The Mathematical Structure of the Equations for Quasi-Static Plane Strain Deformations of Granular Material

  • Ian F. Collins
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 26)


An analysis of the quasi-static, plane strain, deformation of a rigid/plastic material whose yield stress depends on the current material density is presented. Such constitutive equations are widely used to model the initial yielding, final failure and flow of granular materials in what is generally known as “critical state soil mechanics”. It will be shown that the mathematical structure of these problems is rather more complex than seems to have been hitherto realized. It will be shown that there exist families of weak discontinuities other than “the stress and velocity characteristics” and that they provide a key to the resolution of the long standing debate regarding the non-coincidence of stress and velocity characteristics for frictional materials. The role of isotropy, anisotropy, normal flow rules and non-normal flow rules will be discussed.


Critical State Granular Material Yield Surface Velocity Characteristic Flow Rule 
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  1. [1]
    J.H. Atkinson, Foundation and Slopes, McGraw-Hill, London (1981).Google Scholar
  2. [2]
    J.H. Atkinson, and P.L. Bransby, The Mechanics of Soils, McGraw-Hill, London (1978).Google Scholar
  3. [3]
    M. Bolton, A Guide to Soil Mechanics, Macmillan, London (1979).Google Scholar
  4. [4]
    A.M. Britto and M.J. Gunn, Critical State Soil Mechanics via Finite Elements, Ellis Harwood, Chichister (1987).zbMATHGoogle Scholar
  5. [5]
    R. Butterfield and R.M. Harkness, The Kinematics of Mohr-Coulomb Materials, Proc. Roscoe Memorial Symposium “Stress-Strain Behaviour of Soils” ed. by R.H.G. Parry, G.T. Foulis Henley (1972), pp. 220–233.Google Scholar
  6. [6]
    J. Chakarabarty, Theory of Plasticity, McGraw-Hill, London (1987).Google Scholar
  7. [7]
    I.F. Collins, Boundary Value problems in plane strain plasticity, Mechanics of Solids, the Rodney Hill 60th Anniversary Volume, eds H.G. Hopkins and M.J. Sewell, Pergamon Press, Oxford (1981), pp. 135–184.Google Scholar
  8. [8]
    I.F. Collins, Plane strain characteristics theory for soils and granular materials with density dependent yield criteria, J. Mech. Phys. Solids (in press).Google Scholar
  9. [9]
    G. De Josselin De Jong, The double-sliding, free-rotating model for granular assemblies, Géotechnique, 21 (1971), pp. 155–163.CrossRefGoogle Scholar
  10. [10]
    G. De Josselin de Jong, Mathematical elaboration of the double-sliding, free rotating model., Archs. Mech., 29 (1977), pp. 561–591.Google Scholar
  11. [11]
    D.C. Drucker, R.E. Gibson and D.J. Henkel, Soil mechanics and work-hardening theories of plasticity, Trans A.S.C.E. 122 (1957), pp. 338–346.Google Scholar
  12. [12]
    E.H. Davis and J.R. Booker, Some adaptions of classical plasticity theory for soil stability problems, Symposium on Plasticity and Soil Mechanics, Cambridge, (1973), pp. 24–41.Google Scholar
  13. [13]
    G.A. Geniev, Problems of dynamics of granular media, Akad. Stroit. i. Arch. Moscow (1959).Google Scholar
  14. [14]
    R. Hill, The Mathematical Theory of Plasticity, Clarendon Press, Oxford (1950).zbMATHGoogle Scholar
  15. [15]
    G.T. Houlsby and C.P. Wroth, Direct solution of plasticity problems in soils by the method of characteristics, Proc. 4th Int. Conf. Num. Methods in Geomechanics, Edmonton, Canada, 2 (1982), pp. 1059–1071.Google Scholar
  16. [16]
    R. Jackson, Some mathematical and physical aspects of continuum models for the motion of granular materials, Theory of Dispersed Multiphase flow, ed. R.E. Meyer, Academic Press (1983), pp. 291–337.Google Scholar
  17. [17]
    P.V. Lade, Effects of voids and volume changes on the behaviour of frictional materials, Int. J. Num. and Anal. Methods Geomechanics, 12 (1988), pp. 351–370.CrossRefGoogle Scholar
  18. [18]
    J. Mandel, Sur les équations à écoulement des sols idéaux en déformations plane et le concept de double glissement, J. Mech. Phys. Solids, 14 (1966), pp. 303–306.CrossRefGoogle Scholar
  19. [19]
    G. Mandl and R. Fernandez Luque, Fully developed plastic shear flow of granular materials, Géotechnique, 20 (1970), pp. 277–307.CrossRefGoogle Scholar
  20. [20]
    M.M. Mehrabadi and S.C. Cowin, Initial planar deformation of dilatant granular materials, J. Mech. Phys. Solids, 26 (1978), pp. 268–284.CrossRefMathSciNetGoogle Scholar
  21. [21]
    M.M. Mehrabadi And S.C. Cowin, On the double sliding free-rotating model for the deformation of granular materials, J. Mech. Phys. Solids, 29 (1981), pp. 269–282.CrossRefMathSciNetGoogle Scholar
  22. [22]
    Z. Mroz, Proc. 15th IUTAM Congress, Toronto, ed. F.P.J. Rimrott and B. Tabarrok, North Holland, Amsterdam (1980), pp. 119–132, Deformation and flow of granular materials.Google Scholar
  23. [23]
    Z. Mroz and CZ. Szymanski, Non-associated flow rules in description of plastic flow of granular materials, in “Limit analysis and rheological approach to soil mechanics”, CISM Course 217, Udine (1979), pp. 50–94.Google Scholar
  24. [24]
    D.B. Pitman and D.G. Schaeffer, Stability of time dependent compressible granular flow in two dimensions, Comm. Pure Appl. Maths., 40 (1987), pp. 421–447.CrossRefzbMATHMathSciNetGoogle Scholar
  25. [25]
    K.H. Roscoe, A.N. Schofield and C.P. Wroth, On the yielding of soils, Geotechnique, 8 (1958), pp. 22–53.CrossRefGoogle Scholar
  26. [26]
    D.G. Schaeffer, Instability in the evolution equations describing incompressible granular flow, J. Diff. Equations 66 (1987), pp. 19–50.CrossRefzbMATHMathSciNetGoogle Scholar
  27. [27]
    A.N. Schofield and C.P. Wroth, Critical State Soil Mechanics, McGraw-Hill, London (1968).Google Scholar
  28. [28]
    R.T. Shield, Mixed boundary value problems in soil mechanics, Q. Appl. Math., 11 (1953), pp. 61–75.zbMATHMathSciNetGoogle Scholar
  29. [29]
    V.V. Sokolovskii, Statics of Granular Media, Pergamon Press, Oxford (1965).Google Scholar
  30. [30]
    A.J.M. Spencer, A theory of the kinematics of ideal soils under plane strain conditions, J. Mech. Phys. Solids, 12 (1964), pp. 337–351.CrossRefzbMATHMathSciNetGoogle Scholar
  31. [31]
    A.J.M. Spencer, Deformation of ideal granular materials, Mechanics of Solids, The Rodney Hill 60th Anniversary Volume, eds. H.G. Hopkins and M.J. Sewell, Pergamon Press, Oxford (1981), pp. 607–652.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • Ian F. Collins
    • 1
  1. 1.Department of Engineering ScienceUniversity of AucklandAucklandNew Zealand

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