Modelling Grain Fragmentation in Hypoplasticity

  • Erich BauerEmail author
Conference paper
Part of the Springer Series in Geomechanics and Geoengineering book series (SSGG)


The focus of the present paper is on constitutive modelling of the influence of grain fragmentation on the mechanical behaviour of cohesionless granular materials like sand, gravel or broken rock. To this end the so-called solid hardness of a grain assembly is defined within a continuum description and is a key parameter for modelling the effect of both grain fragmentation and grain rearrangement. While in the original version by Bauer the solid hardness is a constant parameter, an extended concept has recently been proposed where the solid hardness is considered as a state dependent quantity. The general format of the evolution equation for the solid hardness allows the modelling of the influence of various factors on grain fragmentation. Such factors are, for instance, an increase in the mean stress, the deviatoric stress and the rotation resistance of particles and also a time dependent process caused by progressive weathering. The embedding of the solid hardness into hypoplasticity follows the consistency condition originally proposed by Gudehus. In this paper the consistency condition is also applied to proposed constitutive equations for time independent as well as rheological material properties. The performance of these different models is verified with experiments.



The author wishes to thank Professor W. Huang and Dr. Z. Fu for the fruitful discussion of extended numerical tools used for the simulations. The assistance of Dr. L. Li and Mr. S. Safikhani in preparing the drawings presented in the paper is gratefully acknowledged.


