Computational Particle Mechanics

, Volume 4, Issue 4, pp 441–450 | Cite as

Three-dimensional bonded-cell model for grain fragmentation

  • D. CantorEmail author
  • E. Azéma
  • P. Sornay
  • F. Radjai


We present a three-dimensional numerical method for the simulation of particle crushing in 3D. This model is capable of producing irregular angular fragments upon particle fragmentation while conserving the total volume. The particle is modeled as a cluster of rigid polyhedral cells generated by a Voronoi tessellation. The cells are bonded along their faces by a cohesive Tresca law with independent tensile and shear strengths and simulated by the contact dynamics method. Using this model, we analyze the mechanical response of a single particle subjected to diametral compression for varying number of cells, their degree of disorder, and intercell tensile and shear strength. In particular, we identify the functional dependence of particle strength on the intercell strengths. We find that two different regimes can be distinguished depending on whether intercell shear strength is below or above its tensile strength. In both regimes, we observe a power-law dependence of particle strength on both intercell strengths but with different exponents. The strong effect of intercell shear strength on the particle strength reflects an interlocking effect between cells. In fact, even at low tensile strength, the particle global strength can still considerably increase with intercell shear strength. We finally show that the Weibull statistics describes well the particle strength variability.


Bonded-cell model Fragmentation Discrete element method Contact dynamics method Voronoi cell Weibull statistics 



This work was financially supported by a research grant awarded by the French Alternative Energies and Atomic Energy Commission (CEA). Farhang Radjai would also like to acknowledge the support of the ICoME2 Labex (ANR-11-LABX-0053) and the A*MIDEX projects (ANR-11-IDEX-0001-02) cofunded by the French program Investissements d’Avenir, managed by the French National Research Agency (ANR).


