Philosophical Magazine Letters,
Vol. 85, No. 7, July 2005, 339–343
Measurement of the size effect in the
yield strength of nickel foils
P. MOREAU, M. RAULIC, K. M. Y. P’NG*, G. GANNAWAY,
P. ANDERSON, W. P. GILLIN, A. J. BUSHBY and D. J. DUNSTAN
Centre for Materials Research, Queen Mary, University of London,
London E1 4NS, UK
(Received 12 September 2004; in final form 5 December 2004)
There is a growing consensus that materials become stronger in small volumes
and in the presence of large strain gradients. It has not been clear whether this is
due to increased resistance to the motion of dislocations, fewer dislocations, or
increased difficulty of multiplying dislocations in these situations. A classic
experiment by Stölken and Evans (J.S. Stölken and A.G. Evans, Acta metall.
46 5109 (1998)) showed that thin nickel foils under bending display increased
strengthening at large plastic strain values and, correspondingly, large plastic
strain gradients. We have adapted their technique to small strains, and report
preliminary data for the stress–strain curves of thin nickel foils through the
elastic–plastic transition. These data show unambiguously that the yield
strength is greater in the thinner foils. The strengthening is additive to the
Hall–Petch effect, and is consistent with a size effect at the onset of plastic
deformation.
Increased yield strength in a material in the form of a small specimen, or under
a highly localized stress field, is often observed and attributed to an effect of the size
of the deformed volume. The size effect is displayed unambiguously in only a few
experimental reports. Nanoindentation provides perhaps the clearest data, with a
trend to increased indentation hardness when indents or indenters are made smaller.
Nevertheless, this data has been the subject of prolonged controversy. There is a
strong dependence on the shape of the indenter and on the state of the free surface
[2–12]. The stress and strain fields in indentation hardness testing are, however, very
complicated. In simpler geometries, Tabata et al. [13] reported an increased tensile
strength in thin aluminium wires. Fleck et al. [5] reported an increased torsional
strength in thin copper wires. Stölken and Evans [1] developed a highly elegant
microbend test method for thin metal foils, and reported increased strength in
thin nickel foils. Very recently, Nix [14] has reported GPa yield strengths in
small gold pillars under uniaxial compression. A wide variety of interpretations
have been given, invoking a wide variety of characteristic lengths, from the
*Corresponding author. Email: [email protected]
Philosophical Magazine Letters
ISSN 0950–0839 print/ISSN 1362–3036 online # 2005 Taylor & Francis
http://www.tandf.co.uk/journals
DOI: 10.1080/09500830500071564
340
P. Moreau et al.
Matthews thermodynamic equilibrium critical thickness familiar in semiconductor
nanostructures [12, 15] to the spacing of geometrically necessary dislocations
in plastic strain gradients which is the key parameter in strain-gradient theory
[1, 5, 9, 16–20].
In this field, the quality of the data is a central issue. With soft metals such as Cu
and Ni, the metallurgy—grain size and texture and dislocation densities, and also the
surface morphology and chemistry may be called into question. When these properties are well controlled at suitable values, the metals are so soft that obtaining data
around the yield point has not been feasible, and stress–strain data are obtained only
after relatively large plastic deformation. Such data may be consistent with a variety
of theoretical interpretations but may be inadequate for distinguishing between
them. For this reason, we have developed the Stölken and Evans [1] microbend
technique to yield high quality data around the yield point of soft annealed nickel
foils.
The Stölken and Evans [1] technique was to bend foils to a known radius of
curvature, r1, by wrapping them round a mandrel, and then to unload them
and measure their radius of curvature, r2, in the unloaded state. Elementary
beam theory gives the surface strain directly from r1, and the surface stress corresponding to that strain is derived from the elastic change of radius in unloading from
r1 to r2,
M
Eh 1
1
¼
ð1Þ
bh2 12 r1 r2
where M is the bending moment, b is the foil width, h is the foil thickness and E is the
Young modulus which is 200 GPa for nickel. Stölken and Evans [1] studied foils of
thickness 12.5, 25 and 50 mm on mandrels of diameter 125, 250, 500, 1000 and
2000 mm, thereby obtaining the stresses for surface strains of 9.09%, 4.76% and
2.4%. Much larger mandrels are required if data embracing the elastic–plastic transition are to be obtained. We therefore built an apparatus in which the foil can be
wrapped around a surface of variable curvature, variable from infinite radius of
curvature to values around 10 mm (figure 1). This apparatus was designed to be
mounted under an optical profilometer (Wolf & Beck Sensorik OTM3), in which
the curvature of the foil could be measured without physical contact and without
dismounting the foil from the apparatus. The procedure was then to start with a
straight foil to measure its surface profile, and then to bend it to progressively higher
curvatures while measuring the unloaded curvatures at each step.
