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Scripta Materialia 60 (2009) 148–151
www.elsevier.com/locate/scriptamat
On the importance of sample compliance in uniaxial microtesting
D. Kiener,a,b W. Grosingerb and G. Dehmb,c,*
a
b
Materials Center Leoben, Forschungs GmbH, A-8700 Leoben, Austria
Erich Schmid Institute of Materials Science, Austrian Academy of Sciences, A-8700 Leoben, Austria
c
Department of Materials Physics, Montanuniversita¨t Leoben, A-8700 Leoben, Austria
Received 1 July 2008; revised 4 September 2008; accepted 19 September 2008
Available online 8 October 2008
Recent investigations have reported signiﬁcantly higher ﬂow stresses for microcompression compared to microtensile testing. To
clarify this point a load reversal test was performed on a microtensile specimen and equal ﬂow stresses in tension and compression
were observed. In contrast, if the lateral sample compliance constrains the deformation, microcompression testing overestimates the
ﬂow stresses. Additional contributions to the measured ﬂow stresses stem from the aspect ratio of the sample and dislocation pile-ups.
Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Copper; Materials with reduced dimensions; Size eﬀects; Plastic deformation
It is well established that the strength of small
single-crystal specimens is inﬂuenced by the sample
dimension [1–8]. This has been extensively studied by a
number of groups using a microcompression technique
originally developed by Uchic and Dimiduk [9], where
a focussed ion beam (FIB) machined sample is compressed by a ﬂat diamond tip attached to a nanoindenter, for a variety of metals (Ni, Au, Cu, Mo, NiTi, Al,
etc.) [1–6,8]. Despite the diﬀerent experimental parameters for the various experiments, a general conclusion
that smaller samples sustain higher stress levels can be
drawn. Nonetheless, the origin of the observed strengthening is still under debate. The two most frequently considered models focus on source truncation and
exhaustion hardening [10–12] and dislocation starvation
[13]. Recently, a new microtensile testing technique was
reported by Kiener et al. [14]. Comparing the strength of
FIB machined single-crystal Cu specimens loaded in
compression [5,15,16] and tension [14] using identical
fabrication parameters and the same in situ testing
set-up at moderate strain rates of the order of 3 103 s1 reveals diﬀerent stress levels between the two
loading modes with signiﬁcantly lower stresses for the
tensile experiments, as shown in Figure 1 at 10% strain.
Power-law ﬁts lead to exponents of 0.38 and 0.47 for
* Corresponding author. Address: Erich Schmid Institute of Materials
Science, Austrian Academy of Sciences, A-8700 Leoben, Austria.
Tel.: +43 3842 804 112; fax: +43 3842 804 116; e-mail: gerhard.
[email protected]
microcompression and microtensile testing, respectively.
The exponent for microtensile tests with an aspect ratio
of 5:1 is dominated by the two data points of the smallest samples (0.5 lm). If the tests presented in Figure 2
with aspect ratios (sample length/side length) P2:1 were
included into the ﬁt, the exponent reduces to 0.42.
Since the values for power exponents reported in the literature vary by ±0.1, this diﬀerence is not considered
further. Much more important, and thus subject to further investigation, is that the measured stress levels diﬀer
by more than a factor of 3.
A similar observation was reported by Gruber et al.
[17] who compared tensile testing of single-crystal Au
ﬁlms on a compliant polyimide substrate to compression
testing of Au pillars [1,4]. This discrepancy between
tension and compression is unexpected. Furthermore,
theoretical models such as exhaustion hardening or starvation are not capable of explaining diﬀerent stress
levels for microtension and microcompression testing.
