International Journal of Fracture

, Volume 196, Issue 1–2, pp 59–98 | Cite as

Why do cracks branch? A peridynamic investigation of dynamic brittle fracture

  • Florin BobaruEmail author
  • Guanfeng Zhang
Special Invited Article Celebrating IJF at 50


In this paper we review the peridynamic model for brittle fracture and use it to investigate crack branching in brittle homogeneous and isotropic materials. The peridynamic simulations offer a possible explanation for the generation of dynamic instabilities in dynamic brittle crack growth and crack branching. We focus on two systems, glass and homalite, often used in crack branching experiments. After a brief review of theoretical and computational models on crack branching, we discuss the peridynamic model for dynamic fracture in linear elastic–brittle materials. Three loading types are used to investigate the role of stress waves interactions on crack propagation and branching. We analyze the influence of sample geometry on branching. Simulation results are compared with experimental ones in terms of crack patterns, propagation speed at branching and branching angles. The peridynamic results indicate that as stress intensity around the crack tip increases, stress waves pile-up against the material directly in front of the crack tip that moves against the advancing crack; this process “deflects” the strain energy away from the symmetry line and into the crack surfaces creating damage away from the crack line. This damage “migration”, seen as roughness on the crack surface in experiments, modifies, in turn, the strain energy landscape around the crack tip and leads to preferential crack growth directions that branch from the original crack line. We argue that nonlocality of damage growth is one key feature in modeling of the crack branching phenomenon in brittle fracture. The results show that, at least to first order, no ingredients beyond linear elasticity and a capable damage model are necessary to explain/predict crack branching in brittle homogeneous and isotropic materials.


Dynamic fracture Crack branching Brittle fracture Peridynamics Nonlocal methods 



This research has been supported by ARO/ARL (Grant Number W911NF1010431), program manager Dr. Asher Rubinstein (ARO) and Dr. Chian-Fong Yen (ARL), and by the AFOSR’s MURI Center for Material Failure Prediction Through Peridynamics, program managers Dr. David Stargel, Dr. Ali Sayir, and Dr. Fariba Fahroo. We are grateful for all their support without which this research would not have been possible.

Supplementary material

Supplementary material 1 (mpg 3170 KB)

Supplementary material 2 (mpg 2319 KB)

Supplementary material 3 (mpg 3476 KB)

Supplementary material 4 (mpg 2501 KB)

Supplementary material 5 (mpg 2898 KB)

Supplementary material 6 (mpg 742 KB)

Supplementary material 7 (mpg 984 KB)

Supplementary material 8 (mpg 417 KB)

Supplementary material 9 (mpg 2069 KB)

Supplementary material 10 (mpg 2069 KB)

Supplementary material 11 (mpg 978 KB)

Supplementary material 12 (mpg 2877 KB)

Supplementary material 13 (mpg 2900 KB)

Supplementary material 14 (mpg 2884 KB)

Supplementary material 15 (mpg 2591 KB)

Supplementary material 16 (mpg 3608 KB)

Supplementary material 17 (mpg 2884 KB)

Supplementary material 18 (mpg 6470 KB)

Supplementary material 19 (mpg 6665 KB)

Supplementary material 20 (mpg 2482 KB)

Supplementary material 21 (mpg 2730 KB)

Supplementary material 22 (mpg 2589 KB)


