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Tensile Instabilities at Large Strains

  • V. Tvergaard
Part of the International Centre for Mechanical Sciences book series (CISM, volume 327)

Abstract

After a short introduction to basic formulations for elastic-plastic and elastic-viscoplastic material behaviour at finite strains the conditions for uniqueness and bifurcation of the incremental solution are discussed. The strong dependence of bifurcation predictions on the constitutive model is emphasized, and specific analyses of tensile instabilities are mentioned. The effect of viscoplastic material behaviour on stability analyses is emphasized. Furthermore, studies of localization in shear bands are discussed, and some recent results for cavitation instabilities in highly constrained plastic flow are presented.

Keywords

Shear Band Yield Surface Void Nucleation Finite Strain Bifurcation Mode 
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.

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References

  1. 1.
    Hutchinson, J.W.: Plastic buckling, Advan. Appl. Mech., 14 (1974), 67–144.Google Scholar
  2. 2.
    Needleman, A. and Tvergaard, V.: Aspects of plastic post—buckling behaviour, Mechanics of Solids, The Rodney Hill 60th Anniversary Volume (eds. H.G. Hopkins and M.J. Sewell ), Pergamon Press, Oxford, (1982), 453–498.Google Scholar
  3. 3.
    Shanley, F.R.: Inelastic column theory, J. Aeronaut. Sci., 14 (1947), 261–267.CrossRefGoogle Scholar
  4. 4.
    Hill, R.: A general theory of uniqueness and stability in elastic—plastic solids, J. Mech. Phys. Solids, 6 (1958), 236–249.CrossRefMATHADSGoogle Scholar
  5. 5.
    Hill, R.: Bifurcation and uniqueness in nonlinear mechanics of continua, Problems of Continuum Mechanics, Soc. Ind. Appl. Math., Philadelphia, Pennsylvania, (1961), 155–164.Google Scholar
  6. 8.
    Ball, J.M.: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Phil. Trans. R. Soc. Lond., A306 (1982), 557–610.CrossRefMATHADSGoogle Scholar
  7. 9.
    Horgan, C.O. and Abeyaratne, R.: A bifurcation problem for a compressible nonlinearly elastic medium: growth of a microvoid, J. of Elasticity, 16 (1986), 189–200.CrossRefMATHMathSciNetGoogle Scholar
  8. 10.
    Huang, Y., Hutchinson, J.W. and Tvergaard, V.: Cavitation instabilities in elastic—plastic solids, J. Mech. Phys. Solids, 39 (1991), 223–241.CrossRefADSGoogle Scholar
  9. 11.
    Tvergaard, V., Huang, Y. and Hutchinson, J.W.: Cavitation instabilities in a power hardening elastic—plastic solid, Danish Center for Appl. Math. and Mech., Report No. 415 (1990).Google Scholar
  10. 12.
    Green, A.E. and Zerna, W.: Theoretical elasticity, Oxford University Press, Oxford (1968).Google Scholar
  11. 13.
    Budiansky, B.: Remarks on theories of solid and structural mechanics, in Problems of Hydrodynamics and Continuum Mechanics (eds. M.A. Lavrent’ev et al.), SIAM, Philadelphia (1969), 77–83.Google Scholar
  12. 14.
    Hutchinson, J.W.: Finite strain analysis of elastic—plastic solids and structures, in Numerical Solution of Nonlinear Structural Problems (ed. R.F. Hartung ), ASME, New York (1973), 17.Google Scholar
  13. 15.
    McMeeking, R.M. and Rice, J.R.: Finite-element formulations for problems of large elastic—plastic deformation, Int. J. Solids Structures, 11 (1975), 601–616.CrossRefMATHGoogle Scholar
  14. 16.
    Hutchinson, J.W.: Postbifurcation behavior in the plastic range, J. Mech. Phys. Solids, 21 (1973), 163–190.CrossRefMATHADSGoogle Scholar
  15. 17.
    Rudnicki, J.W. and Rice, J.R.: Conditions for localization of deformation in pressure—sensitive dilatant materials, J. Mech. Phys. Solids, 23 (1975), 371–394.CrossRefADSGoogle Scholar
  16. 18.
    Tvergaard, V.: Influence of void nucleation on ductile shear fracture at a free surface, J. Mech. Phys. Solids, 30 (1982), 399–425.CrossRefMATHADSGoogle Scholar