An asymmetry between tension and compression is
known from body-centred cubic (bcc) bulk materials
as a consequence of the three-fold dislocation core splitting of screw dislocations. For face-centred cubic (fcc)
bulk materials, there is an asymmetry between tension
and compression in cyclic single-crystal deformation,
with the stresses in compression higher than those in
tension. The main reason for this behaviour is inhomogeneous deformation within persistent slip bands with
higher conﬁnement of the soft persistent slip bands by
the harder matrix in compression. Moreover, the observed diﬀerences are only of the order of several per
1359-6462/$ - see front matter Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.scriptamat.2008.09.024
D. Kiener et al. / Scripta Materialia 60 (2009) 148–151
Figure 1. Comparison of the size-dependent shear stress between
microcompression and microtensile testing data of single-crystal Cu at
10% strain. The data was taken from Refs. [5,15,16] (microcompression testing) and [14] (microtensile testing).
Figure 2. Technical shear stress vs. strain curves for Cuh2 3 4i
microtensile samples with side length of 3 lm and aspect ratios ranging
from 1:1 to 13.5:1. A considerably higher ﬂow stress is only observed
for the sample with an aspect ratio of 1:1.
cent. For a detailed analysis of single-crystal Cu see e.g.
Ref. [18]. However, no diﬀerence between tension and
compression is expected for monotonic loading if frictional inﬂuences and end eﬀects are minimized.
Atomistic simulations of nanowires showed a diﬀerence between tension and compression. Zimmermann
and co-workers [19,20] pointed out the inﬂuence of the
surface stress and surface orientation on the deformation of Au nanowires. In general, and contrary to our
results, they observe higher yield stresses for tensile
loading, since the tensile surface stress induces compressive stresses in the interior. Tschopp and McDowell [21]
reported that Cu nanowires require higher stresses for
dislocation nucleation in compression and suggested
that the resolved stress normal to the slip plane inﬂuences the dislocation nucleation, thus causing the tension–compression asymmetry. It is important to notice
that all these simulations are limited to specimen dimensions of several nanometres and starting conﬁgurations
reﬂecting perfect single crystals, therefore requiring
149
homogeneous dislocation nucleation. In contrast, the
specimens reported here possess micrometre dimensions
and a grown-in dislocation network with a dislocation
density q 1013 m2 [11].
An approach postulated by Gil Sevillano et al. [22]
predicts a dependence of the ﬂow stresses on the sample
dimension. Considering the analogy between dislocation
glide and a ﬂuid percolating a porous medium, Gil Sevillano predicted a transition from dislocation motion by
pinning–depinning to swapping of the entire glide plane.
While the former is size-independent and governed by
the obstacle spacing, the latter scales with a power-law
exponent of 3/4. The transition length dc is dependent
on the dislocation density and leads for the investigated
single-slip conﬁguration to a critical dimension
dc 8.5 lm. There is a diﬀerence in sample length
between microcompression and microtensile testing,
but the critical dimension in terms of the sample diameter is the same for the two types of experiment.
Considering a single-ended dislocation source in a
microcrystal, Rao et al. [23] depicted diﬀerent source
strengths depending on the loading direction of the
source using three-dimensional discrete dislocation
dynamics simulations. In their study the inﬂuence of
the loading direction is mainly given by the diﬀerent critical conﬁgurations required to overcome the dislocation
source, leading to diﬀerent stress values for forward and
reverse activation. For a Poisson’s ratio of 0.33 a diﬀerence by a factor 2 was found for Ni, which could
partially account for the diﬀerent stress levels in microcompression and microtension testing.
In order to shed light on this phenomenon we prepared
single-crystal Cu tensile samples with a h2 3 4i loading
direction which were tested in tension and compression.
The intention of this approach is to investigate whether
the origin of the observed discrepancy in Figure 1 is a
physical eﬀect or caused by the measuring method. To exclude a possible inﬂuence of the sample aspect ratio in this
load reversal test, a systematic study of the inﬂuence of the
aspect ratio on the mechanical response of micron-sized
uniaxially loaded specimens was required in advance.
All investigated specimens had a side length of 3 lm
and aspect ratios ranging from 1:1 to 13.5:1. Sample fabrication was performed under grazing ion impact using a
FIB (Zeiss 1540 XB) operated at 30 kV. For sample ﬁnishing a ﬁnal Ga+ ion current of 100 pA was applied.
