From micro scale to boundary value problem: using a micromechanically based model

  • Hao Xiong
  • François Nicot
  • Zhenyu YinEmail author
Research Paper


A 3D multi-scale approach is presented to investigate the mechanical behavior of a macroscopic specimen consisting of a granular assembly, as a boundary value problem. The core of this approach is a multi-scale coupling, wherein the finite element method is used to solve a boundary value problem and a micromechanically based model is employed as constitutive relationship used at a representative volume element scale. This approach provides a convenient way to link the macroscopic observations with intrinsic microscopic mechanisms. The plane strain triaxial loading condition is selected to simulate the occurrence of strain localization. A series of tests are performed, wherein distinct failure patterns are observed and analyzed. A system of shear band naturally appears in a homogeneous setting specimen. By defining the shear band area, microstructural mechanisms are separately investigated inside and outside the shear band. The normalized second-order work introduced as an indicator of instability occurrence is analyzed not only on the macroscale but also on the micro scale.


FEM Granular materials Mesoscopic scale Micromechanics Multi-scale approach Second-order work Shear band Strain localization 



The authors would like to express their sincere thanks to the scholarship from China Scholarship Council (CSC) under the Grant CSC Number 201406250016, the National Natural Science Foundation of China (Grant No. 51579179), the Region Pays de la Loire of France (Project RI-ADAPTCLIM) and the French Research Network GeoMech (Multi-physics and Multi-scale Couplings in Geo-environmental Mechanics, GDRI CNRS, 2016–2019).


