Bifurcation and Instability of Non-Associative Elastoplastic Solids

  • D. Bigoni
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 414)


Global and local uniqueness and stability criteria for elastoplastic solids with non-associative flow rules are presented. Hill’s general theory is developed in the form generalized by Raniecki to non-associativity. Local stability criteria are presented and systematically discussed in a critical way. These are: positive definiteness and non-singularity of the constitutive operator, and positive definiteness (strong ellipticity) and non-singularity (ellipticity) of the acoustic tensor. The former criteria are particularly relevant for homogeneous deformation of solids subject to all-round controlled nominal surface tractions. Dually, the latter criteria are particularly relevant for homogeneous deformation of solids subject to displacements prescribed on the entire boundary. Flutter instability as related to complex conjugate eigenvalues of the acoustic tensor is also briefly discussed.


Flow Rule Positive Definiteness Associative Flow Rule Strong Ellipticity Exclusion Condition 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. An, L. and Schaeffer, D. (1990). The flutter instability in granular flow. J. Mech. Phys. Solids 40: 683–698.MathSciNetCrossRefGoogle Scholar
  2. Beatty, M.F. (1987). Topics in finite elasticity: hyperelasticity of rubber, elastomers, and biological tissues-with examples. Appl. Mech. Rev. 40: 1699–1734.CrossRefGoogle Scholar
  3. Benallal, A. Billardon, R. and Geymonat, G. (1990). Phènoménes de localisation a la frontière d’un solide. C. R. Acad. Sci., Paris 310: 670–684.MathSciNetGoogle Scholar
  4. Bigoni, D. (1995). On flutter instability in elastoplastic constitutive models. Int. J. Solids Structures 32: 3167–3189.MathSciNetCrossRefMATHGoogle Scholar
  5. Bigoni, D. (1996). On smooth bifurcations in non-associative elastoplasticity. J. Mech. Phys. Solids 44: 1337–1351.CrossRefGoogle Scholar
  6. Bigoni, D. and Hueckel, T. (1990). A note on strain localization for a class of non-associative plasticity rules. Ingenieur-Archiv 60: 491–499.Google Scholar
  7. Bigoni, D. and Hueckel, T. (1991a). Uniqueness and localization-I. Associative and non-associative elastoplasticity. Int. J. Solids Structures 28: 197–213.MathSciNetCrossRefMATHGoogle Scholar
  8. Bigoni, D. and Hueckel, T. (1991b). Uniqueness and localization-II. Coupled elastoplasticity. Int. J. Solids Structures 28: 215–224.MathSciNetCrossRefMATHGoogle Scholar
  9. Bigoni, D. and Laudiero, F. (1989). The quasi-static finite cavity expansion in a non-standard elastoplastic medium. Int. J. Mech. Sci. 31: 825–837.CrossRefGoogle Scholar
  10. Bigoni, D. and Loret, B. (1999). Effects of elastic anisotropy on strain localization and flutter instability in plastic solids. J. Mech. Phys. Solids 47: 1409–1436.MathSciNetCrossRefMATHGoogle Scholar
  11. Bigoni, D., Loret, B. and Radi, E. (2000). Localization of deformation in plane elastic-plastic solids with anisotropic elasticity. J. Mech. Phys. Solids, Special issue dedicated to Prof. J.R. Willis,in press.Google Scholar
  12. Bigoni, D. and Willis, J.R. (1994). A dynamical interpretation of flutter instability. In: Chambon, R., Desrues, J. and Vardoulakis, I., eds., Localisation and Bifurcation of Rocks and Soils Rotterdam: A.A. Balkema Scientific Publishers. 51–58.Google Scholar
  13. Bigoni, D. and Zaccaria, D. (1992a). Strong ellipticity of comparison solids in elastoplasticity with volumetric non-associativity. Int. J. Solids Structures 29: 2123–2136.MathSciNetCrossRefMATHGoogle Scholar