Since microcompression samples are in general shorter
than microtensile specimens in order to prevent plastic
buckling, a systematic variation of the sample length
was performed for microtensile specimens. These experiments were aimed at investigating the inﬂuence of the
aspect ratio and to check whether this boundary condition has a strong eﬀect on the mechanical response. The
resulting engineering shear stress vs. strain curves are
shown in Figure 2. For aspect ratios between 2:1 and
13.5:1 there is no pronounced impact of the aspect ratio
on the ﬂow stress. There is a slight trend that samples
with higher aspect ratio depict lower ﬂow stresses. This
can be explained by an altered stress state of the short
samples due to edge eﬀects or by considering a weaklink statistic. Since the diﬀerences are of the order of
the experimental scatter known for this kind of test
[3], no quantiﬁcation is attempted. Only for an aspect
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D. Kiener et al. / Scripta Materialia 60 (2009) 148–151
ratio of 1:1 is a clear increase in the ﬂow stress in conjunction with a suppression of distinct load drops observed. Comparable results were reported previously
for micro-tensile samples with side lengths ranging from
0.5 to 8 lm [14] and were explained by the limited free
glide of dislocations over the sample cross-section resulting in a dislocation pile-up.
To directly check the inﬂuence of the loading direction, a special tension/compression experiment with a
microtensile sample having an aspect ratio of 2:1, comparable to typically tested microcompression samples,
was performed. The specimen was loaded in situ in displacement-controlled open loop mode with six intentional unloading cycles applied with a microindenter
(ASMEC UNAT) mounted in a scanning electron
microscope (SEM; Zeiss LEO Stereoscan 440) equipped
with a microgripper. The details of this method are described elsewhere [14]. After straining, the sample was
examined in a high-resolution SEM (Zeiss LEO 1525)
and parts of the sample head were removed by FIB milling to simplify the compression test alignment. Subsequently, the microgripper was replaced by a ﬂat
diamond tip with a diameter of 20 lm, and the prestrained sample was compressed in a displacement-controlled open loop without unloading cycles. The
technical shear stress vs. strain curves for both loading
steps are shown in Figure 3. The microcompression data
starts at the strain upon unloading of the tensile sample.
Note that global ﬂow commences at the same stress level
as reached in pure tensile straining. After tensile straining to e = 0.22 the specimen started to deform in multiple slip, which is indicated by an increased apparent
hardening rate. This hardening continues during the
subsequent compression testing. The in situ images taken during deformation reveal that within the ﬁrst few
per cent of compressive loading a visible reduction in
the large glide step which had formed in the centre of
the tensile specimen occurs (see insets in Fig. 3). This implies that, within the resolution limits of the SEM, glide
Figure 3. Technical shear stress vs. strain data for a Cuh2 3 4i
microtensile sample with 3 lm side length and an aspect ratio of 2:1.
After straining to e = 0.27 with six intentional unloading cycles, parts
of the sample head were removed. The same specimen was subsequently compressed without unloading. The compression data starts at
the strain upon unloading the tensile sample.
occurs on the same glide plane but now in the opposite
direction.
In stark contrast to Figure 1, the experiment depicted
in Figure 3 revealed no diﬀerence between microtensile
and microcompression testing. This result was conﬁrmed by another microtension/compression sample of
similar dimensions which was ﬁrst compressed and then
strained in tension. Moreover, our observations do not
support the model of Rao et al. [23] for the given experiment, assuming the same source was operated in opposite directions at a comparable stress level. Additionally,
considering the results from Figure 2, the inﬂuences of
the aspect ratio can be excluded.