  1. 1.
    Alshibli KA, Batiste SN, Sture S (2003) Strain localization in sand: plane strain versus triaxial compression. J Geotech Geoenviron Eng 129:483–494CrossRefGoogle Scholar
  2. 2.
    Andrade JE, Avila CF, Hall SA, Lenoir N, Viggiani G (2011) Multiscale modeling and characterization of granular matter: from grain kinematics to continuum mechanics. J Mech Phys Solids 59:237–250CrossRefGoogle Scholar
  3. 3.
    Andrade JE, Tu XX (2009) Multiscale framework for behavior prediction in granular media. Mech Mater 41:652–669CrossRefGoogle Scholar
  4. 4.
    Cambou B, Dubujet P, Emeriault F, Sidoroff F (1995) Homogenization for granular materials. Eur J Mech A Solids 14:255–276MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chang C, Yin ZY, Hicher PY (2010) Micromechanical analysis for interparticle and assembly instability of sand. J Eng Mech 137:155–168CrossRefGoogle Scholar
  6. 6.
    Chang CS, Yin ZY (2009) Micromechanical modeling for inherent anisotropy in granular materials. J Eng Mech 136:830–839CrossRefGoogle Scholar
  7. 7.
    Christoffersen J, Mehrabadi MM, Nemat-Nasser S (1981) A micromechanical description of granular material behavior. J Appl Mech 48:339–344CrossRefGoogle Scholar
  8. 8.
    Cundall PA, Strack ODL (1979) A discrete numerical model for granular assemblies. Géotechnique 29:47–65CrossRefGoogle Scholar
  9. 9.
    Daouadji A, Darve F, Al Gali H, Hicher P, Laouafa F, Lignon S, Nicot F, Nova R, Pinheiro M, Prunier F (2011) Diffuse failure in geomaterials: experiments, theory and modelling. Int J Numer Anal Methods Geomech 35:1731–1773CrossRefGoogle Scholar
  10. 10.
    Darve F, Servant G, Laouafa F, Khoa H (2004) Failure in geomaterials: continuous and discrete analyses. Comput Methods Appl Mech Eng 193:3057–3085CrossRefGoogle Scholar
  11. 11.
    De Saxcé G, Fortin J, Millet O (2004) About the numerical simulation of the dynamics of granular media and the definition of the mean stress tensor. Mech Mater 36:1175–1184CrossRefGoogle Scholar
  12. 12.
    Desrues J, Chambon R, Mokni M, Mazerolle F (1996) Void ratio evolution inside shear bands in triaxial sand specimens studied by computed tomography. Géotechnique 46:529–546CrossRefGoogle Scholar
  13. 13.
    Desrues J, Viggiani G (2004) Strain localization in sand: an overview of the experimental results obtained in Grenoble using stereophotogrammetry. Int J Numer Anal Methods Geomech 28:279–321CrossRefGoogle Scholar
  14. 14.
    Guo N, Zhao JD (2013) A hierarchical model for cross-scale simulation of granular media. In: AIP conference proceedings, pp 1222–1225Google Scholar
  15. 15.
    Guo N, Zhao JD (2014) A coupled fem/dem approach for hierarchical multiscale modelling of granular media. Int J Numer Methods Eng 99:789–818MathSciNetCrossRefGoogle Scholar
  16. 16.
    Guo N, Zhao JD (2016a) 3D multiscale modeling of strain localization in granular media. Comput Geotech 80:360–372CrossRefGoogle Scholar
  17. 17.
    Guo N, Zhao JD (2016b) Parallel hierarchical multiscale modelling of hydro-mechanical problems for saturated granular soils. Comput Methods Appl Mech Eng 305:37–61MathSciNetCrossRefGoogle Scholar
  18. 18.
    Han C, Drescher A (1993) Shear bands in biaxial tests on dry coarse sand. Soils Found 33:118–132CrossRefGoogle Scholar
  19. 19.
    Hibbitt Karlsson Sorensen K (2001) ABAQUS/explicit: user’s manual, vol 1. Hibbitt Karlsson and Sorenson Incorporated, Plymouth, MIGoogle Scholar
  20. 20.
    Huang W, Huang L, Sheng D, Sloan SW (2015) Dem modelling of shear localization in a plane Couette shear test of granular materials. Acta Geotech 10:389–397CrossRefGoogle Scholar
  21. 21.
    Liu Y, Sun W, Yuan Z, Fish J (2016) A nonlocal multiscale discrete-continuum model for predicting mechanical behavior of granular materials. Int J Numer Methods Eng 106:129–160CrossRefGoogle Scholar
  22. 22.
    Love AEH (2013) A treatise on the mathematical theory of elasticity, vol 1. Cambridge University Press, CambridgezbMATHGoogle Scholar
  23. 23.
    Ma G, Regueiro RA, Zhou W, Liu J (2018) Spatiotemporal analysis of strain localization in dense granular materials. Acta Geotech. CrossRefGoogle Scholar
  24. 24.
    Mehrabadi MM, Nemat-Nasser S, Oda M (1982) On statistical description of stress and fabric in granular materials. Int J Numer Anal Methods Geomech 6:95–108MathSciNetCrossRefGoogle Scholar
  25. 25.
    Meier HA, Steinmann P, Kuhl E (2008) Towards multiscale computation of confined granular media-contact forces, stresses and tangent operators. Tech Mech 28:32–42Google Scholar
  26. 26.
    Meier HA, Steinmann P, Kuhl E (2009) On the multiscale computation of confined granular media. In: ECCOMAS multidisciplinary jubilee symposium. Springer. pp 121–133Google Scholar
  27. 27.