  1. 1.
    Åström J, Herrmann H (1998) Fragmentation of grains in a two-dimensional packing. Eur Phys J B 5(3):551–554CrossRefGoogle Scholar
  2. 2.
    Azéma E, Radjai F (2010) Stress-strain behavior and geometrical properties of packings of elongated particles. Phys Rev E 81:051,304CrossRefGoogle Scholar
  3. 3.
    Azéma E, Estrada N, Radjai F (2012) Nonlinear effects of particle shape angularity in sheared granular media. Phys Rev E 86:041,301CrossRefGoogle Scholar
  4. 4.
    Azéma E, Radjai F (2012) Force chains and contact network topology in sheared packings of elongated particles. Phys Rev E 85:031,303CrossRefGoogle Scholar
  5. 5.
    Azéma E, Radjai F, Peyroux R, Saussine G (2007) Force transmission in a packing of pentagonal particles. Phys Rev E 76(1 Pt 1):011,301CrossRefGoogle Scholar
  6. 6.
    Azéma E, Radjai F, Saint-Cyr B, Delenne JY, Sornay P (2013) Rheology of 3D packings of aggregates: microstructure and effects of nonconvexity. Phys Rev E 87:052,205CrossRefGoogle Scholar
  7. 7.
    Azéma E, Radjai F, Saussine G (2009) Quasistatic rheology, force transmission and fabric properties of a packing of irregular polyhedral particles. Mech Mater 41:721–741CrossRefGoogle Scholar
  8. 8.
    Bagherzadeh Kh A, Mirghasemi A, Mohammadi S (2011) Numerical simulation of particle breakage of angular particles using combined dem and fem. Powder Technol 205(1–3):15–29CrossRefGoogle Scholar
  9. 9.
    Bandini V, Coop MR (2011) The influence of particle breakage on the location of the critical state line of sands. Soils Found 51(4):591–600CrossRefGoogle Scholar
  10. 10.
    Barton N (1976) The shear strength of rock and rock joints. Int J Rock Mech Min Sci Geomech Abstr 13(9):255–279CrossRefGoogle Scholar
  11. 11.
    Barton N (2013) Shear strength criteria for rock, rock joints, rockfill and rock masses: Problems and some solutions. J Rock Mech Geotech Eng 5(4):249–261CrossRefGoogle Scholar
  12. 12.
    Bratberg I, Radjai F, Hansen A (2002) Dynamic rearrangements and packing regimes in randomly deposited two-dimensional granular beds. Phys Rev E 66(3):1–34CrossRefGoogle Scholar
  13. 13.
    Cecconi M, DeSimone A, Tamagnini C, Viggiani G (2002) A constitutive model for granular materials with grain crushing and its application to a pyroclastic soil. Int J Numer Anal Meth Geomech 26(15):1531–1560CrossRefzbMATHGoogle Scholar
  14. 14.
    Chau K, Wei X (1998) Spherically isotropic elastic spheres subject to diametral point load strength test. Int J Solids Struct 25Google Scholar
  15. 15.
    Cheng YP, Nakata Y, Bolton MD (2008) Micro- and macro-mechanical behaviour of dem crushable materials. Géotechnique 58(6):471–480CrossRefGoogle Scholar
  16. 16.
    Ciantia M, Arroyo M, Calvetti F, Gens A (2015) An approach to enhance efficiency of dem modelling of soils with crushable grains. Geotechnique 65(2):91–110CrossRefGoogle Scholar
  17. 17.
    Cundall PA (1988) Formulation of a three-dimensional distinct element model-Part I. A scheme to detect and represent contacts in a system composed of many polyhedral blocks. Int J Rock Mech Min Sci 25(3):107–116CrossRefGoogle Scholar
  18. 18.
    Du Q, Faber V, Gunzburger M (1999) Centroidal Voronoi tessellations: applications and algorithms. SIAM J Numer Anal 41(4):637–676MathSciNetzbMATHGoogle Scholar
  19. 19.
    Estrada N, Azéma E, Radjai F, Taboada A (2011) Identification of rolling resistance as a shape parameter in sheared granular media. Phys Rev E 84(1):011306CrossRefGoogle Scholar
  20. 20.
    Dubois F, Jean M, et al (2016) LMGC90 wiki page. Accessed 7 Mar 2016
  21. 21.
    Fukumoto T (1992) Particle breakage characteristics of granular soils. Soils Found 32(1):26–40CrossRefGoogle Scholar
  22. 22.
    Galindo-Torres S, Pedroso D, Williams D, Li L (2012) Breaking processes in three-dimensional bonded granular materials with general shapes. Comput Phys Commun 183(2):266–277CrossRefGoogle Scholar