  1. Abraham FF, Brodbeck D, Rudge WE, Xu X (1997) A molecular dynamics investigation of rapid fracture mechanics. J Mech Phys Solids 45(9):1595–1619CrossRefGoogle Scholar
  2. Abraham F (2005) Unstable crack motion is predictable. J Mech Phys Solids 53:1071–1078CrossRefGoogle Scholar
  3. Anthony SR, Chubb JP, Congleton J (1970) The crack-branching velocity. Philos Mag 22(180):1201–1216CrossRefGoogle Scholar
  4. Aranson IS, Kalatsky VA, Vinokur VM (2000) Continuum field description of crack propagation. Phys Rev Lett 85:118–121CrossRefGoogle Scholar
  5. Bažant ZP, Jirásek M (2002) Nonlocal integral formulations of plasticity and damage: survey of progress. J Eng Mech 128(11):1119–1149CrossRefGoogle Scholar
  6. Beauchamp EK (1996) Mechanisms for hackle formation and crack branching. In: Fractography of glasses and ceramics III, vol. 64 of ceramic transactions. American Ceramic Society, pp 409–445Google Scholar
  7. Bhate DN, Kumar A, Bower AF (2000) Diffuse interface model for electromigration and stress voiding. J Appl Phys 87(4):1712–1721CrossRefGoogle Scholar
  8. Bobaru F (2007) Influence of van der waals forces on increasing the strength and toughness in dynamic fracture of nanofibre networks: a peridynamic approach. Model Simul Mater Sci Eng 15(5):397CrossRefGoogle Scholar
  9. Bobaru F, Yang M, Alves LF, Silling SA, Askari E, Xu J (2009) Convergence, adaptive refinement, and scaling in 1d peridynamics. Int J Numer Meth Eng 77(6):852–877CrossRefGoogle Scholar
  10. Bobaru F, Ha YD, Hu W (2012) Damage progression from impact in layered glass modeled with peridynamics. Open Eng 2(4):551–561CrossRefGoogle Scholar
  11. Bobaru F, Ha YD (2011) Adaptive refinement and multiscale modeling in 2d peridynamics. Int J Multiscale Comput Eng 9(6):635–660CrossRefGoogle Scholar
  12. Bobaru F, Hu W (2012) The meaning, selection, and use of the peridynamic horizon and its relation to crack branching in brittle materials. Int J Fract 176(2):215–222CrossRefGoogle Scholar
  13. Bolander JE, Saito S (1998) Fracture analyses using spring networks with random geometry. Eng Fract Mech 61(5):569–591CrossRefGoogle Scholar
  14. Bonamy D, Ravi-Chandar K (2003) Interaction of shear waves and propagating cracks. Phys Rev Lett 91(23):235502CrossRefGoogle Scholar
  15. Bonamy D, Ravi-Chandar K (2005) Dynamic crack response to a localized shear pulse perturbation in brittle amorphous materials: on crack surface roughening. Int J Fract 134(1):1–22CrossRefGoogle Scholar
  16. Borden MJ, Verhoosel CV, Scott MA, Hughes TJR, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217:77–95CrossRefGoogle Scholar
  17. Bouchbinder E, Mathiesen J, Procaccia I (2005) Branching instabilities in rapid fracture: dynamics and geometry. Phys Rev E 71:056118CrossRefGoogle Scholar
  18. Bouchbinder E, Livne A, Fineberg J (2010) Weakly nonlinear fracture mechanics: experiments and theory. Int J Fract 162(1–2):3–20CrossRefGoogle Scholar
  19. Bouchbinder E, Tamar Goldman T, Fineberg J (2014) The dynamics of rapid fracture: instabilities, nonlinearities and length scales. Rep Prog Phys 77(4):046501CrossRefGoogle Scholar
  20. Bourdin B, Francfort GA, Marigo JJ (2008) The variational approach to fracture. J Elast 91:5–148CrossRefGoogle Scholar
  21. Bourdin B, Larsen CJ, Richardson CL (2011) A time-discrete model for dynamic fracture based on crack regularization. Int J Fract 168:133–143CrossRefGoogle Scholar
  22. Bowden FP, Brunton JH, Field JE, Heyes AD (1967) Controlled fracture of brittle solids and interruption of electrical current. Nature 216:38–42CrossRefGoogle Scholar
  23. Broberg K (1999) Cracks and fracture. Academic Press, San DiegoGoogle Scholar
  24. Buehler MJ, Gao H (2006) Dynamical fracture instabilities due to local hyperelasticity at crack tips. Nature 439(7074):307–310CrossRefGoogle Scholar
  25. Camacho GT, Ortiz M (1996) Computational modelling of impact damage in brittle materials. Int J Solids Struct 33(20):2899–2938CrossRefGoogle Scholar