The foils were obtained from Goodfellows Cambridge Limited, with purity of
99.9% and better. The thinner foils were known to be electrodeposited with very
small grain size (thickness h ¼ 10, 25 and 50 mm), while the thickest was unannealed
rolled sheet (thickness h ¼ 125 mm). Samples were annealed for short times at high
temperatures in an inert atmosphere in a graphite-strip rapid thermal annealing
furnace [21]. This was preferred over long anneals at lower temperatures in a traditional furnace in order to minimize surface roughening. The three thinnest foils were
annealed for different times at 1000 C. The rolled sheet (thickness h ¼ 125 mm) was
annealed for 30 seconds at 700 C. This gave no measurable increase in grain size, but
it is likely that this short low-temperature anneal reduced excessive dislocation densities left by rolling. We measured the grain size by optical microscopy after etching
in a mixture of acetic and nitric acid at a ratio of five to one for between one and two
minutes, as required to reveal the grains [22]. It was found possible to equalize grain
341
Measurement in the yield strength of nickel foils
Weighted bars
Outer roller
Outer roller
Foil
Shim
Stopping
roller
Screw
Wedge
Middle rollers
Figure 1. Schematic of the apparatus with roller positions shown. The foil is laid on a shim
that is in a 4-point bend and constrained by two weighted rollers at both ends. The screw
drives a wedge that lifts the middle two rollers. This increases the 4-point bend and thus
decreases the radius of curvature on the foil.
size at 3.0 0.1 mm in all the foils. Samples with different grain sizes were also
studied, but in all cases, the grains are significantly smaller than the foil thickness.
We did not control for texture, but here we assume that the annealing has normalized the microstructure and reduced the effect of preferred orientation in the rolled
sheet.
Figure 2 shows stress–strain curves for the four foil thicknesses. The crucial
result is that data can be obtained through the elastic-plastic yield point. There is
a clear increase of yield strength in the thinner foils. The results are consistent with a
size effect already operative at the yield point, although a link with a systematic
change of texture with thickness cannot be ruled out at this stage.
The stress–strain curves of the samples with other grain sizes were also measured.
In figure 3, the yield points are plotted as yield strength against the reciprocal square
root of grain-size, in accordance with the Hall–Petch theory (Kelly and Macmillan
[23], and references therein). We see that the size effect appears to be independent of
the Hall–Petch effect, and extrapolation to infinite grain size under this assumption
gives what we would describe as the intrinsic yield strength of bulk single crystal
under the size effect. This figure shows that the data are consistent with the simple
addition of the Hall–Petch effect and the size effect.
We have recently presented a theoretical account of the size effect in terms of
critical thickness theory. The yield point should be raised in situations of high elastic
strain gradient, and can be predicted without the use of free fitting parameters or
characteristic lengths [15]. We showed that the data of both Fleck et al. [5] and
Stölken and Evans [1] was not inconsistent with our theory. However, since their
data did not extend down to the yield point, it was not possible to prove that the
yield point itself is raised by the size effect. The actual yield stress of bulk material is
a crucial parameter in the theory. We find that the quantitative fit of the theory to
our experimental data is poor. Figure 4 shows, however, that a fit to the extrapolated
342
P. Moreau et al.
20
50µm
10µm
15
M/(bh2) / MPa
125µm
25µm
10
5
0
0.000
0.002
0.004
0.006
ε
Figure 2. The stress–strain curves for the four foils. The stress–strain points of the foil
thicknesses are represented by: g for 10 mm foil thickness, t for 25 mm foil thickness, m
for 50 mm foil thickness and f for 125 mm foil thickness.
Yield strength / MPa
22
18
10µm
14
25µm
50µm
10
125µm
6
0.3
0.4
0.5
0.6
d
-1/2
0.7
(mm
-1/2
0.8
0.9
1
)
Figure 3. The Hall–Petch yield points against grain size. Extrapolation of the lines to infinite
grain size gives the values of the yield strength of a foil with zero Hall–Petch effect. The yield
points are: g for 10 mm foil thickness, t for 25 mm foil thickness, m for 50 mm foil thickness
and f for 125 mm foil thickness.
strengths for infinite grain size is excellent, suggesting that the Hall–Petch effect and
the size effect observed here are additive.
In summary, the experimental data reported here are of a preliminary nature, as
is the theoretical analysis. Much more data need to be gathered together with
Measurement in the yield strength of nickel foils
343
Figure 4. Dunstan and Bushby’s [15] theory (represented by the solid line) compared to our
experimental data (represented by m) and the extrapolation of our experimental data to
infinite grain size (represented by g).
detailed metallurgical analysis of the specimens. In particular, it will be necessary to
control for texture, since this is not necessarily eliminated by annealing, and may
vary in a systematic way with foil thickness. The experiments need to be taken to
much higher plastic strains to test strain-gradient theory. The theoretical analysis
shows agreement with critical thickness theory under the assumption that it and the
Hall–Petch effect are merely additive. The basis of that assumption needs further
investigation. Crucially for these studies, the Stölken and Evans [1] technique can be
extended to reveal very accurate stress–strain data in the region of the yield point of
very soft metals.
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