Flow Relations and Yield Functions for Dissipative Strain-Gradient Plasticity

  • Carsten Carstensen
  • François Ebobisse
  • Andrew T. McBride
  • B. Daya Reddy
  • Paul Steinmann
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 69)


In this work we carry out a theoretical investigation of a dissipative model of rate-independent strain-gradient plasticity. The work builds on the investigation in [1], which in turn was inspired by the investigations in [4] of responses to non-proportional loading in the form of surface passivation. We recall the global nature of the flow relation when expressed in terms of the Cauchy stress and dissipation function. We highlight the difficulties encountered in attempts to obtain dual forms of the flow relation and associated yield functions, for the continuous and discrete problems, and derive upper bounds on the global yield function.


Plastic Strain Yield Function Dissipation Function Flow Relation Plastic Range 
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.



The work reported in this paper was carried out with support through the South African Research Chair in Computational Mechanics to BDR and ATMcB. This support is gratefully acknowledged. PS acknowledges support through the Collaborative Research Center 814.


  1. 1.
    Carstensen, C., Ebobisse, F., McBride, A.T., Reddy, B.D., Steinmann, P.: Some properties of the dissipative model of strain-gradient plasticity. Phil. Mag., in press (2017)Google Scholar
  2. 2.
    Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)zbMATHGoogle Scholar
  3. 3.
    Fleck, N.A., Hutchinson, J.W.: A reformulation of strain gradient plasticity. J. Mech. Phys. Solids 49, 2245–2271 (2001)CrossRefzbMATHGoogle Scholar
  4. 4.
    Fleck, N.A., Hutchinson, J.W., Willis, J.R.: Strain-gradient plasticity under non-proportional loading. Proc. R. Soc. A 470, 20140267 (2015)CrossRefGoogle Scholar
  5. 5.
    Fleck, N.A., Hutchinson, J.W., Willis, J.R.: Guidelines for constructing strain gradient plasticity theories. J. Appl. Mech. 82, 071002-1–10 (2015)CrossRefGoogle Scholar
  6. 6.
    Fleck, N.A., Willis, J.R.: A mathematical basis for strain-gradient plasticity - Part I: scalar plastic multiplier. J. Mech. Phys. Solids 57, 151–177 (2009)zbMATHGoogle Scholar
  7. 7.
    Fleck, N.A., Willis, J.R.: A mathematical basis for strain-gradient plasticity - Part II: tensorial plastic multiplier. J. Mech. Phys. Solids 57, 1045–1057 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gudmundson, P.: A unified treatment of strain gradient plasticity. J. Mech. Phys. Solids 52, 1379–1406 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gurtin, M.E., Anand, L.: A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part I: small deformations. J. Mech. Phys. Solids 53, 1624–1649 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Han, W., Reddy, B.D.: Plasticity: Mathematical Theory and Numerical Analysis. Second Edition, Springer, New York (2013)Google Scholar
  11. 11.
    Reddy, B.D.: The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity. Part 1: polycrystalline plasticity. Contin. Mech. Thermodyn. 23, 527–549 (2011)Google Scholar
  12. 12.
    Reddy, B.D., Ebobisse, F., McBride, A.T.: Well-posedness of a model of strain gradient plasticity for plastically irrotational materials. Int. J. Plast. 24, 55–73 (2008)CrossRefzbMATHGoogle Scholar
  13. 13.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  • Carsten Carstensen
    • 1
  • François Ebobisse
    • 2
  • Andrew T. McBride
    • 3
  • B. Daya Reddy
    • 2
  • Paul Steinmann
    • 4
  1. 1.Humboldt-Universität Zu BerlinBerlinGermany
  2. 2.University of Cape TownCape TownSouth Africa
  3. 3.The University of GlasgowGlasgowUK
  4. 4.University of Erlangen-NurembergErlangenGermany

Personalised recommendations