Advertisement

Granular Matter

, Volume 13, Issue 4, pp 341–364 | Cite as

A discrete element model to describe failure of strong rock in uniaxial compression

  • Christian Ergenzinger
  • Robert Seifried
  • Peter EberhardEmail author
Original Paper

Abstract

A bonded particle model is investigated by means of an extended Discrete Element Method with respect to failure of strong rock in uniaxial compression. A coordination number based inflation scheme is presented, which generates isotropic sphere packings that are featuring a higher average coordination number than conventional procedures. A progressive failure model is proposed, which promotes crack propagation and localization and allows adjusting brittleness of fracture. Failure of the granular solid is discussed in detail. The fracture process is studied in dependence of the introduced failure models. The influence of particle size and bond strength distributions, particle numbers, particle layering in finite sphere packings and end constraints is addressed. Comparison to published experimental results reveals that many of the observed features of rock failure are reproduced.

Keywords

Granular material Discrete Element Method Bonded particles Particle packing Particle bond failure Rock strength and failure 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Achmus M., Abdel-Rahman K.: The influence of  “up-scaling” on the results of particle method calculations of non-cohesive soils. In: Konietzky, H. (eds) Numerical Modeling in Micromechanics via Particle Methods, pp. 183–188. A. A. Balkema, Lisse (2003)Google Scholar
  2. 2.
    Anderson W.F., Fair P.: Behavior of railroad ballast under monotonic and cyclic loading. J. Geotech. Geoenviron. Eng. 134(3), 316–327 (2008)CrossRefGoogle Scholar
  3. 3.
    Aursudkij B., McDowell G.R., Collop A.C.: Cyclic loading of railway ballast under triaxial conditions and in a railway test facility. Granul Matter 11(6), 391–401 (2009)CrossRefGoogle Scholar
  4. 4.
    Bäckström A., Antikainen J., Backers T., Feng X., Jing L., Kobayashi A., Koyama T., Pan P., Rinne M., Shen B., Hudson J.A.: Numerical modelling of uniaxial compressive failure of granite with and without saline porewater. Int. J. Rock Mech. Min. Sci. 45(7), 1126–1142 (2008)CrossRefGoogle Scholar
  5. 5.
    Bagi, K.: From order to chaos: the mechanical behaviour of regular and irregular assemblies. In: Proceedings QuaDPM’03 Workshop on the Quasi-static Deformations of Particulate Materials (2003)Google Scholar
  6. 6.
    Bagi K.: An algorithm to generate random dense arrangements for discrete element simulations of granular assemblies. Gran. Matter 7, 31–43 (2005)zbMATHCrossRefGoogle Scholar
  7. 7.
    Bathurst R.J., Rothenburg L.: Micromechanical aspects of isotropic granular assemblies with linear contact interactions. J. Appl. Mech. 55, 17–23 (1988)ADSCrossRefGoogle Scholar
  8. 8.
    Belytschko T., Black T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Methods Eng. 45, 601–620 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bishop R.E.D., Johnson D.C.: The Mechanics of Vibration. Cambridge University Press, Cambridge (1960)zbMATHGoogle Scholar
  10. 10.
    Blake, A. (eds): Handbook of Mechanics, Materials, and Structures. Wiley Series in Mechanical Engineering Practice. Crystal Dreams Pub, New York (1985)Google Scholar
  11. 11.
    Brady B.H.G., Brown E.T.: Rock Mechanics for Underground Mining, 3rd edn. Kluwer Academic Publishers, Dordrecht, Boston London (2004)Google Scholar
  12. 12.
    Brook N.: Comprehensive Rock Engineering—Volume 3 Rock Testing and Site Characterization, Chapter The Measurement and Estimation of Basic Rock Strength, pp. 41–66. Pergamon Press, Oxford (1993)Google Scholar
  13. 13.
    Brown, E.T. (eds): Rock Characterization, Testing and Monitoring—ISRM Suggested Methods. Pergamon Press, Oxford (1981)Google Scholar
  14. 14.
    Buckingham E.: On physically similar systems; illustrations of the use of dimensional equations. Phys. Rev. 4(4), 345–376 (1914)ADSCrossRefGoogle Scholar
  15. 15.
