Size effects in tensile testing of thin cold rolled and annealed Cu foils

Transcription

Materials Science and Engineering A 416 (2006) 290–299
Size effects in tensile testing of thin cold rolled and annealed Cu foils
G. Simons a,∗ , Ch. Weippert b , J. Dual a , J. Villain b
a
Swiss Federal Institute of Technology Zurich (ETHZ), Institute of Mechanical Systems-Mechanics,
CH-8092 Zurich, Switzerland
b University of Applied Sciences Augsburg, D-86161 Augsburg, Germany
Received in revised form 26 October 2005; accepted 28 October 2005
Abstract
Thin copper foils of varying thickness were tested in tension at room temperature. The thickness of the samples was between 10 and 250 ␮m,
length and width were scaled according to the thickness. Two types of samples were tested: as-received samples and annealed samples. The
microstructure of the foils was determined in detail by means of several methods. If samples of the same processing condition were compared,
there was a clear dependence of the mechanical behavior on the thickness of the foils in the tensile test. When the thickness was reduced from 250
to 10 ␮m, the fracture strain decreased for the as-received foils from approximately 20% to 0.2% and for the samples with heat treatment from
35% to 15%. It has to be stressed that this size effect occurred in the absence of considerable strain gradients and for samples with a comparable
microstructure.
© 2005 Elsevier B.V. All rights reserved.
Keywords: Tensile test; Ductility; Fracture; Size effect; Copper
1. Introduction
In the last decade the interest in the mechanical properties
of materials in small dimensions has grown immensely as by
means of modern process technologies the fabrication of technical systems at the micro- or even at the nanoscale has become
feasible. For a safe and reliable design of such systems it is
mandatory to know the mechanical behavior of the materials
used. These properties, such as critical stresses and strains, cannot necessarily be deduced from macroscopic specimens due
to two reasons. Firstly, the size of microstructural features, for
example the grain size, can be in the order of the size of a structure and, hence, interactions different from that in bulk materials
can be expected. Secondly, small structures can be fabricated by
means of new processes yielding microstructures not available
at the macroscale. These facts generate the need for testing structures in small configurations.
The size dependence of material properties is generally
termed “size effect”. In the following, recent findings of size
effects in elastic and inelastic properties of engineering materials will be reviewed briefly. The focus of the literature overview
∗
Corresponding author. Tel.: +41 44 6323572; fax: +41 44 6321145.
E-mail address: [email protected] (G. Simons).
0921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.msea.2005.10.060
will be on the influence of the size on the mechanical properties
of materials and not on the type of the test methods.
1.1. Size effects in elastic material behavior
No experimental results could be found which show a size
effect on elastic properties in micrometer sized metallic structures. This is confirmed by several studies in which no size
dependence of the Young’s modulus was found [1,2]. Conversely, in [3] measurement results are reported which show a
decrease of Young’s modulus for Al films thinner than 100 nm.
For more complex materials with large structural features
as, for example, foams or bones, there seem to be size effects
for elastic properties (see e.g. [4,5]). Generalized continuum
theories which incorporate a length scale were developed for
explaining these findings such as micropolar and nonlocal elasticity or gradient theories (e.g. in [6–8]).
1.2. Size effects in inelastic material behavior
In inelastic material behavior of crystalline structures other
deformation mechanisms than those relevant for elasticity are
important, such as dislocation motion and grain boundary sliding
[9]. These mechanisms are directly coupled to the microstructure
and in particular to the grain size. If the size of a structure is in the
G. Simons et al. / Materials Science and Engineering A 416 (2006) 290–299
order of its microstructural features it is questionable whether the
same mechanisms hold as those which describe larger structure
sizes. Therefore, it is not surprising that for inelastic material
parameters size effects have been frequently reported, also for
␮m-sized specimens.
