Post-Critical Plastic Deformation Pattern in Incrementally Nonlinear Materials at Finite Strain

  • K. Thermann
Conference paper
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 414)


Deformation pattern in a simple, time-independent material with a yield-surface vertex effect are investigated in plane stress and plane strain beyond the critical instant of ellipticity loss under isothermal, quasistatic loading. The energy criterion of path instability applied to a family of post-critical solutions eliminates unstable deformation paths and enforces the return to the elliptic regime or its boundary. In that way an automatic branch switching algorithm for the discretized problem is used to determine the overall deformation pattern, although the solutions remain locally indeterminate due to the absence of an internal length scale. The incipient volume fraction of multiple planar bands has a well-defined value insensitive to the mesh size in finite element calculations. The observed formation of coarse, differently aligned secondary shear bands in isochoric plain strain compression at later stages of the numerical simulation is theoretically supported by the energy criterion.


Shear Rate Shear Band Localization Zone Critical Instant Deformation Path 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aifantis, E. C. (1995). Pattern formation in plasticity. Int. J. Engng. Sci. 33: 2161–2178.MathSciNetCrossRefMATHGoogle Scholar
  2. Christoffersen, J., and Hutchinson, J. W. (1979). A class of phenomenological corner theories of plasticity. J. Mech. Phys. Solids 27: 465–487.MathSciNetCrossRefMATHGoogle Scholar
  3. de Borst, R., and van der Giessen, E., eds. (1998). Material Instabilities in Solids. Chichester: John Wiley and Sons.Google Scholar
  4. Fiacco, A. V., and McCormick, G. P. (1968). Nonlinear programming. Sequential Unconstrained Minimization Techniques. New York: John Wiley.MATHGoogle Scholar
  5. Fleck, N. A., and Hutchinson, J. W. (1997). Strain gradient plasticity. Advances in Applied Mechanics 33: 295–361.CrossRefGoogle Scholar
  6. Fletcher, R. (1987). Practical Methods for Optimization. Chichester: John Wiley.Google Scholar
  7. Gay, D. M. (1981). Computing optimally constrained steps. SIAM J. Sci. Stat. Comput. 2: 186–197.MathSciNetCrossRefMATHGoogle Scholar
  8. Gay, D. M. (1983). Algorithm 611, subroutines for unconstrained minimization using a model/trust-region approach. ACM Transactions on Mathematical Software 9: 503–524.MathSciNetCrossRefMATHGoogle Scholar
  9. Curtin, M. E. (1972). The linear theory of elasticity. In Flügge, S., ed., Encyclopedia of Physics, Volume VIa/2, 1–295. Berlin, Heidelberg, New York: Springer.Google Scholar
  10. Hill, R., and Hutchinson, J. W. (1975). Bifurcation phenomena in the plane tension test. J. Mech. Phys. Solids 23: 239–264.MathSciNetCrossRefMATHGoogle Scholar
  11. Hill, R. (1958). A general theory of uniqueness and stability in elastic-plastic solids. J. Mech. Phys. Solids 6: 236–249.CrossRefMATHGoogle Scholar
  12. Hill, R. (1959). Some basic principles in the mechanics of solids without a natural time. J. Mech. Phys. Solids 7: 209–225.MathSciNetCrossRefMATHGoogle Scholar
  13. Hill, R. (1962). Acceleration waves in solids. J. Mech. Phys. Solids 10: 1–16.MathSciNetCrossRefMATHGoogle Scholar
  14. Hill, R. (1967). The essential structure of constitutive laws for metal composites and polycrystals. J. Mech. Phys. Solids 15: 79–95.CrossRefGoogle Scholar
  15. Hill, R. (1978). Aspects of invariance in solids mechanics. Advances in Applied Mechanics 18:1–75. Academic Press, New York.Google Scholar
