Strain bursts in plastically deforming Molybdenum micro- and nanopillars
ABSTRACT Plastic deformation of micron and sub-micron scale specimens is characterized by intermittent sequences of large strain bursts (dislocation avalanches) which are separated by regions of near-elastic loading. In the present investigation we perform a statistical characterization of strain bursts observed in stress-controlled compressive deformation of monocrystalline Molybdenum micropillars. We characterize the bursts in terms of the associated elongation increments and peak deformation rates, and demonstrate that these quantities follow power-law distributions that do not depend on specimen orientation or stress rate. We also investigate the statistics of stress increments in between the bursts, which are found to be Weibull distributed and exhibit a characteristic size effect. We discuss our findings in view of observations of deformation bursts in other materials, such as face-centered cubic and hexagonal metals. Comment: 14 pages, 8 figures, submitted to Phil Mag
arXiv:0802.1843v1 [cond-mat.mtrl-sci] 13 Feb 2008
Strain bursts in plastically deforming
Molybdenum micro- and nanopillars
February 14, 2008
M. Zaiser1, J. Schwerdtfeger1, A.S. Schneider2,
C.P. Frick2, B.G. Clark2and P.A. Gruber2,3, and E. Arzt2,4
1The University of Edinburgh, Institute for Materials and Processes,
The King’s Buildings, Sanderson Building, Edinburgh EH9 3JL, UK
2Max-Planck-Institut f¨ ur Metallforschung, Heisenbergstrasse 3,
70569 Stuttgart, Germany
3Universit¨ at Karlsruhe, Institut f¨ ur Zuverl¨ assigkeit
von Bauteilen und Systemen, Kaiserstr. 12, 76131 Karlsruhe, Germany
4Leibniz Institute for New Materials, Campus Building D2 2, 66123
Saarbr¨ ucken, Germany
Plastic deformation of micron and sub-micron scale specimens is char-
acterized by intermittent sequences of large strain bursts (dislocation
avalanches) which are separated by regions of near-elastic loading. In
the present investigation we perform a statistical characterization of strain
bursts observed in stress-controlled compressive deformation of monocrys-
talline Molybdenum micropillars. We characterize the bursts in terms of
the associated elongation increments and peak deformation rates, and
demonstrate that these quantities follow power-law distributions that do
not depend on specimen orientation or stress rate. We also investigate the
statistics of stress increments in between the bursts, which are found to be
Weibull distributed and exhibit a characteristic size effect. We discuss our
findings in view of observations of deformation bursts in other materials,
such as face-centered cubic and hexagonal metals.
On microscopic and mesoscopic scales, plastic deformation of crystalline solids
proceeds as an intermittent series of strain bursts (’slip avalanches’). Indirect
evidence of such bursts has been provided by systematic acoustic emission (AE)
studies of Weiss and co-workers on ice [1, 2], hcp metals , and fcc metals
. These studies indicate that the AE signals of plastically deforming crystals
consist of discrete bursts separated by quiescent intervals of low AE activity.
The energies E (amplitude square integrated over the duration of a burst) and
peak amplitudes A of the AE bursts exhibit a huge scatter; their statistics is
characterized by scale-free (power law) distributions, with probability density
functions p(E) ∝ E−κEand p(A) ∝ A−κAthat are well described as power laws
with material-independent exponents κE≈ 1.5 and κA≈ 2, extending over up
to 8 decades with no apparent cut-off.
Dimiduk and co-workers confirmed temporal intermittency of plastic flow by
direct observation of strain bursts during compressive deformation of micropil-
lars machined out of Ni single crystals . In these experiments, the elongation
vs. time curves observed under stress-controlled loading were characterized by
an intermittent sequence of deformation jumps, with elongation increments ∆l
that exhibited a scale-free distribution p(∆l) ∝ ∆l−κlwhere κl≈ 1.5. Recently,
Ngan observed bursts with similar statistical characteristicsin creep deformation
of aluminum micropillars under constant stress conditions . These findings
can be directly related to the acoustic emission results if one assumes that a
fixed fraction of the work done by the external forces during an elongation jump
is released in the form of acoustic energy.
