Skip to main content

Ordering and stress transmission in packings of straight and curved spherocylinders


In this work we apply the discrete element method (DEM) to model packings of spherocylinders. The so-called composite spheres method was used to construct particles of different aspect ratio, surface shape and curvature. Using the DEM we probe in detail the effect of particle curvature and surface shape on packing morphology and stress transmission. We find that particle shape has a remarkable influence on both the packing morphology (quantified via the solid fraction, particle orientation distribution and radial distribution function) and stress transmission. Specifically, elongated particles have a high preference for horizontal alignment, whereas an increasing particle curvature leads to a more continuous (i.e. less discrete) particle orientation distribution. Generally, we observe that rough and curved particles have a stronger tendency for interlocking (in particular for small particle aspect ratios, i.e. \(AR=2\) and 3) leading to the formation of dense packing structures. In addition packings of rough and curved particles of small aspect ratios favor stress transmission in the gravitational direction, thus, limiting stress saturation with depth.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10


A :

Particle cross-sectional area (m\(^{2})\)

AR :

Particle aspect ratio (\(-\))

\(d_p \) :

Primary sphere diameter (m)

dt :

Time step of DEM simulations (s)

\(E_k \) :

Kinetic energy (J)

\(E_p \) :

Elastic potential energy (J)

\(f_{ij} \) :

Contact force between particles i and j (N)

\(F_n \) :

Normal force (N)

\(F_t \) :

Tangential force (N)

\(G\left( r \right) \) :

Radial distribution function (\(-\))

h :

Dimensionless packing depth (\(-\))

\(h_c \) :

Characteristic depth of packing (\(-\))

H :

Height of container (m)

\(k_n \) :

Normal spring stiffness (N/m)

\(k_t \) :

Tangential spring stiffness (N/m)

\(l_{axial} \) :

Length of particle major axis (m)

m :

Particle mass (kg)

N :

Number of particles (\(-\))

\(P_0 \) :

Stress (N/m)

\(P_j \) :

Saturation stress (N/m)

\(r,\mathbf{r}\) :

Distance/vector relative to a point (m)

\(\delta r\) :

Differential distance (m)

\(v_n \) :

Relative normal velocity (m/s)

\(v_t \) :

Relative tangential velocity (m/s)

W :

Width of container (m)

XY :

Coordinates (m)

\(\delta _n \) :

Particle overlap (m)

\(\delta _t \) :

Tangential displacement (m)

\(\eta _n \) :

Normal damping factor (\(-\))

\(\eta _t \) :

Tangential damping factor (\(-\))

\(\gamma \) :

Particle orientation angle (\(^{\circ })\)

\(\mu \) :

Coefficient of friction (\(-\))

\(\phi \) :

Solid fraction (\(-\))

\(\rho \) :

Number of particles per unit area (1/m\(^{2})\)

\(\rho _p \) :

Particle density (kg/m\(^{3})\)

\(\theta \) :

Angle of curvature (\(^{\circ })\)

\(\omega \) :

Coarse-graining scale (\(-\))

\(\sigma _v \) :

Major eigenvalue of stress tensor (N/m)

\(\sigma _h \) :

Minor eigenvalue of stress tensor (N/m)

\(\bar{\sigma }_{\alpha \beta } \left( \mathbf{r} \right) \) :

Mean stress tensor (N/m)

\(\varphi \left( \mathbf{r} \right) \) :

Gaussian coarse-graining function (1/m\(^{2})\)


  1. Torquato, S., Stillinger, F.H.: Jammed hard-particle packings: from Kepler to Bernal and beyond. Rev. Mod. Phys. 82, 2633–2672 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  2. Müller, C.R., Davidson, J.F., Dennis, J.S., Fennell, P.S., Gladden, L.F., Hayhurst, A.N., Mantle, M.D., Rees, A.C., Sederman, A.J.: Real-time measurement of bubbling phenomena in a three-dimensional gas-fluidized bed using ultrafast magnetic resonance imaging. Phys. Rev. Lett. 96, 154504 (2006)

