APPLIED PHYSICS LETTERS 89, 141909 共2006兲
Compression of stacked niobium bilayers: Void-induced strain
localization at interfaces
Y. Liua兲
Institut für Physik und Zentrum für Mikro- und Nanotechnologien, Technische Universität Ilmenau,
98693 Ilmenau, Germany
D. Brunner and M. Ruehle
Max-Planck-Institut für Metallforschung, 70569 Stuttgart, Germany
共Received 24 July 2006; accepted 16 August 2006; published online 4 October 2006兲
An optical full-field strain mapping technique has been used to provide direct evidence for the
existence of a highly localized strain at the interface of stacked Nb/ Nb bilayers during the
compression tests loaded normal to the interface. No such strain localization is found in the bulk Nb
away from the interface. The strain localization at the interfaces is due to a high void fraction
resulting from the rough surfaces of Nb in contact, which prevents the extension of deformation
bands in bulk Nb crossing the interface, while no distinguished feature from the stress-strain curve
is detected. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2358932兴
When a thin metal piece is bonded to a piece of stiff
ceramics such as in Nb/ Al2O3 joints, its deformation is
highly constrained by the interface. Both experiments and
numerical simulations by the finite element method 共FEM兲
show a high strain localization near the interface when the
metal is loaded uniaxially normal to the interface.1–5 The
elevated stress triaxiality developed in the metal near the
interface is believed to be the main origin for the strain localizations, and it can also trigger cavitation and/or interfacial debonding.3,6–8 Other contributions are the voids or
pores existing at an interface though their fraction is very
small at a fully bonded interface. If the size of the voids at
the interface is larger than that in the bulk materials, as is
typical for diffusion bonded Nb/ Al2O3 joints, a higher
effective strain is developed around the voids under
deformation.9,10 The high strain localization near the fully
bonded metal/ceramic interfaces is, therefore, a coupled effect from the stress triaxiality and the voids. Do the voids
alone, without coupling with the elevated stress triaxiality,
cause strain localization at the interface? We try to answer
this question by studying the deformation of stacked Nb bilayers, whose interface is in a state of stiction and moves
together during deformation. As a consequence, no friction
and no elevated triaxiality can be assumed. Emphasis is put
on the yield locus in terms of strain in the deformation of the
bilayers and its dependence on the strain history.
The Nb sheet 共purity ⬎99.9 wt %, Goodfellow兲 of
1.85 mm thickness was cut into square pieces of 4 ⫻ 4 mm2
and subsequently polished by 800 mesh SiC paper for
⬃2 min. The surface roughness in terms of root mean square
is about ⬃1 ␮m measured by a profilometer 共Dektak兲. Before stacking into bilayers, the Nb pieces were cleaned in
acetone and methanol by ultrasonic agitation for 5 min each
to remove the contamination from polishing. The Nb bilayers
were compressed in a general purpose materials testing machine 共Zwick 1474兲. Unidirectional load was applied through
the moving beam 共cross head兲 to the top end of the stack that
was positioned on the fixed beam. Both the load and the
displacement of the cross head are measured simultaneously
a兲
Electronic mail: yonghe.liu@tu-ilmenau.de
by a force sensor and a linear variable displacement transducer, respectively, and these data are used to calculate
stress-strain curves. Lubricant of MoS2 paste was applied to
the loading platens before each compression test to minimize
friction. A constant displacement rate of 1.67 ␮m / s was employed in all the tests. In addition, we employed an optical
full-field strain mapping 共OFSM兲 technique to measure the
surface strain occurring at the Nb layers during compression.
The details of OFSM technique have been described
elsewhere.1,2 It consists of the specimen, cameras to capture
images of the side surfaces of specimens, a light source for
illumination, and commercial software 共Aramis, GOM,
Braunschweig, Germany兲 to compare and analyze pairs of
these images. Prior to the deformation, a stochastic black and
white pattern is sprayed onto the side surfaces of Nb. During
the deformation the stochastic pattern changes and by comparing these changes of the images under different loads, the
in-plane surface displacement field is obtained and converted
to strain maps by the software.