  1. 1.
    ABAQUS Software. Dassault Systèmes (2017)Google Scholar
  2. 2.
    Alikarami, R., Andò, E., Gkiousas-Kapnisis, M., Torabi, A., Viggiani, G.: Strain localisation and grain breakage in sand under shearing at high mean stress: insights from in situ X-ray tomography. Acta Geotechnica 10, 15–30 (2015)CrossRefGoogle Scholar
  3. 3.
    Alonso, E.E., Cardoso, R.: Behavior of materials for earth and rockfill dams: perspective from unsaturated soil mechanics. Front. Archit. Civ. Eng. China 4(1), 1–39 (2010). Scholar
  4. 4.
    Ando, E., Hall, S.A., Viggiani, G., Desrues, J., Besuelle, P.: Grain-scale experimental investigation of localised deformation in sand: a discrete particle tracking approach. Acta Geotechnica 7, 1–13 (2012)CrossRefGoogle Scholar
  5. 5.
    Bauer, E.: Constitutive modelling of critical states in hypoplasticity. In: Proceedings of the Fifth International Symposium on Numerical Models in Geomechanics, pp. 15–20. Balkema, Davos (1995)Google Scholar
  6. 6.
    Bauer, E.: Calibration of a comprehensive hypoplastic model for granular materials. Soils Found. 36(1), 13–26 (1996)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bauer, E., Herle, I.: Stationary states in hypoplasticity. In: Kolymbas, D. (ed.) Constitutive Modelling of Granular Materials, pp. 167–192. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  8. 8.
    Bauer, E.: Conditions for embedding Casagrande’s critical states into hypoplasticity. Mech. Cohesive-Frictional Mater. 5, 125–148 (2000)CrossRefGoogle Scholar
  9. 9.
    Bauer, E., Huang, W., Wu, W.: Investigation of shear banding in an anisotropic hypoplastic material. Solids Struct. 41, 5903–5919 (2004)CrossRefGoogle Scholar
  10. 10.
    Bauer, E.: Hypoplastic modelling of moisture-sensitive weathered rockfill materials. Acta Geotechnica 4, 261–72 (2009)CrossRefGoogle Scholar
  11. 11.
    Bauer, E., Fu, Z., Liu, S.: Influence of pressure and density on the rheological properties of rockfills. Front. Struct. Civil Eng. 6, 25–34 (2012). Scholar
  12. 12.
    Bauer, E., Li, L., Huang, W.: Hypoplastic constitutive modelling of grain damage under plane shearing. In: Bifurcation and Degradation of Geomaterials in the New Millennium, pp. 181–187 (2015)Google Scholar
  13. 13.
    Bauer, E.: Simulation of the influence of grain damage on the evolution of shear strain localization. In: Albers, B., Kuczma, M. (eds.) Continuous Media with Microstructure, vol. 2, pp. 231–244. Springer, Cham (2016). ISBN 978-3-319-28239-8CrossRefGoogle Scholar
  14. 14.
    Bauer, E., Li, L., Khosravi, M.: Modelling grain damage under plane strain compression using a micro-polar continuum. In: Papamichos, E., Papanastasiou, P., Pasternak, E., Dyskin, A. (eds.) Proceedings of the 11th International Workshop on Bifurcation and Degradation in Geomaterials Dedicated to Hans Muhlhaus, Limassol, Cyprus, May 21–25 2017. Geomechanics and Geoengineering, pp. 539-546. Springer, Heidelberg (2017)., ISBN 978-3-319-56396-1Google Scholar
  15. 15.
    Bolton, M.D., Nakata, Y., Cheng, Y.P.: Micro- and macro-mechanical behavior of DEM crushable materials. Géotechnique 58(6), 471–480 (2008)CrossRefGoogle Scholar
  16. 16.
    Coop, M.R., Sorensen, K.K., Freitas, T.B., Georgoutsos, G.: Particle breakage during shearing of a carbonate sand. Géotechnique 58(6), 471–480 (2004)Google Scholar
  17. 17.
    Daouadji, A., Hicher, P.Y., Rahma, A.: An elastoplastic model for granular materials taking into account grain breakage. Eur. J. Mech. A/Solids 20, 113–137 (2001)CrossRefGoogle Scholar
  18. 18.
    Ebrahimian, B., Bauer, E.: Numerical simulation of the effect of interface friction of a bounding structure on shear deformation in a granular soil. Int. J. Numer. Anal. Methods Geomech. 36, 1486–1506 (2012)CrossRefGoogle Scholar
  19. 19.
    Einav, I.: Breakage mechanics-part I: theory. J. Mech. Phys. Solids 55(6), 1274–1297 (2007)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Einav, I.: Breakage mechanics-Part II: modelling granular materials. J. Mech. Phys. Solids 55(6), 1298–1320 (2007)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Fu, Z.Z., Bauer, E.: Hypoplastic constitutive modelling of the long term behaviour and wetting deformation of weathered granular materials. In: Bauer, E., Semprich, S. Zenz, G. (eds.) Proceedings of the 2nd International Conference on Long Term Behaviour of Dams, Graz, Austria, pp. 473–478 (2009). ISBN 978-3-85125-070-1Google Scholar
  22. 22.
    Fu, R., Hua, X., Zhou, B.: Discrete element modeling of crushable sands considering realistic particle shape effect. Comput. Geotech. 91, 179–191 (2017)CrossRefGoogle Scholar
  23. 23.
    Garga, V.K., Infante Sedano, J.A.: Steady state strength of sands in a constant volume ring shear apparatus. Geotech. Test. J. 25, 414–421 (2002)Google Scholar
  24. 24.
    Gudehus, G.: A comprehensive constitutive equation for granular materials. Soils Found. 36(1), 1–12 (1996)CrossRefGoogle Scholar
  25. 25.
    Gudehus, G.: A visco-hypoplastic constitutive relation. Soils Found 44(4), 11–25 (2004)CrossRefGoogle Scholar
  26. 26.
    Gudehus, G., Nübel, K.: Evolution of shear bands in sand. Géotechnique 54, 187–201 (2004)CrossRefGoogle Scholar
  27. 27.
    Gudehus, G.: Physical Soil Mechanics. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  28. 28.
    Hardin, B.O.: Crushing of soil particles. J. Geotech. Eng. 111(10), 1177–1192 (1985)CrossRefGoogle Scholar
  29. 29.
    Herle, I., Gudehus, G.: Determination of parameters of a hypoplastic constitutive model from grain properties. Mech. Cohesive-Frictional Mater. 4, 461–486 (1999)CrossRefGoogle Scholar
