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Existence of solutions to a phase–field model of dynamic fracture with a crack–dependent dissipation


We propose a phase–field model of dynamic fracture based on the Ambrosio–Tortorelli’s approximation, which takes into account dissipative effects due to the speed of the crack tips. By adapting the time discretization scheme contained in Larsen et al. (Math Models Methods Appl Sci 20:1021–1048, 2010), we show the existence of a dynamic crack evolution satisfying an energy–dissipation balance, according to Griffith’s criterion. Finally, we analyze the dynamic phase–field model of Bourdin et al. (Int J Fract 168:133–143, 2011) and Larsen (in: Hackl (ed) IUTAM symposium on variational concepts with applications to the mechanics of materials, IUTAM Bookseries, vol 21. Springer, Dordrecht, 2010, pp 131–140) with no dissipative terms.

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The author wishes to thank Prof. Gianni Dal Maso for having proposed the problem and for many helpful discussions on the topic. The author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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Correspondence to Maicol Caponi.

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Caponi, M. Existence of solutions to a phase–field model of dynamic fracture with a crack–dependent dissipation. Nonlinear Differ. Equ. Appl. 27, 14 (2020).

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  • Dynamic fracture mechanics
  • Phase–field approximation
  • Elastodynamics
  • Griffith’s criterion
  • Energy balance
  • Crack path

Mathematics Subject Classification

  • 35L53
  • 35Q74
  • 49J40
  • 74R10