1.2.1. Plastic properties
In [10] thin Cu wires were investigated in tension and torsion
(diameters, 12–170 ␮m). There it was found that the amount of
work hardening increased for thinner wires in torsion whereas
there was no such effect in tension. This was explained by an
increase in the density of geometrically necessary dislocations
in torsion, as in torsion there exists a macroscopic strain gradient
responsible for the storage of geometrically necessary dislocations in contrast to tension. Similar results were found by
[11] for bending of thin Ni foils. Thinner foils required a larger
equivalent moment for producing a certain amount of surface
strain. Various authors made indentation experiments with different materials such as Al or Cu [12–18]. The general result of
these experiments was that the hardness decreased with increasing indentation depth (“indentation size effect”). These findings
were explained by hardening due to the strain gradient in the
vicinity of the indenter tip. It must be stressed that due to the
complexity of the stress field in indentation, size effect studies
using this technique are difficult to interpret. An increasing yield
stress with decreasing thickness in thin Cu films was found by
[19–22] which was explained by models which consider interface dislocations and grain size effects in thin films. An influence
of the ratio of thickness to grain size was found for thin foils by
[23], the smaller the ratio the smaller the flow stress. Uchic et
al. [24] investigated micrometer sized single crystals in compression and found an increase in yield stress with decreasing
size. Recent experiments of [25] showed an increase in yield
stress for very thin free standing Al, Au and Cu films (thickness
0.2–1 ␮m) which was interpreted to be independent of the misfit
dislocation formation.
1.2.2. Failure properties
Size effects in fracture strength (ultimate tensile strength in
tensile testing) were found by a number of authors (e.g. for
quasibrittle materials as concrete and rocks by [26–28], for fcc
metals such as Cu by [21,29], for Si by [30]). Although the reason
for this behavior differed with the type of material the general
finding was that the thinner a structure the higher its ultimate
tensile strength. In tensile testing (in a macroscopically uniaxial
stress field) a decrease in the strain at fracture with decreasing
thickness was found for thin Cu foils by [31–36] and for thin Au
and Al foils by [25]. A size dependence of fatigue life was found
in tension–tension fatigue of thin Au and Cu wires by [37] as well
as of thin Cu foils by [38]. The smaller the size of a structure the
higher the number of cycles performed before failure. In [39]
multiple step tests of tension–tension fatigue were performed
showing that thinner Cu foils had higher plastic strain amplitudes
than thicker ones. A change in crack growth rate was observed
by [33] in tensile fatigue of thin Al, Cu and Mo foils: foils
thinner than 150 ␮m showed an intermittent crack arrest. A step
wise change in crack growth rate was found by [40] who tested
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small beams fabricated by the LIGA technique (composition
NiFe 13%) in bending. For completeness, a well known size
effect from fracture mechanics is reported. An influence of the
sample size on the fracture toughness was found first by [41]
who investigated several mm thick Al sheets containing a central
crack. In that paper, the sample size dependence of the fracture
toughness was attributed to a change in fracture mode (shearing
in thin sheets versus flat-tensile fracture mode in thick sheets).
1.3. Size effect theories
Several models were developed for explaining size effects,
e.g. strain gradient theories ([42–44], with a reformulation of
the latter in [45], and [46]), models with dislocations confined
in thin films [47–49], discrete dislocation dynamics theories
[50,51], fracture mechanics theories (especially for concrete,
[26,52]), and models based on microstructural and dimensional
constraints [53,54]. Statistical models were proposed already by
[55] and later by [56,57].
Different kinds of size effects are reported in the literature.
Whereas for standard crystalline materials there are no substantial indications for size effects of the elastic properties, plastic
and failure properties seem to be dependent on the size to a
certain extent. The general trend is that smaller structures have
higher strength and lower ductility than larger ones. However,
the variety of size effects reported in literature indicates that various physical phenomena, operating on different length scales,
govern mechanical material behavior. Therefore, it is difficult to
develop a unified theory which explains all these effects. This
is manifested in the variety of models which are proposed to
explain size effects. Regarding this complexity, an experimental investigation of the influence of the size of a sample on its
mechanical behavior must include a careful determination of
the microstructure. In this study, this was tried to be achieved by
means of modern material characterization methods. The length
scale investigated comprised of Cu foils with thicknesses from
10 to 250 ␮m while typical grain dimensions varied from a few
to about 100 ␮m.
2. Experiments
Thin rolled copper foils (99.97% purity) of varying thickness
(10, 20, 50, 100 and 250 ␮m) were tested in tension. In the
following the experimental key points are discussed, a more
detailed description on the experiments performed can be found
in [58].
2.1. Samples
2.1.1. Geometry
The examination of size effects requires well defined loading
conditions. Therefore, the standard dog bone shape was chosen
as the shape for tensile test samples. To guarantee geometrically
similar flow conditions, the geometry of the samples, in particular the gauge section, was scaled according to the thickness, H.