Figure 1 shows the stress-strain curve, on which the imaging steps are indicated by arrows. We started the imaging
at 600 N to obtain the zeroth image, and the subsequent images were captured by software in every 20 s automatically.
As can be seen, the imaging steps measured by strain are not
equal due to the nonuniformity in plastic flow under deformation. An obvious yielding was observed from the curve,
and the upper and lower yielding strengths were determined
to be 191 and 175 MPa, respectively. These values, especially the lower limit, are the same as were determined from
the single layer Nb pieces with thicknesses ranging between
0.8 and 3 mm.1
In the following, we examine the nonuniformity of plastic deformation on a microscale in detail. Figure 2 shows the
effective strain 共Von Mises strain兲 maps at various deformation stages with colors representing the values according to
the attached scale bar. The strain map for an image of interest
共i兲 is compared with a reference image 共r兲 and denoted as
strain at stage r → i. Following this convention and referring
to the imaging steps displayed in Fig. 1, the elastic deformation occurs at stage 0 → 4, and the plastic deformation occurs
after imaging step 4.
0003-6951/2006/89共14兲/141909/3/$23.00
89, 141909-1
© 2006 American Institute of Physics
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141909-2
Liu, Brunner, and Ruehle
FIG. 1. Stress-strain curve for a typical compression test of a stacked
Nb/ Nb bilayer with arrows indicating the imaging steps. The imaging step
in terms of strain is not uniform due to the nonuniformity in deformation.
Stress and strain values at some selected imaging steps are listed in the
figure.
The strain in Nb at some distance from the interface is
very small, ⬍0.2%, during elastic deformation if compared
with the high strain at the Nb/ Nb interface which is ⬃0.5%
as shown by the strain map for stage 0 → 4 in Fig. 2共a兲. As
the reference image 0 was located at the beginning of the
linear elastic deformation stage in Fig. 1 共step 0兲, the high
strain at the Nb/ Nb interface is, therefore, not from the deformation in the alignment before the linear elastic deforma-
Appl. Phys. Lett. 89, 141909 共2006兲
FIG. 3. Von Mises strain developed at each imaging step at the interface and
in the bulk Nb obtained by statistics. After subtracting the strain in bulk Nb,
a nearly constant strain at the interface is observed during plastic
deformation.
tion. Additionally we can exclude errors from illumination
since the Nb/ Nb interface was located at the center of illumination and was homogeneously illuminated during the
whole deformation process. The strain localization at the
Nb/ Nb interface is, therefore, intrinsic and persistent during
the whole deformation process as documented by the strain
maps of subsequent deformation stages shown in Figs.
2共a兲–2共f兲.
Deformation bands with high strain parallel to each other
were observed immediately after the onset of plastic deformation as shown by strain map for stage 4 → 5 in Fig. 2共b兲.
The band with highest strain is along the diagonal starting
from the upper left corner approximately. Widening of the
deformation bands was observed in the stages 5 → 6 and 6
→ 7 as shown by the strain maps in Figs. 2共c兲 and 2共d兲,
respectively. In the subsequent steps, the deformation bands
vanish and the strain distribution turns random as shown by
the strain maps for the stages 8 → 9 and 9 → 10 in Figs. 2共e兲
and 2共f兲, respectively. These observations are in general
agreement with the dislocation theory for the yielding of
bulk metals.11 However, the strain localization at the interface and its influence on the development of local strain in
bulk Nb require further examination.
A comparison of the mean value of strain at an interface
and in the bulk Nb at each imaging step 共except step 4 for the
cumulative elastic deformation兲 obtained by statistics function provided by the software is presented in Fig. 3. As can
be seen, the strain at the interface is ⬃75% higher than that
in bulk Nb during the whole deformation process. In addition, a similar shape of the two curves indicates a strong
coupling between the deformation at the interface and in
bulk Nb. After subtracting the strain in bulk Nb, we find a
nearly constant strain of ⬃0.3% at the interface with the
increase of stress in the stage of plastic deformation after
step 9.