Theoretically, the formation of intermittent deformation bursts has been
modelled using two-dimensional  and three-dimensional  discrete dislo-
cation dynamics simulations, as well as various types of continuum models
[9, 10, 11, 12]. In the discrete simulations, the stochastic nature of the de-
formation process is directly ’inherited’ from the statistical choice of the initial
dislocation configuration, which in turn reflects the variability of the initial
microstructure of real specimens. In the continuum models, statistical hetero-
geneity needs to be explicitly incorporated into the constitutive equations, e.g.
in terms of fluctuations in the local flow stresses or energy dissipation rates.
While less ’realistic’ than discrete dislocation simulations, such models provide
a conceptual framework for understanding the origin of the scale-free avalanche
dynamics which can be related to a depinning-like transition between an elas-
tic and a plastically deforming phase (‘yielding transition’). A comprehensive
overview of experimental and theoretical results has been given by Zaiser .
In spite of all these investigations, open questions remain. The relations
between different characteristics of strain bursts, such as the associated strain
or elongation increments, the burst durations and the peak strain rates, have
remained largely unexplored. Other open issues concern the statistics of stress
increments between bursts and the correlation between stress increments and
burst sizes. Furthermore, practically all the experimental evidence has been
gathered on materials (fcc and hcp metals, ice close to its melting point) where
the motion of dislocations is governed by their mutual interactions. Molybde-
num (Mo), on the other hand, is a bcc metal where dislocation interactions with
the crystal lattice (Peierls stresses) may have a crucial influence on the defor-
mation behavior: Below the so-called knee temperature (∼ 500-550 K for Mo
), the plastic deformation of bulk bcc metals is controlled by the nucleation
and motion of kinks on screw dislocations. These materials exhibit therefore a
strong temperature and strain-rate dependence of the flow stress (for a detailed
overview of the deformation behavior of bcc metals, see [14, 15]). Since the mo-
tion of dislocations is at low temperatures governed by their interactions with
the crystal lattice, it has been argued that collective behavior, and hence strain
bursts, may be suppressed in this temperature regime .
In the present paper, we adress these open questions by investigating strain
bursts observed in Molybdenum (Mo) micropillars that are deformed at room
temperature in compression under load control. In the following sections we first
describe the experimental procedure and the methods used for characterizing
the strain bursts. We then discuss the results of our analysis in view of the
statistics of burst sizes and stress increments, and the relations between burst
strain and peak strain rate. We conclude with a comparison of our findings with
theoretical and experimental results on strain bursts in other materials.
2.1Specimen preparation and mechanical testing
Zone refined Mo single crystals were oriented using Laue diffraction, and disk-
shaped samples of approximately 3 mm height and 10 mm diameter with disk
normals pointing along  and  lattice directions were cut by spark ero-
sion. The disk surfaces were then mechanically polished using 6, 3 and 1 µm
diamond suspensions and subsequently electro-polished for 60 seconds using a
mixture of 610 ml Methanol and 85 ml H2SO4at a current of 1-2 Amperes. The
samples were mounted on a custom machined aluminum holder for testing, and
their orientation was confirmed by electron backscatter diffraction.
The specific process used in fabricating micropillars from the oriented sam-
ples is very similar to the method of Frick et al. . Free-standing pillars
of tapered shape were fabricated using a dual focused ion beam (FIB) and
scanning electron microscope (SEM) (FEI Nova 600 NanoLab DualBeamTM).
The as-machined pillars were deformed in compression at ambient pressure and
temperature by using a MTS XP nanoindenter system equipped with a sapphire
conical indenter with a flat 10 µm diameter tip. The loading rates varied be-
tween 4 and 60 µN/sec, depending on pillar diameter. Geometrical parameters
(top diameter dt, bottom diameter db, and length l of the analysed pillars) and
loading rates are compiled in Table 1.