    Article  ADS  Google Scholar 

  3. Cundall, P.A., Strack, O.D.L.: A discrete numerical model for granular assemblies. Géotechnique 29, 47–65 (1979)

    Article  Google Scholar 

  4. Zhu, H.P., Zhou, Z.Y., Yang, R.Y., Yu, A.B.: Discrete particle simulation of particulate systems: theoretical developments. Chem. Eng. Sci. 62, 3378–3396 (2007)

    Article  Google Scholar 

  5. Zhu, H.P., Zhou, Z.Y., Yang, R.Y., Yu, A.B.: Discrete particle simulation of particulate systems: a review of major applications and findings. Chem. Eng. Sci. 63, 5728–5770 (2008)

    Article  Google Scholar 

  6. Lu, G., Third, J.R., Müller, C.R.: Critical assessment of two approaches for evaluating contacts between super-quadric shaped particles in DEM simulations. Chem. Eng. Sci. 78, 226–235 (2012)

    Article  Google Scholar 

  7. Lu, G., Third, J.R., Müller, C.R.: Effect of particle shape on domino wave propagation: a perspective from 3D, anisotropic discrete element simulations. Granul. Matter 16, 107–114 (2014)

    Article  Google Scholar 

  8. Lu, G., Third, J.R., Müller, C.R.: Effect of wall rougheners on cross-sectional flow characteristics for non-spherical particles in a horizontal rotating cylinder. Particuology 12, 44–53 (2014)

    Article  Google Scholar 

  9. Börzsönyi, T., Stannarius, R.: Granular materials composed of shape-anisotropic grains. Soft Matter 9, 7401–7418 (2013)

    Article  Google Scholar 

  10. Lu, G., Third, J.R., Müller, C.R.: Discrete element models for non-spherical particle systems: from theoretical developments to applications. Chem. Eng. Sci. 127, 425–465 (2015)

    Article  Google Scholar 

  11. Baule, A., Makse, H.A.: Fundamental challenges in packing problems: from spherical to non-spherical particles. Soft Matter 10, 4423–4429 (2014)

    Article  ADS  Google Scholar 

  12. Langston, P.A., Al-Awamleh, M.A., Fraige, F.Y., Asmar, B.N.: Distinct element modelling of non-spherical frictionless particle flow. Chem. Eng. Sci. 59, 425–435 (2004)

    Article  Google Scholar 

  13. Wouterse, A., Luding, S., Philipse, A.P.: On contact numbers in random rod packings. Granul.Matter 11, 169–177 (2009)

    Article  MATH  Google Scholar 

  14. Kyrylyuk, A.V., van de Haar, M.A., Rossi, L., Wouterse, A., Philipse, A.P.: Isochoric ideality in jammed random packings of non-spherical granular matter. Soft Matter 7, 1671–1674 (2011)

    Article  ADS  Google Scholar 

  15. Zhao, J., Li, S.X., Zou, R.P., Yu, A.B.: Dense random packings of spherocylinders. Soft Matter 8, 1003–1009 (2012)

    Article  ADS  Google Scholar 

  16. Meng, L.Y., Li, S.X., Lu, P., Li, T., Jin, W.W.: Bending and elongation effects on the random packing of curved spherocylinders. Phys. Rev. E 86, 061309 (2012)

    Article  ADS  Google Scholar 

  17. Meng, L.Y., Lu, P., Li, S.X., Zhao, J., Li, T.: Shape and size effects on the packing density of binary spherocylinders. Powder Technol. 228, 284–294 (2012)

    Article  Google Scholar 

  18. Deng, X.L., Davé, R.N.: Dynamic simulation of particle packing influenced by size, aspect ratio and surface energy. Granul. Matter 15, 401–415 (2013)