  26. Chen Z, Bobaru F (2015) Selecting the kernel in a peridynamic formulation: a study for transient heat diffusion. Comput Phys Commun 197:51–60CrossRefGoogle Scholar
  27. Cox BN, Gao H, Gross D, Rittel D (2005) Modern topics and challenges in dynamic fracture. J Mech Phys Solids 53(3):565–596CrossRefGoogle Scholar
  28. Cundall P, Strack O (1979) A discrete numerical model for granular assemblies. Geotechnique 29:47–65CrossRefGoogle Scholar
  29. Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Meth Eng 46(1):131–150CrossRefGoogle Scholar
  30. Döll W (1975) Investigations of the crack branching energy. Int J Fract 11(1):184–186CrossRefGoogle Scholar
  31. Erdogan F, Sih GC (1963) On the crack extension in plates under plane loading and transverse shear. J Basic Eng 85(4):519–525CrossRefGoogle Scholar
  32. Eringen AC (2002) Nonlocal continuum field theories. Springer-Verlag, New YorkGoogle Scholar
  33. Eshelby J (1969) The elastic field of a crack extending non-uniformly under general anti-plane loading. J Mech Phys Solids 17(3):177–199CrossRefGoogle Scholar
  34. Field JE (1971) Brittle fracture: its study and application. Contemp Phys 12(1):1–31CrossRefGoogle Scholar
  35. Fineberg J, Bouchbinder E (2015) Recent developments in dynamic fracture: some perspectives. Int J Fract 1–25. doi: 10.1007/s10704-015-0038-x
  36. Fischer-Cripps AC, Mustafaev I (2000) Introduction to contact mechanics. Springer, BerlinGoogle Scholar
  37. Freund LB (1972) Crack propagation in an elastic solid subjected to general loadingi. constant rate of extension. J Mech Phys Solids 20(3):129–140CrossRefGoogle Scholar
  38. Freund LB (1990) Dynamic fracture mechanics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  39. Gao H (1996) A theory of local limiting speed in dynamic fracture. J Mech Phys Solids 44(9):1453–1474CrossRefGoogle Scholar
  40. Gerstle W, Sau N, Silling SA (2005) Peridynamic modeling of plain and reinforced concrete structures. In: 18th International conference on structural mechanics in reactor technology (SMiRT 18). Biejing, China, no. SMiRT18-B01–2Google Scholar
  41. Ghajari M, Iannucci L, Curtis P (2014) A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media. Comput Methods Appl Mech Eng 276:431–452CrossRefGoogle Scholar
  42. Guan PC, Chi SW, Chen JS, Slawson TR, Roth MJ (2011) Semi-lagrangian reproducing kernel particle method for fragment-impact problems. Int J Impact Eng 38(12):1033–1047CrossRefGoogle Scholar
  43. Ha YD, Bobaru F (2010) Studies of dynamic crack propagation and crack branching with peridynamics. Int J Fract 162(1–2):229–244CrossRefGoogle Scholar
  44. Ha YD, Bobaru F (2010) Studies of dynamic crack propagation and crack branching with peridynamics. Int J Fract 162(1–2):229–244CrossRefGoogle Scholar
  45. Ha YD, Bobaru F (2011a) Characteristics of dynamic brittle fracture captured with peridynamics. Eng Fract Mech 78(6):1156–1168CrossRefGoogle Scholar
  46. Hauch JA, Holland D, Marder MP, Swinney HL (1999) Dynamic fracture in single crystal silicon. Phys Rev Lett 82:3823–3826CrossRefGoogle Scholar
  47. 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 Meth Eng 93(3):276–301CrossRefGoogle Scholar
  48. Hu W, Ha YD, Bobaru F (2011) Modeling dynamic fracture and damage in fiber-reinforced composites with peridynamics. Int J Multiscale Comput Eng 9(6):707–726CrossRefGoogle Scholar
  49. Hu W, Ha Y, Bobaru F, Silling S (2012a) The formulation and computation of the nonlocal J-integral in bond-based peridynamics. Int J Fract 176(2):195–206CrossRefGoogle Scholar
  50. Hu W, Ha YD, Bobaru F (2012b) Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites. Comput Methods Appl Mech Eng 217:247–261CrossRefGoogle Scholar
  51. Hu W, Wang Y, Yu J, Yen C-F, Bobaru F (2013) Impact damage on a thin glass plate with a thin polycarbonate backing. Int J Impact Eng 62:152–165CrossRefGoogle Scholar