Deformation experiments were performed at a control rate of 500 Hz with
a data storage rate of 25 Hz, i.e., data were recorded at intervals ∆t = 0.04 s.
Tests were typically performed with two intermediate unloading and reloading
cycles at about 2.5% and 5% strain in order to observe the linear elastic response
and transient loading/unloading behavior of the pillars. The intermediate un-
loading may influence the strain burst statistics: Most of the deformation occurs
during the largest bursts (see Figure 1) and it is thus very likely that these are
truncated by unloading (in fact, sometimes the burst continued during unload-
ing or burstlike deformation resumed at a reduced stress level upon reloading).
Therefore, a set of  oriented pillars (labelled with the subscript ’nr’ in Table
1) were deformed without intermediate unloading. Analysing these separately
allows us to assess the influence of unloading on the burst statistics.
The flow stresses of the investigated samples increase with decreasing sample
size. A study of this size effect has been published elsewhere . Here, we
focus exclusively on the intermittent nature of the deformation process. As
can be seen from Figure 1, the deformation curves are characterized by an
irregular sequence of large strain bursts visible as steps on the stress vs. strain
or elongation vs. time curves. During the bursts, which typically lasted less
than a second, deformation rates were high (peak strain rates > 1 s−1). The
elongation rate signals have the typical signature of a ’crackling noise’ (Figure
Figure 1: Stress-strain curves of  oriented Mo micropillars  (Specimens
2nr, 5nr, 6nr, 7nr, 9nr, and 10nrin Table 1).
2) , i.e., they are composed of discrete bursts of widely varying magnitude.
Strain bursts were characterized in terms of their size (defined as the elongation
increment between the beginning and the end of the burst), duration, and peak
elongation rate. To define a burst, the elongation vs. time signals l(t) were first
conditioned by performing a running average over an averaging time interval
∆tav. This served to eliminate high-frequency noise resulting from the defor-
mation setup. The averaged signals¯l(t) were then differentiated using a simple
central difference scheme, and the resulting elongation rate signals dt¯l(t) were
broken into bursts by thresholding: A strain burst was associated with a time
for t = ti
2+ ∆t. The burst duration was then defined
as Ti:= ti
peak elongation rate as˙Li
elongation rate as˙Li
was defined as σi:= σ(ti
some rare occasions, bursts occurred during intermediate unloading or reload-
ing, leading to negative stress increments. These bursts were discarded from
the stress increment statistics.
In our analysis we used the standard parameters ∆tav= 0.8 s and˙lthresh=
0.5 nm/s. For these parameters, a typical specimen of 0.5µm diameter yielded
between 50 and 100 bursts, most of them small. Since this is not sufficient for
1− ∆t and for t = ti
2] such that dt¯l(t) >˙lthreshfor all t ∈ [ti
2] and dt¯l(t) <˙lthresh
2, the burst elongation as Si= l(ti
pav:= max[dt¯l(t)] for t ∈ [ti
p:= max[dtl(t)] for t ∈ [ti
1), and the stress increment as ∆σi:= σi− σi−1. On
2) − l(ti
1), the time-averaged
2]. The burst initiation stress
2], and the true peak
Figure 2: Strain rate vs. time signal during deformation of a  oriented Mo
micropillar (Specimen 10nr, lowermost curve in Figure 1).
a meaningful statistical analysis, we grouped specimens of the same orientation
into size classes with typically 6-8 specimens in each class. Those  spec-
imens that were deformed without intermediate stress relaxation are grouped
separately such that the influence of intermediate unloading on the strain burst
statistics can be assessed. The class partition is shown in Table 1. For the
bursts obtained from all specimens in a given class, probability distributions
p(S) were determined by logarithmically binning the Sidata. This is appropri-
ate for power-law distributed data where logarithmic binning may significantly
improve the statistics in the regime of large events without introducing spu-
rious cut-off effects. Stress increments, on the other hand, were found to be
Weibull distributed. In this case, since the data scatter around a characteristic
value, logarithmic binning makes little sense. Instead, we base our statistical
analysis on the cumulative distribution P>(∆σ) as determined from the ordered
sequence of the ∆σi: P>(∆σn) ≈ n/(N + 1) where N is the total number of
stress increments and ∆σnthe nth member in the descending sequence.
