Skip to main content

A bond model for DEM simulation of cementitious materials and deformable structures


There is an increasing use of the discrete element method (DEM) to study cemented (e.g. concrete and rocks) and sintered particulate materials. The chief advantage of the DEM over continuum based techniques is that it does not make assumptions about how cracking and fragmentation initiate and propagate, since the DEM system is naturally discontinuous. The ability for the DEM to produce a realistic representation of a cemented granular material depends largely on the implementation of an inter-particle bonded contact model. This paper presents a new bonded contact model based on the Timoshenko beam theory which considers axial, shear and bending behaviour of the bond. The bond model was first verified by simulating both the bending and dynamic response of a simply supported beam. The loading response of a concrete cylinder was then investigated and compared with the Eurocode equation prediction. The results show significant potential for the new model to produce satisfactory predictions for cementitious materials. A unique feature of this model is that it can also be used to accurately represent many deformable structures such as frames and shells, so that both particles and structures or deformable boundaries can be described in the same DEM framework.

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
Fig. 11


\(A\) :

Area \((\hbox {m}^{2})\)

\(d\) :

Translational displacement (m)

\(e\) :

Coefficient of restitution

\(E\) :

Young’s modulus (Pa)

\(f_{s}\) :

Form factor for shear

\(G\) :

Shear modulus (Pa)

\(I\) :

Second moment of area \((\hbox {m}^{4})\)

\(K\) :

Stiffness \((\hbox {N m}^{-1})\)

\(L\) :

Length (m)

\(m\) :

Mass (kg)

\(M\) :

Moment (N m)

\(N\) :

Random number

\(P\) :


\(r\) :

Radius (m)

\(S\) :

Mean bond strength (Pa)

\(t\) :

Time (s)

\(u\) :

Displacement vector (m)

\(W\) :

Point load (N)

\(x, y, z\) :

Local Cartesian coordinates (m, m, m)

\(X, Y, Z\) :

Global Cartesian coordinates (m, m, m)

\(\gamma \) :

Transformation matrix

\(\delta \) :

Mid-span deflection (m)

\(\Delta \)t:

Time step (s)

\(\varepsilon \) :


\(\eta \) :

Contact radius multiplier

\(\lambda \) :

Bond radius multiplier

\(\mu \) :

Coefficient of friction

\(\rho \) :

Density \((\hbox {kg m}^{-3})\)

\(\varsigma \) :

Coefficient of variation of strength

\(\sigma \) :

Axial stress (MPa)

\(\tau \) :

Shear stress (Pa)

\(v\) :

Poisson’s ratio

\(\alpha , \beta \) :

Ends of a single bond

\(A, B\) :

Particle labels

\(b\) :


\(c\) :


\(C\) :

Compressive stress

crit :


\(g\) :






\(\rho \) :


\(r\) :

Rolling friction

\(s\) :

Static friction

\(S\) :

Shear stress

\(T\) :

Tensile stress

\(x, y, z\) :

Local Cartesian coordinates

\(X, Y, Z\) :

Global Cartesian coordinates


  1. Kuhl, E., D’Addetta, G.A., Herrmann, H.J., Ramm, E.: A comparison of discrete granular material models with continuous microplane formulations. Granul. Matter. 2, 113–121 (2000)

    Article  Google Scholar 

  2. Cundall, P.A.: A computer model for simulating progressive large scale movement in blocky rock systems. In: Proceedings of the Symposium of the International Society of Rock Mechanics, Nancy (1971)

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

    Article  Google Scholar 

  4. Labra, C.: Advances in the development of the discrete element method for excavation processes. Ph.D. thesis, Universitat Politecnica de Catalunya, Barcelona, Spain (2012)

  5. Rojek, J., Labra, C., Su, O., Oñate, E.: Comparative study of different discrete element models and evaluation of equivalent micromechanical parameters. Int. J. Solids Struct. 49, 1497–1517 (2012)

    Article  Google Scholar 

  6. Ergenzinger, C., Seifried, R., Eberhard, P.: A discrete element model to describe failure of strong rock in uniaxial compression. Granul. Matter. 13, 341–364 (2011)

