## Abstract

By taking into account the dislocations that are geometrically necessary for producing a curvature or twist of the atomic lattice in crystals, Gao et al. recently developed a theory of strain-gradient plasticity on the micrometer scale and showed that it agrees relatively well with the tests of hardness, torsion and bending of copper on the micrometer scale. This paper subjects this theory to an asymptotic scaling analysis. It is shown that the small-size asymptotic limit of this theory exhibits (1) an unusually strong size effect in which the corresponding nominal stresses in geometrically similar structures of different sizes D vary as D^{-5/2}, and (2) an asymptotic approach to a load-deflection diagram whose tangent stiffness gradually increases, starting with an infinitely small initial stiffness at infinitely small stress. Although this peculiar small-size asymptotic behavior might not be attainable within the practical applicability range of a continuum theory, it renders questionable any efforts to construct approximations of an asymptotic matching character, with a two-sided asymptotic support, which have previously been proven effective for quasibrittle materials such as concrete, rock, ice and fiber composites. A possible simple modification of the existing theory with respect to the small-size asymptotic properties is suggested. However, the questions of experimental justification of such a modification and its compatibility with the dislocation theory will require further study. The small-size asymptotic properties of other strain gradient theories of plasticity have not been analyzed, except for those of the previous Fleck-Hutchinson theory, which have been found reasonable.

Original language | English (US) |
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Pages (from-to) | 435-448 |

Number of pages | 14 |

Journal | Journal of the Mechanics and Physics of Solids |

Volume | 50 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2002 |

### Funding

Partial support under U.S. National Science Foundation Grant CMS-9732791 to Northwestern University is gratefully acknowledged. Thanks are due to Milan Jirásek of EPFL, Lausanne, for pointing out that, even asymptotically for very small D , the load-deflection curve starts with a vertical slope, and to Zaoyang Quo of Northwestern University for pointing out the exceptional asymptotic scaling for pure bending.

## Keywords

- Asymptotic methods
- Dislocations
- Metal plasticity
- Micromechanics
- Scaling
- Size effect
- Strain gradient

## ASJC Scopus subject areas

- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering