Structural Investigation of Size Effects in Plasticity using Indentation ...

A.4 Discussion
the experimental observations (see Figures A.6 and A.7). The suggested model is
also supported by results found by Minor et al., 24 where dislocation loops nucleated
in a defect-free volume are not contained in a predefined plastic zone. Instead, they
propagate far into the bulk, producing a plastic zone differing from those proposed
by classical continuum mechanics models. Atomistic simulation studies of the initial
stages of nanoindentation show similar results. As the simulations demonstrate, dislocation
loops are nucleated in regions directly beneath the surface and propagate
towards the undeformed crystal as the load increases. 31–34
The bilinear behavior of the hardness data (Figure A.9) as well as the misorientation
maps (Figure A.3) indicate a change in the deformation mechanism. For large
indentations it seems that the pile-up model described in Figure A.10 is responsible
for the indentation size effect. Examining the suggested model (Figure A.10) reveals
similarities to the Hall-Petch effect. However, there are also some differences: in a
polycrystalline material the pile-ups that occur at a boundary have to trigger plasticity
in the neighboring grains, whereas the pile-ups in this case have to accommodate
the shape of the indenter. Contrary to polycrystalline materials, the deformation
zone below an indentation (region A in Figure A.10) is highly bounded on only one
side, namely to the indenter flank. The other sides are less bounded, due to either a
change in the shear stress field, which forms only a weak barrier (regions 1 and 3),
or the necessity of pushing previously generated dislocations into the bulk material
(region 4). Due to the similarities between both models, the hardness should follow
the Hall-Petch relation 35,36
σy = σ0 + kHP
1
√ D
(A.2)
where the grain size is substituted by D, the diameter of region A (see Figure
A.10). Both, kHP the Hall-Petch parameter and σ0 the intrinsic yield strength in
the absence of grain size effects are constants that depend on the nature and state of
the crystal. Using Tabors rule 37 to convert the hardness into the corresponding flow
stress σy and assuming that the size of region A is proportional to the indentation
depth h, the Hall-Petch relation can be easily rewritten as
(H − 3σ0) 2 1
= k1
h
(A.3)
where k1 is a constant. As can be seen, Eq. A.4 seems to be similar to the
Nix-Gao relation
H 2 − H 2 0 = k 1
h
(A.4)
where k is the slope of the hardness in the Nix-Gao plot and H0 is the macroscopic
hardness. It should be noted that the hardness can follow both relations only if σ0
29