As a standard, the width was chosen to be W = 20H, the length
L = 200H and the radius of the transition arc R = 100H. Fig. 1
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Fig. 1. Geometry of small tensile test samples (top view). The dimensions of the gauge section (L, R and W) were scaled according to the thickness.
shows as an example the basic layout of the 10 and 20 ␮m thick
samples, the large plates at the end were used for handling and
fixation purposes.
2.1.2. Fabrication
Rolled copper foils of 99.97% purity and of 10, 20, 50, 100
and 250 ␮m thickness served as a base material for the tensile
test samples. As the exact rolling procedure was not disclosed to
the authors by the manufacturer detailed information on this process cannot be reported. Whereas foils thicker than 50 ␮m were
structured by conventional wet etching techniques, thinner foils
had to be fixed to a substrate for stability reasons before processing. Thin Cu foils of 50 mm × 50 mm size were glued on 3 in. Si
wafers. For this step, particular care had to be taken to ensure that
the foils were perfectly flat. This could be achieved by pressing
them overnight with another wafer and a weight. Once the foils
were glued on the wafer, the foils could be patterned by standard
photolithography (spin-coating, exposing, and developing) taking care of the orientation of the specimen’s axes with respect
to the rolling direction of the Cu foil. These axes coincided for
the samples tested in this study. The etching of the Cu foils was
performed in a 50 ◦ C sodium persulfate (Na2 S2 O8 ) solution for
several seconds to a few minutes depending on the thickness of
the foil. Finally, the foil was removed from the silicon wafer by
putting it into a solvent.
With this procedure bars of 50 ␮m width could be produced.
The under etching was negligible in comparison with the width
which is 20 times the thickness in the standard design.
2.1.3. Heat treatment
After etching, the samples were ready to be used in a tensile
test; these samples were termed “as-received”. In addition, some
samples were heat treated in a vacuum at 300 ◦ C for 2 h to study
the influence of a change in microstructure (“heat treated” or
“annealed” samples).
2.2. Characterization of microstructure and geometry
The microstructure of the samples was analyzed in detail by
means of several methods. X-ray diffraction was used for the
study of the macrotexture (Seifert XRD3000 PTS Goniometer).
Electron backscatter diffraction (EBSD [59]) was applied for
a detailed characterization of the microstructure, i.e. the size,
the distribution and the orientation of the grains (CAMSCAN
CS44LB, OIM 3.5). By means of ion beam slope cutting (IBSC
[60]) different sections of the copper foils could be accessed
which allowed for a three-dimensional characterization of the
microstructure (details in [61]). For confirmation of the EBSD
results metallographic samples were made and analyzed by standard light microscopy.
Width and thickness were tested for each sample. The width
was measured with a non-contact laser profilometer (Autofokusprofilometer 2010, UBM Messtechnik) and the thickness with a
contact height gauge (Cary Compare).
2.3. Tensile testing
Tensile tests were performed at room temperature at a strain
rate of ε̇ = 10−4 1/s. Samples thinner than 50 ␮m were tested
with a home built device based on a setup of [62]. The upper
part of a specimen was fixed to a vertical translation stage and
the lower part to a weight on a precision balance. An upwards
movement of the stage resulted in a reduction in weight and
hence yielded the force acting on the specimen. The strain was
measured optically by means of a least square template matching
algorithm (LSM algorithm, deformation resolution 20 nm, see
[63]).
The larger specimens were tested with a commercial tensile
test machine (Zwick 1445) with a 10 and 200 N load cell and
a specially designed clamping apparatus to allow clamping free
of initial tension (details in [35]).
3. Results
In the following, a summary of the results of the microstructural investigations and the mechanical behavior under loading
is given. Further details can be found in [58].
3.1. Geometry of samples
Thickness and width of each tensile test sample were determined prior to the tensile test. Within one sample these values
varied in average less then 1.5%. Variation in both thickness and
width was in average less than 6% between different samples.
Table 1 lists the surface roughness parameters of undeformed
foils tested parallel to the rolling direction. The surface roughness was in the same order of magnitude for all samples tested
meaning that the relative surface roughness was higher for thinner foils. The 20 ␮m foils had the smallest surface roughness.