The strain localization at the interface is found to prevent the elongation of the deformation band. As shown by
Figs. 2共a兲 and 2共b兲, the two bands neighboring the main diagonal band starting from the upper and lower ends of the
FIG. 2. 共Color online兲 Von Mises strain maps obtained by OFSM technique
for 共a兲 cumulative elastic deformation at stage 0 → 4; for 关共b兲, 共c兲, 共d兲, 共e兲,
and 共f兲兴 stepwise plastic deformation at steps of 4 → 5, 5 → 6, 6 → 7, 8 → 9,
and 9 → 10, respectively, and referring to the imaging steps displayed in Fig.
1. The arrows attached indicate the approximate position of interfaces. Colors refer to strains as indicated in the scale bar 共%兲 at the bottom.
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141909-3
Appl. Phys. Lett. 89, 141909 共2006兲
Liu, Brunner, and Ruehle
stack, respectively, terminated at the Nb/ Nb interface at
stage 4 → 5, but went across the interface at stage 5 → 6.
Several factors have to be taken into account for the
deformation near an interface. Among others there are the
length scales, morphology of the structures, and the mode of
deformation. For the interface in stacked Nb/ Nb bilayers, the
surface morphology 共roughness兲, which is closely related
with the void fraction at the interface, must be considered.
Here we demonstrate that the high void fraction resulting
from rough surfaces leads to high strain localization at an
interface during deformation. We find that the Gurson model
for porous medium can explain our results reasonably well.
According to this model,12 the yielding criterion is expressed
as
⌽
冉
冊冉 冊
␴m ␴h
␴m
, ,f =
␴Y ␴Y
␴Y
2
冉 冊
+ 2f cosh
3␴h
− 共1 + f 2兲 = 0,
2␴Y
共1兲
where ␴h 共␴h = 共␴1 + ␴2 + ␴3兲 / 3兲, ␴Y and ␴m are stress triaxiality, uniaxial yielding strength, and Von Mises stress, respectively, and f is the void fraction. Considering the Nb/ Nb
interface, where the stress triaxiality does not exist, but with
a high void fraction due to the high rms roughness of Nb
pieces, high strain localization is thus expected at the interfaces according to Eq. 共1兲. This explains our findings on
strain localization at the interface in Fig. 2.
For most rough contacts, the deformation at the interface
is usually achieved by yielding of microasperities in contact
while the bulk materials maintain elastic.13,14 In this case the
fraction of voids at the interface becomes smaller with the
increase of load due to the increased real contact area. However, our results were obtained when the plastic deformation
of bulk Nb took place. Assuming ␴m / ␴Y , ␴h / ␴Y in Eq. 共1兲
are constant in the linear hardening stage, and a constant
void fraction f is expected to maintain the same yielding
criterion. This might be the cause for a nearly constant strain
at the interface with the increase of stress in Fig. 3. We must
notice that the appearance of strain localization at the interface is random in position, indicating a random appearance
of microasperity contact with the increase of stress, which
obviously needs further studies on a smaller length scale.
In contrast to many available studies on the deformation
of constrained metal layers in fully bonded state,3–5,15–17 this
work distinguishes itself by studying the deformation of
stacked Nb bilayers. The results obtained by the in situ
OFSM technique are considered important in materials design, in the cross examination of FEM simulations3,9,10 and
in the diagnosis of advanced materials processing such as
precision metal forming.18 However, the OFSM technique is
at most getting to the resolution of approximately micrometers, as shown in Fig. 2. In order to clarify the details of the
influence of voids at interfaces on the deformation behavior,
other techniques with a higher resolution are necessary.
This research is supported by the Max Planck Society
and the German Science Foundation 共DFG, SFB622兲. The
authors would like to acknowledge J. A. Schäfer from Technical University of Ilmenau for discussions.
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