  52. Hu Y, Yu Y, Wang H (2014) Peridynamic analytical method for progressive damage in notched composite laminates. Compos Struct 108:801–810CrossRefGoogle Scholar
  53. Hu Y, Yu Y, Wang H (2014) Peridynamic analytical method for progressive damage in notched composite laminates. Compos Struct 108:801–810CrossRefGoogle Scholar
  54. Hull D (1994) The effect of mixed mode I/III on crack evolution in brittle solids. Int J Fract 70(1):59–79CrossRefGoogle Scholar
  55. Hull D (1999) Fractography: observing, measuring and interpreting fracture structure topography. Cambridge University Press, CambridgeGoogle Scholar
  56. Jirásek M, Bazǎnt ZP (1995) Particle model for quasibrittle fracture and application to sea ice. J Eng Mech 121(9):1016–1025CrossRefGoogle Scholar
  57. Johnson E (1992) Process region changes for rapidly propagating cracks. Int J Fract 55(1):47–63CrossRefGoogle Scholar
  58. Karma A, Kessler DA, Levine H (2001) Phase-field model of mode III dynamic fracture. Phys Rev Lett 87:045501CrossRefGoogle Scholar
  59. Kermode JR, Albaret T, Sherman D, Bernstein N, Gumbsch P, Payne MC, Csanyi G, De Vita A (2008) Low-speed fracture instabilities in a brittle crystal. Nature 455:1224–1227CrossRefGoogle Scholar
  60. Kunin IA (1982) Elastic media with microstructure I. One-dimensional models. Springer, BerlinCrossRefGoogle Scholar
  61. Le Q, Bobaru F (2015) Surface corrections in peridynamic models for elasticity and fracture. In reviewGoogle Scholar
  62. Livne A, Ben-David O, Fineberg J (2007) Oscillations in rapid fracture. Phys Rev Lett 98:124301CrossRefGoogle Scholar
  63. Macek RW, Silling S (2007) Peridynamics via finite element analysis. Finite Elem Anal Des 43(15):1169–1178CrossRefGoogle Scholar
  64. Madenci E, Oterkus E (2014) Peridynamics theory and its applications. Springer, New YorkCrossRefGoogle Scholar
  65. Marder M, Gross S (1995) Origin of crack tip instabilities. J Mech Phys Solids 43(1):1–48CrossRefGoogle Scholar
  66. Meyers M (1994) Dynamic behavior of materials. Wiley, New YorkCrossRefGoogle Scholar
  67. Morrissey J, Rice J (2000) Perturbative simulations of crack front waves. J Mech Phys Solids 48(6–7):1229–1251CrossRefGoogle Scholar
  68. Negri M (2006) A non-local approximation of free discontinuity problems in SBV and SBD. Calc Var Partial Differ Equ 25(1):33–62CrossRefGoogle Scholar
  69. Ortiz M, Pandolfi A (1999) Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis. Int J Numer Meth Eng 44(9):1267–1282CrossRefGoogle Scholar
  70. Oterkus S, Madenci E, Agwai A (2014) Peridynamic thermal diffusion. J Comput Phys 265:71–96CrossRefGoogle Scholar
  71. Oterkus S, Madenci E (2012) Peridynamic analysis of fiber-reinforced composite materials. J Mech Mater Struct 7(1):45–84CrossRefGoogle Scholar
  72. Ožbolt J, Sharma A, Reinhardt H-W (2011) Dynamic fracture of concrete compact tension specimen. Int J Solids Struct 48:1534–1543CrossRefGoogle Scholar
  73. Ožbolt J, Bošnjak J, Sola E (2013) Dynamic fracture of concrete compact tension specimen: experimental and numerical study. Int J Solids Struct 50:4270–4278CrossRefGoogle Scholar
  74. Pandolfi A, Li B, Ortiz M (2013) Modeling fracture by material-point erosion. Int J Fract 184:3–16CrossRefGoogle Scholar
  75. Pandolfi A, Ortiz M (2012) An eigenerosion approach to brittle fracture. Int J Numer Meth Eng 92(8):694–714CrossRefGoogle Scholar
  76. Pons AJ, Karma A (2010) Helical crack-front instability in mixed-mode fracture. Nature 464(7285):85–89CrossRefGoogle Scholar
  77. Procaccia I, Zylberg J (2013) Propagation mechanism of brittle cracks. Phys Rev E 87:012801CrossRefGoogle Scholar
  78. Rabczuk T (2013) Computational methods for fracture in brittle and quasi-brittle solids: state-of-the-art review and future perspectives. ISRN Appl Math 2013(849231):38Google Scholar
  79. Rabczuk T, Belytschko T (2004) Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int J Numer Meth Eng 61(13):2316–2343CrossRefGoogle Scholar
  80. Rahman M, Michelitsch T (2006) A note on the formula for the Rayleigh wave speed. Wave Motion 43(3):272–276CrossRefGoogle Scholar