The applied heat treatment did not have a measurable influence
on the surface roughness.
Table 1
Surface roughness parameters determined in rolling direction for undeformed
foils of different thickness H
H (␮m)
Ra (␮m)
Rpm (␮m)
Rz (␮m)
Ra /H
(%)
Rpm /H
(%)
Rz /H
(%)
10
20
50
100
250
0.27
0.15
0.32
0.33
0.32
1.11
0.70
1.51
1.55
1.88
2.24
1.76
2.69
2.77
3.50
2.70
0.75
0.64
0.33
0.13
11.10
3.50
3.02
1.55
0.75
22.40
8.80
5.38
2.77
1.40
The relative surface roughness parameters Ra /H, Rpm /H and Rz /H are higher
for thinner foils. In transverse direction the roughness parameters were 10–20%
higher than in rolling direction. The annealing did not change the surface roughness measurably. The values were determined according to DIN 4776 (Ra is the
arithmetic average of the absolute values of all points of a profile with respect to
its mean line, Rpm the arithmetic average of the maximum distance between the
highest profile point and the mean line of a profile determined in five consecutive sections, Rz the arithmetic average of the maximum peak to valley height
determined in five consecutive sections).
3.2. Microstructure
All as-received samples had a strong (1 0 0)[0 0 1] cube texture with some rests of a rolling texture which was carried by
highly deformed grains. Fig. 2 shows an inverse pole figure map
acquired by EBSD for an as-received 20 ␮m thick sample. The
grains were highly elongated in the rolling direction and had a flat
ellipsoidal shape in a plane normal to the rolling direction. This
grain structure was found for all as-received samples irrespective
of thickness. As the orientation of the crystallites changed only
gradually within a cross-section, it was difficult to determine
a meaningful grain size for the as-received samples. A rough
estimate for the extension of an average grain yielded: length of
100 ␮m in rolling direction, 20 ␮m in transverse direction and
5 ␮m in normal direction (representative values for samples of
all thicknesses tested). Thus, there were only a few grains per
thickness in the thinnest samples. Twins could not be detected
in the as-received samples.
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The heat treatment induced major changes in the microstructure of all samples. An example is shown in Fig. 3 for an annealed
20 ␮m thick sample. The grain shapes were no longer elongated
but were rather equi-axed. As for the as-received samples, the
microstructure after annealing was similar for foils of different
thickness. In average, the grains had a cubic grain shape and
an extension of approximately 15 ␮m in all directions. Thus, the
very thin foils often consisted of only one or two grains per thickness. Annealing twins could be observed frequently. Only a few
grains were left in cube orientation. Rolling texture components
with 1 1 1 parallel to the rolling direction formed the preferred
orientations. However, the texture index was rather low. The
difference in texture before and after annealing for the 20 ␮m
thick samples is shown in the pole figures of Fig. 4. Whereas
the as-received samples showed high maxima indicating a strong
texture, this was not the case any more for the annealed samples.
The results found by EBSD were in good agreement with
the metallographic analysis and the X-ray measurements. More
detailed information on the microstructure of the as-received
and annealed Cu foils can be found in [61].
3.3. Tensile tests
Tensile tests were performed using the setups described in
Section 2.3. Typically five samples were tested for each thickness and processing condition (more for thinner samples). Fig. 5
shows an engineering stress strain diagram for samples of different thickness and processing conditions. The very thin asreceived foils (10 and 20 ␮m thick) displayed hardly any plastic
deformation whereas the 100 ␮m thick foils showed a ductile
behavior typical for copper. The heat treatment induced a strong
softening of the material, i.e. the ultimate tensile strength was
reduced significantly and the ductile part increased strongly, in
particular for the 10 and 20 ␮m foils. The tensile test results
showed good reproducibility: the variation of the stress values
between different tests was in the order of 5%. The strain values,
in particular the fracture strain, showed higher variation which
Fig. 2. Result of an EBSD analysis of a 20 ␮m thick as-received, unloaded sample: inverse pole figure map [1 0 0] (different colors mean different orientations as
indicated by the triangle on the right). The plane shown represents a cut parallel to the rolling (RD) and normal direction (ND). Most grains had the same orientation
which indicated a strong texture (cube texture (1 0 0)[0 0 1]). Grains which were highly elongated in rolling direction could be observed. Due to gradual changes in
orientation a clear identification of grain boundaries was not possible for a substantial part of the cross-section.
