Continuum Mechanics and Thermodynamics

, Volume 31, Issue 1, pp 341–359 | Cite as

Multiscale modeling of unsaturated granular materials based on thermodynamic principles

  • Chao-Fa Zhao
  • Younes Salami
  • Pierre-Yves HicherEmail author
  • Zhen-Yu Yin
Original Article


The effect of water on the hydromechanical behavior of unsaturated granular materials has been studied with a micromechanical model based on thermodynamic principles. A general framework based on the theory of thermodynamics with internal variables for constructing thermodynamically consistent multiscale constitutive relations for unsaturated granular materials has been developed. Within this framework, the microscopic total Helmholtz free energy has been separated between a mechanical and a hydraulic part, each of which is a function of either the elastic displacement or the capillary bridge volume and the distance between particles at the microscale. The inter-particle dissipation of energy, assumed to be frictional in origin, is a function of the incremental plastic displacements at the microscale. Both the microscale Helmholtz free energy and the dissipative energy have been volumetrically averaged to obtain the homogenized energy functions at the macroscale. In accordance with the suggested multiscale thermomechanical framework, a micromechanical model has been constructed to describe the behavior of partially saturated granular soils. This model has considered the deformation of soil skeleton by applying a Coulomb-type criterion at the inter-particle contacts. The hydraulic potential is made to be dependent on the size of the particles and is derived through use of the expression for the water retention curve by assuming that liquid bridges are isotropically distributed within the specimen. The performance of the suggested model has been demonstrated through numerical simulations of the behavior of sand under various degrees of saturation and a wide range of mechanical loadings.


Granular material Multiscale modeling Unsaturated soil Micromechanical model Thermodynamic principles 


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Chao-Fa Zhao acknowledges the financial support by the international scientific network GDRI GeoMech (Multi-Physics and Multiscale Couplings in Geo-Environmental Mechanics), and the helpful suggestions from Prof. Anil Misra (The University of Kansas) and Dr. Jian Li (Beijing Jiaotong University). Zhen-Yu Yin acknowledges the financial support by the Natural Science Foundation of China (No. 51579179).


