International Journal of Fracture

, Volume 178, Issue 1–2, pp 113–129 | Cite as

Continuum phase field modeling of dynamic fracture: variational principles and staggered FE implementation

Original Paper


The modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations of complex crack topologies including branching. This drawback can be overcome by a diffusive crack modeling based on the introduction of a crack phase field as proposed in Miehe et al. (Comput Methods Appl Mech Eng 19:2765–2778, 2010a; Int J Numer Meth Eng 83:1273–1311, 2010b), Hofacker and Miehe (Int J Numer Meth Eng, 2012). In this work, we summarize basic ingredients of a thermodynamically consistent, variational-based model of diffusive crack propagation under quasi-static and dynamic conditions. It is shown that all coupled field equations, in particular the balance of momentum and the gradient-type evolution equation for the crack phase field, follow as the Euler equations of a mixed rate-type variational principle that includes the fracture driving force as the mixed field variable. This principle makes the proposed formulation extremely compact and provides a perfect basis for the finite element implementation. We then introduce a local history field that contains a maximum energetic crack source obtained in the deformation history. It drives the evolution of the crack phase field. This allows for the construction of an extremely robust operator split scheme that updates in a typical time step the history field, the crack phase field and finally the displacement field. We demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples, which show the evolution of complex crack patterns under dynamic loading.


Dynamic fracture Crack branching Phase field modeling Coupled multi-field problems Incremental variational principles 