Fig. 3. Result of an EBSD analysis of a 20 ␮m thick annealed, unloaded sample: inverse pole figure map [1 0 0] (different colors mean different orientations as
indicated by the triangle on the right). The plane shown represents a cut parallel to the rolling (RD) and normal direction (ND). There was no preferred orientation
of the grains. The grain shape was no longer elongated but rather equi-axed. Annealing twins could be observed frequently.
294
Fig. 4. Contoured pole figures for as-received and annealed 20 ␮m thick unloaded foils (equal contour level values for both). The as-received samples showed a
strong cube texture which was hardly present in the annealed samples. The major texture component of the heat treated samples was a weak rolling texture with
1 1 1 parallel to the rolling direction.
demonstrated the statistical nature of the fracture process. The
10 ␮m as-received samples had the highest variation of the fracture strain with 50% of the average measurement value. For the
other samples the variation was significantly lower (typically
considerably smaller than 25%).
Fig. 6 summarizes the size dependence of the mechanical
behavior of the thin Cu foils tested. The fracture strain, A and
the ultimate tensile strength, Rm were chosen as values for comparison for both the as-received and the annealed samples. A
clear trend could be observed regardless of the scattering of the
measurement values. Thinner foils had a smaller fracture strain
for both the as-received and the annealed samples. When the
thickness was reduced from 250 to 10 ␮m, the fracture strain
decreased for the as-received foils from approximately 20% to
0.2% and for the samples with heat treatment from 35% to 15%.
The situation was not so clear for the ultimate tensile strength.
The annealed samples did not show a strong size dependency
at all. The as-received samples, however, showed the tendency
that thinner samples were stronger with the 20 ␮m foil being an
exception. It has to be stressed at this point that the microstructure of all samples tested was comparable in the as-received
state.
For the same Cu foils as presented in this study, the influence of other parameters such as width and length of a sample,
strain rate, surface oxidation and orientation with respect to the
rolling direction were examined in [58]. In comparison with the
thickness, they only had a minor influence on the mechanical
behavior of the foils tested.
Fig. 5. Engineering stress–strain diagram for copper foils of the thicknesses 10, 20 and 100 ␮m and different processing conditions (sample curves; “annealed”
samples were heat treated for 2 h at 300 ◦ C in a vacuum). There was a size dependence of the mechanical behavior. Thinner foils had smaller fracture strains. The
annealing induced strong softening, in particular for the 10 and 20 ␮m foils.
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Fig. 6. Size dependence of fracture strain, A (left) and ultimate tensile strength, Rm (right) of thin copper foils tested in tension. Thinner foils had a smaller fracture
strain. For the ultimate tensile strength there was no clear trend for the annealed samples. The 10 ␮m thick foils displayed by far the strongest variation between
different samples (S.D. for A 50% and for Rm 10%), for the other foils it was significantly lower (typically for A 15% and for Rm 5%).
Fig. 7. Top view of surfaces of an as-received and annealed 20 ␮m thick foil after loading. The samples were pulled in the vertical direction (parallel to rolling
direction). Deformation structures which were induced by the rolling process prior to the tensile test (e.g. grooves from rolling) were clearly visible. Slip lines at an
angle of 45◦ to the tensile axis could be observed indicating dislocation activity. The as-received foils had a relatively even surface after testing in comparison to the
annealed samples.
3.4. Analysis of post-mortem samples
After loading, slip lines could be observed in the sheet plane
surfaces for both the as-received and annealed samples (Fig. 7)
indicating dislocation motion. In addition, deformation structures, which originated in the rolling process (e.g. grooves from
rolling) and which were present before the tensile test, were
detectable. The major geometrical modifications took place in
the thickness direction what could be observed in SEM-images
of fractured specimens. The samples failed by forming a neck
in thickness direction with a knife edge type of fracture surface.
This type of failure was observed for all samples irrespective of
thickness and processing condition. No substantial reduction in
thickness could be observed outside the necking region for the
10 and 20 ␮m as-received foils in contrast to all other samples.
