Sample size effect and microcompression of Mg Cu Gd metallic glass

APPLIED PHYSICS LETTERS 91, 161913 共2007兲
Sample size effect and microcompression of Mg65Cu25Gd10 metallic glass
C. J. Lee and J. C. Huanga兲
Institute of Materials Science and Engineering, Center for Nanoscience and Nanotechnology, National Sun
Yat-Sen University, Kaohsiung, Taiwan 804, Republic of China
T. G. Nieh
Department of Materials Science and Engineering, the University of Tennessee, Knoxville,
Tennessee 37996-2200, USA
共Received 18 July 2007; accepted 27 September 2007; published online 19 October 2007兲
Micropillars with diameters of 1 and 3.8 ␮m were fabricated from Mg-based metallic glasses using
focus ion beam, and then tested in compression at strain rates ranging from 6 ⫻ 10−5 to 6
⫻ 10−1 s−1. The apparent yield strength of the micropillars is 1342– 1580 MPa, or 60%–100%
increment over the bulk specimens. This strength increase can be rationalized using the Weibull
statistics for brittle materials, and the Weibull modulus of the Mg-based metallic glasses is estimated
to be about 35. Preliminary results indicated that the number of shear bands increased with the
sample size and strain rates. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2800313兴
The strengths of face-center-cubic single crystals are
known to be strongly dependent on the sample dimensions,
especially when the dimension is close to micron or submicron scales.1–5 For example, the strengths of Ni or Au microsized single crystal specimens are several times higher than
that of the bulk specimens, and the strength can be even
further increased by more than one order of magnitude in
submicron pillar specimens. This dramatic effect occurs
when the size of samples is smaller than the characteristic
length for multiplication, which results in the dislocation
starvation.
In contrast, bulk metallic glasses 共BMGs兲 do not have a
long range ordered structure. The plastic deformation of
BMGs at room temperature proceeds in the form of highly
localized shear bands.6,7 Yielding is induced by shear band
nucleation and propagation. BMGs are not expected to work
harden because localized shear bands do not exert long range
interactive force like dislocations do. Thus, once yielding
starts shear band propagates very rapidly and leads to catastrophic failure.
It has been recently shown that the strength of Mg-based
BMGs is sensitive to the specimen size.8 Specifically, the
strength is higher in a smaller size sample. There is yet another sample size effect in BMGs. Schuh et al.9 recently
employed nanoindentation to study the shear band dynamics
and identified a region 共homogeneous region II兲, which was
characterized by extensive emission of multiple shear bands.
Gao et al.10 also developed a free-volume-based, thermoviscoplastic constitutive law to describe the inhomogeneous deformation in amorphous alloys and concluded that homogeneous region II actually corresponded to a region with fine
shear bands, and the shear band spacing decreases with increasing strain rate. The question then arises as to what
would happen when the specimen size is reduced to a range
where the specimen can no longer support even one single
shear band. A preliminary result10 indicated that the size
scale is probably around 1 ␮m or less. In the current paper,
we fabricated microscale pillar specimens and subsequently
tested them in compression to study their mechanical response as a function of specimen size and strain rate.
We selected a Mg65Cu25Gd10 共in at. %兲 amorphous alloy
for this study.
Compression samples were prepared using the dual focus ion beam system 共FIB兲 of Seiko, SMI3050 SE, following
the method developed previously.5 Firstly, the 30 keV and
7 – 12 nA currents Ga beam in FIB was faced to the surface
of the BMG disk to mill a crater with a bigger size island
being located in the center. Secondly, the same voltage and
smaller currents of 3 down to 0.2 nA were used to refine the
preserved island in the center to a desired diameter and
height of pillars. Two kinds of micropillars with diameters of
3.8 and 1 ␮m, and the respective height-to-diameter ratio of
2–2.5 and 2.5–3 were fabricated. These pillars, as shown in
Fig. 1, are tapered because the divergence of ion beam, and
the convergence angle is about 2.5°.