    Article  Google Scholar 

  7. Potyondy, D.O., Cundall, P.A.: A bonded-particle model for rock. Int. J. Rock Mech. Min. Sci. 41, 1329–1364 (2004)

    Article  Google Scholar 

  8. Cho, N., Martin, C.D., Sego, D.C.: A clumped particle model for rock. Int. J. Rock Mech. Min. Sci. 44, 997–1010 (2007)

    Article  Google Scholar 

  9. Su, O., Ali Akcin, N.: Numerical simulation of rock cutting using the discrete element method. Int. J. Rock Mech. Min. Sci. 48, 434–442 (2011)

    Article  Google Scholar 

  10. D’Addetta, G.A., Ramm, E.: A microstructure-based simulation environment on the basis of an interface enhanced particle model. Granul. Matter. 8, 159–174 (2006)

    Article  MATH  Google Scholar 

  11. Schneider, B., Bischoff, M., Ramm, E.: Modeling of material failure by the discrete element method. In: Proceedings in Applied Mathmatics and Mechanics, Stuttgart (2010)

  12. André, D., Iordanoff, I., Charles, J., Néauport, J.: Discrete element method to simulate continuous material by using the cohesive beam model. Comput. Methods Appl. Mech. Eng. 213–216, 113–125 (2012)

    Article  Google Scholar 

  13. Carmona, H.A., Wittel, F.K., Kun, F., Herrmann, H.J.: Fragmentation processes in impact of spheres. Phys. Rev. E 77, 051302 (2008)

    Article  ADS  Google Scholar 

  14. D’Addetta, G.A., Kun, F., Ramm, E.: On the application of a discrete model to the fracture process of cohesive granular materials. Granul. Matter. 4, 77–90 (2002)

    Article  MATH  Google Scholar 

  15. Schlangen, E., Garboczi, E.J.: Fracture simulations of concrete using lattice models: computational aspects. Eng. Fract. Mech. 57(2/3), 319–332 (1997)

    Article  Google Scholar 

  16. Ostoja-Starewski, M.: Lattice models in micromechanics. Appl. Mech. Rev. 50(1), 35–59 (2002)

    Article  ADS  Google Scholar 

  17. Lilliu, G., van Mier, J.G.M.: 3D lattice type fracture model for concrete. Eng. Fract. Mech. 70, 927–941 (2003)

    Article  Google Scholar 

  18. Azevedo, N.M., Lemos, J.V., De Almeida, J.R.: Influence of aggregate deformation and contact behaviour on discrete particle modelling of fracture of concrete. Eng. Fract. Mech. 75, 1569–1586 (2008)

    Article  Google Scholar 

  19. Camborde, F., Mariotti, C., Donze, F.V.: Numerical study of rock and concrete behaviour by discrete element modelling. Comput. Geotech. 27, 225–247 (2000)

    Article  Google Scholar 

  20. Hentz, S., Donzé, F.V., Daudeville, L.: Discrete element modelling of concrete submitted to dynamic loading at high strain rates. Comput. Struct. 82, 2509–2524 (2004)

    Article  Google Scholar 

  21. Qin, C., Zhang, C.: Numerical study of dynamic behavior of concrete by meso-scale particle element modeling. Int. J. Impact Eng. 38, 1011–1021 (2011)

    Google Scholar 

  22. Magnier, S.A., Donze, F.V.: Numerical simulations of impacts using a discrete element method. Mech. Cohes-Frict. Mater. 3, 257–276 (1998)

  23. Sawamoto, Y., Tsubota, H., Kasai, Y., Koshika, N., Morikawa, H.: Analytical studies on local damage to reinforced concrete structures under impact loading by discrete element method. Nuclear Eng. Design. 179, 157–177 (1998)

    Article  Google Scholar 

  24. Przemieniecki, J.S.: Theory of Matrix Structural Analysis. McGraw-Hill, New York (1968)

    MATH  Google Scholar 

  25. Wittel, F.K., Carmona, H.A., Kun, F., Herrmann, H.J.: Mechanics in impact fragmentation. Int. J. Fract. 154, 105–117 (2008)