  1. 1.
    Cho, S.E., Lee, S.R.: Instability of unsaturated soil slopes due to infiltration. Comput. Geotech. 28(3), 185–208 (2001)Google Scholar
  2. 2.
    Fredlund, D.G., Rahardjo, H., Fredlund, M.D.: Unsaturated Soil Mechanics in Engineering Practice. Wiley, New York (2012)Google Scholar
  3. 3.
    Ng, C.W.W., Shi, Q.: A numerical investigation of the stability of unsaturated soil slopes subjected to transient seepage. Comput. Geotech. 22(1), 1–28 (1998)Google Scholar
  4. 4.
    Gens, A.: Soil–environment interactions in geotechnical engineering. Géotechnique 60(1), 3 (2010)Google Scholar
  5. 5.
    Sheng, D.: Review of fundamental principles in modelling unsaturated soil behaviour. Comput. Geotech. 38(6), 757–776 (2011)Google Scholar
  6. 6.
    Alonso, E.E., Gens, A., Josa, A.: A constitutive model for partially saturated soils. Géotechnique 40(3), 405–430 (1990)Google Scholar
  7. 7.
    Cui, Y.J., Delage, P.: Yeilding and plastic behaviour of an unsaturated compacted silt. Géotechnique 46(2), 291–311 (1996)Google Scholar
  8. 8.
    Sheng, D., Sloan, S.W., Gens, A.: A constitutive model for unsaturated soils: thermomechanical and computational aspects. Comput. Mech. 33(6), 453–465 (2004)zbMATHGoogle Scholar
  9. 9.
    Sun, D., Sheng, D., Sloan, S.W.: Elastoplastic modelling of hydraulic and stress–strain behaviour of unsaturated soils. Mech. Mater. 39(3), 212–221 (2007)Google Scholar
  10. 10.
    Wheeler, S.J., Sivakumar, V.: An elasto-plastic critical state framework for unsaturated soil. Géotechnique 45(1), 35–53 (1995)Google Scholar
  11. 11.
    Bishop, A.W., Blight, G.: Some aspects of effective stress in saturated and partly saturated soils. Géotechnique 13(3), 177–197 (1963)Google Scholar
  12. 12.
    Buscarnera, G., Einav, I.: The yielding of brittle unsaturated granular soils. Géotechnique 62(2), 147 (2012)Google Scholar
  13. 13.
    Li, J., Yin, Z.-Y., Cui, Y., Hicher, P.-Y.: Work input analysis for soils with double porosity and application to the hydromechanical modeling of unsaturated expansive clays. Can. Geotech. J. 54(2), 173–187 (2016)Google Scholar
  14. 14.
    Zhao, C.G., Liu, Y., Gao, F.P.: Work and energy equations and the principle of generalized effective stress for unsaturated soils. Int. J. Numer. Anal. Methods Geomech. 34(9), 920–936 (2010)zbMATHGoogle Scholar
  15. 15.
    Alonso, E.E., Pereira, J.-M., Vaunat, J., Olivella, S.: A microstructurally based effective stress for unsaturated soils. Géotechnique 60(12), 913–925 (2010)Google Scholar
  16. 16.
    Nuth, M., Laloui, L.: Effective stress concept in unsaturated soils: clarification and validation of a unified framework. Int. J. Numer. Anal. Methods Geomech. 32(7), 771–801 (2008)zbMATHGoogle Scholar
  17. 17.
    Chang, C.S., Hicher, P.Y.: An elasto-plastic model for granular materials with microstructural consideration. Int. J. Solids Struct. 42(14), 4258–4277 (2005)zbMATHGoogle Scholar
  18. 18.
    Hicher, P.-Y., Chang, C.S.: A microstructural elastoplastic model for unsaturated granular materials. Int. J. Solids Struct. 44(7–8), 2304–2323 (2007)zbMATHGoogle Scholar
  19. 19.
    Scholtès, L., Hicher, P.-Y., Nicot, F., Chareyre, B., Darve, F.: On the capillary stress tensor in wet granular materials. Int. J. Numer. Anal. Methods Geomech. 33(10), 1289–1313 (2009)zbMATHGoogle Scholar
  20. 20.
    Zhao, C.-F., Salami, Y., Yin, Z.-Y., Hicher, P.-Y.: A micromechanical model for unsaturated soils based on thermodynamics. In: Poromechanics VI, pp. 594–601 (2017)Google Scholar
  21. 21.