To ensure that the burst statistics do not significantly depend on the sig-
nal conditioning and thresholding parameters ∆tav and˙lthresh, we performed
a systematic parameter study by varying these parameters in the ranges 0.2 s
≤ ∆tav ≤ 2.4 s and 0.25 nm/s ≤˙lthresh≤ 2.5 nm/s, and studying the corre-
sponding changes in the p(S) probability distribution for ’small’  oriented
pillars (class S, Table 1).
3 Results and Discussion
We first investigate to which extent the statistics of strain burst sizes is influ-
enced by the parameters used for smoothing and thresholding the raw elongation
rate signals. Figure 3 shows distributions obtained for ’small’  crystals us-
ing three different sizes of the averaging window. For short averaging windows,
the distribution of burst sizes exhibits two distinct regimes: At small burst sizes,
the burst size distribution has a ’hump’ which decays exponentially, whereas at
large sizes, the exponential decay is replaced by a power-law tail. We may as-
sociate these two regimes with two different physical processes, viz on the one
hand the high-frequency noise of the deformation setup which produces a large
number of small ’bursts’ – in fact, just irregular oscillations of the deformation
machine – and on the other hand the collective dynamics of the dislocation
system which produces intermittent large bursts of plastic deformation activity
with a power-law size distribution. What is important is that the high-frequency
noise of the machine does not mask the power-law scaling since the amplitude of
the machine-induced elongation fluctuations is limited to values less than 1 nm.
By increasing the length of the averaging window, we can suppress this expo-
nential ’hump’ while the power-law part of the distribution remains unchanged
– in fact, the length of the scaling regime increases and reaches a maximum at
a window length of 0.8 s which we choose as our default value. If the original
signal is averaged over even larger times, the size distribution of large bursts
remains unchanged but the length of the scaling regime decreases again since
smaller bursts are ’washed out’ as their peak elongation rates fall below the
Figure 4 shows the dependence of the burst size distribution on the imposed
elongation rate threshold. For small thresholds, the distribution is practically
independent on threshold, while a threshold substantially above our default
value of 0.5 nm/s eliminates small bursts but leaves the size distribution of
large bursts practically unchanged. Crucially, neither the size of the averaging
window nor the choice of the threshold seem to have any appreciable influence
on the power-law scaling of the burst size distribution in the large-burst regime
(elongation increments larger than approximately 1 nm). This robustness of
the procedure indicates that it is indeed viable to envisage our elongation rate
signals as ’crackling noise’ composed of discrete events.
Probability distributions of strain burst sizes for the different specimen
classes are shown in Figure 5 (left). All distributions can with reasonable accu-
racy be described as power laws:
p(S) ∝ S−κ.(1)
Least-square fits to the logarithmically binned data yield values of 1.34 ≤ κ ≤
1.76, and compiling bursts from all specimens and determining the overall size
distribution as shown in the inset of Figure 5 (left) yields κ = 1.6 ± 0.03. No
systematic dependency of the exponent κ on pillar orientation or pillar size can
be detected. Even though intermediate unloading is expected to truncate some
of the largest bursts,  oriented specimens deformed without unloading do
not exhibit larger bursts than those from the other groups - if anything, the
above average exponent κ ≈ 1.76 for the NR class suggests the opposite.