After the tensile tests, an increase in surface roughness in the
gauge section could be observed for both the as-received and
the annealed samples as shown in Fig. 7. The surface of the asreceived samples was relatively even outside the necking zone
containing some slip lines, whereas the surface of the annealed
samples contained slip lines and was relatively uneven not only
in the necking zone but also in the entire gauge section (“orange
peel effect”). Fig. 8 shows roughness scans for both as-received
and annealed 20 ␮m thick samples after loading which were
taken parallel to the tensile axis. For both samples an increase in
surface roughness could be observed in the gauge section as there
the stresses were maximal. The increase in surface roughness
was moderate for the as-received samples (typical roughness
parameters increased by 10–20%) and very pronounced for the
annealed samples (typical roughness parameters doubled).
4. Discussion
When the tensile test results are analyzed in detail several aspects attract attention: firstly, the strong microstructural
changes induced by the heat treatment and the consequential
softening of the mechanical response; secondly, the nearly brittle behavior of the 10 and 20 ␮m as-received samples; thirdly,
the size dependence of the mechanical behavior and fourthly, the
increase in surface roughness after tensile testing. These facets
will be discussed in the following paragraphs.
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Fig. 8. Roughness scans of as-received and annealed 20 ␮m thick samples (in gray) after loading. The measurements were taken at the middle lines of the samples
as indicated in the sketches in the left upper corner of the graphs. The surface roughness of the annealed samples increased strongly in the gauge section where the
stresses were maximal.
4.1. Softening induced by heat treatment
As can be seen by a comparison of Figs. 2 and 3, the applied
heat treatment at 300 ◦ C for 2 h causes a recrystallization of
the grain structure. Thus, there are changes in grain size and
shape, texture and dislocation distribution. These changes in
microstructure yield a completely different mechanical behavior
of the annealed samples. In contrast to the as-received samples,
the annealed samples show a stress–strain behavior with a relatively low yield stress and a large amount of work hardening
followed by failure caused by necking. The large plastic deformations accompanied by work hardening cause microstructural
changes. This can be observed by EBSD analyses of loaded samples [58]. Annealed samples show a moderate elongation of the
grains after tensile testing and a strong increase in surface roughness. Furthermore, a thickness reduction in loaded samples can
be noticed in the analysis of the fracture surfaces. Obviously,
these samples have the ability to work harden substantially as
the dislocation density is much lower than in the as-received
samples.
Moreover, as the grain size is considerably larger, the fraction
of grains at the surface is higher in the annealed samples. These
grains can deform more easily due to the reduced constraints at
the surface [23]. This is another reason for the increased ductility
of the annealed samples.
Thus, comparing as-received and annealed samples shows
that, at the length scale investigated (thickness, 10–250 ␮m), the
microstructure has a dominant effect on the mechanical behavior. Besides the well known effects of microstructural parameters
such as grain size or dislocation density the importance of nanotwins was recently shown for Cu in [64,65]. There it was found
that in samples with nano-twins the ultimate tensile strength was
in the order of 1000 MPa whereas the material still showed substantial ductility. However, as these strength values are far larger
than the ones found in this study, nano-twins are not considered
to play an important role for the deformation behavior of the Cu
foils tested.
4.2. Low fracture strain of very thin as-received
foils
If the stress–strain curves are analyzed for both, the 10 and
20 ␮m thick as-received samples, only a very small amount of
work hardening can be noticed (see Fig. 5). From a macroscopical point of view, this behavior could be described as being
nearly brittle as there hardly seems to be any plastic deformation.
Microscopically however, the failure is by no means brittle as
there is considerable plastic yielding in the necking zone (Fig. 7).
At moderate distances away from the crack tip no microstructural changes are detectable by means of EBSD. This indicates
that the plastic deformation field is very inhomogeneous along
the gauge section with the plastic strain being localized and maximal at the neck and very small outside of this region. The strain
measured in the tensile test is the average strain of the entire
gauge section, however. Thus, as the longitudinal extension of
the neck is only in the order of the thickness, the large deformations present in this region are averaged out along the total
length which is 200 times the thickness. This fact explains the
low overall fracture strain, A, which lies in the order of only
0.5%. As can be shown by a simple geometrical model [58], the
strain caused by necking is the main contribution to the plastic
strain measured in the thin as-received samples. A homogeneous
thickness reduction of the gauge section cannot be detected in
these samples. The reason for the immediate formation of a
neck can be seen in a relatively high dislocation density present
in cold rolled copper foils [66]. At very high dislocation densities the work hardening rate is limited by the lower fraction of
mobile dislocations [67] and, thus, the material can hardly compensate for geometrical softening as caused by the formation of
a neck.