  17. 19.
    Raniecki, B. and Bruhns, O.T.: Bounds to bifurcation stresses in solids with non—associated plastic flow law at finite strain, J. Mech. Phys. Solids, 29 (1981), 153–172.CrossRefMATHMathSciNetADSGoogle Scholar
  18. 20.
    Hill, R.: Generalized constitutive relations for incremental deformation of metal crystals by multislip, J. Mech. Phys. Solids, 14 (1966), 95–102.CrossRefADSGoogle Scholar
  19. 21.
    Hutchinson, J.W.: Elastic—plastic behavior of polycrystalline metals and composites, Proc. Roy. Soc. London, A318 (1970), 247–272.CrossRefADSGoogle Scholar
  20. 22.
    Sewell, M.J.: A survey of plastic buckling, in Stability (ed. H. Leipholz ), University of Waterloo Press (1972), 85.Google Scholar
  21. 23.
    Christoffersen, J. and Hutchinson, J.W.: A class of phenomenological corner theories of plasticity, J. Mech. Phys. Solids, 27 (1979), 465–487.CrossRefMATHMathSciNetADSGoogle Scholar
  22. 24.
    Tvergaard, V.: Plastic buckling of axially compressed circular cylindrical shells, Int. J. Thin-Walled Struct., 1 (1983), 139–163.CrossRefGoogle Scholar
  23. 25.
    Nguyen, S.Q. and Triantafyllidis, N.: Plastic bifurcation and postbifurcation analysis foreneralized standard continua, Ecole Polytechnique, Palaiseau, France (1988).Google Scholar
  24. 26.
    Considere, M.: Annales des Ponts et Chaussees, 9 (1885), 574.Google Scholar
  25. 27.
    Hutchinson, J.W. and Miles, J.P.: Bifurcation analysis of the onset of necking in an elastic-plastic cylinder under uniaxial tension, J. Mech. Phys. Solids, 22 (1974), 61.CrossRefMATHADSGoogle Scholar
  26. 28.
    Tvergaard, V.: Bifurcation in elastic-plastic tubes under internal pressure, European Journal of Mechanics, A/Solids, 9 (1990), 21–35.MATHGoogle Scholar
  27. 29.
    Larsson, M., Needleman, A., Tvergaard, V. and Storâkers, B.: Instability and failure of internally pressurized ductile metal cylinders, J. Mech. Phys. Solids, 30 (1982), 121–154.CrossRefADSGoogle Scholar
  28. 30.
    Sewell, M.J.: On the calculation of potential functions defined on curved boundaries, Proc. R. Soc. London, A286 (1965), 402–411.CrossRefMATHMathSciNetADSGoogle Scholar
  29. 31.
    Storâkers, B.: Bifurcation and instability modes in thick-walled rigid plastic cylinders under pressure, J. Mech. Phys. Solids, 19 (1971), 339–351.CrossRefMATHADSGoogle Scholar
  30. 32.
    Strifors, H. and Storâkers, B.: Uniqueness and stability at finite-deformation of rigid plastic thick-walled cylinders under hydrostaticressure, in Foundations of Plasticity (ed. A. Sawczuk), Nordhoff, Leiden, 1 (1973), 327.Google Scholar
  31. 33.
    Chu, C.-C.: Bifurcation of elastic-plastic circular cylindrical shells under internal pressure, J. Appl. Mech., 46 (1979), 889–894.CrossRefMATHADSGoogle Scholar
  32. 34.
    Stilren, S. and Rice, J.R.: Localized necking in thin sheets, J. Mech. Phys. Solids, 23 (1975), 421–441.CrossRefADSGoogle Scholar
  33. 35.
    Needleman, A. and Tvergaard, V.: Necking of biaxially stretched elastic-plastic circular plates, J. Mech. Phys. Solids, 25 (1977), 159–183.CrossRefMATHADSGoogle Scholar
  34. 36.
    Tvergaard, V. and Needleman, A.: On the localization of buckling patterns, J. Appl. Mech., 47 (1980), 613–619.CrossRefADSGoogle Scholar
  35. 37.
    Tvergaard, V.: On the burst strength and necking behaviour of rotating disks, Int. J. Mech. Sci., 20 (1978), 109–120.CrossRefMATHGoogle Scholar
  36. 38.
    Percy, M.J., Ball, K. and Mellor, P.B.: Ana erimental study of the burst strength of rotating disks, Int. J. Mech. Sci., 16 (1974), 809–817.CrossRefGoogle Scholar
  37. 39.
    Hutchinson, J.W. and Neale, K.W.: Influence of strain—rate sensitivity on necking under uniaxial tension, Acta Metallurgica, 25 (1977), 839–846.CrossRefGoogle Scholar
  38. 40.