  14. Bigoni, D. and Zaccaria, D. (1992b). Loss of strong ellipticity in non-associative elastoplasticity. J. Mech. Phys. Solids 40: 1313–1331.MathSciNetCrossRefMATHGoogle Scholar
  15. Bigoni, D. and Zaccaria, D. (1994a). On eigenvalues of the acoustic tensor in elastoplasticity. Eur. J. Mechanics-A/Solids 13: 621–638.MathSciNetMATHGoogle Scholar
  16. Bigoni, D. and Zaccaria, D. (1994b). Eigenvalues of the elastoplastic constitutive operator. ZAMM 74: 355–357.MathSciNetCrossRefMATHGoogle Scholar
  17. Biot, M.A. (1965) Mechanics of incremental deformations. New York: Wiley.Google Scholar
  18. Boehler, J.P. and Willis J.R. (1991). An analysis of localization in highly pre-deformed sheet steel. Unpublished.Google Scholar
  19. Brannon, R.M. and Drugan, W.J. (1993). Influence of non-classical elastic-plastic constitutive features on shock wave existence and spectral solutions. J. Mech. Phys. Solids 41: 297–330.MathSciNetCrossRefMATHGoogle Scholar
  20. Bruhns, O. and Raniecki, B. (1982). Ein Schrankenverfahren bei Verzweigungsproblemen in-elastischer Formänderungen. ZA MM 62: T111–T113.MATHGoogle Scholar
  21. Cattaneo, C. (1946). Su un teorema fondamentale nella teoria delle onde di discontinuità. Atti Acad. Naz. Lincei (parts I and II) I:67–72 and 728–734.Google Scholar
  22. Chadwick, P. and Powdrill B. (1965). Singular surfaces in linear thermoelasticity. Int. J. Eng. Science 3: 561–595.MathSciNetCrossRefGoogle Scholar
  23. Chau, K.T. (1992). Non-normality and bifurcation in a compressible pressure-sensitive circular cylinder under axisymmetric tension and compression. Int. J. Solids Structures 29: 801–824.CrossRefMATHGoogle Scholar
  24. Chau, K.T. (1995). Buckling, barrelling, and surface instabilities of a finite, transversely isotropic circular cylinder. Quart. Appl. Math. 53: 225–244.MathSciNetMATHGoogle Scholar
  25. Chau, K.T. and Rudnicki, J.W. (1990). Bifurcations of compressible pressure-sensitive materials in plane strain tension and compression. J. Mech. Phys. Solids 38: 875–898.CrossRefMATHGoogle Scholar
  26. Cheng, Y.S. and Lu, W.D. (1993). Uniqueness and bifurcation in elastic-plastic solids. Int. J. Solids Structures 30: 3073–3084.MathSciNetCrossRefMATHGoogle Scholar
  27. Christoffersen, J. (1991). Hyperelastic relations with isotropic rate forms appropriate for elastoplasticity. Eur. J. Mechanics-A/Solids 10: 91–99.MATHGoogle Scholar
  28. Piero, G. (1979). Some properties of the set of fourth-order tensors, with applications to elasticty. J. Elasticity 9: 245–261.MathSciNetCrossRefMATHGoogle Scholar
  29. Drucker, D.C. (1954). Coulomb friction plasticity and limit ‘dads. J. Appl. Mech. 76: 71–74.Google Scholar
  30. Curtin, M.E. (1972). The linear theory of Elasticity. In Fliigge, S., ed., Encyclopedia of Physics VIa/2. Berlin: Springer. 1–295.Google Scholar
  31. Curtin, M.E. (1981). An introduction to continuum mechanics. New York: Academic Press.Google Scholar
  32. Hayes, M. (1966). On the displacement boundary-value problem in linear elastostatics. Quart. J. Mech. Appl. Math. XIX: 151–155.MathSciNetCrossRefGoogle Scholar
  33. Hadamard, J. (1903). Leçons sur la Propagation des Ondes et les Équations de l’ Hydrodynamique. Paris: Hermann.MATHGoogle Scholar
  34. Hill, R. (1950). The mathematical theory of plasticity. Oxford: Clarendon Press.MATHGoogle Scholar
  35. Hill, R. (1952). On discontinuous plastic states, with special reference to localized necking in thin sheets. J. Mech. Phys. Solids 1: 19–30.MathSciNetCrossRefGoogle Scholar
  36. Hill, R. (1958). A general theory of uniqueness and stability in elastic-plastic solids. J. Mech,. Phys. Solids 6: 236–249.CrossRefMATHGoogle Scholar
  37. Hill, R. (1959). Some basic principles in the mechanics of solids without a natural time. 1. Mech. Phys. Solids 7: 209–225.CrossRefMATHGoogle Scholar