    Cheng Y.P., Nakata Y., Bolton M.D.: Discrete element simulation of crushable soil. Géotechnique 53(7), 633–641 (2003)CrossRefGoogle Scholar
  16. 16.
    Cho N., Martin C.D., Sego D.C.: A clumped particle model for rock. Int. J. Rock Mech. Min. Sci. 44, 997–1010 (2007)CrossRefGoogle Scholar
  17. 17.
    Cook N.G.W.: The failure of rock. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 2, 389–403 (1965)CrossRefGoogle Scholar
  18. 18.
    Coulomb C.A.: Essai sur une application des règles de maximis et minimis à quelques problèmes de statique, relatifs à l’architecture (in French). Mémoires de Mathématique & de Physique, présentés à l’Académie Royale des Sciences par divers Savants, & lûs dans ses Assamblées 7, 343–382 (1773) (published 1776)Google Scholar
  19. 19.
    Cui L., O’Sullivan C.: Analysis of a triangulation based approach for specimen generation for discrete element simulations. Gran. Matter 5, 135–145 (2003)zbMATHCrossRefGoogle Scholar
  20. 20.
    Cundall P.A., Strack O.D.L.: A discrete numerical model for granular assemblies. Géotechnique 29(1), 47–65 (1979)CrossRefGoogle Scholar
  21. 21.
    D’Addetta, G.A.: Discrete Models for Cohesive Frictional Materials. Doctoral thesis, Universität Stuttgart (2004)Google Scholar
  22. 22.
    D’Addetta G.A., Kun F., Ramm E.: On the application of a discrete model to the fracture process of cohesive granular materials. Gran. Matter 4, 77–90 (2002)zbMATHCrossRefGoogle Scholar
  23. 23.
    D’Addetta G.A., Ramm E.: A microstructure-based simulation environment on the basis of an interface enhanced particle model. Gran. Matter 8, 159–174 (2006)zbMATHCrossRefGoogle Scholar
  24. 24.
    Davie C.T., Bićanic N.: Failure criteria for quasi-brittle materials in lattice-type models. Commun. Numer. Methods Eng. 19(9), 703–713 (2003)zbMATHCrossRefGoogle Scholar
  25. 25.
    Delaplace A., Desmorat R.: Discrete 3D model as complimentary numerical testing for anisotropic damage. Int. J. Fract. 148(2), 115–128 (2007)CrossRefGoogle Scholar
  26. 26.
    Diederichs, M.S.: Instability of Hard Rockmasses: The Role of Tensile Damage and Relaxation. PhD thesis, University of Waterloo (2000)Google Scholar
  27. 27.
    Diederichs M.S.: Rock fracture and collapse under low confinement conditions. Rock Mech. Rock Eng. 36(5), 339–381 (2003)CrossRefGoogle Scholar
  28. 28.
    DIN 18136: Baugrund, Untersuchung von Bodenproben— Einaxialer Druckversuch (in German). In: Deutsche Norm. Beuth Verlag, Berlin (2003)Google Scholar
  29. 29.
    DIN 18137-2: Baugrund, Versuche und Versuchsgeräte— Bestimmung der Scherfestigkeit—Teil 2: Triaxialversuch (in German). In: Deutsche Norm. Beuth Verlag, Berlin (1990)Google Scholar
  30. 30.
    Donev A., Cisse I., Sachs D., Variano E.A., Stillinger F.H., Connelly R., Torquato S., Chaikin P.M.: Improving the density of jammed disordered packings using ellipsoids. Science 303, 990–993 (2004)ADSCrossRefGoogle Scholar
  31. 31.
    Ergenzinger C., Seifried R., Eberhard P.: Modelling of crushable ballast using an extended discrete element method. In: Oñate, E., Owen, D.R.J. (eds) Particle-Based Methods: Fundamentals and Applications, pp. 134–137. CIMNE, Barcelona (2009)Google Scholar
  32. 32.
    Fairhurst, C., Cook, N.G.W.: The phenomenon of rock splitting parallel to the direction of maximum compression in the neighbourhood of a surface. In: Proceedings of the First Congress of the International Society of Rock Mechanics, vol. 1, pp. 687–692, Lisboa (1966)Google Scholar
  33. 33.
    Feng Y.T., Han K., Owen D.R.J.: Filling domains with disks: an advancing front approach. Int. J. Numer. Methodes Eng. 56, 699–713 (2003)zbMATHCrossRefGoogle Scholar
  34. 34.