    Article  MATH  Google Scholar 

  26. DEM Solutions: EDEM 2.1 User Guide (2008)

  27. Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1987)

    Google Scholar 

  28. Timoshenko, S.P.X.: On the transverse vibrations of bars of uniform cross-section. Philos. Mag. Ser. 6 43, 125–131 (1922)

    Article  Google Scholar 

  29. Gere, J.M., Timoshenko, S.P.: Mechanics of materials. PWS-KENT (1990)

  30. Johnstone, M.W.: Calibration of DEM models for granular materials using bulk physical tests (2010)

  31. Misra, A., Cheung, J.: Particle motion and energy distribution in tumbling ball mills. Powder Technol. 105, 222–227 (1999)

    Article  Google Scholar 

  32. Tsuji, Y., Tanaka, T., Ishida, T.: Lagrangian numerical simulation of plug flow of cohesionless particles in a horizontal pipe. Powder Technol. 71, 239–250 (1992)

    Article  Google Scholar 

  33. DEM.Solutions: EDEM 2.3 (2010)

  34. Bažant, Z.P., Tabbara, M.R., Kazemi, M.T., Pyaudier-cabot, G.: Random particle model for fracture of aggregate or fiber composites. J. Eng. Mech. 116(8), 1686–1705 (1990)

    Google Scholar 

  35. Hentz, S., Daudeville, L., Donzé, F.-V.: Identification and validation of a discrete element model for concrete. J. Eng. Mech. 130, 709–719 (2004)

    Article  Google Scholar 

  36. Tavarez, F.A., Plesha, M.E.: Discrete element method for modelling solid and particulate materials. Int. J. Numer. Methods Eng. 70, 379–404 (2007)

    Article  MATH  Google Scholar 

  37. Ross, C.T.F., Case, J., Chilver, A.: Strength of Materials and Structures. Arnold, London (1999)

    Google Scholar 

  38. CIMNE: GiD, (2012)

  39. Labra, C., Escolano, E., Pasenau, M.: GiD features for discrete element simulations. In: 5th Conference on Advances and Applications of GiD., Barcelona (2010)

  40. O’Sullivan, C., Bray, J.D.: Selecting a suitable time step for discrete element simulations that use the central difference time integration scheme. Eng. Comput. 21(2/3/4), 278–303 (2004)

    Article  MATH  Google Scholar 

  41. Cundall, P.A.: Distinct element models of rock and soil structure. In: Brown, E.T., Bray, J. (eds.) Analytical and Computational Methods in Engineering Rock Mechnics, pp. 129–163. Allen and Unwin, London (1987)

    Google Scholar 

  42. Brown, N.J.: Discrete Element Modelling of Cementitious Materials. Ph.D. thesis, The University of Edinburgh, Edinburgh, UK (2013)

  43. Dhir, R.K., Sangha, R.M.: Development and propagation of microcracks in plain concrete. Matériaux et Constr. 7, 17–23 (1974)

    Article  Google Scholar 

  44. Mehta, P.K., Monterio, P.J.M.: Concrete: Structure, Properties and Materials. Prentice Hall, Englewood Cliffs (1993)

    Google Scholar 

  45. Zhou, Y.-W., Wu, Y.-F.: General model for constitutive relationships of concrete and its composite structures. Compos. Struct. 94, 580–592 (2012)

    Article  Google Scholar 

  46. BS EN 1992-1-1: Eurocode 2: Design of concrete structures—Part 1–1: general rules and rules for buildings (2004)

  47. Neville, A.M., Brooks, J.J.: Concrete Technology. Longman Scientific & Technical, Harlow (1987)

    Google Scholar 

Download references


The authors would like to thank EPSRC and DEM Solutions Ltd for the funding and sponsorship. We are also grateful for the assistance and discussion with DEM solutions.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Jin Y. Ooi.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Brown, N.J., Chen, JF. & Ooi, J.Y. A bond model for DEM simulation of cementitious materials and deformable structures. Granular Matter 16, 299–311 (2014).

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI:


  • Discrete element method (DEM)
  • Bond model
  • Cementitious materials
  • Numerical modelling