    Article  Google Scholar 

  19. Nan, W.G., Wang, Y.S., Ge, Y., Wang, J.Z.: Effect of shape parameters of fiber on the packing structure. Powder Technol. 261, 210–218 (2014)

    Article  Google Scholar 

  20. Alonso-Marroquin, F.: Spheropolygons: a new method to simulate conservative and dissipative interactions between 2D complex-shaped rigid bodies. Europhys. Lett. 83, 14001 (2008)

    Article  ADS  Google Scholar 

  21. Hidalgo, R.C., Zuriguel, I., Maza, D., Pagonabarraga, I.: Role of particle shape on the stress propagation in granular packings. Phys. Rev. Lett. 103, 118001 (2009)

    Article  ADS  Google Scholar 

  22. Hidalgo, R.C., Zuriguel, I., Maza, D., Pagonabarraga, I.: Granular packings of elongated faceted particles deposited under gravity. J. Stat. Mech. Theory Exp. 6, 06025 (2010)

    Article  Google Scholar 

  23. Hidalgo, R.C., Kadau, D., Kanzaki, T., Herrmann, H.J.: Granular packings of cohesive elongated particles. Granul. Matter 14, 191–196 (2012)

    Article  Google Scholar 

  24. Azéma, E., Radjaï, F.: Stress-strain behavior and geometrical properties of packings of elongated particles. Phys. Rev. E 81, 051304 (2010)

    Article  ADS  Google Scholar 

  25. Azéma, E., Radjaï, F.: Force chains and contact network topology in sheared packings of elongated particles. Phys. Rev. E 85, 031303 (2012)

    Article  ADS  Google Scholar 

  26. Kanzaki, T., Acevedo, M., Zuriguel, I., Pagonabarraga, I., Maza, D., Hidalgo, R.C.: Stress distribution of faceted particles in a silo after its partial discharge. Eur. Phys. J. E 34, 133 (2011)

    Article  Google Scholar 




  30. Goldhirsch, I.: Stress, stress asymmetry and couple stress: from discrete particles to continuous fields. Granul. Matter 12, 239–252 (2010)

    Article  MATH  Google Scholar 

  31. Acevedo, M., Zuriguel, I., Maza, D., Pagonabarraga, I., Alonso-Marroquin, F., Hidalgo, R.C.: Stress transmission in systems of faceted particles in a silo: the roles of filling rate and particle aspect ratio. Granul. Matter 16, 411–420 (2014)

    Article  Google Scholar 

  32. Weinhart, T., Thornton, A.R., Luding, S., Bokhove, O.: From discrete particles to continuum fields near a boundary. Granul. Matter 14, 289–294 (2012)

    Article  Google Scholar 

  33. Zou, R.P., Yu, A.B.: Evaluation of the packing characteristics of mono-sized non-spherical particles. Powder Technol. 88, 71–79 (1996)

    Article  Google Scholar 

  34. Janssen, H.A.: Versuche über Getreidedruck in Silozellen, Verein Deutscher Ingenieure. Zeitschrift (Dusseldorf) 39, 1045–1049 (1895)

    Google Scholar 

Download references


The authors are grateful to the Swiss National Science Foundation (200021_132657/1) and the China Scholarship Council (Guang Lu) for partial financial support of this work. R.C. Hidalgo acknowledges the financial support from Ministerio de Economía y Competitividad (Spanish Government) through FIS2011-26675 and FIS2014-57325 Projects.

Author information

Authors and Affiliations


Corresponding author

Correspondence to C. R. Müller.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lu, G., Hidalgo, R.C., Third, J.R. et al. Ordering and stress transmission in packings of straight and curved spherocylinders. Granular Matter 18, 34 (2016).

Download citation

  • Received:

  • Published:

  • DOI:


  • Particle packing
  • Non-spherical particles
  • Particle ordering
  • Stress transmission
  • Discrete element method (DEM)