Microcompression tests were performed with an MTS
nanoindenter XP under the continuous stiffness measurement
mode using a flat punch, which was machined out of a standard Berkovich indenter also by FIB. The projected area of
the tip of the punch is an equilateral triangle of 13.5 ␮m. A
prescribed displacement rate of 0.25– 500 nm/ s, which corresponds to the initial strain rates of 6 ⫻ 10−5 – 6 ⫻ 10−2 s−1,
was used to deform the pillars. The load-displacement data
are presented in Fig. 2. Traditionally, the curves are readily
converted into the engineering stress and strain curves, with
the assumption that the sample is uniformly deformed. However, the assumption is obviously violated since a bulk metallic glass is deformed by the emission of highly localized
shear bands, as will be shown later. Thus, we choose to
present the load-displacement data for the convenience of
discussion.
Author to whom correspondence should be addressed. Tel.: ⫹886-7-5252000 Ext 4063. Electronic mail: jacobc@mail.nsysu.edu.tw
FIG. 1. Mg-BMG micropillars fabricated by focus ion beam technique: 共a兲
1 ␮m in diameter and 共b兲 3.8 ␮m in diameter.
a兲
0003-6951/2007/91共16兲/161913/3/$23.00
91, 161913-1
© 2007 American Institute of Physics
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161913-2
Appl. Phys. Lett. 91, 161913 共2007兲
Lee, Huang, and Nieh
TABLE I. Summary of the compressive stress 共in unit of MPa兲 of
Mg65Cu25Gd10 glass at different strain rates.
1 ␮m ⌽
3.8 ␮m ⌽
Bulk 3 mm ⌽
FIG. 2. 共Color online兲 Compression load-displacement curves of Mg-BMG
at different strain rates: 共a兲 1 ␮m and 共b兲 3.8 ␮m.
Since the pillars are uniformly tapered, the total displacement ⌬h is
⌬h =
1
h0
冕
h0
⌬u共x兲dx,
共1兲
0
where h0 is the height of the pillar ⌬u共x兲 is the local displacement. In the elastic range, with ␴ = E␧ and ␴共x兲
= P / A共x兲, where P and A are instantaneous load and sample
cross section area, respectively, Eq. 共1兲 is deduced to
⌬h =
冕
h0
0
P
dx,
␲共d0 + x sin ␪兲2E
共2兲
where d0 is the radius of the pillar and ␪ = 2.5° is the taper
angle. As sin ␪ ⬇ ␪, we obtain
冉 冊冤
P
E=
⌬h
冉
2␪h0
d0
␲ d 02 ␪
ln 1 +
冊
冥
.
共3兲
Substitute d0 = 0.5 ␮m and h0 = 2.5 ␮m, we finally have
E = 2.64 ⫻ 1012
冉 冊
P
,
⌬h
共4兲
where P and ⌬h are in the units of Newton and meter, respectively. Thus, the elastic modulus is the slope of loaddisplacement curve multiplied by a constant of 2.64⫻ 1012,
which is about 41 GPa. However, the above calculation was
based on a rigid substrate. In our case, the substrate is also
the same Mg-BMG. Under such a condition, according to a
finite element analysis of microcompression,11 the actual
Young’s modulus should be corrected by a factor of about
1.25. This results in a value of 51.3 GPa, which is in excellent agreement with the literature value of 50.6 GPa measured from bulk Mg-BMG sample.12
Since the sample is tapered, yielding will, in principle,
start from the top of the sample because this is the location
with the minimum cross section. The departure load divided
by the top area is then the “apparent” yield strength. Despite
a slight variation, the apparent yield strengths of the 3.8 and
1 ␮m pillars range from 1342 to 1580 MPa, which are much
higher than that for the bulk Mg65Cu25Gd10 samples, which
is about 800 MPa.13 These data are summarized in Table I.
The current result of improved strength caused by a decrease
in sample size is in accord with a previous study on
millimeter-scaled BMG.8 Also noted in Table I is the fact that
the average strength of the 1 ␮m pillar 共⬃1500 MPa兲 is
higher than that of the 3.8 ␮m pillar 共⬃1390 MPa兲. This
demonstrates that the sample size effect can extend from the
millimeter to micron range. The strength improvement from
10−4 s−1
10−3 s−1
10−2 s−1
6 ⫻ 10−2 s−1
1400
1342
800
1488
1362
1580
1384
1525
1464
millimeter-scaled sample to micron-scaled pillar is noted to
be about 60%–100%, which is quite remarkable.