  16. Hutchinson, J., and Tvergaard, V. (1981). Shear band formation in plane strain. hit. J. Solids Structures 17: 451–470.CrossRefMATHGoogle Scholar
  17. Lulla, C. (1999). Phd thesis, Dortmund University, to appear. Private Communication.Google Scholar
  18. More, J. J., and Sorensen, D. C. (1981). Computing a trust region step. SIAM J. Sci. Stat. Comput. 4: 553–572.MathSciNetCrossRefGoogle Scholar
  19. Petryk, H., and Thermann, K. (1992). On discretized plasticity problems with bifurcations. Int. J. Solids Structures 29: 745–765.MathSciNetCrossRefMATHGoogle Scholar
  20. Petryk, H., and Thermann, K. (1995). On plastic strain localisation in the non-elliptic range under plane stress. In Owen, D., and Onate, E., eds., Computational Plasticity: Fundamentals and Applications, 647–658. Swansea: Pineridge Press.Google Scholar
  21. Petryk, H., and Thermann, K. (1996). Post-critical plastic deformation of biaxially stretched sheets. In. J. Solids Structures 33: 689–705.CrossRefMATHGoogle Scholar
  22. Petryk, H., and Thermann, K. (1997). A yield-vertex modification of two-surface models of metal plasticity. Arch. Mech. 49: 847–863.MATHGoogle Scholar
  23. Petryk, H., and Thermann, K. (1999a). Post-critical deformation pattern in plane strain plastic flow with yield-surface vertex effect. Int. J. Mech. Sciences,in press.Google Scholar
  24. Petryk, H., and Thermann, K. (1999h). Post-critical plastic deformation in incrementally nonlinear materials. to appear.Google Scholar
  25. Petryk, H. (1982). A consistent energy approach to defining stability of plastic deformation processes. In Schroeder, F., ed., Stability in the Mechanics of Continua, Proc. IUTAM Symposium, Niinnbrecht 1981, 262–272. Berlin: Springer.CrossRefGoogle Scholar
  26. Petryk, H. (1989). On constitutive inequalities and bifurcation in elastic-plastic solids with a yield-surface vertex. J. Mech. Phys. Solids 37: 265–291.MathSciNetCrossRefMATHGoogle Scholar
  27. Petryk, H. (1991). The energy criteria of instability in time-independent inelastic solids. Arch. Mech. 43: 519–545.MathSciNetMATHGoogle Scholar
  28. Petryk, H. (1992). Material instability and strain-rate discontinuities in incrementally non-linear continua. J. Mech. Phys. Solids 40: 1227–1250.MathSciNetCrossRefMATHGoogle Scholar
  29. Petryk, H. (1997). Instability of plastic deformation processes. In et al., T. T., ed., Proceedings XIXth Int. Congress of Theoretical and Applied Mechanics, Kyoto 1996, 497–516. Amsterdam: Elsevier.Google Scholar
  30. Petryk, H. (1998). Evolution of the localization zone in an icrementally non-linear plastic material. Private Communication.Google Scholar
  31. Rice, J. R. (1977). The localization of plastic deformation. In Koiter, W. T., ed., Theoretical and Applied Mechanics, 207–220. Amsterdam: North-Holland.Google Scholar
  32. Rudnicki, J. W., and Rice, J. R. (1975). Conditions for the localization of deformation in pressure-sensitive materials. J. Mech. Phys. Solids 23: 371–394.CrossRefGoogle Scholar
  33. Shanley, F. R. (1947). Inelastic column theory. J. Aero. Sci. 14: 261–267.CrossRefGoogle Scholar
  34. Stören, S., and Rice, J. R. (1975). Localized necking in thin sheets. J. Mech. Phys. Solids 23: 421–441.CrossRefMATHGoogle Scholar
  35. Tomita, Y. (1994). Simulation of plastic instabilities in solid mechanics. Appl. Mech, Rev. 47: 171–205.Google Scholar
  36. Tvergaard, V., Needleman, A., and Lo, K. K. (1981). Flow localization in the plane strain tensile test. J. Mech. Phys. Solids 29: 115–147.CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 2000

Authors and Affiliations

  • K. Thermann
    • 1
  1. 1.University of DortmundDortmundGermany

Personalised recommendations