Length scales interaction in nonlocal plastic strain localization of bars of varying section

Xilin Lu, Jean Pierre Bardet, Maosong Huang

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


In computational mechanics, strain softening generates ill-posed boundary value problems, which cannot be solved without being regularized, e.g., through the introduction of an internal length scale. This paper investigates how the internal length scale that is introduced by regularization may interact with the external length scale arising from boundary conditions in the particular case of a strain-softening bar of varying cross section and a nonlocal averaging regularization. The interaction of internal and external length scales is examined using an analytical closed-form solution for overnonlocal softening plasticity that derives from a Fredholm equation of the second kind. In the absence of external length (bars of constant section), the analysis shows that the overnonlocal averaging confines and smoothly distributes plastic strain into a localized band. The localization width, plastic strain distribution inside the band, and load-displacement response are controlled by the internal length of the averaging function and the overnonlocal weighting factor. In the presence of external length (bar of varying section), the analytical solution shows that the localization width is controlled by the interaction of external and internal length scales. This interaction is significant when the external and internal lengths are of comparable magnitude, and decreases when the external length becomes large compared to the internal length. The bandwidth is found to depend on the internal and external lengths and stress level while strain localizes, and to relate only to the internal length when the bar collapses.

Original languageEnglish (US)
Article number009008QEM
Pages (from-to)1036-1042
Number of pages7
JournalJournal of Engineering Mechanics
Issue number8
StatePublished - Aug 2010
Externally publishedYes


  • Analytical solution
  • Nonlocal material
  • Regularization
  • Strain localization
  • Strain softening plasticity

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering


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