After yielding, the load remains essentially constant at
different strain rates, as evident in Fig. 2. The corresponding
engineering stress-strain curves, not shown here, have a similar shape. This particular shape of stress-strain curve usually,
according to the conventional wisdom, leads to a conclusion
that BMG behaves like a perfectly plastic material. However,
the raw displacement-time data, as given in Fig. 3, seem to
reveal a different message. Specifically, data in Fig. 3 indicate the displacement 共or strain兲 takes place almost instantly.
In other words, strain does not occur in a gradual fashion but
in the form of burst, even though the tests were conducted
under strain rate control conditions. Every strain burst event,
regardless of the strain rate, proceeds within about 1 s, suggesting that the strain rate during these bursts was at least
10−1 s−1. There is also a notable trend that, for both micropillars, a faster compression rate results in a larger strain
burst. The instant strain burst is peculiar but is usually indicative of a sudden propagation of a localized shear band.9
The morphology of compressed pillar samples is shown
in Fig. 4. It is well known that bulk Mg-BMGs generally
fracture into fragments at the onset of yielding since they
have a low Poisson’s ratio of ⬃0.32.14 However, the present
micropillar samples do not exhibit such a catastrophic fracture mode even at a “macroscopic” strain of 10%–15%. In
fact, all of them deformed by macroscopic shear, presumably
resulted from localized shear band propagation. In principle,
shear band is anticipated to initiate from the corner of contact between specimen and compression punch. This is the
location where the sample experiences the maximum stress,
not only because it has the minimum cross section as a result
of sample taper, but also due to the large constraint caused by
the friction between the test specimen and punch. Indeed, it
is clearly shown in Fig. 4, in which all pillars are observed to
deform by this shear mode after yielding. Furthermore, immediately following the yielding, the upper part of the specimen begins to slide along a major shear plane. However, the
effective load-bearing cross-sectional area always remains
constant, as the punch impresses onto the bottom part of the
FIG. 3. 共Color online兲 Scanning electron microscopy 共SEM兲 micrographs
showing the appearance of deformed pillars: 共a兲 3.8 ␮m, ⬃6 ⫻ 10−4 s−1; 共b兲
1 ␮m, ⬃2 ⫻ 10−4 s−1; 共c兲 3.8 ␮m, ⬃1.2⫻ 10−3 s−1; 共d兲 1 ␮m, ⬃1
⫻ 10−3 s−1; 共e兲 3.8 ␮m, ⬃6 ⫻ 10−2 s−1; and 共f兲 1 ␮m, ⬃6 ⫻ 10−2 s−1.
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161913-3
Appl. Phys. Lett. 91, 161913 共2007兲
Lee, Huang, and Nieh
An increase in strength with decreasing sample size can
be rationalized in the following. For brittle materials, the
variability of their strengths is expected based on their flaw
sensitivity and may be analyzed using the Weibull statistics.
The Weibull equation describes the fracture probability P f as
a function of a given uniaxial stress ␴ in the form of17
冋 冉 冊册
P f = 1 − exp − V
␴ − ␴u
␴0
m
,
共5兲
where ␴0 is a scaling parameter, m is the Weibull modulus,
and V is the volume of the tested sample. The parameter ␴u
denotes the stress at which there is a zero failure probability,
and is usually taken to be zero. Assume the characteristic
flaw causing fracture in both millimeter and micron samples
are the same, then, at a fixed fracture probability, i.e.,
P f = constant, the above equation reduces to
V
冉 冊
␴
␴0
m
= const.
共6兲
Since V ⬀ d3, where d is the diameter of the compression
sample, we obtain
FIG. 4. Displacement as a function of loading time during microcompression for the 共a兲 1 ␮m and 共b兲 3.8 ␮m pillars.
sample. This explains the observation of “lips” on the top of
deformed samples and the constancy of load in Fig. 2.
Additional remarks must be made from the examination
of Fig. 4. First, the recorded displacement 共or strain burst兲
comes from a sudden macroscopic sliding, which is triggered
by microscopic shear band propagation in the specimen, not
from a uniform plastic deformation per se. It is, in fact,
equivalent to the situation of rigid-body sliding or bicrystal
sliding.15,16 The measured strain should not be interpreted as
general plasticity, at least in the conventional sense, since
neither the two “crystals” deform. Furthermore, the constancy of load 共or engineering stress兲 is not indicative of a
perfectly plastic material, or has any thing to do with work
hardening or softening. Rather, it is a measure of friction of
sliding between two rigid bodies. In the current case, it represents the self frictional force of the Mg-BMG.