    Chalak, C., Chareyre, B., Nikooee, E., Darve, F.: Partially saturated media: from dem simulation to thermodynamic interpretation. Eur. J. Environ. Civ. Eng. 21(7–8), 798–820 (2017)Google Scholar
  22. 22.
    Duriez, J., Wan, R.: Stress in wet granular media with interfaces via homogenization and discrete element approaches. J. Eng. Mech. 142(12), 04016099 (2016)Google Scholar
  23. 23.
    Wang, K., Sun, W.: Anisotropy of a tensorial bishops coefficient for wetted granular materials. J. Eng. Mech. 143(3), B4015004 (2015)Google Scholar
  24. 24.
    Wang, J.-P., Li, X., Yu, H.-S.: Stress–force–fabric relationship for unsaturated granular materials in pendular states. J. Eng. Mech. 143(9), 04017068 (2017)Google Scholar
  25. 25.
    Yuan, C., Chareyre, B., Darve, F.: Deformation and stresses upon drainage of an idealized granular material. Acta Geotech. 13(4), 961–972 (2018)Google Scholar
  26. 26.
    Jiang, Y., Einav, I., Liu, M.: A thermodynamic treatment of partially saturated soils revealing the structure of effective stress. J. Mech. Phys. Solids 100, 131–146 (2017)ADSMathSciNetGoogle Scholar
  27. 27.
    Li, X.S.: Effective stress in unsaturated soil: a microstructural analysis. Géotechnique 53(2), 273–277 (2003)Google Scholar
  28. 28.
    Lu, N.: Is matric suction a stress variable? J. Geotech. Geoenviron. Eng. 134(7), 899–905 (2008)Google Scholar
  29. 29.
    Duriez, J., Wan, R., Pouragha, M., Darve, F.: Revisiting the existence of an effective stress for wet granular soils with micromechanics. Int. J. Numer. Anal. Methods Geomech. 42(8), 959–978 (2018)Google Scholar
  30. 30.
    Duriez, J., Wan, R.: Subtleties in discrete-element modelling of wet granular soils. Géotechnique 67(4), 365–370 (2016)Google Scholar
  31. 31.
    Wan, R., Khosravani, S., Pouragha, M.: Micromechanical analysis of force transport in wet granular soils. Vadose Zone J. 13(5), 1–12 (2014)Google Scholar
  32. 32.
    Houlsby, G.T., Puzrin, A.M.: Principles of Hyperplasticity: An Approach to Plasticity Theory Based on Thermodynamic Principles. Springer, Berlin (2007)Google Scholar
  33. 33.
    Coussy, O., Pereira, J.-M., Vaunat, J.: Revisiting the thermodynamics of hardening plasticity for unsaturated soils. Comput. Geotech. 37(1–2), 207–215 (2010)Google Scholar
  34. 34.
    Dangla, P., Pereira, J.-M.: A thermodynamic approach to effective stresses in unsaturated soils incorporating the concept of partial pore deformations. Vadose Zone J. 13(5), 1–11 (2014)Google Scholar
  35. 35.
    Li, X.S.: Thermodynamics-based constitutive framework for unsaturated soils. 1: theory. Géotechnique 57(5), 411–422 (2007)Google Scholar
  36. 36.
    Houlsby, G.: The work input to an unsaturated granular material. Géotechnique 47(1), 193–6 (1997)Google Scholar
  37. 37.
    Borja, R.I.: On the mechanical energy and effective stress in saturated and unsaturated porous continua. Int. J. Solids Struct. 43(6), 1764–1786 (2006)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Zhang, Y.: Effect of water-particle interactions on the crushing of granular materials. Ph.D. thesis, Northwestern University (2016)Google Scholar
  39. 39.
    Hassanizadeh, M., Gray, W.G., et al.: General conservation equations for multi-phase systems: 3. Constitutive theory for porous media flow. Adv. Water Resour. 3(1), 25–40 (1980)ADSGoogle Scholar
  40. 40.
    Lewis, E.R., Morgan, K., Schrejler, B., Hinton, E., Bettess, P., Zienkiewicz, O., Desai, E.C., Gallagher, R., Wood, R., Alex, J., et al.: The finite element method in the deformation and consolidation of porous media. Wiley, NewYork (1987)Google Scholar