Differences between the distributions obtained for different specimen classes
Figure 3: Strain burst size distributions for ’small’  oriented Mo micropillars
(class S in Table 1), determined with an elongation rate threshold of 0.02
nm/s and different sizes of the averaging window; full line: fit function p(x) =
100exp[−x/0.12] + 0.05x−1.5
should not be over-interpreted – they may simply reflect statistical scatter in-
herent in the not very large size of the datasets which comprise typically some
400 bursts for each specimen class. To illustrate this point, we show on the
right-hand side of Figure 5 simulated p(S) distributions determined from 6 sets
of surrogate data, each consisting of 400 random numbers drawn independently
from a distribution p(S) ∝ S−1.5. As can be seen, the scatter of the exponents
determined from these sets, the scatter in the data ranges, and the error of the
linear least-square fits are all comparable with the corresponding values for the
experimental datasets. This illustrates the intrinsic problems encountered in
determining distribution parameters from limited sets of data.
We now proceed to investigate other burst characteristics, viz the burst du-
rations and peak elongation rates. Unfortunately, the intrinsic burst durations
may be well below the size ∆tav of our averaging window. As a consequence,
all large bursts determined from the averaged elongation rate signal have ap-
proximately the same duration which is roughly proportional to ∆tav. Hence,
the burst durations as determined from the averaged signals are no longer good
characterizersof the bursts and, for evident reasons, the same is true for the peak
rates of the time-averaged signals which decrease with increasing ∆tav. The true
peak elongation rate˙Lp, on the other hand, represents an intrinsic property of
the bursts that is not affected by time averaging. There is a strong statistical
correlation between˙Lpand burst size S (correlation coefficient r > 0.9) but it
it is not easy to establish a clear-cut mathematical relation between the two
quantities. This is seen from Figure 6 which shows˙Lpvs. S values for all large
Figure 4: Strain burst size distributions for ’small’  oriented Mo micropil-
lars (class S in Table 1), determined with different threshold values of the
elongation rate and a window size of 0.8 s.
bursts. While part of the observed bursts seem to exhibit peak rates that are
approximately proportional to the burst sizes (˙Lp ∝ s, upper straight line in
Figure 6), other data seem to suggest a proportionality to the square root of the
burst sizes˙Lp∝ S1/2, lower straight line in Figure 6). Fitting a power law to all
the data yields˙Lp∝ S0.8which badly represents either group. Interpretation
of these findings is further complicated by the fact that the two behaviors do
not represent different specimen classes – rather, bursts from one and the same
specimen may be found both near the upper and the lower straight lines.
To assess the degree of correlation between the burst size and the magnitude
of the preceding and following stress increments, the respective correlation coef-
ficients were evaluated separately for the 6 datasets in class S, and the mean
correlation coefficient as well as the variance of the r values were determined.
The results (r = 0.03±0.24 for the correlation coefficient between burst size and
magnitude of the preceding stress increment, and r = 0.11±0.28 for the correla-
tion between burst size and magnitude of the following stress increment) do not
indicate any statistically significant correlation. There is also no statistically
significant correlation between the sizes of successive bursts (r = 0.02 ± 0.05).
The statistics of stress increments differs substantially from the statistics
of burst sizes: Instead of scale-free power laws we find distributions with a
characteristic scale that depends on specimen size. Figure 7 shows cumulative
distributions P>(∆σ) (probability to find a stress increment larger than ∆σ)
for the specimen classes S, M, L, and XL. The data can be
Figure 5: Left: Compilation of strain burst size distributions for the different
specimen classes, the legend shows the κ values for each class; inset: aggre-
gated distribution of burst sizes from all classes; right: surrogate data (distribu-
tions determined from sets of 400 random numbers S drawn from a distribution
p(S) ∝ S−1.5with Smin= 0.02); full lines: κ = 1.5.
Figure 6: Relation between burst size S and true peak elongation rate˙Lp; upper
straight line:˙Lp∝ S, lower straight line:˙Lp∝ S1/2.
