Skip to main content
Springer Nature Link
Log in

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

  • Original Paper
  • Published:
International Journal of Fracture Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

Notes

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

References

  • Ambati M, Gerasimov T, De Lorenzis L (2015) Phase-field modeling of ductile fracture. Comput Mech 55(5):1017–1040

    Article  Google Scholar 

  • Ambrosio L, Tortorelli VM (1990) Approximation of functional depending on jumps by elliptic functional via t-convergence. Commun Pure Appl Math 43(8):999–1036

    Article  Google Scholar 

  • Amor H, Marigo JJ, Maurini C (2009) Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J Mech Phys Solids 57(8):1209–1229

    Article  Google Scholar 

  • Balay S, Abhyankar S, Adams MF, Brown J, Brune P, Buschelman K, Dalcin L, Eijkhout V, Gropp WD, Kaushik D, Knepley MG, McInnes LC, Rupp K, Smith BF, Zampini S, Zhang H, Zhang H (2016) PETSc web page. http://www.mcs.anl.gov/petsc

  • Barenblatt GI (1962) The mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech 7:55–129

    Article  Google Scholar 

  • 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(12):1873–1905

    Article  Google Scholar 

  • Bobaru F, Zhang G (2015) Why do cracks branch? A peridynamic investigation of dynamic brittle fracture. Int J Fracture 196(1–2):59–98

    Article  Google Scholar 

  • Borden MJ, Verhoosel CV, Scott MA, Hughes TJ, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217:77–95

    Article  Google Scholar 

  • Bouchbinder E, Goldman T, Fineberg J (2014) The dynamics of rapid fracture: instabilities, nonlinearities and length scales. Rep Prog Phys 77(4):046,501

    Article  Google Scholar 

  • Bouchbinder E, Mathiesen J, Procaccia I (2005) Branching instabilities in rapid fracture: dynamics and geometry. Phys Rev E 71(5):056,118

    Article  Google Scholar 

  • Bourdin B, Francfort GA, Marigo JJ (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48(4):797–826

    Article  Google Scholar 

  • Bourdin B, Francfort GA, Marigo JJ (2008) The variational approach to fracture. J Elast 91(1–3):5–148

    Article  Google Scholar 

  • Bourdin B, Larsen CJ, Richardson CL (2011) A time-discrete model for dynamic fracture based on crack regularization. Int J Fracture 168(2):133–143

    Article  Google Scholar 

  • Broberg K (1996) How fast can a crack go? Mater Sci 32(1):80–86

    Article  Google Scholar 

  • Cazes F, Moës N (2015) Comparison of a phase-field model and of a thick level set model for brittle and quasi-brittle fracture. Int J Numer Methods Eng 103(2):114–143

    Article  Google Scholar 

  • Dalmas D, Guerra C, Scheibert J, Bonamy D (2013) Damage mechanisms in the dynamic fracture of nominally brittle polymers. Int J Fracture 184(1):93–111

    Article  Google Scholar 

  • Dugdale DS (1960) Yielding of steel sheets containing slits. J Mech Phys Solids 8(2):100–104

    Article  Google Scholar 

  • Falk ML, Needleman A, Rice JR (2001) A critical evaluation of cohesive zone models of dynamic fracture. J Phys IV 11(PR5):Pr5–Pr43

    Google Scholar 

  • Fineberg J, Marder M (1999) Instability in dynamic fracture. Phys Rep 313(1):1–108

    Article  Google Scholar 

  • Fisher DS, Dahmen K, Ramanathan S, Ben-Zion Y (1997) Statistics of earthquakes in simple models of heterogeneous faults. Phys Rev Lett 78(25):4885

    Article  Google Scholar 

  • Francfort GA, Marigo JJ (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46(8):1319–1342

    Article  Google Scholar 

  • Freund LB (1998) Dynamic fracture mechanics. Cambridge University Press, Cambridge

    Google Scholar 

  • Gao H (1993) Surface roughening and branching instabilities in dynamic fracture. J Mech Phys Solids 41(3):457–486

    Article  Google Scholar 

  • Goldman T, Cohen G, Fineberg J (2015) Origin of the microbranching instability in rapid cracks. Phys Rev Lett 114(5):054,301

    Article  Google Scholar 

  • Guerra C, Scheibert J, Bonamy D, Dalmas D (2012) Understanding fast macroscale fracture from microcrack post mortem patterns. Proc Natl Acad Sci 109(2):390–394

    Article  Google Scholar 

  • Guerra Amaro, CM (2009) Dynamic fracture in brittle amorphous materials : dissipation mechanisms and dynamically-induced microcracking in PMMA. Theses, Ecole Polytechnique

  • Henry H (2008) Study of the branching instability using a phase field model of inplane crack propagation. EPL (Europhys Lett) 83(1):160,04