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  1. Ambrosio L, Tortorelli VM (1990) Approximation of functionals depending on jumps by elliptic functionals via Γ- convergence. Commun Pure Appl Math 43: 999–1036CrossRefGoogle Scholar
  2. Armero F, Linder C (2009) Numerical simulation of dynamic fracture using finite elements with embedded discontinuities. Int J Fract 160: 119–141CrossRefGoogle Scholar
  3. Belytschko T, Chen H, Xu J, Zi G (2003) Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment. Int J Numer Methods Eng 58: 1873–1905CrossRefGoogle Scholar
  4. Borden MJ, Verhoosel CV, Scott MA, Hughes TJR, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217(220): 77–95CrossRefGoogle Scholar
  5. Bourdin B, Francfort GA, Marigo JJ (2008) Special invited exposition: the variational approach to fracture. J Elast 91: 5–148CrossRefGoogle Scholar
  6. Bourdin B, Larsen C, Richardson CL (2001) A time-discrete model for dynamic fracture based on crack regularization. Int J Fract 168: 133–143CrossRefGoogle Scholar
  7. Braides DP (1998) Approximation of free discontinuity problems. Springer, BerlinGoogle Scholar
  8. Braides DP (2002) Γ-convergence for beginners. Oxford University Press, New YorkCrossRefGoogle Scholar
  9. Buliga M (1999) Energy minimizing brittle crack propagation. J Elast 52: 201–238CrossRefGoogle Scholar
  10. Camacho GT, Ortiz M (1996) Computational modelling of impact damage in brittle materials. Int J Solids Struct 33: 2899–2938CrossRefGoogle Scholar
  11. Capriz G (1989) Continua with microstructure. Springer, BerlinCrossRefGoogle Scholar
  12. Dal Maso G (1993) An introduction to Γ-convergence. Birkhuser, BostonCrossRefGoogle Scholar
  13. Dal Maso G, Toader R (2002) A model for the quasistatic growth of brittle fractures: existence and approximation results. Arch Ration Mech Anal 162: 101–135CrossRefGoogle Scholar
  14. Eastgate LO, Sethna JP, Rauscher M, Cretegny T (2002) Fracture in mode I using a conserved phase-field model. Phys Rev E 65: 036117CrossRefGoogle Scholar
  15. Fagerström M, Larsson R (2006) Theory and numerics for finite deformation fracture modeling using strong discontinuities. Int J Numer Methods Eng 66: 911–948CrossRefGoogle Scholar
  16. Francfort GA, Marigo JJ (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46: 1319–1342CrossRefGoogle Scholar
  17. Frémond M (2002) Non-smooth thermomechanics. Springer, BerlinGoogle Scholar
  18. Frémond M, Nedjar B (1996) Damage, gradient of damage and principle of virtual power. Int J Solids Struct 33: 1083–1103CrossRefGoogle Scholar
  19. Freund LB (1990) Dynamic fracture mechanics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  20. Griffith AA (1920) The phenomena of rupture and flow in solids. Philos Trans R Soc Lond A 221: 163–198CrossRefGoogle Scholar
  21. Gürses E, Miehe C (2009) A computational framework of three dimensional configurational force driven brittle crack propagation. Comput Methods Appl Mech Eng 198: 1413– 1428CrossRefGoogle Scholar
  22. Hakim V, Karma A (2009) Laws of crack motion and phase-field models of fracture. J Mech Phys Solids 57: 342–368CrossRefGoogle Scholar
  23. Hofacker M, Miehe C (2012) A phase field model of dynamic fracture: robust field updates for the analysis of complex crack patterns. Int J Numer Meth Eng. doi:10.1002/nme.4387
  24. Kalthoff JF, Winkler S (1987) Failure mode transition at high rates of shear loading. In: Impact loading and dynamic behavior of materials. DGM Informationsgesellschaft, Oberursel, pp 185–195Google Scholar
  25. Kane C, Marsden J, Ortiz M, West M (2000) Variational integrators and the newmark algorithm for conservative and dissipative mechanical systems. Int J Numer Methods Eng 49: 1295–1325CrossRefGoogle Scholar
  26. Karma A, Kessler DA, Levine H (2001) Phase-field model of mode III dynamic fracture. Phys Rev Lett 87: 045501/1–045501/4Google Scholar
  27. Kuhn C, Müller R (2011) A new finite element technique for a phase field model of brittle fracture. J Theor Appl Mech 49: 1115–1133Google Scholar
  28. Linder C, Armero F (2009) Finite elements with embedded branching. Finite Elem Anal Des 45: 280–293CrossRefGoogle Scholar
  29. Mariano PM (2001) Multifield theories in mechanics of solids. Adv Appl Mech 38: 1–93CrossRefGoogle Scholar
  30. Miehe C (1998) Comparison of two algorithms for the computation of fourth-order isotropic tensor functions. Comput Struct 66: 37–43CrossRefGoogle Scholar
  31. Miehe C (2011) A multi-field incremental variational framework for gradient-extended standard dissipative solids. J Mech Phys Solids 59: 898–923CrossRefGoogle Scholar
  32. Miehe C, Gürses E (2007) A robust algorithm for configurational-force-driven brittle crack propagation with R-adaptive mesh alignment. Int J Numer Methods Eng 72: 127–155CrossRefGoogle Scholar
  33. Miehe C, Lambrecht M (2001) Algorithms for computation of stresses and elasticity moduli in terms of Seth–Hill’s family of generealized strain tensors. Commun Numer Methods Eng 17: 337–353CrossRefGoogle Scholar
  34. Miehe C, Hofacker M, Welschinger F (2010a) A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng 199: 2765–2778CrossRefGoogle Scholar
  35. Miehe C, Welschinger F, Hofacker M (2010b) Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations. Int J Numer Meth Eng 83: 1273–1311CrossRefGoogle Scholar
  36. Mumford D, Shah J (1989) Optimal approximations by piecewise smooth functions and associated variational problems. Commun Pure Appl Math 42: 577–685CrossRefGoogle Scholar
  37. Pandolfi A, Ortiz M (2002) An efficient adaptive procedure for three-dimensional fragmentation simulations. Eng Comput 18: 148–159CrossRefGoogle Scholar
  38. Peerlings RHJ, de Borst R, Brekelmans WAM, de Vree JHP (1996) Gradient enhanced damage for quasi-brittle materials. Int J Numer Methods Eng 39: 3391–3403CrossRefGoogle Scholar
  39. Radovitzky R, Ortiz M (1999) Error estimation and adaptive meshing in strongly nonlinear dynamic problems. Comput Methods Appl Mech Eng 172: 203–240CrossRefGoogle Scholar
  40. Ramulu M, Kobayashi A (1985) Mechanic of crack curving and branching—a dynamic fracture analysis. Int J Frac Mech 27: 187–201CrossRefGoogle Scholar
  41. Ravi-Chandar K, Knauss W (1984a) An experimental investigation into dynamic fracture: I. Crack initiation and crack arrest. Int J Fract 25: 247–262CrossRefGoogle Scholar
  42. Ravi-Chandar K, Knauss W (1984b) An experimental investigation into dynamic fracture: II. Microstructural aspects. Int J Fract 26: 65–80CrossRefGoogle Scholar
  43. Ravi-Chandar K, Knauss W (1984c) An experimental investigation into dynamic fracture: III. On steady-state crack propagation and crack branching. Int J Fract 26: 141–154CrossRefGoogle Scholar
  44. Ravi-Chandar K, Knauss W (1984d) An experimental investigation into dynamic fracture: IV. On the interaction of stress waves with propagation cracks. Int J Fract 26: 189–200CrossRefGoogle Scholar
  45. Song JH, Areias P, Belytschko T (1984) A method for dynamic crack and shear band propagation with phantom nodes. Int J Numer Methods Eng 67: 868–893CrossRefGoogle Scholar
  46. Song JH, Belytschko T (2009) Cracking node method for dynamic fracture with finite elements. Int J Numer Methods Eng 77: 360–385CrossRefGoogle Scholar
  47. Song JH, Wang H, Belytschko T (2008) A comparative study on finite element methods for dynamic fracture. Comput Methods Appl Mech Eng 42: 239–250Google Scholar
  48. Xu XP, Needleman A (1994) Numerical simulations of fast crack growth in brittle solids. J Mech Phys Solids 42: 1397–1434CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Chair 1, Institute of Applied MechanicsUniversity of StuttgartStuttgartGermany

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