Strain Gradient Plasticity: Theory and Implementation

Lorenzo Bardella*, Christian F. Niordson

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingBook chapterResearchpeer-review


This chapter focuses on the foundation and development of various higher-order strain gradient plasticity theories, and it also provides the basic elements for their finite element implementation. To this aim, we first refer to experiments exhibiting size-effects in metals and explain them by resorting to the concept of geometrically necessary dislocations. We then bring this concept to the continuum level by introducing Nye’s dislocation density tensor and by postulating the existence of higher-order stresses associated with dislocation densities. This provides the motivation for the development of higher-order strain gradient plasticity theories. For this purpose, we adopt the generalized principle of virtual work, initially illustrated for conventional crystal plasticity and subsequently extended to both crystal and phenomenological strain gradient plasticity theories.
Original languageEnglish
Title of host publicationMechanics of Strain Gradient Materials. CISM International Centre for Mechanical Sciences (Courses and Lectures)
EditorsA. Bertram, S. Forest
Place of PublicationCham
Publication date2020
ISBN (Print)978-3-030-43829-6
ISBN (Electronic)978-3-030-43830-2
Publication statusPublished - 2020
SeriesInternational Centre for Mechanical Sciences. Courses and Lectures


  • Micron-scale metal plasticity
  • Geometrically necessary dislocations
  • Dislocation density tensor
  • Strain gradient crystal plasticity
  • Strain gradient plasticity


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