Gao et al.10 predicted a finer shear band spacing at a
faster strain rate. Although the strain rate differs by only two
orders of magnitude in the present experiments, there is a
general tendency of having more shear bands in pillars deformed at a higher strain rate; this is clearly demonstrated in
Fig. 4. It is also noted that there are very limited number of
shear bands in these micropillar samples, especially in the
1 ␮m pillar sample. In fact, at the slowest strain rate of
2 ⫻ 10−4 s−1, only one shear band is present. This suggests
that 1 ␮m is close to the scale limit of shear band spacing.
Should it be the case, then, the measured apparent yield
strength of the 1 ␮m pillar 共1580 MPa兲 would represent the
strength for a material which does not contain any volumetric defect with size greater than the critical value that can
lead to catastrophic fracture. The critical strength of the
present Mg BMG is about G / 12, where G is the shear modulus of 19.3 GPa. Although there is no existing theory to predict the theoretical strength of amorphous metals, it is interesting to note that G / 共10– 12兲 is also coincidentally the
predicted value for polycrystalline metals.
d3␴m = const.
共7兲
As listed in Table I, the apparent strengths of the 3 mm and
1 ␮m Mg-BMG samples are 800 and 1580 MPa, respectively. Insert these values into Eq. 共7兲, the Weibull modulus
is calculated to be about 35, which is within the range of the
m values recently reported for Zr48Cu45Al7 共m = 73.4兲 and
共Zr48Cu45Al7兲98Y2 共m = 25.5兲.18 The above analysis supports
the observation that the compression stress increases with
decreasing specimen size.
The authors gratefully acknowledge the sponsorship
from the National Science Council of Taiwan, Republic of
China, under the Project No. NSC 96-2218-E-110-001. This
work was also partially supported by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, U.S. Department of Energy under Contract No. DEFG02-06ER46338 with the University of Tennessee.
1
M. D. Uchick, D. M. Dimiduk, J. N. Florando, and W. D. Nix, Science
305, 986 共2004兲.
2
D. M. Dimiduk, M. D. Uchic, and T. A. Parthasarathy, Acta Mater. 53,
4065 共2005兲.
3
J. R. Greer, W. C. Oliver, and W. D. Nix, Acta Mater. 53, 1821 共2005兲.
4
C. A. Volkert and E. T. Lilleodden, Philos. Mag. 86, 5567 共2006兲.
5
M. D. Uchick and D. M. Dimiduk, Mater. Sci. Eng., A 400-401, 268
共2005兲.
6
F. Spaepen, Acta Metall. 25, 407 共1976兲.
7
A. S. Argon, Acta Metall. 27, 47 共1979兲.
8
Q. Zheng, S. Cheng, J. H. Strader, E. Ma, and J. Xu, Scr. Mater. 56, 161
共2007兲.
9
C. A. Schuh, A. C. Lund, and T. G. Nieh, Acta Mater. 52, 5879 共2004兲.
10
Y. F. Gao, B. Yang, and T. G. Nieh, Acta Mater. 55, 2319 共2004兲.
11
H. Zhang, B. E. Schuster, Q. Wei, and K. T. Ramesh, Scr. Mater. 54, 181
共2006兲.
12
X. K. Xi, R. J. Wang, D. Q. Zhao, M. X. Pan, and W. H. Wang,
J. Non-Cryst. Solids 344, 105 共2004兲.
13
H. M. Chen, Y. C. Chang, T. H. Hung, X. H. Du, J. C. Huang, J. S. C.
Jang, and P. K. Liaw, Mater. Trans. 48, 1802 共2007兲.
14
J. Schroers and W. L. Johnson, Phys. Rev. Lett. 93, 255506 共2004兲.
15
T. Watanabe, M. Yamada, S. Shima, and S. Karashina, Philos. Mag. A 40,
667 共1979兲.
16
H. Miura, K. Hamashima, and K. T. Aust, Acta Metall. 28, 1591 共1980兲.
17
W. J. Weibull, J. Appl. Mech. 18, 293 共1951兲.
18
W. F. Wu, Y. Li, and C. A. Schuh, 共to be published兲.
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