    Ferellec J., McDowell G.R.: Modelling realistic shape and particle inertia in DEM. Géotechnique 60(3), 227–232 (2010)CrossRefGoogle Scholar
  35. 35.
  36. 36.
    Fleissner, F.: Parallel Object Oriented Simulation with Lagrangian Particle Methods. Doctoral thesis, Institute of Engineering and Computational Mechanics, University of Stuttgart, Shaker Verlag, Aachen (2010)Google Scholar
  37. 37.
    Fleissner, F., Eberhard, P.: Product Engineering: Tools and Methods Based on Virtual Reality, Chapter Examples for Modeling, Simulation and Visualization with the Discrete Element Method in Mechanical Engineering, pp. 419–426. Springer, Dordrecht (2008)Google Scholar
  38. 38.
    Fleissner F., Gaugele T., Eberhard P.: Applications of the discrete element method in mechanical engineering. Multibody Syst. Dyn. 18(1), 81–94 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    García-Rojo R., Herrmann H.J.: Shakedown of unbound granular material. Gran. Matter 7(2–3), 109–118 (2005)zbMATHCrossRefGoogle Scholar
  40. 40.
    Gaugele T., Fleissner F., Eberhard P.: Simulation of material tests using meshfree Lagrangian particle methods. Proc. IMechE, Part K: J. Multi-body Dyn. 222(4), 327–338 (2008)Google Scholar
  41. 41.
    Georgalli G.A., Reuter M.A.: A particle packing algorithm for packed beds with size distribution. Gran. Matter 10, 257–262 (2008)zbMATHCrossRefGoogle Scholar
  42. 42.
    Gonthier Y., McPhee J., Lange C., Piedboeuf J.-C.: A regularized contact model with asymmetric damping and dwell-time dependent friction. Multibody Syst. Dyn. 11, 209–233 (2004)zbMATHCrossRefGoogle Scholar
  43. 43.
    Griffith A.A.: The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. A221, 163–198 (1921)ADSGoogle Scholar
  44. 44.
    Griffith, A.A.: The theory of rupture. In: Biezeno, C.B., Burgers, J. M., (eds.) Proceedings of the International Congress for Applied Mechanics, pp. 55–63, Delft (1924). Technische Boekhandel en Drukkerij J. Waltman JrGoogle Scholar
  45. 45.
    Guises R., Xiang J., Latham J.-P., Munjiza A.: Granular packing: numerical simulation and the characterisation of the effect of particle shape. Gran. Matter 11, 281–292 (2009)CrossRefGoogle Scholar
  46. 46.
    Hairer E., Wanner G.: Solving Ordinary Differential Equations II—Stiff and Differential-Algebraic Problems. Springer, Berlin (1991)zbMATHGoogle Scholar
  47. 47.
    Hairer E., Wanner G.: Stiff differential equations solved by Radau methods. J. Comput. Appl. Math. 111, 93–111 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Hentz, S., Daudeville, L., Donzé, F.: Modeling of reinforced concrete structures subjected to impacts by the discrete element method. In: Proceedings 16th ASCE Engineering Mechanics Conference (2003)Google Scholar
  49. 49.
    Hentz S., Daudeville L., Donzé F.: Identification and validation of a discrete element model for concrete. J. Eng. Mech. 130(6), 709–719 (2004)CrossRefGoogle Scholar
  50. 50.
    Herbst O., Luding S.: Modeling particulate self-healing materials and application to uni-axial compression. Int. J. Fract. 154(1–2), 87–103 (2008)zbMATHCrossRefGoogle Scholar
  51. 51.
    Hertz H.: Über die Berührung fester elastischer Körper (in German). J. für die reine und angewandte mathematik 92, 156–171 (1882)CrossRefGoogle Scholar
  52. 52.
    Holtzendorff, K.: Untersuchung des Setzungsverhaltens von Bahnschotter und der Hohllagenentwicklung auf Schotterfahrbahnen (in German). Doctoral thesis, Technische Universität Berlin (2003)Google Scholar
  53. 53.