  30. 30.
    Huang, W.: Hypoplastic modelling of shear localisation in granular materials. Ph.D. thesis, Graz University of Technology, Austria (2000)Google Scholar
  31. 31.
    Huang, W., Nübel, K., Bauer, E.: A polar extension of hypoplastic model for granular material with shear localization. Mech. Mater. 34, 563–576 (2002)CrossRefGoogle Scholar
  32. 32.
    Huang, W., Bauer, E.: Numerical investigations of shear localization in a micro-polar hypoplastic material. Int. J. Numer. Anal. Methods Geomech. 27, 325–352 (2003)CrossRefGoogle Scholar
  33. 33.
    Jaky, J.: Pressure in silos. In: 2nd ICSMFE, London, vol. 1, pp. 103–107 (1948)Google Scholar
  34. 34.
    Khosravi, M., Linke Li, L., Bauer E.: Numerical simulation of post construction deformation of a concrete face rockfill dam. In: Noorzad, A., Bauer, E., Ghaemian, M., Ebrahimian, B. (eds.) Proceedings of the 4th International Conference on Long-Term Behaviour and Environmentally Friendly Rehabilitation Technologies of Dams, LTBD 2017, pp. 307–314. Verlag der Technischen Universität Graz (2017). ISBN 978-3-85125-564-5Google Scholar
  35. 35.
    Kolymbas, D., Bauer, E.: Soft oedometer – a new testing device and its application for the calibration of hypoplastic constitutive laws. Geotech. Test. J. GTJODJ 16(2), 263–270 (1993)CrossRefGoogle Scholar
  36. 36.
    Laufer, I.: Grain crushing and high-pressure oedometer tests simulated with the discrete element method. Granul. Matter 17, 389 (2015). Scholar
  37. 37.
    Li, G., X.: Triaxial wetting experiments on rockfill materials used in Xiaolangdi earth dam. Research report from Tsinghua University (1988)Google Scholar
  38. 38.
    Li, L., Wang, Z., Liu, S., Erich Bauer, E.: Calibration and performance of two different constitutive models for rockfill materials. Water Sci. Eng. 9(3), 227–239 (2016). Scholar
  39. 39.
    Luzzani, L., Coop, M.R.: On the relationship between particle breakage and the critical state of sand. Soils Found. 42(2), 71–82 (2002)CrossRefGoogle Scholar
  40. 40.
    Matsuoka, H., Nakai, T.: Stress-strain relationship of soil based on the ‘SMP’. In: Proceedings of Speciality Session 9, IX International Conference on Soil Mechanics and Foundation Engineering, Tokyo, pp. 153–162 (1977)Google Scholar
  41. 41.
    McDowell, G.R., Bolton, M.D.: On the micromechanics of crushable aggregates. Geóeteehnique 4S(5), 667–679 (1998)Google Scholar
  42. 42.
    Nakata, Y., Hyodo, M., Hyde, A.F., Kato, Y., Murata, H.: Microscopic particle crushing of sand subjected to high pressure one-dimensional compression. Soils Found. 41(1), 69–82 (2001)CrossRefGoogle Scholar
  43. 43.
    Nguyen, G.D., Einav, I.: Numerical regularization of a model based on breakage mechanics for granular materials. Int. J. Solids Struct. 47(10), 1350–1360 (2010)CrossRefGoogle Scholar
  44. 44.
    Niemunis, A., Herle, I.: Hypoplastic model for cohesionless soils with elastic strain range. Mech. Cohesive-Frictional Mater. 2(4), 279–299 (1997)CrossRefGoogle Scholar
  45. 45.
    Ohde, J.: Zur Theorie der Druckverteilung im Baugrund. Bauingenieur 20, 451–459 (1939)Google Scholar
  46. 46.
    Oldecop, L.A., Alonso, E.E.: Theoretical investigation of the time-dependent behaviour of rockfill. Géotechnique 57(3), 289–301 (2007)CrossRefGoogle Scholar
  47. 47.
    Ovalle, C., Frossard, E., Dano, C., Hu, W., Maiolino, S., Hicher, P.-Y.: The effect of size on the strength of coarse rock aggregates and large rockfill samples through experimental data. Acta Mechanica 225(8), 2199–2216 (2014)CrossRefGoogle Scholar
  48. 48.
    Ovalle, C., Dano, C., Hicher, P.Y., Cisternas, M.: An experimental framework for evaluating the mechanical behavior of dry and wet crushable granular materials based on the particle breakage ratio. Can. Geotech. J. 52, 1–12 (2015)CrossRefGoogle Scholar
  49. 49.
    Sadrekarimi, A., Olson, S.M.: Particle damage observed in ring shear tests on sands. Can. Geotech. J. 47(5), 497–515 (2010)CrossRefGoogle Scholar
  50. 50.
    Sadrekarimi, A., Olson, S.M.: Critical state friction angle of sands. Géotechnique 61(9), 771–783 (2011)CrossRefGoogle Scholar
  51. 51.
    Salim, W., Indraratna, B.: A new elastoplastic constitutive model for coarse granular aggregates incorporating particle breakage. Can. Geotech. J. 41, 657–671 (2004)CrossRefGoogle Scholar
  52. 52.
    Tejchman, J., Bauer, E.: Numerical simulation of shear band formation with a polar hypoplastic constitutive model. Comput. Geotech. 19, 221–44 (1996)CrossRefGoogle Scholar
  53. 53.
    von Wolffersdorff, P.A.: A hypoplastic relation for granular materials with a predefined limit state surface. Mech. Cohesive-Frictional Mater. 1, 251–271 (1996)CrossRefGoogle Scholar
  54. 54.
    Wu, W., Bauer, E.: A hypoplastic model for barotropy and pyknotropy of granular soils. In: Kolymbas, D. (ed.) Proceedings of the International Workshop on Modern Approaches to Plasticity, pp. 225–245. Elsevier Press, Amsterdam (1992–1993)CrossRefGoogle Scholar
  55. 55.
    Wu, W., Bauer, E., Kolymbas, D.: Hypoplastic constitutive model with critical state for granular materials. Mech. Mat. 23, 45–69 (1996)CrossRefGoogle Scholar
  56. 56.
    Yamamuro, J.A., Bopp, P.A., Lade, P.V.: One-dimensional compression of sands at high pressures. J. Geotech. Eng. ASCE 122(2), 147–154 (1996)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Applied MechanicsGraz University of TechnologyGrazAustria

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