On the Bifurcation and Postbifurcation Theory for a General Class of Elastic-Plastic Solids

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


The present work is concerned with the bifurcation and postbifurcation analysis of a class of rate independent plasticity models obeying Hill’s maximum dissipation principle. A variational inequality approach, which differs from the classical formulation of the plastic bifurcation problem, is employed. The rate n bifurcation problem is formulated and sufficient conditions for uniqueness of the corresponding boundary value problem are given. A connection is made with Hill’s nonbifurcation criterion. In addition the issue of the postbifurcation behavior of the solid is addressed in this more general context showing the possiblity of angular as well as smooth bifurcations of rate n > 1.

Finally an example, capable of exhibiting both an angular as well as a smooth bifurcation is analyzed using the general formulation derived in this work. The presentation is concluded with some comments and comparisons of the present methodology with the classical approach.


Variational Inequality Positive Definitness Bifurcation Problem Bifurcate Solution Plastic Solid 
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. (1).
    CONSIDERE, A. Resistance des Pieces Comprimees. Congr. Intl. Proc. Const. (1891), p. 371, ParisGoogle Scholar
  2. (2).
    VON KARMAN, T. Untensuchungen Uder Knickfestigkeit, Mitteilungen Uder Forchungsarbeiten. Ver. Deut. Ing.,(1910), Vol. 81 Google Scholar
  3. (3).
    SHANLEY, F.R. Inelastic Column Theory. J. Aeronaut. Sci., (1947), Vol. 14, p. 261CrossRefGoogle Scholar
  4. (4).
    HILL, R. On the Problem of Uniquness in the Theory of a Rigid - Plastic Solid. J. Mech. Phys. Solids, (1956), Vol. 4, p. 247ADSCrossRefMATHMathSciNetGoogle Scholar
  5. (5).
    HILL, R A General Theory for Uniquness and Stability in Elastic - Plastic Solids. J. Mech. Phys. Solids, (1958), Vol. 6, p. 236ADSCrossRefGoogle Scholar
  6. (6).
    HUTCHINSON, J.W. Post - Bifurcation Behavior in the Plastic Range. J. Mech. Phys. Solids, (1973), Vol. 21, p. 163ADSCrossRefMATHGoogle Scholar
  7. (7).
    HUTCHINSON, J.W. Plastic Buckling. Advances Appl. Mech.,(1974), Vol. 14, p. 67 (8) KOTTER, W.T. On the stability of Elastic Equilibrium. Doctoral Thesis (1945), Delft.Google Scholar
  8. (9).
    BUDIANSKY, B. Theory of Buckling and Post - Buckling Behavior of Elastic Structures. Advances Appl. Mech. (1974), Vol. 14, p. 1CrossRefGoogle Scholar
  9. (10).
    NGUYEN, Q.S. Bifurcation and Post - Bifurcation Analysis in Plasticity and Brittle Fracture. J. Mech. Phys. Solids, (1987), Vol. 35, p. 123CrossRefGoogle Scholar
  10. (11).
    NGUYEN, Q.S. Contribution a la Theorie Macroscopique de l’ Elastoplasticite avec Ecrouissage. Doctoral Thesis,(1973) ParisGoogle Scholar
  11. (12).
    HALPHEN, B.and NGUYEN, Q.S. Sur les Materiaux Standards Generalises. J. Mecanique, (1975), Vol. 14, p. 39MathSciNetGoogle Scholar
  12. (13).
    TRIANTAFYLLIDIS, N. Bifurcation and Postbifurcation Analysis of Elastic - Plastic Solids Under General Prebifurcation Conditions. J. Mech. Phys. Solids, (1983), Vol. 31, p. 499ADSCrossRefMATHGoogle Scholar
  13. (14).
    CHRISTOFFERSEN, J. and HUTCHINSON, J.W. A Class of Phenomenological Corner Theories of Plasticity. J. Mech. Phys. Solids, (1979), Vol. 27, p. 465ADSCrossRefMATHMathSciNetGoogle Scholar
  14. (15).
    NGUYEN,Q.S.and TRIANTAFYLLIDIS, N. Plastic Bifurcation and Post - Bifurcation Analysis for Generalized Standard Continua. J. Mech. Phys. Solids, (1989), Vol. 27, p. 465Google Scholar
  15. (16).
    NGUYEN, Q.S. and STOLZ, C. Sur la Methode du Developpement Asymptotique en Flambage Plastique.C.R. Acad. Sci. Paris, (1985), V. 300, Ser. II, No. 7, p. 235Google Scholar
  16. (17).
    NGUYEN, Q.S. Problemes de Plasticite et de Rupture, Course Notes,(1980), Univ. of Orsay.Google Scholar
  17. (18).
    PETRYK, H. and THERMANN, K. Second Order Bifurcations in Elastic — Plastic Solids. J. Mech. Phys. Solids, (1985), Vol. 33, p. 577ADSCrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Wien 1993

Authors and Affiliations

  • N. Triantafyllidis
    • 1
  1. 1.The University of MichiganAnn ArborUSA

Personalised recommendations