, Volume 59, Issue 2, pp 148–159 | Cite as

Numerical experiments on localization in frictional materials

  • P. A. Cundall


Two types of numerical experiment are performed in order to elucidate the nature of localization in a frictional material. In the first type, a continuum calculation is done with a strain-hardening constitutive model. Localization is shown to occur when the value of the strength parameter has a random distribution in space. In the second type of numerical experiment, the distinct element method is used to conduct a shear test on a simulated sample of 1000 disks. Localization is seen to occur: measurements are made of shear band thickness, distribution of particle spins, contact forces and stress components.


Neural Network Numerical Experiment Shear Band Contact Force Constitutive Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Numerische Simulation der Lokalisierung in reibungsbehaftetem Material


Zwei Arten von numerischer Simulation zur AufklÄrung der Natur von Lokalisierungen in reibungsbehaftetem Material werden durchgespielt. Im ersten Fall wird eine kontinuumsmechanische Rechnung mit einem verfestigenden Materialverhalten vorgenommen. Es wird gezeigt, da\ Lokalisierung auftritt, wenn der Grenzreibwert rÄumlich eine Zufallsverteilung besitzt. Im zweiten Fall wird eine Methode mit diskreten Elementen benutzt und mit einer durch 1000 Scheiben simulierten Probe ein Scherversuch durchgeführt. Auch hier tritt Lokalisierung auf: Messungen der Scherzonendicke, der KontaktkrÄfte und der gemittelten Spannungen werden angegeben.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Rudnicki, J. W.; Rice, J. R.: Conditions for the localization of the deformation in pressure-sensitive dilatant materials. J. Mech. Phys. Solids 23 (1975) 371–394Google Scholar
  2. 2.
    Vardoulakis, I.: Shear band inclination and shear modulus of sand in biaxial tests. Int. J. Numer. Anal. Methods Geomech. 4 (1980) 103–119Google Scholar
  3. 3.
    Vermeer, P. A.: A simple shear-band analysis using compliances. In: Vermeer, P. A.; Luger, H. J. (eds.) Deformation and failure of granular materials. Proc. IUTAM conference deformation and failure of gra- nular materials, Delft, pp. 493–499. Rotterdam: Balkema 1982Google Scholar
  4. 4.
    Ortiz, M.; Leroy, Y.; Needleman, A.: A finite element method for localized failure analysis. Comput. Methods Appl. Mech. Eng. 61 (1987) 189–214Google Scholar
  5. 5.
    de Borst, R.: Bifurcations in finite element models with a non-associated flow law. Int. J. Numer. Anal. Methods Geomech. 12 (1988) 99–116Google Scholar
  6. 6.
    Mühlhaus, H.-B.; Vardoulakis, I.: The thickness of shear bands in granular materials. Géotechnique 37 (1987) 271–283Google Scholar
  7. 7.
    Cundall, P. A.; Board, M.: A microcomputer program for modelling large-strain plasticity problems. In: Swoboda, C. (ed.) Numerical methods in geomechanics. Proc. 6th Int. Conf. on numer. meth. in geomechanics, Innsbruck, pp. 2101–2108. Rotterdam: Balkema 1988Google Scholar
  8. 8.
    Otter, J. R. H.; Cassell, A. O.; Hobbs, R. E.: Dynamic relaxation. Proc. Inst. Civ. Eng. 35 (1966) 633–656Google Scholar
  9. 9.
    Cundall, P. A.: Distinct element models of rock and soil structure. In: Brown, E. T. (ed.) Analytical and computational methods in engineering rock mechanics, pp. 129–163. London: Allen & Unwin 1986Google Scholar
  10. 10.
    Vermeer, P. A.; de Borst, R.: Non-associated plasticity for soils, concrete and rock. Heron 29 (1984) 1–64Google Scholar
  11. 11.
    Cundall, P. A.; Strack, O. D. L.: A discrete numerical model for granular assemblies. Géotechnique 29 (1979) 47–65Google Scholar
  12. 12.
    Cundall, P. A.: Computer simulations of dense sphere assemblies. In: Satake, M.; Jenkins, J. T. (eds.) Micromechanics of granular material. Proc. U.S.-Japan Seminar, Sendai/Zao (1988, in press)Google Scholar
  13. 13.
    Mindlin, R. D.: Compliance of elastic bodies in contact. J. Appl. Mech. 16 (1949) 259–268Google Scholar
  14. 14.
    Cundall, P. A.: Formulation of a three-dimensional distinct element model — Part I. A scheme to detect and represent contacts in a system composed of many polyhedral blocks. Int. J. Rock Mech. Min. Sci. Geomech. Abstracts 25 (1988) 107–116Google Scholar
  15. 15.
    Hart, R. J.; Cundall, P. A.; Lemos, J.: Formulation of a three-dimensional distinct element model — Part II. Mechanical calculations and interaction of a system composed of many polyhedral blocks. Int. J. Rock Mech. Min. Sci. Geomech. Abstracts 25 (1988) 117–125Google Scholar
  16. 16.
    Strack, O. D. L.: Cundall, P. A.: The distinct element method as a tool for research in granular media, Part II. Report to National Science Foundation concerning NSF Grant ENG76–20711. Dept. of Civil and Mineral Eng., Univ. of Minnesota (1978)Google Scholar
  17. 17.
    Vardoulakis, I.; Graf, B.: Imperfection sensitivity of the biaxial test on dry sand. In: Vermeer, P. A.; Luger, H. J. (eds.) Deformation and failure of granular materials. Proc. IUTAM conf. deformation and failure of granular materials, Delft, pp. 485–491. Rotterdam: Balkema 1982Google Scholar
  18. 18.
    Vardoulakis, I.: Shear-banding and liquefaction in granular materials on the basis of a Casserat continuum theory. Ing. Arch. 59 (1989) 106–113Google Scholar
  19. 19.
    Wilkins, M. L.: Fundamental methods in hydrodynamics. In: Alder, B.; Fernbach, S.; Rotenberg, M. (eds.) Methods in computational physics, Vol. 3, pp. 211–263. New York: Academic Press 1964Google Scholar
  20. 20.
    Nagtegaal, J. C.; Parks, D. M.; Rice, J. R.: On numerically accurate finite element solutions in the fully plastic range. Comput. Methods Appl. Mech. Eng. 4 (1974) 153–177Google Scholar
  21. 21.
    Marti, J.; Cundall, P.: Mixed discretization procedure for accurate modelling of plastic collapse. Int. J. Numer. Anal. Methods Geomech. 6 (1982) 129–139Google Scholar
  22. 22.
    Lin, M.: Analytical and numerical solutions in limit load plasticity problems. MSc thesis in Civil Eng., Univ. of Minnesota (1987)Google Scholar
  23. 23.
    Triantafyllidis, N.; Aifantis, E. C.: A gradient approach to localization of deformation, I. Hyperelastic materials. J. Elasticity 16 (1986) 225–237Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • P. A. Cundall
    • 1
  1. 1.Marine on St. CroixUSA

Personalised recommendations