Advertisement

Strain Localization in Granular Materials

  • I. Vardoulakis
Conference paper
Part of the International Centre for Mechanical Sciences book series (CISM, volume 385)

Abstract

Contemporary localization theory [99] is a natural extension of Mohr’s [50] original strength theory published in the year 1900 in a milestone paper with the title, “Welche Umstände bedingen die Elastizitätsgrenze and den Bruch eines Materials?”. Mohr’s question cannot be answered without resorting to experiments carefully and systematically run. Experiments, however do not give definite answers, since they are always subject to theoretical interpretation: In order to arrive to some conclusion one needs a theoretical framework within which the experiment is run and interpreted. In that sense Mohr’s fundamental geometrical theory of stress analysis provided a useful tool for engineering design. Fig. 1 is taken from Mohr’s paper and is usually referred to as the graphical representation of the ‘Mohr-Coulomb’ failure criterion, although in Mohr’s original paper no explicit reference to Coulomb’s work [16] is made.

Keywords

Shear Band Granular Material Strain Localization Yield Surface Granular Medium 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arthur, J.R.F., Dunstan, T., Al-Ani, Q.A.J. and Assadi, A.: Plastic deformation and failure of granular media. Géotechnique, 27 (1977), 53–74.CrossRefGoogle Scholar
  2. 2.
    Bardet, J.P.:Orientation of shear bands in frictional soils. J. Eng. Mech. ASCE, 117, (1991), 1466–1484.CrossRefGoogle Scholar
  3. 3.
    Bardet, J.-P., Proubet, J.: A numerical investigation of the structure of persistent shear bands in granular media. Géotechnique, 41 (1991), 599–613.CrossRefGoogle Scholar
  4. 4.
    Beatty, M.F.: Some static and dynamic Implications of the general theory of elastic stability. Arch. Rat. Mech. Anal., 10 (1966), 167–186.MathSciNetGoogle Scholar
  5. 5.
    Benallal, A.. Bilardon, R. and Geymonat G.: Conditions de bifurcation à l’interieur et aux frontières pour une classe de matériaux non standards. Acad. Sci., Paris, 308, série II (1989), 893–898.Google Scholar
  6. 6.
    Bigoni, D. and Hueckel, T.: A note on strain localization for a class of non-associative plasticity ru;es. Ingenieur Archiv, 60 (1990), 491–499.Google Scholar
  7. 7.
    Bigoni, D. and Hueckel, T.: Uniqueness and localization-I. Associative and non-associative plasticity. Int. J. Soilds Structures, 28 (1991), 197–213.CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Bigoni, D. and Zaccaria, D.: Loss of strong ellipticity in non-associativive elastoplasticity. J. Mech. Phys. Solids, 40 (1992), 1313–1331.CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Bigoni, D. and Zaccaria, D.: Strong ellipticity of comparison solids in elastoplasticity with volumetric non-associativity. Int. J. Soilds Structures, 29 (1992), 2123–2136.CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Biot, M.A.: Mechanics of Incremental Deformations, Wiley, 1965.Google Scholar
  11. 11.
    Bogdanova-Boncheva, N. and Lipmann, H.: Rotationsymmetrisches ebenes Fliessen eines granularen Modellmaterials. Acta Mechanica, 21 (1975), 93–113.CrossRefGoogle Scholar
  12. 12.
    Boulon, M.: Basic features of soil-structure interface bahavior. Computers and Geotechnics, 7 (1989).Google Scholar
  13. 13.
    Bransby, P.L. and Milligan G.W.E.: Soil deformations near cantilever sheet pile walls, Géotechnique, 25 (1975), 175–195.CrossRefGoogle Scholar
  14. 14.
    Chambon, R., Caillerie, D. and El Hassan N.: Etude de la localisation unidimensionelle à l’aide d’un modèle de second gradient. C.R. Acad. Sci. Paris, Série II b (1996), 231–238.Google Scholar
  15. 15.