  23. 23.
    Guimaraes M, Valdes J, Palomino AM, Santamarina J (2007) Aggregate production: Fines generation during rock crushing. Int J Miner Process 81(4):237–247CrossRefGoogle Scholar
  24. 24.
    Hardin BO (1985) Crushing of soil particles. J Geotech Eng 111(10):1177–1192CrossRefGoogle Scholar
  25. 25.
    Hégron L, Sornay P, Favretto-Cristini N (2014) Compaction of a bed of fragmentable UO2 particles and associated acoustic emission. IEEE Trans Nucl Sci 61(4):2175–2181CrossRefGoogle Scholar
  26. 26.
    Jaeger H (2015) Celebrating soft matter’s 10th anniversary: Toward jamming by design. Soft Matter 11:12CrossRefGoogle Scholar
  27. 27.
    Jean M (1999) The non-smooth contact dynamics method. Comput Methods Appl Mech Eng 177(3–4):235–257MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kun F, Herrmann H (1996) A study of fragmentation processes using a discrete element method. Comput Methods Appl Mech Eng 7825(96)Google Scholar
  29. 29.
    Lade PV, Yamamuro J, Bopp P (1997) Significance of particle crushing in granular materials. J Geotech Geoenviron Eng 123(9):889–890CrossRefGoogle Scholar
  30. 30.
    Lobo-guerrero S, Vallejo LE (2005) Discrete element method evaluation of granular crushing under direct shear test conditions. J Geotech Geoenviron Eng 131(10):1295–1300CrossRefGoogle Scholar
  31. 31.
    Ma G, Zhou W, Chang XL (2014) Modeling the particle breakage of rockfill materials with the cohesive crack model. Comput Geotech 61:132–143CrossRefGoogle Scholar
  32. 32.
    McDowell G, Bolton M (1998) On the micromechanics of crushable aggregates. Géotechnique 48(5):667–679CrossRefGoogle Scholar
  33. 33.
    McDowell G, Bolton M, Robertson D (1996) The fractal crushing of granular materials. J Mech Phys Solids 44(12):2079–2101CrossRefGoogle Scholar
  34. 34.
    Miura N, Murata H, Yasufuku N (1984) Stress-strain characteristics of sand in a particle-crushing region. Soils Found 24(1):77–89CrossRefGoogle Scholar
  35. 35.
    Moreau J (1994) Some numerical methods in multibody dynamics: application to granular. Eur J Mech A Solids 13:93–114MathSciNetzbMATHGoogle Scholar
  36. 36.
    Nezamabadi S, Radjai F, Averseng J, Delenne J (2015) Implicit frictional-contact model for soft particle systems. J Mech Phys Solids 83:72–87MathSciNetCrossRefGoogle Scholar
  37. 37.
    Nezami EG, Hashash YMA, Zhao D, Ghaboussi J (2004) A fast contact detection algorithm for 3-D discrete element method. Comput Geotech 31(7):575–587CrossRefGoogle Scholar
  38. 38.
    Nezami EG, Hashash YMA, Zhao D, Ghaboussi J (2006) Shortest link method for contact detection in discrete element method. Int J Numer Anal Meth Geomech 30(8):783–801CrossRefzbMATHGoogle Scholar
  39. 39.
    Nguyen DH, Azéma E, Radjai F (2015) Evolution of particle size distributions in crushable granular materials. Geomechanics from Micro to Macro (Md), pp 275–280Google Scholar
  40. 40.
    Nguyen DH, Azéma E, Sornay P, Radjai F (2015) Bonded-cell model for particle fracture. Phys Rev E 91(2):022,203MathSciNetCrossRefGoogle Scholar
  41. 41.
    Nouguier C, Bohatier C, Moreau JJ, Radjai F (2000) Force fluctuations in a pushed granular material. Granular Matter 2:171–178CrossRefGoogle Scholar
  42. 42.
    Okabe A, Boots B, Sugihara K, Chiu SN (1992) Spatial tessellations: concepts and applications of voronoi diagrams. Wiley, New YorkzbMATHGoogle Scholar
  43. 43.
    Quey R, Dawson P, Barbe F (2011) Large-scale 3D random polycrystals for the finite element method: Generation, meshing and remeshing. Comput Methods Appl Mech Eng 200(17–20):1729–1745CrossRefzbMATHGoogle Scholar
  44. 44.
    Quezada JC, Breul P, Saussine G, Radjai F (2012) Stability, deformation, and variability of granular fills composed of polyhedral particles. Phys Rev E 86(3):1–11CrossRefGoogle Scholar