  41. 41.
    Zhao, C.-F., Yin, Z.-Y., Misra, A., Hicher, P.-Y.: Thermomechanical formulation for micromechanical elasto-plasticity in granular materials. Int. J. Solids Struct. 138, 64–75 (2018)Google Scholar
  42. 42.
    Misra, A., Singh, V.: Nonlinear granular micromechanics model for multi-axial rate-dependent behavior. Int. J. Solids Struct. 51(13), 2272–2282 (2014)Google Scholar
  43. 43.
    Misra, A., Poorsolhjouy, P.: Granular micromechanics model for damage and plasticity of cementitious materials based upon thermomechanics. Math. Mech. Solids (2015).
  44. 44.
    Misra, A., Singh, V.: Thermomechanics-based nonlinear rate-dependent coupled damage-plasticity granular micromechanics model. Contin. Mech. Thermodyn. 27(4–5), 787–817 (2015)ADSMathSciNetzbMATHGoogle Scholar
  45. 45.
    Bagi, K.: Analysis of microstructural strain tensors for granular assemblies. Int. J. Solids Struct. 43(10), 3166–3184 (2006)zbMATHGoogle Scholar
  46. 46.
    Kruyt, N.P., Rothenburg, L.: Micromechanical definition of the strain tensor for granular materials. J. Appl. Mech. 63(3), 706–711 (1996)ADSzbMATHGoogle Scholar
  47. 47.
    Liao, C.-L., Chang, T.-P., Young, D.-H., Chang, C.S.: Stress-strain relationship for granular materials based on the hypothesis of best fit. Int. J. Solids Struct. 34(31–32), 4087–4100 (1997)zbMATHGoogle Scholar
  48. 48.
    Rabinovich, Y.I., Esayanur, M.S., Moudgil, B.M.: Capillary forces between two spheres with a fixed volume liquid bridge: theory and experiment. Langmuir 21(24), 10992–10997 (2005)Google Scholar
  49. 49.
    Willett, C.D., Adams, M.J., Johnson, S.A., Seville, J.P.: Capillary bridges between two spherical bodies. Langmuir 16(24), 9396–9405 (2000)Google Scholar
  50. 50.
    Kruyt, N.P., Millet, O.: An analytical theory for the capillary bridge force between spheres. J. Fluid Mech. 812, 129–151 (2017)ADSMathSciNetzbMATHGoogle Scholar
  51. 51.
    Lian, G., Seville, J.: The capillary bridge between two spheres: new closed-form equations in a two century old problem. Adv. Colloid Interface Sci. 227, 53–62 (2016)Google Scholar
  52. 52.
    Zhao, C.-F., Kruyt, N.P., Millet, O.: Capillary bridge force between non-perfectly wettable spherical particles: an analytical theory for the pendular regime. Powder Technol. 339, 827–837 (2018)Google Scholar
  53. 53.
    Israelachvili, J.N.: Intermolecular and Surface Forces. Academic Press, New York (2011)Google Scholar
  54. 54.
    Yuan, Y., Lee, T.R.: Contact angle and wetting properties. In: Bracco, G., Holst, B. (eds.) Surface Science Techniques, pp. 3–34. Springer, Berlin (2013)Google Scholar
  55. 55.
    Ziegler, H.: An Introduction to Thermomechanics, vol. 21. Elsevier, Amsterdam (2012)zbMATHGoogle Scholar
  56. 56.
    Zhao, C.-F., Yin, Z.-Y., Hicher, P.-Y.: A multiscale approach for investigating the effect of microstructural instability on global failure in granular materials. Int. J. Numer. Anal. Methods Geomech. 42(17), 2065–2094 (2018)Google Scholar
  57. 57.
    Biarez, J., Hicher, P.-Y.: Elementary Mechanics of Soil Behaviour: Saturated Remoulded Soils. AA Balkema, Amsterdam (1994)Google Scholar
  58. 58.
    Yin, Z.-Y., Wu, Z.-X., Hicher, P.-Y.: Modeling monotonic and cyclic behavior of granular materials by exponential constitutive function. J. Eng. Mech. 144(4), 04018014 (2018)Google Scholar
  59. 59.
    Yin, Z.-Y., Chang, C.S., Hicher, P.-Y.: Micromechanical modelling for effect of inherent anisotropy on cyclic behaviour of sand. Int. J. Solids Struct. 47(14–15), 1933–1951 (2010)zbMATHGoogle Scholar