    Christoffersen, J.: Hyperelastic relations with isotropic forms appropriate for elastoplasticity. Eur. J. Mech. A/Solids., 10 (1991), 91–99.MATHGoogle Scholar
  16. 16.
    Coulomb Charles-Augustin: Essai Sur une application des règles de Maximis & Minimis à quelques Problèmes de Statique, relatifs à l’Architecture, 1773. In: J. Heyman, Coulomb’s Memoir on Statics, Cambridge University Press, 1972.Google Scholar
  17. 17.
    Cundall, P.A. and Strack, O.D.L.: A discrete numerical model for granular assemblies. Géotechnique, 29 (1979), 47–65.CrossRefGoogle Scholar
  18. 18.
    De Josselin de Jong: Rowe’s stress-dilatancy relation based on friction. Géotechnique, 26 (1976), 527–534.CrossRefGoogle Scholar
  19. 19.
    Desrues, J.: La Localization de la Déformation dans les Matériaux Granulaires. Thése de Doctorat es Science, USMG & INPG, Grenoble, 1984.Google Scholar
  20. 20.
    Desrues J.: Shear band initiation in granular materials: Experimentation and theory. In: Geomaterials: Constitutive Equations and Modelling (Ed. F. Darve ) 1987.Google Scholar
  21. 21.
    Desrues, J. and Hammad, W.: Etude expérimentale de la localisation de la déformation sur sable:Influence de la contrainte moyenne. 12th I.C.S.M.F.E., 1/9 (1989), 31–32.Google Scholar
  22. 22.
    Desrues, J., Mokni, M. and Mazerolle, F.: Tomodensitométrie et la localisation sur les sables. 10th E.C.S.M.F.E., (1991), 61–64.Google Scholar
  23. 23.
    Desrues, J., Chambon, R., Mokni, M. and Mazerolle, F.: Void ratio evolution insideshear bands in triaxial sand specimens studied by computed tomography. Géotechnique, 46 (1996), 529–546.CrossRefGoogle Scholar
  24. 24.
    Dietrich, Th.: Der Psammische Stoff als mechanisches Modell des Sandes. Dissertation Universität Karlsruhe 1976.Google Scholar
  25. 25.
    Doris, J.F. and Nemat-Nasser, S.: Instability of a layer on a half space. J. Appl. Mech., 102 (1980), 304–312.CrossRefGoogle Scholar
  26. 26.
    Drescher, A., Vardoulakis, I. and Chunhua Han: A Biaxial Apparatus for testing Soils. Geotechnical Testing Journal, GTJODJ, 13 (1990), 226–234.Google Scholar
  27. 27.
    Drucker, D.C.: A more fundamental approach to stress-strain relations. Proc. U.S. National Congress of Applied Mechanics, ASME (1951), 487–491.Google Scholar
  28. 28.
    Emeriault, F. and Cambou, B.: Micromechanical modelling of anisotropie non-linear elasticity of granular medium. Int. J. Solids Structures, 33 (1996), 2591–2607.CrossRefMATHGoogle Scholar
  29. 29.
    Germain P.: La méthode des puissances vituelles en mécanique des milieux continus. Part I, J. de Mécanique, 12 (1973), 235–274.MATHMathSciNetGoogle Scholar
  30. 30.
    Germain P.: The Method of virtual power in continuum mechanics. Part 2: Microstructure. SIAM J. Appl. Math, 25 (1973), 556–575.Google Scholar
  31. 31.
    Gudehus, G.: Elastic-plastic constitutive equations for dry sand. Arch. Mech. Stosowanej, 24 (1972), 395–402.MATHGoogle Scholar
  32. 32.
    Hadamard, J.: Leçons sur la propagation des ondes at les équations de l’hydrodynamique. Paris:Hermann 1903. Reprinted New York: Chelsea Publishing Co. 1949.Google Scholar
  33. 33.
    Hill, R.: Mathematical Theory of Plasticity, Clarendon Press, Oxford, 1950.MATHGoogle Scholar
  34. 34.
    Hill, R.: Eigenmodal deformation in elastic/plastic continua. J. Mech. Phys. Solids, 15, (1958), 371–386.CrossRefGoogle Scholar
  35. 35.
    Hill R.: Acceleration waves in solids. J. Mech. Phys. Solids, 10 (1962), 1–16CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Hill, R. and Hutchinson, J.W.: Bifurcation phenomena in the plane tension test. J. Mech. Phys. Solids, 23 (1975), 239–264.CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Hubbert, M.K.: Mechanics of deformation of crustal rocks: Historical development. In: Mechanical Behavior of Crustal Rocks-The Handin Volume (Ed. N.L. Carter et al.) Americal Geophysical Union, 1981, 1–9.Google Scholar
