A boundary constraint energy balance criterion for small volume deformation

William W Gerberich, M. J. Cordill, W. M. Mook, N. R. Moody, C. R. Perrey, C. B. Carter, R. Mukherjee, Steven L Girshick

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21 Scopus citations


A concept for the deformation resistance of small volumes under contact subjected to displacements in the 2-100 nm regime is proposed. In terms of an energy balance criterion, the external work minus stored elastic energy is consumed by surface energy and plastic energy absorption. The surface energy is interpreted in terms of new area of contact or new area created by slip step emergence or oxide film fracture. The plastic energy absorption is interpreted in terms of dislocation work. As such, this is analogous to a contact mechanics Griffith criterion for the hardness or flow strength of crystalline metals, semiconductors or oxides. Derivations of supporting loads and stresses provided for nanoindenter tips, nanospheres, nanocubes and thin-walled nanoboxes are presented. For comparison, measured hardness and flow strengths of Al, Au, Fe-3%Si, Ti, W and Si nanostructures are reported in the 2-100 nm length scale regime. It is proposed that Si and Ti nanospheres experience a minimum in contact resistance as dictated by a balance in surface energy dominance at small contact and dislocation hardening dominance at large contact. The variation of a strength controlling coefficient, α, with material suggests a length scale dependence on shear and/or pressure activation volumes.

Original languageEnglish (US)
Pages (from-to)2215-2229
Number of pages15
JournalActa Materialia
Issue number8
StatePublished - May 2005

Bibliographical note

Funding Information:
The authors greatly appreciated discussions with Professors David Bahr, Perry Leo, Kanji Ono, Kristen Van Vliet, and Stan Veprek during the preparation of this manuscript. This work was supported by the National Science Foundation under Grants DMI 0103169, CMS-0322436, an NSF-IGERT program through Grant DGE-0114372 and the United States Department of Energy Office of Science, DE-AC04-94AL85000.


  • Dislocation
  • Hardness
  • Nanoindentation
  • Surface energy
  • Yield phenomena


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