    Hudson, J.A., Brown, E.T., Fairhurst, C.: Shape of the complete stress–strain curve for rock. In: Cording, E.J. (ed.) Stability of Rock Slopes: Proceedings 13th Symposium on Rock Mechanics, pp. 773–795 (1972)Google Scholar
  54. 54.
    Indraratna B., Vinod J.S., Lackenby J.: Influence of particle breakage on the resilient modulus of railway ballast. Géotechnique 59(7), 643–646 (2009)CrossRefGoogle Scholar
  55. 55.
    Jerier J.-F., Imbault D., Donze F.-V., Doremus P.: A geometric algorithm based on tetrahedral meshes to generate a dense polydisperse sphere packing. Gran. Matter 11, 43–52 (2009)CrossRefGoogle Scholar
  56. 56.
    Kruggel-Emden H., Simsek E., Rickelt S., Wirtz S., Scherer V.: Review and extension of normal force models for the discrete element method. Powder Technol. 171, 157–173 (2007)CrossRefGoogle Scholar
  57. 57.
    Kuhl E., D’Addetta G.A., Herrmann H.J., Ramm E.: A comparison of discrete granular material models with continuous microplane formulations. Gran. Matter 2, 113–121 (2000)CrossRefGoogle Scholar
  58. 58.
    Kun F., Herrmann H.J.: A study of fragmentation processes using a discrete element method. Comput. Methods Appl. Mech. Eng. 138, 3–18 (1996)ADSzbMATHCrossRefGoogle Scholar
  59. 59.
    Lackenby J., Indraratna B., McDowell G.R., Christie D.: Effect of confining pressure on ballast degradation and deformation under cyclic triaxial loading. Géotechnique 57, 527–536 (2007)CrossRefGoogle Scholar
  60. 60.
    Lade P.V.: Comprehensive Rock Engineering—Volume 1 Fundamentals, Chapter Rock Strength Criteria: The Theories and the Evidence, pp. 255–284. Pergamon Press, Oxford (1993)Google Scholar
  61. 61.
    Lankarani H.M., Nikravesh P.E.: A contact force model with hysteresis damping for impact analysis of multibody systems. J. Mech. Des. 112, 369–376 (1990)CrossRefGoogle Scholar
  62. 62.
    Lim W.L., McDowell G.R.: Discrete element modelling of railway ballast. Gran. Matter 7, 19–29 (2005)zbMATHCrossRefGoogle Scholar
  63. 63.
    Lim W.L., McDowell G.R.: The importance of coordination number in using agglomerates to simulate crushable particles in the discrete element method. Géotechnique 57(8), 701–705 (2007)CrossRefGoogle Scholar
  64. 64.
    Lim W.L., McDowell G.R., Collop A.C.: The application of weibull statistics to the strength of railway ballast. Gran. Matter 6, 229–237 (2004)Google Scholar
  65. 65.
    Lobo-Guerrero S., Vallejo L.E.: Crushing a weak granular material: experimental numerical analyses. Géotechnique 55(3), 245–249 (2005)CrossRefGoogle Scholar
  66. 66.
    Lobo-Guerrero S., Vallejo L.E.: DEM analysis of crushing around driven piles in granular materials. Géotechnique 55, 617–623 (2005)CrossRefGoogle Scholar
  67. 67.
    Lobo-Guerrero S., Vallejo L.E.: Discrete element method analysis of railtrack ballast degradation during cyclic loading. Gran. Matter 8, 195–204 (2006)CrossRefGoogle Scholar
  68. 68.
    Lockner D.A., Byerlee J.D., Kusenko V., Ponomarev A., Sidorin A.: Quasi-static fault growth and shear fracture energy in granite. Nature 350, 39–42 (1991)ADSCrossRefGoogle Scholar
  69. 69.
    Lu M., Mcdowell G.R.: Discrete element modelling of ballast abrasion. Géotechnique 56, 651–655 (2006)CrossRefGoogle Scholar
  70. 70.
    Lu M., McDowell G.R.: The importance of modelling ballast particle shape in the discrete element method. Gran. Matter 9, 69–80 (2007)CrossRefGoogle Scholar
  71. 71.
    Lu M., McDowell G.R.: Discrete element modelling of railway ballast under monotonic and cyclic triaxial loading. Géotechnique 60(6), 459–467 (2010)CrossRefGoogle Scholar
  72. 72.