  45. 45.
    Radjai F, Richefeu V, Jean Mm, Moreau JJ, Roux S (1996) Force Distributions in Dense Two-Dimensional Granular Systems. Phys Rev Lett 77(2):274–277CrossRefGoogle Scholar
  46. 46.
    Radjai F, Richefeu V (2009) Contact dynamics as a nonsmooth discrete element method. Mech Mater 41(6):715–728CrossRefGoogle Scholar
  47. 47.
    Radjai F, Dubois F (2011) Discrete-element modeling of granular materials. ISTE Ltd and Wiley, LondonGoogle Scholar
  48. 48.
    Renouf M, Dubois F, Alart P (2004) A parallel version of the non smooth contact dynamics algorithm applied to the simulation of granular media. J Comput Appl Math 168(1–2):375–382MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Ries A, Wolf DE, Unger T (2007) Shear zones in granular media: Three-dimensional contact dynamics simulation. Phys Rev E 76(5):1–9CrossRefGoogle Scholar
  50. 50.
    Russell AR, Muir Wood D, Kikumoto M (2009) Crushing of particles in idealised granular assemblies. J Mech Phys Solids 57(8):1293–1313CrossRefzbMATHGoogle Scholar
  51. 51.
    Saussine G, Cholet C, Gautier PE, Dubois F, Bohatier C, Moreau JJ (2011) Modelling ballast behaviour under dynamic loading. Part 1: A 2D polygonal discrete element method approach. Comput Methods Appl Mech Eng 195(19–22):2841–2859zbMATHGoogle Scholar
  52. 52.
    Saint-Cyr B, Delenne J, Voivret C, Radjai F, Sornay P (2011) Rheology of granular materials composed of nonconvex particles. Phys Rev E 84(4):041302CrossRefGoogle Scholar
  53. 53.
    Staron L, Radjai F, Vilotte J (2005) Multi-scale analysis of the stress state in a granular slope in transition to failure. Eur. Phys. J. E 18:311–320CrossRefGoogle Scholar
  54. 54.
    Staron L, Vilotte JP, Radjai F (2002) Preavalanche instabilities in a granular pile. Phys Rev Lett 89(1):204,302CrossRefGoogle Scholar
  55. 55.
    Stoller RE, Zinkle SJ (2000) On the relationship between uniaxial yield strength and resolved shear stress in polycrystalline materials. J Nucl Mater 283–287(PART I):349–352CrossRefGoogle Scholar
  56. 56.
    Taboada A, Chang KJ, Radjai F, Bouchette F (2005) Rheology, force transmission, and shear instabilities in frictional granular media from biaxial numerical tests using the contact dynamics method. J Geophys Res B 110(9):1–24Google Scholar
  57. 57.
    Topin V, Monerie Y, Perales F, Radjai F (2012) Collapse dynamics and runout of dense granular materials in a fluid. Phys Rev Lett 109(18):1–5CrossRefGoogle Scholar
  58. 58.
    Tsoungui O, Vallet D, Charmet JC (1999) Numerical model of crushing of grains inside two-dimensional granular materials. Powder Technol 105(1–3):190–198CrossRefGoogle Scholar
  59. 59.
    Moreau JJ (1997) Numerical investigation of shear zones in granular materials. In: Wolf D, Grassberger P (eds) Friction, arching, contact dynamics. World Scientific, SingaporeGoogle Scholar
  60. 60.
    Wu S, Chau K (2006) Dynamic response of an elastic sphere under diametral impacts. Mech Mater 38:1039–1060CrossRefGoogle Scholar
  61. 61.
    Zhou W, Yang L, Ma G, Chang X, Cheng Y, Li D (2015) Macro-micro responses of crushable granular materials in simulated true triaxial tests. Granular Matter 17(4):497–509CrossRefGoogle Scholar
  62. 62.
    Zubelewicz A, Bažant ZP (1987) Interface element modeling of fracture in aggregate composites. J Eng Mech 113(11):1619–1629CrossRefGoogle Scholar

Copyright information

© OWZ 2016

Authors and Affiliations

  1. 1.Laboratoire de Mécanique et Génie Civil (LMGC)Université de Montpellier, CNRSMontpellierFrance
  2. 2.CEA, DEN, DEC, SFER, LCUSaint Paul lez DuranceFrance
  3. 3.Department of Mechanical Engineering, Faculty of EngineeringChiang Mai UniversityChiang MaiThailand
  4. 4.Multi-Scale Materials Science for Energy and Environment<MSE>2, UMI 3466 CNRS-MIT Energy InitiativeMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations