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International Journal for Multiscale Computational Engineering
Facteur d'impact: 1.016 Facteur d'impact sur 5 ans: 1.194 SJR: 0.554 SNIP: 0.68 CiteScore™: 1.18

ISSN Imprimer: 1543-1649
ISSN En ligne: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.v8.i1.40
pages 37-60

Softening Gradient Plasticity: Analytical Study of Localization under Nonuniform Stress

Milan Jirasek
Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Czech Republic
Jan Zeman
Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Thakurova 7,166 29 Prague 6, Czech Republic; Centre of Excellence IT4Innovations, VSB-TU Ostrava, 17 listopadu 15/2172 708 33 Ostrava-Poruba, Czech Republic
Jaroslav Vondrejc
Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Czech Republic


Localization of plastic strain induced by softening can be objectively described by a regularized plasticity model that postulates a dependence of the current yield stress on a nonlocal softening variable defined by a differential (gradient) expression. This paper presents analytical solutions of the one-dimensional localization problem under certain special nonuniform stress distributions. The one-dimensional problem can be interpreted as describing either a tensile bar with a variable cross section or a beam subjected to a nonuniform bending moment. Explicit as well as implicit gradient formulations are considered. The evolution of the plastic strain profile and the shape of the load-displacement diagram are investigated. It is shown that even if the local constitutive law exhibits softening right from the onset of yielding, the global load-displacement diagram has a hardening part. The interplay between the internal length scales characterizing the material and the geometry is discussed.


  1. Aifantis, E. C., On the microstructural origin of certain inelastic models. DOI: 10.1115/1.3225725

  2. Bazant, Z. P., Belytschko, T. B., and Chang, T.-P., Continuum model for strain softening.

  3. Bazant, Z. P. and Oh, B.-H., Crack band theory for fracture of concrete. DOI: 10.1007/BF02486267

  4. Challamel, N., A regularization study of some softening beam problems with an implicit gradient plasticity model. DOI: 10.1007/s10665-008-9233-3

  5. Engelen, R. A. B., Geers, M. G. D., and Baaijens, F. P. T., Nonlocal implicit gradient-enhanced elasto-plasticity modelling of softening behaviour. DOI: 10.1016/S0749-6419(01)00042-0

  6. Geers, M. G. D., Finite strain logarithmic hyperelasto-plasticity with softening: a strongly non-local implicit gradient framework. DOI: 10.1016/j.cma.2003.07.014

  7. Geers, M. G. D., Engelen, R. A. B., and Ubachs, R. J. M., On the Numerical modelling of ductile damage with an implicit gradient-enhanced formulation.

  8. Jirásek, M. and Rolshoven, S., Localization properties of strain-softening gradient plasticity models. Part II: Theories with gradients of internal variables. DOI: 10.1016/j.ijsolstr.2008.12.018

  9. Mühlhaus, H. B. and Aifantis, E. C., A variational principle for gradient plasticity. DOI: 10.1016/0020-7683(91)90004-Y

  10. Peerlings, R. H. J., de Borst, R., Brekelmans, W. A. M., and de Vree, J. H. P., Gradient-enhanced damage for quasi-brittle materials. DOI: 10.1002/(SICI)1097-0207(19961015)39:19<3391::AID-NME7>3.0.CO;2-D

  11. Peerlings, R. H. J., On the role of moving elastic-plastic boundaries in strain gradient plasticity. DOI: 10.1088/0965-0393/15/1/S10

  12. Pietruszczak, S. and Mróz, Z., Finite element analysis of deformation of strain-softening materials. DOI: 10.1002/nme.1620170303

  13. Strömberg, L. and Ristinmaa, M., FE-formulation of a nonlocal plasticity theory.

  14. Vermeer, P. A. and Brinkgreve, R. B. J., A new effective non-local strain measure for softening plasticity.

  15. Zbib, H. M. and Aifantis, E. C., On the localization and postlocalization behavior of plastic deformation.

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