From the previous results, no mechanisms inherent to
the material itself seem to be responsible for the diﬀerences depicted in Figure 1. As we have thus excluded
eﬀects due to the material itself, the question arises
whether the fabrication process aﬀects the material
properties. Recent publications by Bei et al. pointed
out that FIB milling introduced dislocation sources into
formerly defect-free Mo whiskers [24,25], thus signiﬁcantly lowering their strength. In contrast, Shan et al.
observed a process termed mechanical annealing [26]
in which compressive loading removed the FIB-generated defects from Ni samples. In both cases no diﬀerence
is expected between compression and tension, but experimental proof is so far lacking. Additionally, several
hardening mechanisms could be attributed to the FIBgenerated defects [27]. But again, this should not lead
to diﬀerences in the ﬂow stresses measured in tension
and compression.
Instead, it appears more advisable to consider how
the experimental constraints aﬀect the depicted results.
Evidently, the presented short microcompression samples are still connected to the stiﬀ bulk material, while
the microtensile specimens are situated on the tip of a
laterally compliant needle. A schematic drawing depicting the diﬀerent situations is given in Figure 4. Three
important diﬀerences are obvious: (i) the low lateral stiﬀness of a microtensile specimen on a long needle; (ii) the
adjustable aspect ratio in microtensile testing; (iii) the
diﬀerent contact situation between sample and ﬂat
punch/gripper, respectively.
Figure 4. Schematic drawing depicting the diﬀerent macroscopic
sample dimensions resulting in diﬀerent lateral compliances between
microtensile testing (a) and microcompression testing (b).
D. Kiener et al. / Scripta Materialia 60 (2009) 148–151
(i) It was shown by Diehl for macroscopic Cu single
crystals that the lateral compliance of the testing
machine has a strong inﬂuence on the measured
stress vs. strain curve. While free crosshead movement resulted in a lower critical shear stress and
single-slip deformation, restricted crosshead movement suppressed stage I glide, leading to higher critical shear stress values and higher hardening rates
[28]. A similar situation is present in the case of common microcompression testing, since friction
between sample top and ﬂat punch is unavoidable.
Furthermore, free lateral movement is hard to realize for conventional nanoindenter systems, which
are designed to be as stiﬀ as possible. The microtensile approach reﬂects a specimen on a compliant
sample base tested by a stiﬀ machine. This statement
also holds true for the observations reported by Gruber et al. [17]. Reducing the constraint of lateral ﬁxation results naturally in lower stresses required for
deformation. The inﬂuence of the system stiﬀness
on the mechanical response of microcompression
samples was demonstrated by placing a microcompression specimen on a needle tip [16]. This also
resulted in clearly reduced ﬂow stresses which nearly
match the ﬂow stresses of microtensile samples. It is
worth noting that, while the results from Diehl are in
qualitative agreement with our experimental ﬁndings, there is a signiﬁcant diﬀerence in the values of
the determined shear stresses. This can be explained
by the three orders of magnitude diﬀerence in sample
size, leading to larger operating dislocation sources
for the bulk single crystals.
(ii) The higher aspect ratios possible for microtensile
samples provide additional lateral compliance.
This eﬀect is not very pronounced for the results
presented here due to the compliant needle, but is
expected to be more pronounced for tensile specimens connected to a stiﬀ bulk material.
(iii) Using in situ Laue [29] and postmortem electron
backscatter diﬀraction [15], crystal rotations indicating a dislocation pile-up were observed for microcompression samples close to the sample top, while
only slight crystal rotations caused by the ﬁxed sample ends were observed for microtensile specimens
[14]. The back-stress exerted by such a dislocation
pile-up will inﬂuence the stresses required for further
deformation. Therefore, before compressing the
specimen shown in Figure 3, only parts of the sample
head were removed to minimize this interfacial
eﬀect.
From the current results we conclude that the lateral
stiﬀness of the loading set-up is the key to understanding
the diﬀerent stress levels observed between microtension
and microcompression testing. A direct analogy was
drawn with investigations of bulk single crystals performed in the 1950s. For a quantitative understanding
151
of this eﬀect, ﬁnite element modelling and three-dimensional discrete dislocation dynamic simulations are required in order to study the inﬂuence of the boundary
conditions.
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