Granular Matter

, 21:28 | Cite as

Micro-scale investigation of the role of finer grains in the behavior of bidisperse granular materials

  • Habib Taha
  • Ngoc-Son NguyenEmail author
  • Didier Marot
  • Abbas Hijazi
  • Khalil Abou-Saleh
Original Paper


This paper presents a numerical study of the effect of fine content on the mechanical behavior of bidisperse granular materials using the discrete element method. Triaxial compression tests are performed on different samples with fine contents varied from 0 to 40%. It was found that, starting from 20%, fine content has a visible effect on the shear strength. The optimal fine content is about 30%, at which the shear strength is the best. An investigation into the granular micro-structure showed that the fine particles, on one hand, come into contact with coarse particles, but on the other hand, separate the latter ones as fine content increases beyond 20%. Thus, the part of the shear stress carried by the coarse–fine contacts increases, while the part carried by the coarse–coarse contacts decreases. For fine content ≤ 30%, the coarse–coarse contacts primarily carry the shear stress. Above this optimal fine content, the fine–coarse contacts overtake the coarse–coarse ones. The fine–fine contacts have little contribution to supporting the shear stress. For the studied range of fine content, the coarse particles primarily carry the shear stress, leaving the fine particles under relatively low stresses. Moreover, the matrix composed of fine particles is greatly softened by the shear loading. A classification of binary mixtures depending on their micro-structure was also proposed.


Bidisperse materials Discrete element method Shear loading Micro-structure Stress transmission Contact network 

List of symbols


Fine content

\(f_n\), \(f_t\)

Normal and tangential contact forces

\(K_n\), \(K_t\)

Normal and tangential contact stiffnesses

\(k_n\), \(k_t\)

Normal and tangential particle stiffnesses


Young’s modulus of the particle material


Contact friction angle

\(D_{\mathrm{min}}\), \(D_{\mathrm{max}}\)

Minimum and maximum diameters of coarse particles

\(d_{\mathrm{min}}\), \(d_{\mathrm{max}}\)

Minimum and maximum diameters of fine particles


Gap ratio

\(\varvec{\sigma }\)

Stress tensor


Mean stress


Deviatoric stress

\(\varvec{\varepsilon }\)

Strain tensor

\(\varepsilon _{11}\)

Axial strain

\(\varepsilon _{\mathrm{v}}\)

Volumetric strain


Sample size


Number of coarse particles


Number of fine particles


Coefficient of variation


Global void ratio


Global porosity


Intergranular void ratio


Interfine void ratio

\(V_{\mathrm{v}}\), \(V_{\mathrm{s}}\)

Void and solid volumes

F, C

Fine and coarse fractions


Coarse–coarse contacts


Coarse–fine contacts


Fine–fine contacts


Coordination number


Average number of \(C{-}C\) contacts per coarse particle


Average number of \(C{-}F\) contacts per coarse particle


Average number of \(F{-}F\) contacts per fine particle


Force at a given contact k


Branch vector joining two particle centers at a given contact k

\(\varvec{\sigma }^{C{-}C}\)

Contribution of the \(C{-}C\) contacts to the macro-stress

\(\varvec{\sigma }^{C{-}F}\)

Contribution of the \(C{-}F\) contacts to the macro-stress

\(\varvec{\sigma }^{F{-}F}\)

Contribution of the \(F{-}F\) contacts to the macro-stress

\(\phi ^\alpha\)

Volume fraction of a given phase \(\alpha\)

\(\varvec{\sigma }^\alpha\)

Intrinsic averaged stress of a given phase \(\alpha\)

\(\widehat{\varvec{\sigma }}^\alpha\)

Partial stress of a given phase \(\alpha\)


Internal moment tensor of a given particle p

\(p^F\), \(q^F\)

Mean and deviatoric stresses carried by the fine fraction


Stress reduction factor

\(\alpha _p^F\)

Mean stress ratio for the fine fraction

\(\alpha _q^F\)

Deviatoric stress ratio for the fine fraction


Contribution of the fine fraction to the macroscopic mean stress


Contribution of the fine fraction to the macroscopic deviatoric stress



The authors would like to thank the charitable and cultural association of Nabatieh-Lebanon and Cedre program of the French and Lebanese scientific cooperation for the financial support for this research project.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.GeM InstituteUniversity of NantesSaint-Nazaire CedexFrance
  2. 2.MPLAB-Multisciplinary Physics Laboratory, Faculty of SciencesLebanese UniversityHadat-BaabdaLebanon

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