  38. Hill, R. (1961). Discontinuity relations in mechanics of solids. In Sneddon, I.N. and 1Ii11, R., eds., Progress in Solid Mechanics II. Amsterdam: North-Holland. 247–276.Google Scholar
  39. Hill, R. (1962). Acceleration waves in solids. J. Mech. Phys. Solids 10: 1–16.MathSciNetCrossRefMATHGoogle Scholar
  40. Hill, R. (1967a). Eigenmodal deformations in elastic/plastic continua.1. Mech. Phys. Solids 15: 371–386.CrossRefGoogle Scholar
  41. Hill, R. (1967b). On the classical constitutive laws for elastic/plastic solids. In Broberg, B., ed., Recent Progress in Applied Mechanics, The Folke Odkvist Volume Stocklrolrn:Alingvist and Wiksell. 241–249.Google Scholar
  42. Hill, R. (1968). On constitutive inequalities for simple materials. J. Mech. Phys. Solids 16: 229–242.CrossRefMATHGoogle Scholar
  43. Hill, R. (1978) Aspects of invariance in solid mechanics. In Yih, C.-S., ed., Advances in. Applied Mechanics 18. New York: Academic Press. 1–75.Google Scholar
  44. Hill, R. and Hutchinson, J. W. (1975). Bifurcation phenomena in the plane tension test. J. Mech. Phys. Solids 23: 239–264.MathSciNetCrossRefMATHGoogle Scholar
  45. Hill, R. and Rice, J. R. (1973). Elastic potentials and the structure of inelastic constitutive laws. SIAM J. Appl. Math. 25: 448–461.MathSciNetCrossRefMATHGoogle Scholar
  46. Horgan, C.O. and Polignone, D.A. (1995). Cavitation in nonlinearly elastic solids: A review. Appi. Mech. Rev. 48: 471–485.CrossRefGoogle Scholar
  47. Huang, K., Hutchinson, J.W. and Tvergaard, V. (1991). Cavitation instabilities in elastic-plastic solids. J. Mech. Phys. Solids 39: 223–241.CrossRefGoogle Scholar
  48. Hueckel, T. (1976). Coupling of elastic and plastic deformation of bulk solids. Meccanica 11: 227–235.CrossRefMATHGoogle Scholar
  49. Hutchinson, J. W. (1973). Post-bifurcation behavior in the plastic range. J. Mech. Phys. Solids 21: 163–190.CrossRefMATHGoogle Scholar
  50. Hutchinson, J. W. and Miles, J.P. (1974). Bifurcation analysis of the onset of necking in an elastic/plastic cylinder under uniaxial tension. J. Mech. Phys. Solids 22: 61–71.CrossRefMATHGoogle Scholar
  51. Kleiber, M. (1984) Numerical study on necking-type bifurcations in void-containing elasticplastic material. Int. J. Solids Structures 20: 191–210.CrossRefMATHGoogle Scholar
  52. Kleiber, M. (1986) On plastic localization and failure in plane strain and round void containing tensile bars. Int. J. Plasticity 2: 205–221.CrossRefMATHGoogle Scholar
  53. Loret, B. (1992). Does deviation from deviatoric associativity lead to the onset of flutter instability?. 1. Mech. Phys. Solids 40: 1363–1375.MathSciNetCrossRefMATHGoogle Scholar
  54. Loret, B., Martins, J.A.C. and Simes, F.M.F. (1995). Surface boundary conditions trigger flutter instability in non-associative elastic-plastic solids. Int. J. Solids Structures 32: 2155–2190.CrossRefMATHGoogle Scholar
  55. Loret, B., Prevost, J.H. and Harireche, O. (1990). Loss of hyperbolicity in elastic-plastic solids with deviatoric associativity. Ear. J. Mechanics-A/Sol ds 9: 225–231.MathSciNetMATHGoogle Scholar
  56. Maier, G. and Hueckel, T. (1979). Non associated and coupled flow-rules of elastopla,sticity for rock-like materials. Int., I. Rock 1llech. Min. Sci. 16: 77–92.CrossRefGoogle Scholar
  57. Mandel, J. (1966). Conditions de stabilité et postulat de Drucker. In Kravtchenko, J. and Sirieys. P.M., eds., Rheology and Soil Mechanics. Berlin: Springer. 58–68.Google Scholar
  58. Mclan, E. (1938). Zur Plastizitiit des räumliche Kontiunuuns. Ingcnicus-.4rchie 9: 116–126.Google Scholar