Figure 7: Distributions of stress increments for specimen classes S, M,
L, and XL; full lines: Weibull fits, for parameters see inset in figure.
well fitted by Weibull distributions,
P>(∆σ) = exp
where m is the Weibull modulus and the stress parameter ∆σ0defines the char-
acteristic stress increment. Parameters for the distributions are shown in the
legend; the Weibull modulus m which determines the width of the distribution is
approximately the same for all distributions, but the stress parameter decreases
with increasing specimen size.
This is further illustrated in Figure 8 where parameters m and ∆σ0of stress
increment distributions obtained from individual specimens are plotted against
specimen size. It is clearly seen that the Weibull moduli m ≈ 0.75 do not de-
pend significantly on specimen size, whereas the stress parameters ∆σ0 (and,
accordingly, the average stress increments between bursts) decrease approxi-
mately in inverse proportion with specimen diameter dt. This implies that, in
larger specimens, smaller stress increments are needed to trigger strain bursts
– an obvious result since in larger specimens we expect to find a larger number
of weak regions or sources that can be activated in any given stress interval. A
more quantitative analysis is, however, hampered by the fact that the charac-
teristic stress increments depend on the averaging and thresholding parameters
used in our data analysis: shorter averaging times ∆tav and smaller threshold
values˙lthreshlead to the identification of a larger number of small ’bursts’ and
a proportional reduction of the characteristic stress increment ∆σ0. Therefore,
Figure 8: Parametersof Weibull fits to stress increment distributions determined
for individual specimens; different symbol shapes distinguish different specimen
classes (⋄ S, ? M, △ L, ▽ XL); open symbols: Weibull
moduli; cross-center symbols: stress parameters.
without a method to clearly distinguish between machine-induced noise and the
smaller bursts that result from collective dislocation motion, it is difficult to
draw quantitative conclusions from the observed size dependence of the P(∆σ)
Our investigation provides an example of plasticity behaving as a ’crackling
noise’ , with intermittent bursts of activity characterized by scale-free size
distributions. For the burst sizes (elongation increments) we find a distribution
p(S) ∝ S−1.5which is in line with experimental findings on Ni micropillars 
as well as theoretical predictions based on continuum and discrete dislocation
models . The same theoretical models predict power-law relationships˙Lp∝
T ∝ S1/2to hold between the peak rate, duration, and size of strain bursts.
Unfortunately, owing to the need for conditioning the signal by time averaging,
no useful information about the burst durations could be obtained in the present
investigation, while the information regarding the relationship between burst
size and peak elongation rate turned was found to be ambiguous.
In line with previous investigations, the power-law characteristics of strain
bursts seem to be little affected by specimen orientation, size, or imposed defor-
mation rate. While theoretical investigations [8, 19] suggest an intrinsic cut-off
to the power-law scaling regime, no such cut-off could be identified in our inves-
tigation. This may be due to the fact that establishing a cut-off requires good
statistics in the region of very large strain bursts, which could not be achieved
in the present investigation as the total number of bursts obtained from each
individual specimen was small (< 100).
The distributions of stress increments between subsequent strain bursts differ
substantially from the burst size distributions. Instead of scale-free power laws,
we find Weibull distributions with a characteristic stress scale (the stress pa-
rameter ∆σ0) that decreases approximately in inverse proportion with specimen
size. However, the very presence of a characteristic scale makes the distribution
parameters depend on the number of identified bursts. This dependency raises
the problem of distinguishing between the effects of collective dislocation motion
and the effects of machine noise, which may increase the apparent burst number
by adding spurious ’bursts’ of small size into the statistics. For the same rea-
son, any conclusions based upon the observed lack of correlation between burst
sizes and strain increments, or between the sizes of subsequent bursts, must be
regarded with caution.
