Journal of the Mechanics and Physics of Solids 54 (2006) 1668–1686 A model of size effects in nano-indentation Y. Huang a, , F. Zhang b , K.C. Hwang b , W.D. Nix c , G.M. Pharr d,e , G. Feng c a Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, IL 61801, USA b FML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PR China c Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305, USA d Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996, USA e Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Received 20 October 2005; received in revised form 7 February 2006; accepted 9 February 2006 Abstract The indentation hardness–depth relation established by Nix and Gao [1998. Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 46, 411–425] agrees well with the micro-indentation but not nano-indentation hardness data. We establish an analytic model for nano-indentation hardness based on the maximum allowable density of geometrically necessary dislocations. The model gives a simple relation between indentation hardness and depth, which degenerates to Nix and Gao [1998. Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 46, 411–425] for micro-indentation. The model agrees well with both micro- and nano-indentation hardness data of MgO and iridium. r 2006 Elsevier Ltd. All rights reserved. Keywords: Nano-indentation hardness; Maximum allowable density of geometrically necessary dislocations; Taylor dislocation model; Indenter tip radius 1. Introduction Micro-indentation hardness experiments have repeatedly shown strong indentation size effects, i.e., the increase of indentation hardness with the decrease of indentation depth (e.g., Nix, 1989; Oliver and Pharr, 1992; de Guzman et al., 1993; Stelmashenko et al., 1993; ARTICLE IN PRESS www.elsevier.com/locate/jmps 0022-5096/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2006.02.002 Corresponding author E-mail address: huang9@uiuc.edu (Y. Huang).