Size Effect in Single Crystal Copper Examined with Spherical Indenters
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THE size effect, sometimes referred to as ’’smaller is stronger’’, has been investigated using different experiments including micro-beam tension, micro pillar compression, and indentation. A review concerning materials mechanical size effect was presented by Zhu et al.[1] In the study two categories of size effect were distinguished: the intrinsic, resulting from microstructural constraints, and the extrinsic, caused by dimensional constraints (e.g., small sample size). The indentation size effect (ISE) was categorized as the extrinsic size effect with three-dimensional constraints. The interaction of extrinsic and intrinsic size effects was considered. In the case of ISE, two types of experiments can be distinguished: sharp indentation (using Berkovich and Vickers tips) and spherical indentation. In a majority of papers sharp indentation was analyzed where the size effect manifested as an increase of hardness with diminishing penetration depth. This relationship was often presented as a diagram of square of hardness vs
STANISŁAW KUCHARSKI and STEFANIA WOZ´NIACKA are with the Institute of Fundamental Technological Research, Pawin´skiego 5b, 02-106, Warsaw, Poland. Contact e-mail: [email protected] Manuscript submitted July 10, 2018. Article published online March 7, 2019 METALLURGICAL AND MATERIALS TRANSACTIONS A
the inverse of penetration depth and according to earlier investigations had a linear character. This linear form was correctly captured by the first ISE model proposed by Nix and Gao[2] where ISE was attributed to the density of geometrically necessary dislocations (GND) generated beneath the indenter. In more recent papers, where very small penetration depths were considered, results showed that the relationship between square of hardness and inverse of penetration depth deviates from linearity and should be approximated by a bilinear form.[3,4] Therefore, the Nix-Gao model was later corrected using two different approaches. In the first it was assumed that there was an upper limit of GND density (saturation of GND) that can be achieved.[5] In the second[3,6,7] it was assumed that the storage volume for GND was greater than previously estimated in the original Nix-Gao model. The approach was applied to model ISE in polycrystalline copper,[6] and nickel,[7] where both spherical and pyramidal indentation were analyzed. These corrections enabled an agreement of the Nix-Gao model with bilinear behavior of squared hardness. The newest experimental observations indicated that the density of GND is not constant; GND are not uniformly distributed, and constitute a type of sub-grain structure.[8,9] Constitutive models of small scale plasticity were also developed, where the GND were associated with a plastic strain gradient. The well-known constitutive relations for classical plasticity were enhanced with the
VOLUME 50A, MAY 2019—2139
term that corresponded to the strain gradient. An exhaustive review of experiments and models concerning ISE were presented by Pharr et al.[10] In the majority of papers on ISE, the
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