  59. Miles, J.P. (1973). Fluid-pressure cigeastates and bifurcation in tension specimens under lateral pressure. J. Mech. Phys. Solids 21: 145–162.CrossRefMATHGoogle Scholar
  60. Miles, J.P. and Nnwayhid, U.A. (1985). Bifurcation in compressible elastic/plastic cylinders under nniaxial tension. Appl. Sci. lies. 42: 33–514.MATHGoogle Scholar
  61. Mrdz, Z. (1963). Non-associated flow laws in plasticity.1. de Mechaniguc 2: 21–42.Google Scholar
  62. Mrdz, Z. (1966). On fornns of constitutive laws for elastic-plastic solids. Arch. Alcch. Stesowane) 18: 1–34.Google Scholar
  63. Nadai, A. (1931) New York:MrGi:nv-Ilill.Google Scholar
  64. Nadai, A. (1950) Theory of flow and frn.elon of solids. New York: McGraw-Hill.Google Scholar
  65. Neale, K.W. (1981). Phenomenological constitutive laws in finite plasticity SM Archives 6: 79–128.MATHGoogle Scholar
  66. Needleman, A. (1979). Non-normality and bifurcation in plane strain tension or compression. 1. Mech. Phys. Solids 27: 231–2514.MathSciNetCrossRefMATHGoogle Scholar
  67. Needleman, A. and Ortiz, M. (1991). Elfects of ularies ant interfaces on shear-band localization. Int. J. Solids Structures 28: 859–877.CrossRefMATHGoogle Scholar
  68. Nguyen, S.Q. and Ttiantafyllidis, N. (1989). Plastic bifurcation and postbifurca,tion analysis for generalized standard continua. 1. Alcch. Plrys. Solids 37: 515–566.Google Scholar
  69. Nikolaevskii, V.N. and Rice, H. (1979). Current. topics in non-elastic deformation of geological materials. lu ‘l’inunerhans, N.D. and Barber, ALS., eds., Proocedirrgs of the Smith. AIR:I PT C.’onfercnce: lligh. Iiïssarc Science and Technology. New York: Plenuum. 2: 455–464.Google Scholar
  70. Ogden, R.W. (1984). Non-linear elastic deformations. Chichester:Ellis Norwood Google Scholar
  71. Ogden, W. (1985). Local:unl global bifurcation phenomena in plane-strain finite elasticity. ha. J. Solids Structures 21: 121–132.CrossRefMATHGoogle Scholar
  72. Ottosen, N.S. and Bunesson, H. (1991). Acceleration waves in elastoplasticit.y. Irrt. J. Solids Stractnres 28: 135–159.CrossRefMATHGoogle Scholar
  73. Petryk, H. (1985a). On energy criteria of plastic instability. In Plastic Instability, Proc. Considère Memorial. Paris: Ecole Nat. Ponts Chauss. Press. 215–226.Google Scholar
  74. Petryk, H. (1985b). On stability and symmetry conditions in time-independent plasticity. Arch. Mech. 37: 503–520.MATHGoogle Scholar
  75. Petryk, H. (1991). The energy criteria of instability in time-independent inelastic solids. Arch. Mech. 43: 519–545.MathSciNetMATHGoogle Scholar
  76. Petryk, H. (1992). Material instability and strain-rate discontinuities in incrementally nonlinear continua. J. Mech. Phys. Solids 40: 1227–1250.MathSciNetCrossRefMATHGoogle Scholar
  77. Petryk, H. (1993a). Theory of bifurcation and instability in time-independent plasticity. In Nguyen, Q.S., ed., CISM Lecture Notes No. 327, Udine 1991. Wien: Springer. 95–152.Google Scholar
  78. Petryk, H. (1993b). Stability and constitutive inequalities in plasticity. In Muschik, W., ed., CISM Lecture Notes No. 336, Udine 1992. Wien: Springer. 255–329.Google Scholar
  79. Petryk, H. (1999). General conditions for uniqueness in materials with multiple mechanisms of inelastic deformation. J. Mech. Phys. Solids in press.Google Scholar
  80. Petryk, H. and Thermann, K. (1985). Second-order bifurcation in elastic-plastic solids. J. Mech. Phys. Solids 33: 577–593.MathSciNetCrossRefMATHGoogle Scholar
  81. Prager, W. (1954). Discontinuous fields of plastic stress and flow. In 2nd Nat. Congr. Appl. Mech., Ann Arbor, Michigan, 21–32.Google Scholar
  82. Radi, E., Bigoni, D. and Tralli, A. (1999). On uniqueness for frictional contact rate problems. J. Mech. Phys. Solids 47: 275–296.MathSciNetCrossRefMATHGoogle Scholar