  60. 60.
    Yin, Z.-Y., Chang, C.S.: Stress-dilatancy behavior for sand under loading and unloading conditions. Int. J. Numer. Anal. Methods Geomech. 37(8), 855–870 (2013)Google Scholar
  61. 61.
    Zhang, Y., Buscarnera, G.: Implicit integration under mixed controls of a breakage model for unsaturated crushable soils. Int. J. Numer. Anal. Methods Geomech. 40(6), 887–918 (2016)Google Scholar
  62. 62.
    Brooks, R.H., Corey, A.T.: Hydraulic properties of porous media and their relation to drainage design. Trans. ASAE 7(1), 26–0028 (1964)Google Scholar
  63. 63.
    Fredlund, D.G., Xing, A.: Equations for the soil–water characteristic curve. Can. Geotech. J. 31(4), 521–532 (1994)Google Scholar
  64. 64.
    Van Genuchten, M.T.: A closed-form equation for predicting the hydraulic conductivity of unsaturated soils 1. Soil Sci. Soc. Am. J. 44(5), 892–898 (1980)Google Scholar
  65. 65.
    Kruyt, N.P.: Statics and kinematics of discrete cosserat-type granular materials. Int. J. Solids Struct. 40(3), 511–534 (2003)zbMATHGoogle Scholar
  66. 66.
    Kruyt, N.P., Millet, O., Nicot, F.: Macroscopic strains in granular materials accounting for grain rotations. Granul. Matter 16(6), 933–944 (2014)Google Scholar
  67. 67.
    Li, X.S., Dafalias, Y.F.: Dissipation consistent fabric tensor definition from DEM to continuum for granular media. J. Mech. Phys. Solids 78, 141–153 (2015)ADSMathSciNetzbMATHGoogle Scholar
  68. 68.
    Zhao, C.-F., Yin, Z.-Y., Hicher, P.-Y.: Integrating a micromechanical model for multiscale analyses. Int. J. Numer. Methods Eng. 114(2), 105–127 (2018)MathSciNetGoogle Scholar
  69. 69.
    Chang, C.S., Misra, A.: Packing structure and mechanical properties of granulates. J. Eng. Mech. 116(5), 1077–1093 (1990)Google Scholar
  70. 70.
    Yin, Z.-Y., Zhao, J., Hicher, P.-Y.: A micromechanics-based model for sand–silt mixtures. Int. J. Solids Struct. 51(6), 1350–1363 (2014)Google Scholar
  71. 71.
    Bardet, J.P., Choucair, W.: A linearized integration technique for incremental constitutive equations. Int. J. Numer. Anal. Methods Geomech. 15(1), 1–19 (1991)zbMATHGoogle Scholar
  72. 72.
    Klotz, E.U., Coop, M.R.: An investigation of the effect of soil state on the capacity of driven piles in sands. Géotechnique 51(9), 733–751 (2001)Google Scholar
  73. 73.
    Kuwajima, K., Hyodo, M., Hyde, A.F.: Pile bearing capacity factors and soil crushabiity. J. Geotech. Geoenviron. Eng. 135(7), 901–913 (2009)Google Scholar
  74. 74.
    Coop, M.R.: The mechanics of uncemented carbonate sands. Géotechnique 40(4), 607–626 (1990)Google Scholar
  75. 75.
    Fern, E.J., Robert, D.J., Soga, K.: Modeling the stress–dilatancy relationship of unsaturated silica sand in triaxial compression tests. J. Geotech. Geoenviron. Eng. 142(11), 04016055 (2016)Google Scholar

Copyright information

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

Authors and Affiliations

  • Chao-Fa Zhao
    • 1
    • 2
  • Younes Salami
    • 1
    • 3
  • Pierre-Yves Hicher
    • 1
    Email author
  • Zhen-Yu Yin
    • 1
    • 4
  1. 1.Institut de Recherche en Génie Civil et Mécanique (GeM), UMR CNRS 6183Ecole Centrale de NantesNantes cedex 3France
  2. 2.LaSIE-UMR CNRS 7356Université de La RochelleLa RochelleFrance
  3. 3.Euro-Mediterranean University of Fès (UEMF)FèsMorocco
  4. 4.Department of Civil and Environmental EngineeringThe Hong Kong Polytechnic UniversityHung Hom, KowloonHong Kong

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