  38. 38.
    Hutchinson, J.W.: Finite strain analysis of elastic-plastic solids and structures.In: Numerical Solution of Nonlinear Structural Problems (Ed. R.F. Hartung) ASME, 1973.Google Scholar
  39. 39.
    Jean, M.: Frictional contact in collections of rigid or deformable bodies: a numerical simulation of geomaterial motion. In: Mechanics of Geomaterials Interfaces, (Ed. A.P.S. Selvadurai ) 1995, Elsevier.Google Scholar
  40. 40.
    Ken-Ichi Kanatani: A micropolar continuum theory for the flow of granular matereals. Int. J. Engng. Sci., 17 (1979), 419–432.CrossRefMATHGoogle Scholar
  41. 41.
    Lerat, P., Schlosser, F., and Vardoulakis, I.: Nouvel appareil de ciseillement pour l’étude des interfaces matériau granulaire-structure. 14t ICSMFE, ( 1997, in print.Google Scholar
  42. 42.
    Loret, B., Prevost, J.H. and Harireche, O.: Loss of hyperbolicity in elastic-plastic solids with deviatoric associativity. Eur. J. Mech., A/Solids, 9 (1990), 225–231.MATHMathSciNetGoogle Scholar
  43. 43.
    Lloyd, G.E.R.: Early Greek Science, Norton, 1970.Google Scholar
  44. 44.
    Mandel, J.: Ondes plastiques dans un milieu indéfini à trois dimensions. Journal de Méchanique, 1 (1962), 3–30.MathSciNetGoogle Scholar
  45. 45.
    Mandel, J.: Conditions de stabilité et postulat de Drucker. In: Rheology and Soil Mechanics, Springer, 58–67, 1966.Google Scholar
  46. 46.
    Mandel, J.: Propagation des surfaces de discontinuite dans un milieu elasto-plastique. In: Stress Waves in Anelastic Solids, Springer, Berlin, 1964, 331–341.CrossRefGoogle Scholar
  47. 47.
    McMeeking, R. M. and Rice, J.R.: Finite-element formulations for problems of large elastic-plastic deformation. Int. J. Solids Struct„ 11 (1975), 601–616CrossRefMATHGoogle Scholar
  48. 48.
    Mindlin, R D • Micro-structure in linear elasticity. Arch. Rat. Mech. Anal., 10 (1964), 5177.Google Scholar
  49. 49.
    Mohr, O. Welche Umstände bedingen die Elastizitätsgrenze und den Bruch eines Materials? Zeitschrift des Vereines deutscher Ingenieure, 44 (1900), 1–12.Google Scholar
  50. 50.
    Molenkamp, F.: Comparisn of frictional material models with resoect to shear band initiation. Géotechnique, 35 (1985), 127–143.CrossRefGoogle Scholar
  51. 51.
    Mroz, Z. (1963). Non-associate flow laws in plasticity. J. de Mécanique, 2, 21–42.MathSciNetGoogle Scholar
  52. 52.
    Mroz, Z. Mathematical Models of Inelastic Behavior. University of Waterloo Press, 1973Google Scholar
  53. 53.
    Mühlhaus, H.-B. and Aifantis, E.C.: A variational principle for gradient plasticity. Int. J. Solids Structures, 28 (1991), 845–857.CrossRefMATHGoogle Scholar
  54. 54.
    Mühlhaus, H.-B. and Vardoulakis, I.: The thickness of shear bands in granular materials. Géotechnique, 37 (1987), 271–283.CrossRefGoogle Scholar
  55. 55.
    Nguyen, Q.S. and Bui H.D.: Sur les matériaux élastoplastiques à écrouissage positif ou négatif. Journal de Méchanique, 3 (1974), 322–432.Google Scholar
  56. 56.
    Ockendon, H. and Ockendon, J.R.: Viscous Flow. Cambridge University Press, 1995.Google Scholar
  57. 57.
    Oda, M., Kazama, H. and Konishi, J.: Effects of induced anisotropy on the development of shear bands in granular materials. Mechanics of Materials, 25 (1997), in print.Google Scholar
  58. 58.
    Ord, A., Vardoulakis, I. and Kajewski, R.: Shear band formation in Gosford Sandstone. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr,, 28 (1991), 397–409.CrossRefGoogle Scholar
  59. 59.
    Ortiz, M., Leroy, Y. and Needleman, A.: A finite element method for localized failure analysis. Comput.Meth. Appl. Mech. Eng., 61 (1987), 189–194.CrossRefMATHGoogle Scholar