    Article  Google Scholar 

  • Henry H, Adda-Bedia M (2013) Fractographic aspects of crack branching instability using a phase-field model. Phys Rev E 88(6):060,401

    Article  Google Scholar 

  • Hofacker M, Miehe C (2012) Continuum phase field modeling of dynamic fracture: variational principles and staggered FE implementation. Int J Fracture 178(1–2):113–129

    Article  Google Scholar 

  • Hofacker M, Miehe C (2013) A phase field model of dynamic fracture: Robust field updates for the analysis of complex crack patterns. Int J Numer Methods Eng 93(3):276–301

    Article  Google Scholar 

  • Jirasek M (1998) Nonlocal models for damage and fracture: comparison of approaches. Int J Solids Struct 35(31):4133–4145

    Article  Google Scholar 

  • Johnson E (1992) Process region changes for rapidly propagating cracks. Int J Fracture 55(1):47–63

    Article  Google Scholar 

  • Karma A, Kessler DA, Levine H (2001) Phase-field model of mode III dynamic fracture. Phys Rev Lett 87(4):045,501

    Article  Google Scholar 

  • Karma A, Lobkovsky AE (2004) Unsteady crack motion and branching in a phase-field model of brittle fracture. Phys Rev Lett 92(24):245,510

    Article  Google Scholar 

  • Lapusta N, Liu Y (2009) Three-dimensional boundary integral modeling of spontaneous earthquake sequences and aseismic slip. J Geophys Res Solid Earth 114(B9). doi:10.1029/2008JB005934

  • Li T, Marigo JJ, Guilbaud D, Potapov S (2015) Variational approach to dynamic brittle fracture via gradient damage models. In: Applied mechanics and materials, vol. 784. Trans Tech Publ, pp 334–341

  • Li T, Marigo JJ, Guilbaud D, Potapov S (2016) Gradient damage modeling of brittle fracture in an explicit dynamics context. Int J Numer Methods Eng (in press), Nme.5262

  • Li T, Maurini C (2015) FEniCS (dynamic) gradient damage. https://bitbucket.org/litianyi/dynamic-gradient-damage

  • Livne A, Bouchbinder E, Fineberg J (2008) Breakdown of linear elastic fracture mechanics near the tip of a rapid crack. Phys Rev Lett 101(26):264,301

    Article  Google Scholar 

  • Logg A, Mardal KA, Wells G (2012) Automated solution of differential equations by the finite element method: the FEniCS book, vol 84. Springer, New York

    Google Scholar 

  • Lorentz E, Cuvilliez S, Kazymyrenko K (2012) Modelling large crack propagation: from gradient damage to cohesive zone models. Int J Fracture 178(1–2):85–95

    Article  Google Scholar 

  • Marder M (1991) New dynamical equation for cracks. Phys Rev Lett 66(19):2484

    Article  Google Scholar 

  • May S, Vignollet J, De Borst R (2015) A numerical assessment of phase-field models for brittle and cohesive fracture: \(\varGamma \)-convergence and stress oscillations. Eur J Mech-A/Solids 52:72–84

    Article  Google Scholar 

  • Miehe C, Hofacker M, Welschinger F (2010) A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng 199(45):2765–2778

    Article  Google Scholar 

  • Miehe C, Welschinger F, Hofacker M (2010) Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int J Numer Methods Eng 83(10):1273–1311

    Article  Google Scholar 

  • Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1):131–150

    Article  Google Scholar 

  • Moës N, Stolz C, Bernard PE, Chevaugeon N (2011) A level set based model for damage growth: the thick level set approach. Int J Numer Methods Eng 86(3):358–380

    Article  Google Scholar 

  • Murali P, Guo T, Zhang Y, Narasimhan R, Li Y, Gao H (2011) Atomic scale fluctuations govern brittle fracture and cavitation behavior in metallic glasses. Phys Rev Lett 107(21):215,501

    Article  Google Scholar 

  • Pandolfi A, Ortiz M (2012) An eigenerosion approach to brittle fracture. Int J Numer Methods Eng 92(8):694–714

    Article  Google Scholar 

  • Patinet S, Vandembroucq D, Roux S (2013) Quantitative prediction of effective toughness at random heterogeneous interfaces. Phys Rev Lett 110(16):165,507

    Article  Google Scholar 

  • Peerlings R, De Borst R, Brekelmans W, Geers M (1998) Gradient-enhanced damage modelling of concrete fracture. Mech Cohesive-Frictional Mater 3(4):323–342

    Article  Google Scholar 

  • Pham K, Amor H, Marigo JJ, Maurini C (2011) Gradient damage models and their use to approximate brittle fracture. Int J Damage Mech 20(4):618–652

    Article  Google Scholar 

  • Pijaudier-Cabot G, Bazant ZP (1987) Nonlocal damage theory. J Eng Mech 113(10):1512–1533

