Plastic strain and strain gradients at very small indentation depths

N. I. Tymiak, D. E. Kramer, D. F. Bahr, T. J. Wyrobek, William W Gerberich

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Plastic strains and their respective strain gradients produced by nanoindentation have been theoretically interpreted and experimentally measured at shallow indentation depths. Existing data for 〈100〉 tungsten with four different conical tip radii varying from 85 to 5000 nm and new data for four conical tips (R = 0.5 to 20 μm) into 〈100〉 aluminum are presented. Theoretical results based on geometrically necessary dislocations and semi-empirical experimental continuum calculations are compared for spherical and wedge indenters. For a sharp wedge, both experimental continuum based and theoretical geometrical approaches suggest strain gradient decreasing with the increasing indentation depth, δ. In contrast, theoretical geometrical analysis for a spherical contact yields a depth independent strain gradient proportional to 1/R and continuum calculations suggest a slight increase of a strain gradient proportional to δ1/4/R3/4. Both single crystals exhibit about a factor of two decrease in hardness with increasing depth, irrespective of either increasing or decreasing average strain gradients. Implications to strain gradient plasticity and indentation size effect interpretations at very shallow depths are discussed.

Original languageEnglish (US)
Pages (from-to)1021-1034
Number of pages14
JournalActa Materialia
Issue number6
StatePublished - Apr 2 2001

Bibliographical note

Funding Information:
We greatly appreciate the use of a number of diamond tips, a high-load transducer and the aluminum single crystal as provided by Hysitron, Inc. Research support from ONR Grants E-25-746-51 and N00014-91-J-1998 for three of us (DK, DB and WG) and NSF DMI-99871863 for two of us (NT and WG), are gratefully acknowledged.

Copyright 2007 Elsevier B.V., All rights reserved.


  • Atomic force microscopy (AFM)
  • Dislocations
  • Hardness
  • Metallic
  • Yield phenomena


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