  60. 60.
    Papamichos, E. Labuz, J.F. and Vardoulakis, I.: A surface instability detection apparatus. Rock Mech. And Rock Eng., 27 (1994), 37–56.CrossRefGoogle Scholar
  61. 61.
    Papamichos, E. and Vardoulakis, I.: Effect of confining pressure in shear band formation in sand.Géotechnique, 45 (1994), 649–661.Google Scholar
  62. 62.
    Raniecki, B. and Bruhns, O.T.: Bounds to bifurcation stress in solids with non-associated plastic flow law at finite strain. J. Mech. Phys. Solids, 29 (1981), 153–172.CrossRefMATHMathSciNetGoogle Scholar
  63. 63.
    Reynolds, O.: On the dilatancy of media composed of rigid particles in contact. With experimental illustrations. Phil. Mag. (2) 20 (1885), 469–481. Also: Truesdell, C. and Noll, W.: The Non-Linear Field Theories of Mechanics, Handbuch der Physik Band I1I/3, section 119, Springer 1965.Google Scholar
  64. 64.
    Rice, J.R.: The localization of plastic deformation. Theoretcal and Applied Mechanics (Ed. by W.T. Koiter), Proc. 14th IUTAM Congress, Delft, 1977, 207–221.Google Scholar
  65. 65.
    Rice, J.R. and Rudnicki, J.: A note on some features of the theory of localization of deformation. Int. J. Solids Structures, 16 (1980), 597–605.CrossRefMATHMathSciNetGoogle Scholar
  66. 66.
    Roscoe, K.H.: The influence of strains in Soil Mechanics. Géotechnique, 20 (1970), 129–170.CrossRefGoogle Scholar
  67. 67.
    Rowe, P.W.: The stress-dilatancy relation for static equilibrium of an assembly of particles in contact. Proc. Roy. Soc., 269 (1962), 500–527.CrossRefGoogle Scholar
  68. 68.
    Rowe, P.W.: Theoretical meaning and observed values of deformation parametrs for soil. Proc. Roscoe Mem. Symp., Cambridge, (1971), 143–194.Google Scholar
  69. 69.
    Rudnicki, J. and Rice, J.R.: Conditions for the localization of deformation in pressure-sensitive dilatant materials. J. Mech. Phys. Solids, 23 (1975), 371–394.CrossRefGoogle Scholar
  70. 70.
    Scarpelli, G. and Wood, D.M.: Experimental observations of shear band patterns in direct shear tests. In: Deformation and Failure of Granular Materials, Balkema, (1982), 473–484.Google Scholar
  71. 71.
    Shanley, R.F.: Inelastic column theory. J. Aeronaut., 4 (1947), 101–104.Google Scholar
  72. 72.
    Shield, R.T.: Mixed boundary value problems in soil mechanics, Q. Appl. Math., 11 (1953), 61–75.MATHMathSciNetGoogle Scholar
  73. 73.
    Sommerfeld, A.: Mechanik der deformierbaren Medien. Verlag Harri Deutsch (1948) 1992.Google Scholar
  74. 74.
    Tatsuoka F., Nakamura S., Huang C.-C. and Tani K.: Strength anisotropy and shear band direction in plane strain tests on sands. Soils and Foundations, 30 (1990), 35–54.CrossRefGoogle Scholar
  75. 75.
    Taylor, D.W.: Fundamentals of Soil Mechanics, John Wiley, 1948.Google Scholar
  76. 76.
    Tejchman, J.: Scherzonenbildung und Verspanungseffekte in Granulaten unter Berücksichtigung von Korndrehung. Dissertation Univesität Krlsruhe, 1990.Google Scholar
  77. 77.
    Tejhman, J. and Wei Wu: Experimental and numerical study of sand-stell interfaces. Int. J. Num. Anal. Meth. Geomech., 19 (1995), 513–536.CrossRefGoogle Scholar
  78. 78.
    Thomas T.Y.: Plastic Flow and Fracture, Vol. 2., Academic Press, 1961.Google Scholar
  79. 79.
    Truesdell C. and Noll W.: Nonlinear Field Theories of Mechanics, Handbuch der Physik, Vol. III/3, Sections 68, 68bis, 69, 99, 100, Springer 1965.Google Scholar
  80. 80.
    Unterreiner, F. and Vardoulakis, I.: Interfacial localisation in granular media. Computer Methods in Geomechanics (Ed. Siriwardane & Zaamn ), Balkema, (1994), 1711–1715Google Scholar
  81. 81.
    Vardoulakis I.: Berechnungsverfahren fair Erdkörper mit plastischer Ver-and Entfestigung: Entstehung and Ausbreitung von Scherfugen. DFG Report GU 103 /16, 1974.Google Scholar
  82. 82.
    Vardoulakis I.: Die lineare Näherung des Prinzipes der virtuellen Verschiebungen. Int. Conf. on Num. Meth. in Soil and Rock Mech., Karlsruhe 1975, Universität Karlsruhe, (Ed. G. Borm and H. Meissner ) (1975), 39–46.Google Scholar
  83. 83.
    Vardoulakis I.: Equilibrium theory of the shear bands in plastic bodies. Mech. Res. Comm., 3 (1976), 209–214.CrossRefMATHGoogle Scholar
  84. 84.
    Vardoulakis I.: Shear band inclination and shear modulus of sand in biaxial test. Int. J. Num. Anal. Meth. in Geomechanics, 4 (1980), 103–119.CrossRefMATHGoogle Scholar
  85. 85.
    Vardoulakis, I.: Stability and bifurcation of undrained plane rectilinear deformations on water-saturated granular soils. Int. J. Num. Anal. Meth. Geomechanics, 9 (1985), 399414.Google Scholar
  86. 86.
    Vardoulakis, I.: Dynamic stability of undrained simple shear on water saturated granular soils. Int. J. Num. Anal. Meth. Geomechanics, 10 (1986), 177–190.CrossRefMATHGoogle Scholar
  87. 87.
    Vardoulakis I.: Theoretical and experimental bounds for shear-band bifurcation strain in biaxial tests on dry sand. Res Mechanica, 23 (1988), 239–259.Google Scholar
  88. 88.
    Vardoulakis, I.: Shear-banding and liquefaction in granular materials on the basis of a Cosserat continuum theory. Ingenieur Archiv, 59 (1989), 106–113.CrossRefGoogle Scholar
  89. 89.
    Vardoulakis, I.: Potentials and limitations of softening models in Geomechanics (the role of second order work). Eur. J. Mech. A/Solids, 13 (1994), 195–226.MATHMathSciNetGoogle Scholar
  90. 90.
    Vardoulakis, I.: Deformation of water saturated sand: I. Uniform undrained deformation and shear banding. Géotechnique, 46 (1995), 441–456.CrossRefGoogle Scholar
  91. 91.
    Vardoulakis, I.: Deformation of water saturated sand: II. The effect of pore-water flow and shear banding. Géotechnique, 46 (1995), 457–472.CrossRefGoogle Scholar
  92. 92.
    Vardoulakis, I. and Aifantis, E.C.: A gradient flow theory of plasticity for granular materials. Acta Mechanica, 87 (1991), 197–217.CrossRefMATHMathSciNetGoogle Scholar
  93. 93.
    Vardoulakis, I. and Frantziskonis, G.: Micro-structure in kinematic-hardening plasticity. Eur. J. Mech./Solids, 11 (1992), 467–486.MATHGoogle Scholar
  94. 94.
    Vardoulakis I. and Goldscheider M.: A biaxial apparatus for testing shear bands in soils. 10th Int. Conf. Soil Mech. Found. Engineering, Stockholm 1981, Vol. 4/61, (1981), 819–824, A.A. BalkemaGoogle Scholar
  95. 95.
    Vardoulakis, I. and Graf B.: Imperfection sensitivity of the biaxial test on dry sand. IUTAM Conf. on Deformation and Failure of Granular Materials, Delft 1982, 485–491, A.A. Balkema.Google Scholar
  96. 96.
    Vardoulakis, I. and Graf, B.: Calibration of constitutive models for granular materials using data from biaxial experiments, Géotechnique, 35 (1985), 299–317.CrossRefGoogle Scholar
  97. 97.
    Vardoulakis I., Graf B. and Hettler A.: Shear-band formation in a fine-grained sand. 5th Int. Conf. Num. Methods in Geomechanics, Nagoya 1985, 517–522, A.A. Balkema.Google Scholar
  98. 98.
    Vardoulakis, I., Shah, K.R. and Papanastasiou P.: Modelling of tool-rock interfaces using gradient-dependent flow theory of plasticity. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 29 (1992), 573–582.CrossRefGoogle Scholar
  99. 99.
    Vardoulakis I. and Sulem J.: Bifurcation Analysis in Geomechanics, Blackie Academic and Professional, 1995.Google Scholar

Copyright information

© Springer-Verlag Wien 1998

Authors and Affiliations

  • I. Vardoulakis
    • 1
  1. 1.National Technical University of AthensAthensGreece

Personalised recommendations