    Article  Google Scholar 

  • Ravi-Chandar K (2004) Dynamic fracture. Elsevier, Amsterdam

    Google Scholar 

  • Ravi-Chandar K, Knauss W (1984) An experimental investigation into dynamic fracture II microstrutural aspects. Int J Fracture 26(1):65–80

    Article  Google Scholar 

  • Ravi-Chandar K, Knauss W (1984) An experimental investigation into dynamic fracture: III. On steady-state crack propagation and crack branching. Int J Fracture 26(2):141–154

    Article  Google Scholar 

  • Ravi-Chandar K, Yang B (1997) On the role of microcracks in the dynamic fracture of brittle materials. J Mech Phys Solids 45(4):535–563

    Article  Google Scholar 

  • Rice JR (2001) Physical aspects of fracture, chap. Some studies of crack dynamics. Springer, Dordrecht

    Google Scholar 

  • Seelig T, Gross D (1999) On the interaction and branching of fast running cracksa numerical investigation. J Mech Phys Solids 47(4):935–952

    Article  Google Scholar 

  • Sharon E, Gross SP, Fineberg J (1996) Energy dissipation in dynamic fracture. Phys Rev Lett 76(12):2117

    Article  Google Scholar 

  • Sicsic P, Marigo JJ (2013) From gradient damage laws to Griffiths theory of crack propagation. J Elast 113(1):55–74

    Article  Google Scholar 

  • Stolarska M, Chopp D, Moës N, Belytschko T (2001) Modelling crack growth by level sets in the extended finite element method. Int J Numer Methods Eng 51(8):943–960

    Article  Google Scholar 

  • Stroh AN (1957) A theory of the fracture of metals. Adv Phys 6(24):418–465

    Article  Google Scholar 

  • Vasoya M, Unni AB, Leblond JB, Lazarus V, Ponson L (2016) Finite size and geometrical non-linear effects during crack pinning by heterogeneities: an analytical and experimental study. J Mech Phys Solids 89:211–230

    Article  Google Scholar 

  • Verhoosel CV, Borst R (2013) A phase-field model for cohesive fracture. Int J Numer Methods Eng 96(1):43–62

    Article  Google Scholar 

  • Washabaugh PD, Knauss W (1994) A reconciliation of dynamic crack velocity and Rayleigh wave speed in isotropic brittle solids. Int J Fracture 65(2):97–114

    Google Scholar 

  • Wolff C, Richart N, Molinari JF (2015) A non-local continuum damage approach to model dynamic crack branching. Int J Numer Methods Eng 101(12):933–949

    Article  Google Scholar 

  • Xu D, Liu Z, Liu X, Zeng Q, Zhuang Z (2014) Modeling of dynamic crack branching by enhanced extended finite element method. Comput Mech 54(2):489–502

    Article  Google Scholar 

  • Xu XP, Needleman A (1994) Numerical simulations of fast crack growth in brittle solids. J Mech Phys Solids 42(9):1397–1434

    Article  Google Scholar 

  • Yoffe EH (1951) The moving Griffith crack. Lond Edinb Dublin Philos Mag J Sci 42(330):739–750

    Article  Google Scholar 

  • Yu C, Pandolfi A, Ortiz M, Coker D, Rosakis A (2002) Three-dimensional modeling of intersonic shear-crack growth in asymmetrically loaded unidirectional composite plates. Int J Solids Struct 39(25):6135–6157

    Article  Google Scholar 

  • Zhou F (1996) Study on the macroscopic behavior and the microscopic process of dynamic crack propagation. Ph.D. thesis, PhD dissertation. The University of Tokyo, Tokyo

  • Zhou F, Molinari JF, Shioya T (2005) A rate-dependent cohesive model for simulating dynamic crack propagation in brittle materials. Eng Fracture Mech 72(9):1383–1410

    Article  Google Scholar 

Download references

Acknowledgments

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

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Department of Materials Science, Ecole Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland

    Jérémy Bleyer, Clément Roux-Langlois & Jean-François Molinari

  2. Mechanics and Glasses Department, Institute of physics of Rennes, UMR UR1-CNRS 6251, University of Rennes 1, Campus de Beaulieu, 35042, Rennes Cedex, France

    Clément Roux-Langlois

Authors
  1. Jérémy Bleyer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Clément Roux-Langlois
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jean-François Molinari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jérémy Bleyer.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (mov 480 KB)

Supplementary material 2 (mov 348 KB)

Supplementary material 3 (mov 377 KB)

Supplementary material 4 (mov 349 KB)

Supplementary material 5 (mov 368 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bleyer, J., Roux-Langlois, C. & Molinari, JF. Dynamic crack propagation with a variational phase-field model: limiting speed, crack branching and velocity-toughening mechanisms. Int J Fract 204, 79–100 (2017). https://doi.org/10.1007/s10704-016-0163-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10704-016-0163-1

Keywords

Search

Navigation