Granular Matter

, 21:13 | Cite as

Discrete element modeling of first shear strain gradient effects on mechanical behaviors in granular materials

  • Shu-Mei Yang
  • Wen-Ping WuEmail author
  • Ming-Xiang ChenEmail author
Original Paper


The effects of strain gradient on the mechanical behavior of granular materials have attracted attention from many researchers. In this paper, the effects of the first shear strain gradient in granular materials are focused on. Granular assemblies with various particle radii and porosities are built through a two-dimensional discrete element method (2D-DEM) simulations. The results indicate that the macro shear stress is insensitive to the first shear strain gradient at low strains, but is indeed affected at high strains; the third order stress conjugated with the first shear strain gradient is sensitive to the first shear strain gradient in all deformation cases; the third order work done by the third order stress and the ratio of it versus the total work are affected by the first shear strain gradient; the evolutions of the invariants of the third order stresses present severe change of the local relative vertical displacement of particles, which lead to localization occurrence. Based on DEM simulations, it is found that only the effect consideration of the first shear strain gradient on the macro shear stress is not sufficient, whereas the effect on work should also be taken into account and the third order work as a part of strain energy cannot be ignored, which is falsely classified as a portion of the accumulated energy dissipated by frictional sliding.


First shear strain gradient Macro shear stress Third order stress Third order work Granular materials 

List of symbols

\( A(t) \)

Shape parameter of the primary term of displacement

\( B(t) \)

Shape parameter of the quadratic term of displacement

\( c \)

Ratio of \( B(t) \) versus \( A(t) \)

\( \overline{\sigma }_{12} \)

Macro shear stress

\( \overline{\varSigma }_{221} \)

Third order stress which is conjugated with the first shear strain gradient

\( W_{1} \)

Second order work, the integral of the macro shear stress times the macro shear strain in total volume

\( W_{2} \)

Third order work, the integral of the third order stress times the first shear strain gradient in total volume

\( W \)

Summation of \( W_{1} \) and \( W_{2} \)

\( \alpha \)

Ratio of \( W_{2} \) versus \( W \)

\( F_{1} \), \( F_{2} \), \( F_{3} \), \( F_{4} \), \( F_{5} \)

Five invariants of the conjugated stresses



The work was supported by National Natural Science Foundation of China (NSFC) (Grant Nos. 11711530643 and 11772236).

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.Department of Engineering Mechanics, School of Civil EngineeringWuhan UniversityWuhanChina
  2. 2.State Key Laboratory of Water Resources and Hydropower Engineering ScienceWuhan UniversityWuhanChina

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