4.3. Size dependence of material behavior
If the left graph of Fig. 6 is analyzed, a clear size dependence can be observed for samples with the same processing
condition (as-received or annealed, respectively): thinner foils
display a smaller fracture strain than thicker ones. For the tensile strength, however, no such unambiguous size effect as for
the fracture strain can be found (right graph of Fig. 6). For the
annealed samples, there does not seem to be a size dependence
at all, whereas for the as-received samples the trend is that thinner samples are stronger with the 20 ␮m thick foils being an
exception.
For the explanation of these experimental findings, the combination of several mechanisms and effects has to be considered: differences in geometry and microstructure between
samples of different thickness and effects of the thickness on
the mechanical behavior. The different rolling processes, necessary for the fabrication of foils of varying thickness, are
responsible for differences in surface roughness (Table 1) and
in microstructure. However, as already reported in Section 3.2,
samples of one processing condition turned out to have relatively
similar microstructures irrespective of thickness. Variations in
microstructure could hardly be educed from the experimental
data due to the complicated grain structure present in the asreceived foils. Nevertheless, some variance in microstructure
for foils of different thickness has to be expected, in particular
for the arrangement of dislocation barriers (e.g. grain boundaries
and as a consequence grain size) and for the dislocation density.
With respect to the yield stress, the influence of these differences can be roughly estimated by standard models, such as
the Hall–Petch relation for the grain size (yield stress inversely
proportional to the square root of the grain size) and the Taylor model for the dislocation density (yield stress proportional
to the square root of the dislocation density) [67]: a reduction
of the grain size by 30% increases the yield stress by 20%,
an increase of the dislocation density by 100% increases the
yield stress by 40%. Due to the experimental difficulties in the
determination of microstructural parameters for the as-received
foils, the differences in these can be expected in the order of
magnitude mentioned. The calculated differences in yield stress
correspond to the variation of the tensile strength found for asreceived foils of varying thickness (right graph of Fig. 6). As no
pronounced size dependence can be identified for the annealed,
recrystallized samples, slight variations in the microstructure of
the as-received foils seem to be (at least partially) responsible
for the size dependence found for the tensile strength. However,
there are other effects which influence the yield stress and hence
the tensile strength: in [23,68] it is reported that the flow stress
of thin foils is dependent on the number of grains per thickness
if there are <10 grains per thickness. This is explained by the
relaxed deformation constraint of grains at the surface. Thus,
for samples with a constant grain size but varying thickness this
results in a size dependent material behavior. In addition, for
very thin films (in the order of a few microns and below) there
seem to be additional hardening mechanisms depending on the
behavior of dislocations in films with an oxide layer [48] and
on the number of dislocation sources, active slip systems and
dislocation motion paths per thickness [25].
Strain gradient effects are not believed to cause substantial
hardening for the samples tested in this study as a tensile load
is applied and, hence, the deformation is homogeneous until
297
the formation of a neck (no substantial storage of geometrically
necessary dislocations).
The size dependence of the fracture strain can only be partially attributed to the presumed differences in the microstructure
of foils with varying thickness. It is much more pronounced as
the size dependence observed for the tensile strength and it is
also present in the annealed samples. Due to the complexity of
the grain structure it is believed that the decrease of the fracture
strain with decreasing thickness is a consequence of the combination of various factors. Firstly, as already stated in Section
4.1, the reason for the very low ductility of the thin as-received
foils is seen in the limited hardening rate due to a high dislocation density. However, this does not explain the differences
in fracture strain for the annealed samples of varying thickness.
Secondly, as listed in Table 1 the absolute surface roughness is in
the same order of magnitude for all foils tested in the unloaded
state. This means that for the samples tested in this study the
relative surface roughness is higher for thinner foils. A flaw of a
given size, originating e.g. from the rolling process, is a stronger
weakening for thinner foils. As a consequence, thinner samples
break at smaller macroscopic stresses and smaller strains, either
through stress concentration at the flaw or through a reduction of
the effective cross-section of the tensile test sample. This finding is supported by the fact, that the variance in the fracture
strain is higher for thinner samples which shows the statistical
influence of the surface roughness. Thirdly, the smaller the number of grains per cross-section the lower is the probability that
a gliding system is oriented properly such that it can be activated. Hence, for samples with similar grain size but different
thickness, which is the case for this study, the ability to undergo
plastic deformations is limited in thinner samples and, therefore,
smaller fracture strains can be expected.