    Lubachevsky B.D., Stillinger F.H.: Geometric properties of random disk packings. J. Stat. Phys. 60(5–6), 561–583 (1990)MathSciNetADSzbMATHCrossRefGoogle Scholar
  73. 73.
    Luding S.: Anisotropy in cohesive, frictional granular media. J. Phys. condens. Matter 17, 2623–2640 (2005)ADSCrossRefGoogle Scholar
  74. 74.
    Luding S.: Cohesive frictional powders: Contact models for tension. Gran. Matter 10, 235–246 (2008)zbMATHCrossRefGoogle Scholar
  75. 75.
    Luding S., Suiker A.S.J.: Self-healing of damaged particulate materials through sintering. Philos. Mag. 88(28&29), 3445–3457 (2008)ADSCrossRefGoogle Scholar
  76. 76.
    Martin C.D.: The Strength of Massive Lac du Bonnet Granite Around Underground Openings. PhD thesis, University of Manitoba, Winnipeg, Canada (1993)Google Scholar
  77. 77.
    Martin C.D., Chandler N.A.: The progressive fracture of Lac du Bonnet granite. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 31(6), 643–659 (1994)CrossRefGoogle Scholar
  78. 78.
    McDowell G.R., Harireche O.: Discrete element modelling of soil particle fracture. Géotechnique 52(2), 131–135 (2002)CrossRefGoogle Scholar
  79. 79.
    McDowell G.R., Harireche O.: Discrete element modelling of yielding and normal compression of sand. Géotechnique 52(4), 299–304 (2002)CrossRefGoogle Scholar
  80. 80.
    McDowell G.R., Lim W.L., Collop A.C., Armitage R., Thom N.H.: Laboratory simulation of train loading and tamping on ballast. Proc. Inst. Civ. Eng.Trans. 158(TR2), 89–95 (2005)Google Scholar
  81. 81.
    Paterson M.S., Wong T.-F.: Experimental Rock Deformation—the Brittle Field, 2nd edn. Springer, Berlin (2005)Google Scholar
  82. 82.
    Pells P.J.N.: Comprehensive Rock Engineering—Volume 3 Rock Testing and Site Charcterization, chapter Uniaxial Strength Testing, pp. 67–85. Pergamon Press, Oxford (1993)Google Scholar
  83. 83.
    Petzold, L.R.: A description of DASSL: a differential/algebraic system solver. Technical Report SAND82-8637, Sandia National Laboratories, Livermore, California (1982)Google Scholar
  84. 84.
    Plassiard J.-P., Belheine N., Donzé F.: A spherical discrete element model: calibration procedure and incremental response. Gran. Matter 11(5), 293–306 (2009)CrossRefGoogle Scholar
  85. 85.
    Potyondy D.O., Cundall P.A.: A bonded–particle model for rock. Int. J. Rock Mech. Min. Sci. 41, 1329–1364 (2004)CrossRefGoogle Scholar
  86. 86.
    Saussine G., Cholet C., Gautier P.E., Dubois F., Bohatier C., Moreau J.J.: Modelling ballast behaviour under dynamic loading. Part 1: A 2D polygonal discrete element method approach. Comput. Methods Appl. Mech. Eng. 195(19–22), 2841–2859 (2006)ADSzbMATHCrossRefGoogle Scholar
  87. 87.
    Schäfer J., Dippel S., Wolf D.E.: Force schemes in simulations of granular materials. J. de Physique I 6(1), 5–20 (1996)ADSCrossRefGoogle Scholar
  88. 88.
    Schiehlen, W., Eberhard, P.: Technische Dynamik—Modelle für Regelung und Simulation (in German). Teubner, Wiesbaden (2004)Google Scholar
  89. 89.
    Schinner A.: Fast algorithms for the simulation of polygonal particles. Gran. Matter 2, 35–43 (1999)CrossRefGoogle Scholar
  90. 90.
    Scott G.D., Kilgour D.M.: The density of random close packing of spheres. J. Phys. D 2(6), 863–866 (1969)ADSCrossRefGoogle Scholar
  91. 91.
    Seifried R., Schiehlen W., Eberhard P.: The role of the coefficient of restitution on impact problems in multibody dynamics. Proc. IMechE Part K: J. Multi-body Dyn. 224(3), 279–306 (2010)Google Scholar
  92. 92.