  83. Raniecki, B. (1979). Uniqueness criteria in solids with non-associated plastic flow laws at finite deformations, Bull. Acad. Polon. Sci. ser. sci. tech. XXVII: 391–399.MathSciNetGoogle Scholar
  84. Raniecki, B. and Bruhns, O.T. (1981). Bounds to bifurcation stresses in solids with nonassociated plastic flow law at finite strain. J. Mech. Phys. Solids 29: 153–171.MathSciNetCrossRefMATHGoogle Scholar
  85. Rice, J. R. (1977). The localization of plastic deformation. In Koiter, W.T., ed., Theoretical and Applied Mechanics. Amsterdam: North-Holland. 207–220.Google Scholar
  86. Rice, J.R. and Rudnicki, J.W. (1980). A note on some features of the theory of localization of deformation. Int. J. Solids Structures 16: 597–605.MathSciNetCrossRefMATHGoogle Scholar
  87. Rudnicki, J.W. and Rice, J.R. (1975). Conditions for the localization of deformations in pressure-sensitive dilatant materials. J. Mech. Phys. Solids 23: 371–394.CrossRefGoogle Scholar
  88. Runesson, K. and Mróz, Z. (1989). A note on non-associated plastic flow rules. Int. J. Plasticity 5: 639–658.CrossRefMATHGoogle Scholar
  89. Ryzhak, E. I. (1987). Necessity of Hadamard conditions for stability of elastic-plastic solids. Izv. AN SSSR MTT (Mechanics of Solids) 99–102.Google Scholar
  90. Ryzhak, E. I. (1993). On stable deformation of “unstable” materials in a rigid triaxial testing machine. J. Mech. Phys. Solids 41: 1345–1356.CrossRefMATHGoogle Scholar
  91. Ryzhak, E. I. (1994). On stability of homogeneous elastic bodies under boundary conditions weaker than displacement conditions. Q. Jl. Mech. appl. Math. 47: 663–672.MathSciNetCrossRefMATHGoogle Scholar
  92. Simoes, F.M.F. (1997). Instabilities in non-associated problems of solid mechanics. Ph.D. Thesis, Technical University of Lisbon, in Portuguese.Google Scholar
  93. Szabo, L. (1994). Shear band formulation in finite elastoplasticity. Int. J. Solids Structures 31: 1291–1308.CrossRefMATHGoogle Scholar
  94. Thomas, T.Y. (1953). The effect of compressibility on the inclination of plastic slip bands in flat bars. Proc. Nat. Acad. Sci. 39: 266–273.MathSciNetCrossRefMATHGoogle Scholar
  95. Thomas, T.Y. (1961) Plastic flows and fracture of solids. New York: Academic Press.Google Scholar
  96. Tomita, Y., Shindo, A. and Fatnassi, A. (1988). Bounding approach to bifurcation point of annular plates with nonassociated flow law subjected to uniform tension at their outer edges. Int. J. Plasticity 4: 251–263.CrossRefMATHGoogle Scholar
  97. Truesdell, C. and Noll, W. (1965). The non-linear field theories of mechanics. In Flügge, S., ed., Encyclopedia of Physics:III/3. Berlin: Springer-Verlag.Google Scholar
  98. Tvergaard, V. (1982). Influence of void nucleation on ductile shear fracture at a free surface. J. Mech. Phys. Solids 30: 399–425CrossRefMATHGoogle Scholar
  99. Hove, L. (1947). Sur l’extension de la condition de Legendre du calcul des variations aux intégrales multiples à plusieurs fonctions inconnues. Proc. Sect. Sci. K. Akad. van Wetenschappen, Amsterdam, 50: 18–23.MATHGoogle Scholar
  100. Vardoulakis, I. (1981). Bifurcation analysis of the plane rectilinear deformation on dry sand samples. Int. J. Solids Structures 11: 1085–1101.CrossRefGoogle Scholar
  101. Vardoulakis, I. (1983). Rigid granular plasticity model and bifurcation in the triaxial test. Acta Mechanica 49: 57–79.CrossRefMATHGoogle Scholar
  102. Young, N.J.B. (1976). Bifurcation phenomena in the plane compression test. J. Mech. Phys. Solids 24: 77–91.CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 2000

Authors and Affiliations

  • D. Bigoni
    • 1
  1. 1.University of TrentoTrentoItaly

Personalised recommendations