Thus, there are several reasons for a decrease in the fracture
strain with decreasing thickness. However, the importance of
each mechanism is difficult to state and has to be investigated
separately by varying the grain size and the surface properties.
Similar results, i.e. a decrease of the fracture strain with decreasing thickness, were also found by [31,32,34]. However, there are
some important differences between these studies and the current one, in particular, from an experimental point of view. In this
work, geometrically similar samples were tested whereas in the
other studies the samples only differed in thickness (the importance of testing geometrically similar specimens is discussed
in [58]). Moreover, no information on the surface roughness is
given in the other studies. Thus, the importance of this factor
cannot be assessed.
To sum up, it is believed that the combination of various
effects is responsible for the size dependence of the fracture
strain in tensile testing determined in this study: slight differences in microstructure of samples, influence of surface roughness and number of activated gliding systems. The following
effects are not seen as important for the interpretation of the size
effects found in this study: strain gradients (homogeneous deformation in tensile testing), stress state (assumed to be plane stress,
details in [58]) and grain boundary sliding (no grain boundary
sliding could be observed in as-received samples, only minor
grain boundary sliding in annealed samples).
298
4.4. Surface roughening through loading
Acknowledgements
The surface roughening in the gauge section after tensile
testing (Fig. 8) can be explained by two factors. Firstly, many
slip bands can be observed on the surfaces of the deformed
as-received and annealed samples. This indicates that plastic
deformation is accommodated by dislocation slip in a substantial
amount. These slip bands increase microscopically the surface
roughness. For the annealed samples there is a second factor. As
there are only a few grains per cross-section these grains are not
strongly constrained in their deformation as they are bounded
by free surfaces. As shown in Figs. 3 and 4, these grains are,
in general, oriented differently. Thus, in order to accommodate
plastic deformations these grains can rotate out of their initial
position by gliding along the grain boundaries (uneven plastic
flow). This mechanism yields the surface roughening which can
be observed on a larger scale than the roughening due to slip
bands.
The authors want to express their gratitude to Dr. K. Kunze,
Geological Institute ETH Zurich, for performing the EBSD measurements and to S. Stauss and F. Schmid, EMPA Thun, for the
analysis of the metallographic samples.
5. Conclusions
Thin rolled copper foils of varying thickness with a scaled
geometry and comparable microstructure were tested in tension. Some samples were annealed for 2 h at 300 ◦ C in a vacuum prior to the tensile tests. The most important result of
the tensile tests is that the thickness of the foils has an influence on the mechanical behavior in the size regime studied.
When the thickness is reduced from 250 to 10 ␮m, the fracture
strain decreases for the as-received foils from approximately
20% to 0.2% and for the samples with heat treatment from
35% to 15%. In terms of stresses there was no clear trend for
the annealed samples, the as-received samples showed the tendency that thinner samples were stronger. The annealing induced
a strong softening meaning a strong increase in ductility for
the very thin foils (e.g. for the 10 ␮m thick foils the fracture
strain increased from 0.2% to 15%). The size dependence of
the fracture strain can be qualitatively explained by the combination of several factors. Firstly, the work hardening rate is
limited in thinner samples due to a high dislocation density.
Secondly, thinner samples are more sensitive to geometrical
imperfections stemming from the surface roughness. Thirdly,
for a constant grain size there are fewer grains in thinner samples which could result in a smaller number of activated gliding
systems.
It has to be stressed that the size dependence found in this
work resulted from a tensile test, i.e. a test where no considerable
strain gradients occur. Experimental verification of size effects
in loading situations, where no strain gradients are present, is
scarce (e.g. [25,39]). This work also shows that, for the explanation of the effects observed, a thorough examination of the
microstructure of the samples tested is mandatory. As the influence of many parameters has to be taken into account in detail,
the experimental study of size effects turns out to be a complicated topic. There are several mechanisms which influence
the mechanical behavior and these mechanisms can even have
opposite effects.
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Size effects in tensile testing of thin cold rolled and annealed Cu foils

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