1. science
2. mathematics
3. geometry

International Journal for Multiscale Computational Engineering, 7(3)187–194(2009)
Effects of Sample Geometry on the Uniaxial
Tensile Stress State at the Nanoscale
S. Brinckmann,∗ J.-Y. Kim, A. Jennings, & J. R. Greer
Department of Materials Science, California Institute of Technology, Pasadena, CA 91125
ABSTRACT
Uniaxial compression of micro- and nanopillars is frequently used to elicit plastic size effects
in single crystals. Uniaxial tensile experiments on nanoscale materials have the potential to
enhance the understanding of the experimentally widely observed strength increase. Furthermore, these experiments allow for investigations into the in-strength and to help to study
tension-compression asymmetry. The sample geometry might influence mechanical properties, and to investigate this dependence, we demonstrate two methods of uniaxial nanotensile
sample fabrication. We compare the experimentally obtained tensile stress-strain response for
cylindrical and square nanopillars and provide finite element method simulation results and
discuss the initiation of plastic yielding in these nanosamples.
KEYWORDS
nanoscale, plasticity, mechanical, tension
*Address all correspondence to Steffen.Brinckmann@rub.de.
c 2009 by Begell House, Inc.
1543-1649/08/$35.00 °
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1. INTRODUCTION
Continued miniaturization of devices used in bioand nanoapplications requires a thorough understanding of crystal deformation at small scales. In
recent years, uniaxial compression experiments on
micro- and nanopillars have been used to study the
size effect of materials [1–8]. These studies show
that single crystals are capable of sustaining significantly higher stresses than bulk crystals when their
diameters are reduced below 10 microns. The discrete dislocation framework is a multiscale model
that has recently been used to investigate the deformation in nanosamples [9, 10]. Molecular dynamics
simulations of single crystalline Au nanowire deformation [11] predict a strength increase as well
as tension-compression asymmetry and claim that
the surface tension of nanometer-sized structures
leads to a domain of compressive forces in the core.
The applied stress would compensate the surfaceinduced compressive stress in tension, but the surface induced stress in the core aids the applied stress
during compression. It is proposed that the core
compression leads to a higher magnitude of yield
stress in tension than in compression for samples
of identical size. Classical tensile experiments on
copper, iron, and silver pristine whiskers in the micron regime [12] have shown that the flow stress can
reach values close to the theoretical strength of the
material. Recently, Kiener et al. [13] performed tensile experiments on square cross-sectional samples
milled out of Cu single crystal via focused ion beam
(FIB). They observed a power law size dependence
with the exponent of −0.4 and stresses significantly
below the theoretical strength of Cu. The differences in sample geometry might have contributed
to such disagreement. Pillars with a circular cross
section have lower surface-to-volume ratios compared to those with a square cross section, which
might lead to additional surface-induced stresses in
the pillar core for square-shaped samples. The additional compressive stresses might, in turn, contribute to the tension-compression asymmetry and
cause higher tensile yield stresses for the square
shaped samples. Because the surface/volume ratio
decreases with increasing sample dimensions, the
effects of surface stress on the overall flow stress
become negligible for macrosized samples. Atoms
at the corners in the square cross-sectional geometry are less confined by their neighbors, leading to a
BRINCKMANN ET AL.
significantly lower energy for homogeneous defect
nucleation from the corner than for nucleation on a
flat surface, as shown by the computations of Zhu
et al. [11] and Li [14]. This difference in free energy leads to the lower yield stresses for corner dislocation nucleation (observed in rectangular crosssectional samples) than at a homogeneous surface
(circular cross-sectional pillars). In this work, we report experimental and finite element method (FEM)
results investigating the influence of square versus
round cross-sectionally shaped samples on uniaxial
tensile deformation behavior of single crystal h001ioriented gold nanopillars. We also describe two
new fabrication methods for creating uniaxial tension samples via FIB.
2. NUMERICAL STUDIES
We utilize FEM simulations to quantify the elastic stress distribution within the sample to optimize the sample geometry such that the inhomogeneities in stresses and strains within the sample
are minimized. The stress concentrations might lead
to localized incipient plasticity, and the experiment
would not be representative of the general material response at small scales. Flow stresses are often
assumed to be homogeneous for nanopillars compressed along high-symmetry crystallographic orientations. However, geometrical constraints at the
top and base of the pillar may lead to the elevated
local stresses. At the base, the deformation is constrained by the substrate, and friction between the
pillar and the flat punch indenter tip restricts the deformation of the pillar top. We use the conventional
eight-node linear finite elements (ABAQUS C3D8)
and simulate the deformation of square and round
cross-sectional samples on a substrate. These elements allow us to quantify the geometrical effects,
while allowing for time-efficient simulations. Due
to the double symmetry in the sample geometry, it is
sufficient to model a quarter of the pillar/substrate
combination with 10 three-dimensional elements in
the radial direction and 30 in the axial. We assume linear isotropic elasticity to evaluate the overall stress state at 10% strain. Isotropic properties of
Au are represented by Young’s modulus 79 GPa and
Poisson’s ratio 0.42. Since the constraint at the pillar top cannot be determined unambiguously, we do
not include it in the model. However, the findings
due to the constraint pillar base can used to infer a
International Journal for Multiscale Computational Engineering
EFFECTS OF SAMPLE GEOMETRY ON THE UNIAXIAL TENSILE STRESS STATE
qualitative understanding of the constraints at the
pillar top.