    Shimizu H., Koyama T., Ishida T., Chijimatsu M., Fujita T., Nakama S.: Distinct element analysis for class II behavior of rocks under uniaxial compression. Int. J. Rock Mech. Min. Sci. 4(2), 323–333 (2010)Google Scholar
  93. 93.
    Silbert L.E., Ertas D., Grest G.S., Halsey T.C., Levine D.: Geometry of frictionless and frictional sphere packings. Phys. Rev. E. 65(031304), 1–6 (2002)MathSciNetGoogle Scholar
  94. 94.
    Suiker A.S.J., de Borst R.: A numerical model for the cyclic deterioration of railway tracks. Int. J. Numer. Methods Eng. 57, 441–470 (2003)zbMATHCrossRefGoogle Scholar
  95. 95.
    Suiker A.S.J., Selig E., Frenkel R.: Static and cyclic triaxial testing of ballast and subballast. J. Geotech. Geoenviron. Eng. 131(6), 771–782 (2005)CrossRefGoogle Scholar
  96. 96.
    Tejchman J., Wu W.: Boundary effects on behaviour of granular material during plane strain compression. Eur. J. Mech. A/Solids 29(1), 18–27 (2010)CrossRefADSGoogle Scholar
  97. 97.
    Teschner, M., Heidelberger, B., Mueller, M., Pomeranets, D., Gross, M.: Optimized spatial hashing for collision detection of deformable objects. In: Proceedings of Vision, Modeling, Visualization VMV’03, pp. 47–54 (2003)Google Scholar
  98. 98.
    Thompson B.D., Young P., Lockner D.A.: Fracture in westerly granite under AE feedback and constant strain rate loading: nucleation, quasi-static propagation, and the transition to unstable fracture propagation. Pure Appl. Geophys. 163, 995–1019 (2006)ADSCrossRefGoogle Scholar
  99. 99.
    Thornton C.: Mechanics of granular materials: an introduction, chapter Three dimensional behaviour of granular materials, pp. 187–196. Balkema, Rotterdam (1999)Google Scholar
  100. 100.
    Torquato S., Truskett T.M., Debenedetti P.G.: Is random close packing of spheres well defined?. Phys. Rev. Lett. 84(10), 2064–2067 (2000)ADSCrossRefGoogle Scholar
  101. 101.
    von Wolffersdorff P.-A.: A hypoplastic relation for granular materials with a predifined limit state surface. Mech. Cohes. Frict. Mater. 1, 251–271 (1996)CrossRefGoogle Scholar
  102. 102.
    Walton K.: The effective elastic moduli of a random packing of spheres. J. Mech. Phys. Solids 35, 213–226 (1987)ADSzbMATHCrossRefGoogle Scholar
  103. 103.
    Wang Y., Alonso-Marroquin F.: A finite deformation method for discrete modeling: particle rotation and parameter calibration. Gran. Matter 11(5), 331–343 (2009)CrossRefGoogle Scholar
  104. 104.
    Wang Y., Mora P.: Macroscopic elastic properties of regular lattices. J. Mech. Phys. Solids 56(12), 3459–3474 (2008)MathSciNetADSzbMATHCrossRefGoogle Scholar
  105. 105.
    Wang Y., Tonon F.: Modeling Lac du Bonnet granite using a discrete element model. Int. J. Rock Mech. Min. Sci. 46(7), 1124–1135 (2009)CrossRefGoogle Scholar
  106. 106.
    Wawersik W.R., Brace W.F.: Post-failure behavior of a granite and diabase. Rock Mech. Rock Eng. 3, 61–85 (1971)Google Scholar
  107. 107.
    Wawersik W.R., Fairhurst C.: A study of brittle rock fracture in laboratory compression experiments. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 7, 561–575 (1970)CrossRefGoogle Scholar
  108. 108.
    Zaccone A., Lattuada M., Wu H., Morbidelli M.: Theoretical elastic moduli for disordered packings of interconnected spheres. J. Chem. Phys. 127(174512), 1–9 (2007)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Christian Ergenzinger
    • 1
  • Robert Seifried
    • 1
  • Peter Eberhard
    • 1
    Email author
  1. 1.Institute of Engineering and Computational MechanicsUniversity of StuttgartStuttgartGermany

Personalised recommendations