    Nguyen HN, Prunier F, Djeran-Maigre I, Nicot F (2016) Kinetic energy and collapse of granular materials. Granul Matter 18:1–10CrossRefGoogle Scholar
  28. 28.
    Nicot F (2003) Constitutive modelling of a snow cover with a change in scale. Eur J Mech A Solids 22:325–340CrossRefGoogle Scholar
  29. 29.
    Nicot F, Darve F (2011) The H-microdirectional model: accounting for a mesoscopic scale. Mech Mater 43:918–929CrossRefGoogle Scholar
  30. 30.
    Nicot F, Darve F, Dat Vu Khoa H (2007) Bifurcation and second-order work in geomaterials. Int J Numer Anal Methods Geomech 31:1007–1032CrossRefGoogle Scholar
  31. 31.
    Nicot F, Darve F, Group R (2005) A multi-scale approach to granular materials. Mech Mater 37:980–1006Google Scholar
  32. 32.
    Nicot F, Xiong H, Wautier A, Lerbet J, Darve F (2017) Force chain collapse as grain column buckling in granular materials. Granul Matter 19:18CrossRefGoogle Scholar
  33. 33.
    Prunier F, Laouafa F, Lignon S, Darve F (2009) Bifurcation modeling in geomaterials: From the second-order work criterion to spectral analyses. Int J Numer Anal Methods Geomech 33:1169–1202CrossRefGoogle Scholar
  34. 34.
    Radjaï F, Dubois F (2011) Discrete-element modeling of granular materials. Wiley, HobokenGoogle Scholar
  35. 35.
    Rechenmacher AL (2006) Grain-scale processes governing shear band initiation and evolution in sands. J Mech Phys Solids 54:22–45CrossRefGoogle Scholar
  36. 36.
    Rudnicki JW, Rice J (1975) Conditions for the localization of deformation in pressure-sensitive dilatant materials. J Mech Phys Solids 23:371–394CrossRefGoogle Scholar
  37. 37.
    Shi J, Guo P (2018) Fabric evolution of granular materials along imposed stress paths. Acta Geotech. CrossRefGoogle Scholar
  38. 38.
    Vardoulakis I, Goldscheider M, Gudehus G (1978) Formation of shear bands in sand bodies as a bifurcation problem. Int J Numer Anal Methods Geomech 2:99–128CrossRefGoogle Scholar
  39. 39.
    Voigt W (1889) Ueber die beziehung zwischen den beiden elasticitätsconstanten isotroper körper. Ann Phys 274:573–587CrossRefGoogle Scholar
  40. 40.
    Wan R, Nicot F, Darve F (2017) Failure in geomaterials: a contemporary treatise. Elsevier, New YorkGoogle Scholar
  41. 41.
    Wan R, Pinheiro M, Daouadji A, Jrad M, Darve F (2013) Diffuse instabilities with transition to localization in loose granular materials. Int J Numer Anal Methods Geomech 37:1292–1311CrossRefGoogle Scholar
  42. 42.
    Wang K, Sun W (2016) A semi-implicit discrete-continuum coupling method for porous media based on the effective stress principle at finite strain. Comput Methods Appl Mech Eng 304:546–583MathSciNetCrossRefGoogle Scholar
  43. 43.
    Wang K, Sun W, Salager S, Na S, Khaddour G (2016) Identifying material parameters for a micro-polar plasticity model via X-ray micro-Ct images: lessons learned from the curve-fitting exercises. Int J Multiscale Comput Eng 14:389–413CrossRefGoogle Scholar
  44. 44.
    Wang P, Arson C (2018) Energy distribution during the quasi-static confined comminution of granular materials. Acta Geotech 13(5):1075–1083CrossRefGoogle Scholar
  45. 45.
    Xiong H, Nicot F, Yin Z (2017) A three-dimensional micromechanically based model. Int J Numer Anal Methods Geomech 41:1669–1686CrossRefGoogle Scholar
  46. 46.
    Yin ZY, Chang CS, Hicher PY (2010) Micromechanical modelling for effect of inherent anisotropy on cyclic behaviour of sand. Int J Solids Struct 47:1933–1951CrossRefGoogle Scholar
  47. 47.
    Yin ZY, Chang CS, Hicher PY, Karstunen M (2009) Micromechanical analysis of kinematic hardening in natural clay. Int J Plast 25:1413–1435CrossRefGoogle Scholar
  48. 48.
    Yin ZY, Hattab M, Hicher PY (2011) Multiscale modeling of a sensitive marine clay. Int J Numer Anal Methods Geomech 35:1682–1702CrossRefGoogle Scholar
  49. 49.
    Yin ZY, Zhao J, Hicher PY (2014) A micromechanics-based model for sand-silt mixtures. Int J Solids Struct 51:1350–1363CrossRefGoogle Scholar
  50. 50.
    Zhang Y, Shao J, Liu Z, Shi C, De Saxcé G (2018) Effects of confining pressure and loading path on deformation and strength of cohesive granular materials: a three-dimensional dem analysis. Acta Geotech. CrossRefGoogle Scholar
  51. 51.
    Zhou W, Liu J, Ma G, Chang X (2017) Three-dimensional dem investigation of critical state and dilatancy behaviors of granular materials. Acta Geotech 12:527–540CrossRefGoogle Scholar
  52. 52.
    Zhu H, Nguyen HN, Nicot F, Darve F (2016) On a common critical state in localized and diffuse failure modes. J Mech Phys Solids 95:112–131CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringThe Hong Kong Polytechnic UniversityHung Hom, KowloonChina
  2. 2.Université Grenoble Alpes, IRSTEA, Geomechanics Group, ETNAGrenobleFrance

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