Figure 1 shows the von Mises yield stress distribution inside the pillars of the two described geometries, which is the most appropriate criterion
for the incipient yield position in the continuum
framework. While the stresses in the two geometries are nearly identical in the plane of symmetry,
we find that the maximum axial stress in the circular column is 1.14 GPa, while that in the square
one is 1.80 GPa and is localized in the corner of
the square sample decaying away from it. We find
that the substrate constraint leads to the formation
of higher local stresses in the square-shaped sample, even though the average applied stress is the
same. Assuming that the yield is governed by the
nucleation of dislocations from the region where the
stress state exceeds that of the von Mises yield criterion, this inhomogeneous elastic stress distribution
leads to an inhomogeneous dislocation nucleation
from the areas of elevated shear stresses, assuming
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that the dislocation sources are homogenously distributed and require equal activation energy. The
post yield stress-strain behavior of the nanopillars
consists of multiple discrete slip events indicative
of dislocation avalanche propagation, rendering the
continuum theory inapplicable to describe this behavior. To model plasticity in these nanoscale volumes with free surfaces, powerful 3-D meso-scopic
simulations, such as discrete dislocation simulations
[9, 10, 15–20] as well as molecular dynamic simulations [11], are better suited to investigate the post
yield behavior of nanopillars.
3. SAMPLE DESIGN AND FABRICATION
Kiener et al. [21] have also reported a method for
tensile sample fabrication from single crystal wire
tips, which were also orthogonally etched in the FIB.
Their experimental procedure requires subsequent
remounting of the entire sample onto a different
sample holder for each tensile experiment, result-
FIGURE 1. Von Mises equivalent stress in MPa at 10% strain determined by finite element method for (a) circular
and (b) rectangular cross-sectional samples on substrate due to tension
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190
ing in frequent misalignment issues and activation
of preferential single slip even for high-symmetryoriented samples. The experimental approach described here is an improvement over the previously
reported work because it allows for (1) the fabrication of several samples in the same bulk single crystal, (2) the compression and tension testing of the
same sample, and (3) a greatly reduced misalignment between the sample and the tensile grips. Figure 2 shows the sequence of tensile sample fabrication steps performed in FEI Nova Nanolab 200 FIB.
The resulting structure (Fig. 2(d)) is a circular pillar
with rectangular handles on either side. The handle thickness, b, was designed to be equal to the pillar radius, r, and their height, h, was optimized to
induce lower shear stresses in the handles than in
the sample. Since the maximum shear stresses in
the sample are induced at the angle α = 45◦ from
the axial direction, and the area of the handles is re-
BRINCKMANN ET AL.
quired to be greater than the cross-sectional sample
area, 2bh > πr2 cos α, the handle height has to satisfy the following criteria:
π
h > √ ' 2.22r
2r
(1)
The samples were prepared by the use of a FIB
from a h100i-oriented Au single crystal, which was
cleaned with acetone and isopropyl alcohol (IPA)
and then mounted on a scanning electron microscope (SEM)-sample holder with carbon paint. In
the first fabrication step, we use a perpendicular incident beam and a 7 nA current at 30 kV acceleration voltage to mill out a large ring with the outer
diameter sufficiently large to allow the tensile grips
to be lowered into the cut out volume. The diameter of the remaining material is 2 µm for ∼ 300 nmdiameter samples and 5 µm for 1 µm-diameter ones.
In the second step (Fig. 2(b)), we lower the current
FIGURE 2. Schematic of sample fabrication in focused ion beam: (a) milling out ring, leaving core in the middle
behind; (b) define milling pattern on surface of core using polygons; (c) mill out center piece and handles; and (d)
make undercuts using an i-beam at 45◦
International Journal for Multiscale Computational Engineering
EFFECTS OF SAMPLE GEOMETRY ON THE UNIAXIAL TENSILE STRESS STATE
to 0.1 nA and define the handles by using the polygon drawing tool in the FIB software. To achieve
a smooth surface on the central part of the sample,
15 polygons on either side were milled in parallel.
Special attention is paid to the raster direction such
that the polygons are milled from the outside to the
inside, which reduces the redeposition of atoms. In
the third step (Fig. 2(c)), the sample is tilted to 45◦ ,
and equally sized rectangles are used to make the
under cuts with 10 pA ion current. Special attention
is paid to align the rectangular undercuts at the sample side and at the underside of the handles. The
rectangular shaped pillars are prepared in a similar
fashion. Initially, a large pedestal in the center is created. Using rectangular milling patterns, a rectangular pillar is produced. Afterward, the single crystal is transferred onto a 86◦ -wedged sample holder,
and the undercuts are milled. Postfabrication, the
samples are inspected and measured by SEM.