    Hutchinson, J.W. and Neale, K.W.: Sheet necking — III. Strain rate effects, in Mechanics of Sheet Metal Forming (eds. D.P. Koistinen and N.—M. Wang ), Plenum Publ. Corp., New York (1978), 111–126.CrossRefGoogle Scholar
  39. 41.
    Peirce, D., Asaro, R.J. and Needleman, A.: Material rate dependence and localized deformation in crystalline solids, Acta Metallurgica, 31 (1983), 1951–1976.CrossRefGoogle Scholar
  40. 42.
    Tvergaard, V.: Rate—sensitivity in elastic—plastic panel buckling, in Aspects of the Analysis of Plate Structures, A volume in honour of W.H. Wittrick (eds. D.J. Dawe et al.), Clarendon Press, Oxford (1985), 293–308.Google Scholar
  41. 43.
    Bodner, S.R., Naveh, M. and Merzer, A.M.: Deformation and buckling of axisymmetric viscoplastic shells under thermomechanical loading, Int. J. Solids Structures, 27 (1991), 1915–1924.CrossRefGoogle Scholar
  42. 44.
    Obrecht, H.: Creep buckling and postbuckling of circular cylindrical shells under axial compression, Int. J. Solids Structures, 13 (1977), 337–355.CrossRefGoogle Scholar
  43. 45.
    Hoff, N.J.: Creep buckling of plates and shells, in Proc. 13th Int. Congr. Appl. Mech. (eds. E. Becker and G.K. Mikhailov ), Springer—Verlag (1973), 124–140.Google Scholar
  44. 46.
    Hill, R.: Acceleration waves in solids, J. Mech. Phys. Solids, 10 (1962), 1–16.CrossRefMATHMathSciNetADSGoogle Scholar
  45. 47.
    Rice, J.R.: The localization of plastic deformation, in Theoretical and Appl. Mech. (ed. W.T. Koiter ), North—Holland (1977), 207–220.Google Scholar
  46. 48.
    Hadamard, J.: Lecons sur la propagation des ondes et les équations de l’hydrodynamique, Libraire Scientifique A, Hermann, Paris (1903).Google Scholar
  47. 49.
    Hutchinson, J.W. and Tvergaard, V.: Shear band formation in plane strain, Int. J. Solids Structures, 17 (1981), 451–470.CrossRefMATHGoogle Scholar
  48. 50.
    Marciniak, K. and Kuczynski, K.: Limit strains in the process of stretch forming sheet metal, Int. J. Mech. Sci., 9 (1967), 609–620.CrossRefGoogle Scholar
  49. 51.
    Tvergaard, V.: Effect of yield surface curvature and void nucleation on plastic flow localization, J. Mech. Phys. Solids, 35 (1987), 43–60.CrossRefADSGoogle Scholar
  50. 52.
    Needleman, A. and Tvergaard. V.: Limits to formability in rate—sensitive metal sheets, in Mechanical Behavior of Materials — IV (eds. J. Carlsson and N.G. Ohlson ), Pergamon Press, Oxford (1984), 51–65.CrossRefGoogle Scholar
  51. 53.
    Pan, J., Saje, M. and Needleman, A.: Localization of deformation in rate sensitive porous solids, Int. J. Fracture, 21 (1983), 261–278.CrossRefGoogle Scholar
  52. 54.
    McClintock, F.A.: A criterion for ductile fracture by growth of holes, J. Appl. Mech., 35 (1968), 363–371.CrossRefADSGoogle Scholar
  53. 55.
    Rice, J.R. and Tracey, D.M.: On the ductile enlargement of voids in triaxial stress fields, J. Mech. Phys. Solids, 17 (1969), 201–217.CrossRefADSGoogle Scholar
  54. 56.
    Bishop, R.F., Hill, R. and Mott, N.F.: The theory of indentation and hardness tests, Proc. Phys. Soc., 57 (1945), 147–159.CrossRefADSGoogle Scholar
  55. 57.
    Ashby, M.F., Blunt, F.J. and Bannister, M.: Flow characteristics of highly constrained metal wires, Acta Metallurgica, 37 (1989), 1857.Google Scholar
  56. 58.
    McMeeking, R.M.: Finite deformation analysis of crack tip opening in elastic—plastic materials and implications for fracture, J. Mech. Phys. Solids, 25 (1977), 357–381.CrossRefADSGoogle Scholar
  57. 59.
    Tvergaard, V.: Failure by ductile cavity growth at a metal/ceramic interface, Acta Metall. Mater., 39 (1991), 419–426.Google Scholar

Copyright information

© Springer-Verlag Wien 1993

Authors and Affiliations

  • V. Tvergaard
    • 1
  1. 1.The Technical University of DenmarkLyngbyDenmark

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