Our investigation demonstrates for the first time the occurrence of scale-free
strain bursts in a bcc metal deforming below the transition temperature, and
we find that the burst characteristics are similar to those in fcc metals. This
implies that even a significant Peierls stress, which is of crucial importance for
the deformation properties of bulk Mo at ambient tempeature, is not sufficient to
inhibit burst-like deformation. In this sense, we may conclude that the Peierls
potential is irrelevant as far as the dynamics and statistics of strain bursts
are concerned, and that the observed behaviour constitutes a truly universal
feature of dislocation plasticity that can be observed in all kinds of crystal
lattice structures. It may be mentioned that the same is not true for the size
dependence of the flow stress: The size effects observed in fcc micropillars (for
reference, see, e.g. ) differ substantially from those observed in the present
samples . The universality of strain burst behavior is a key result of the
present study, and it would be desirable to obtain further corroboration of this
result from acoustic emission measurements on bulk bcc metals.
Acknowledgements: We acknowledge support of the Commission of the
European Communities under contract NEST-2005-PATH-COM-043386 and of
EPSRC under Grant No. EP/E029825.
 J. Weiss, J.-R. Grasso, M.-C. Miguel, A. Vesipignani and S. Zapperi, Mater.
Sci. Engng. A 309-310, 360 (2001).
 T. Richeton, J. Weiss and F. Louchet, Acta Mater. 53, 4463 (2005).
 T. Richeton, P. Dobron, F. Chmelik, J. Weiss and F. Louchet, Mater. Sci.
Engng. A 424, 190 (2006).
 J. Weiss, T. Richeton, F. Louchet, F. Chmelik, et al., Phys. Rev. B, in
 D.M. Dimiduk, C. Woodward, R. LeSar and M.D. Uchic, Science 26, 1188
 A.H.W. Ngan, Phil. Mag. Letters 87, 967 (2007).
 M.-C. Miguel, A. Vespignani, S. Zapperi, J. Weiss and J.-R. Grasso, Nature
410, 667 (2001).
 F. F. Csikor, C. Motz, D. Weygand, M. Zaiser and S. Zapperi, Science 318,
 M. Zaiser and E.C. Aifantis, J. Mech. Behavior Mater. 14, 255 (2003).
 M. Koslowski, R. LeSar, and R. Thomson, Phys. Rev. Lett. 93 125502
 M. Zaiser and P. Moretti, J. Stat. Mech., P08004 (2005).
 M. Zaiser and E.C. Aifantis, Int. J. Plasticity 22, 1432 (2006).
 M. Zaiser, Adv. Physics 55, 185 (2006).
 B. Sestak and A. Seeger, Z. Metallkde. 69, 195, 355 (1978).
 A. Seeger, Z. Metallkde. 93, 760 (2002).
 C.P. Frick, B.G. Clark, S. Orso, A.S. Schneider, and E. Arzt, Mater. Sci.
Engng. A, in press.
 A.S. Schneider, C.P. Frick, B.G. Clark, P.A. Gruber and E. Arzt, Appl.
Phys. Letters, submitted.
 J.P. Sethna, K.A. Dahmen and C.R. Myers, Nature 410, 242 (2001).
 M. Zaiser and N. Nikitas, J. Stat. Mech., P04013, (2007).
 D.M. Dimiduk, M.D. Uchic and T.A. Parthasarathy, Acta Mater. 53, 4065
Table 1: Specimen orientations ( or ), geometries and loading rates.
d=180-300 nm 16237282
?˙ σ? = 58.6 MPa/s 17300 363
d=340-600nm 13515 572
?˙ σ? = 36.9 MPa/s 14345 390
?˙ σ? = 29.9 MPa/s61230 1370
d > 2000nm229603440
?˙ σ? = 6.4 MPa/s332203520
?˙ σ? = 53.3 MPa/s4nr
d = 200-600nm9 324436
?˙ σ? = 49.6 MPa/s 10 308500
d = 650-1500nm213701690
?˙ σ? = 28.1 MPa/s312501650
d > 1500nm634704510
?˙ σ? = 14.2 MPa/s734604530
˙ σ [MPa/s]