4. RESULTS AND DISCUSSION
We fabricated nanopillars with diameters ranging
from ∼ 300 nm to ∼ 900 nm out of Au bulk single crystal in h100i orientation. SEM images of the
smallest samples of each geometry are shown in
Fig. 3. The handles are 2 × 1.4 µm, and the pillar tensile gage length is 4.74 µm for the circular
cross-sectional pillar. The rectangular one generally has a gage length of 3 µm, and the handles ex-
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tend ∼ 1µm on either side and have a height of 1–
1.5 µm. Tension experiments were carried out inside
the SEMentor, a one-of-a-kind in situ mechanical deformation instrument with a specifically fabricated
tensile/compressive tip, as shown in Fig. 4. The
SEMentor provides simultaneous visualization and
video capturing capability with mechanical data
collection. The applied force was determined by
the feedback loop such that the displacement rate
is kept constant at 2 nm/s. The applied forces and
the measured displacements are recorded instantaneously throughout the test, while the deformation
is observed and recorded in the SEM. Still images
are extracted during the plastic deformation of the
pillar. These images are used to directly measure the
gage length, L, which is assumed to be equal to the
plastic length, Lp , since elastic deformation is significantly smaller than the plastic contribution. Assuming plastic volume conservation, the true stress
is evaluated by σ = P Lp /(A0 L0 ), where P is the
applied force, and L0 and A0 are the initial gage
length and cross-sectional area, respectively. The
true plastic strain is given by ε = ln(L/L0 ). The obtained stress-strain data are not averaged since the
discrete slip events, which characterize the plastic
deformation, are different from sample to sample
due to the differences in the initial dislocation distribution and the stochastic nature of the deformation. During compression, the statistical distribution of slip burst size follows a power law, as de-
FIGURE 3. Tensile sample at 45◦ tilt angle. The sample has a diameter of 334 nm and a length of 4.74 µm in the gage
length
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BRINCKMANN ET AL.
FIGURE 4. Tensile/compressive grips milled into a diamond nanoindenter tip
rived from the statistical model of Csikor et al. [22]
and the experimental investigations of Dimiduk et
al. [23] and Brinckmann et al. [6]. This power law
appears to be independent of the material and the
sample size. Detailed statistical studies of the tensile behavior will be executed in the future. The experiments indicate that the slip lines form primarily in the middle section of the sample, with clear,
evenly distributed offsets. No indication of deformation near the substrate was observed, most likely
because plasticity at the nanoscale does not necessarily initiate at the regions with highest von Mises
stress but is rather controlled by the local environment in the sample. Figure 5 compares the representative stress-strain curves for a circular and for two
rectangular-shaped pillars of similar dimension.
We
p
evaluate an effective diameter deff = 4/πa1 a2 for
the rectangular pillars to compare pillars of equivalent cross sections. Both pillar shapes exhibit similar
elastic stress-strain responses, and in the following
plastic regime, it appears that below ∼ 7% strain,
the flow stress of the rectangular pillar is more continuous compared with the discrete burst-ridden
signature of the circular one. The difference between the two shown rectangular-shaped samples
is evidence of the experimental scatter observed in
nanopillar experiments. At higher strains, however,
the stress-strain curves are similar, consisting of numerous discrete slip events intermitted by elastic
loading segments. This signature is similar to the
previously reported compression results (e.g., [2]).
Our gold nanopillars show pronounced ductility in
tension, with the deformation intentionally aborted
at ∼ 30% strain, where necking occurred prior to
fracture.
5. CONCLUSIONS
Our studies indicate that the pillar geometry appears to play a role in the deformation response of
nanosized samples, as has been suggested by molecular dynamics simulations [11, 14]. These simulations show that defect nucleation from the corners
requires less activation energy than nucleation from
a flat surface. The FEM results presented here reveal regions of confinement-induced higher localized stresses in the rectangular-shaped pillars compared with the circular ones at the same overall
applied stress. Our uniaxial tensile experiments
on nanosized crystals indicate that unlike cylindrical ones, rectangular cross-sectional pillars exhibit
a continuous increase in flow stress during early
stages plasticity. However, at 10% strain, a value
commonly used for comparison of the flow stress
International Journal for Multiscale Computational Engineering
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EFFECTS OF SAMPLE GEOMETRY ON THE UNIAXIAL TENSILE STRESS STATE
250
stress [MPa]
200
d*=695nm
150
d*=682nm
100
d=692nm
50
0
0
0.05
0.1
0.15
strain [−]
0.2
0.25
0.3
FIGURE 5. Stress-strain curves for circular and square cross-sectional samples under tensile deformation. The crosssectional sample has a diameter of 692 nm. The effective diameter deff =
with the side lengths a1 and a2 is defined for the rectangular sample
for pillars of different diameters, the geometry influence on flow stress becomes insignificant. Further tensile experiments on pillars of different dimensions are necessary to study the size dependence. Furthermore, to shed more light on the discrete events that govern plastic deformation, atomistic simulations of nanosized pillars are necessary.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the NSF CAREER grant (DMR-0748267).
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International Journal for Multiscale Computational Engineering