crack branching and velocity-toughening mechanisms. Int J Fract 204, 79–100 (2017).

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