  81. Ramulu M, Kobayashi AS (1983) Dynamic crack curving—a photoelastic evaluation. Exp Mech 23(1):1–9CrossRefGoogle Scholar
  82. Ramulu M, Kobayashi AS (1985) Mechanics of crack curving and branchinga dynamic fracture analysis. Int J Fract 27:187–201CrossRefGoogle Scholar
  83. Ravi-Chandar K (2004) Dynamic fracture. Elsevier, AmsterdamGoogle Scholar
  84. Ravi-Chandar K, Knauss WG (1982) Dynamic crack tip stresses under stress wave loading: a comparison of theory and experiment. Int J Fract 20:209–222CrossRefGoogle Scholar
  85. Ravi-Chandar K, Knauss WG (1984a) An experimental investigation into dynamic fracture: III. On steady-state crack propagation and crack branching. Int J Fract 26(2):141–154CrossRefGoogle Scholar
  86. Ravi-Chandar K, Knauss WG (1984b) An experimental investigation into dynamic fracture: IV. On the interaction of stress waves with propagating cracks. Int J Fract 26(3):189–200CrossRefGoogle Scholar
  87. Ravi-Chandar K, Yang B (1997) On the role of microcracks in the dynamic fracture of brittle materials. J Mech Phys Solids 45(4):535–563CrossRefGoogle Scholar
  88. Rogula D (1982) Nonlocal theory of material media. Springer, BerlinCrossRefGoogle Scholar
  89. Rösch F, Trebin H-R (2009) Brittle fracture in a complex metallic compound from an atomistic viewpoint: Nbcr 2, a case study. Europhys Lett 85(5):56002CrossRefGoogle Scholar
  90. Seleson P (2014) Improved one-point quadrature algorithms for two-dimensional peridynamic models based on analytical calculations. Comput Methods Appl Mech Eng 282:184–217CrossRefGoogle Scholar
  91. Sharon E, Gross SP, Fineberg J (1995) Local crack branching as a mechanism for instability in dynamic fracture. Phys Rev Lett 74(25):5096Google Scholar
  92. Sharon E, Cohen G, Fineberg J (2002) Crack front waves and the dynamics of a rapidly moving crack. Phys Rev Lett 88:085503CrossRefGoogle Scholar
  93. Sharon E, Fineberg J (1996) Microbranching instability and the dynamic fracture of brittle materials. Phys Rev B 54(10):7128CrossRefGoogle Scholar
  94. Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48(1):175–209CrossRefGoogle Scholar
  95. Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48(1):175–209CrossRefGoogle Scholar
  96. Silling SA, Askari E (2005) A meshfree method based on the peridynamic model of solid mechanics. Comput Struct 83(17):1526–1535CrossRefGoogle Scholar
  97. Silling SA, Bobaru F (2005) Peridynamic modeling of membranes and fibers. Int J Non Linear Mech 40(2–3):395–409CrossRefGoogle Scholar
  98. Sommer E (1969) Formation of fracture ’lances’ in glass. Eng Fract Mech 1(3):539–546CrossRefGoogle Scholar
  99. Song JH, Wang H, Belytschko T (2008) A comparative study on finite element methods for dynamic fracture. Comput Mech 42(2):239–250CrossRefGoogle Scholar
  100. Spatschek R, Brener E, Karma A (2011) Phase field modeling of crack propagation. Phil Mag 91(1):75–95CrossRefGoogle Scholar
  101. Streit R, Finnie I (1980) An experimental investigation of crack-path directional stability. Exp Mech 20(1):17–23CrossRefGoogle Scholar
  102. Stroh AN (1957) A theory of the fracture of metals. Adv Phys 6(24):418–465CrossRefGoogle Scholar
  103. Wiederhorn SM (1969) Fracture surface energy of glass. J Am Ceram Soc 52(2):99–105CrossRefGoogle Scholar
  104. Xu J, Askari E, Weckner O, Silling SA (2008) Peridynamic analysis of impact damage in composite laminates. J Aerosp Eng 21(3):187–194CrossRefGoogle Scholar
  105. Xu XP, Needleman A (1994) Numerical simulations of fast crack growth in brittle solids. J Mech Phys Solids 42(9):1397–1434Google Scholar
  106. Yoffe EH (1951) The moving griffith crack. Phil Mag 42(330):739–750Google Scholar
  107. Zhou SJ, Lomdahl PS, Thomson R, Holian BL (1996) Dynamic crack processes via molecular dynamics. Phys Rev Lett 76(13):2318CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.University of Nebraska-LincolnLincolnUSA

Personalised recommendations