International Journal of Fracture 107: 307–327, 2001.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
Fatigue and fracture properties of thin metallic foils
A. HADRBOLETZ, B. WEISS and G. KHATIBI
Institute of Materialphysics, University of Vienna, Strudlhofg. 4 , 1090 Vienna, Austria.
E-mail: weissb@ap.univie.ac.at
Received 9 February 2000; accepted in revised form 9 August 2000
Abstract. Metallic thin foils are essential structural parts in microsystems ,which may be subjected to fatigue
loading caused by thermal fluctuations and mechanical vibrations influencing their reliability in numerous engineering applications. It is well known that the fatigue properties of bulk material cannot be adopted for small
scaled structures. For a better understanding of the ‘size-effect’ in the present investigation fatigue crack growth
near threshold in the high cycle fatigue regime and associated fracture processes were studied. Free- standing rolled
and electrodeposited Cu-, Mo- and Al foils of thickness from 20 µm to 250 µm in different conditions have been
tested in a special experimental set up operating at R = −1 and a testing frequency of 20 kHz. At a given constant
strain value the fatigue crack growth behaviour has been recorded accompanied by intermittent observation of
the change of the dislocation structure in the vicinity of the growing crack by use of the electron channeling
contrast imaging (ECCI)-technique in a scanning electron microscope (SEM). In a load shedding technique fatigue
threshold stress intensity factor values have been derived and compared with data of bulk material. Typical crack
growth features were detected depending on thickness and grain sizes of the foils. Various criteria (compliance ,
extent of plastic zones and plastic strain gradients) were selected for the explanation of this anomalous behaviour.
Additionally fractomicrographs of uniaxial strained and fatigued foils have been studied to obtain further insight
of the effect of dimensional constraint.
Key words: Dislocation structure, fatigue crack growth, fatigue threshold metallic foils, fracture topography, size
effect special, fatigue testing device.
1. Introduction
Thin foils, sheets or wires of metallic, semiconducting or non metallic materials resemble
essential parts of microsystems. The progressive trend of miniaturisation results in a rapidly
increasing demand of the precise knowledge of mechanical behaviour and properties of such
components to assure safe operation. This demand of knowledge is not restricted to tensile
properties only.
In a variety of bilayered structures (e.g., conductor films ranging in thickness from less
than one micron up to about 35 microns deposited on substrates as typical components in
electronic packaging) the mismatch of thermal expansion coefficients of both layers results in
the occurrence of compressive strains on heating and of tensile strains during cooling. This
thermally induced fatigue may be accompanied by fatigue caused by vibrations since most of
these parts are essential components in automotive- and aircraft structures.
In numerous microdevices thin foils of different thickness act as switches operating at
frequencies from less than 1 Hz up to 1 MHz. Therefore the fatigue behaviour of thin foils
becomes of considerable technical interest in order to assure safe operation of microsystems.
Although the materials used in microdevices mostly consist of conventional metals , semiconductors, ceramics and composites numerous investigations revealed that tensile properties
of the appropriate bulk material can not be adopted to structures of small scale. This com-
308 A. Hadrboletz et al.
monly known ‘size-effect’ is considerably important in fatigue loading. A study of the fatigue
behaviour of components in small scaled structures – preferably free-standing thin foils – is
therefore of outstanding interest.
It is well known that mechanical testing of parts being small in all dimensions is a difficult
task. Specimen preparation, loading, handling and alignment in the testing set-up causes considerable experimental efforts. Furthermore a high degree of accuracy for the measurement of
stress and strain is required.
In most of the cases tests are carried out in out of plane bending, the specimens acting as
cantilever beams .There also exist elaborate systems testing an array of identical specimens
orientated parallel to each other in a common frame and subjected to an increasing amount
of torsion (Judelewicz et al., 1994). In a more recent development triangular shaped planar
specimens were excited by electrostatic combs, thus representing a testing-mode of ‘in plane
deformation’. A more detailed overview of different testing systems is given in (Sharpe and
Turner, 1999).
The aim to test free- standing foils was met in different ways. Often foils are mounted on
a high modulus substrate which behaves elastically even at loading levels where the specimen
material is deformed plastically (Arzt, 1998). In a different approach the tested thin foil is
essentially free-standing in a zone where the substrate has been removed by etching and the
specimen itself has been gripped in the testing equipment in a way that only the material of
the substrate had been fixed (Read, 1998).
In most of the cases the specimen shape is rectangular . To ensure the precise alignment
of the specimen in the testing apparatus they are commonly prestrained by stress values not
exceeding 20% of the yield stress of the appropriate bulk material resulting in a loading mode
of R > 0. Sometimes additional preconditioning procedures are necessary for proper testing,
their influence on the derived material properties have to be taken into account (Merchant
et al., 1999).
In the following an overview of fatigue data of thin foils and films (this term is commonly
used for foils of thickness smaller than a few microns) is given. Most of the investigations
have been performed with Cu- and Al foils. Usually the fatigue behaviour of foils has been
studied by S-N curves. In comparison to the corresponding bulk material different results are
reported. (Judelewicz et al., 1994) investigated Cu foils of 20 µm and 100 µm thickness in
tension-tension fatigue in an elaborate testing system and reported an increase of fatigue life
with decreasing thickness. This was interpreted by the fact that the thinner specimens were
almost free of extrusions and may have lost most of the dislocations due to the short migration
distance to the free surface and the effects of image forces. Hong and Weil, (1996) studied the
cyclic stress- strain behaviour of Cu foils under low cycle fatigue conditions and reported only
a minor effect of thickness on fatigue life. A reduced fatigue resistance of Cu foils (thickness
in the magnitude of one micron) is reported by Read, (1998) possibly due to the peculiarities
of both the testing procedure and the microstructure of the electron beam-evaporated Cu film
covered with a thin layer (0.05 µm) of Ti.
From this it may be summarised that fatigue data of thin foils may not be extrapolated from
data of macrospecimens.
In more recent investigations the elastic-plastic fatigue behaviour of Cu foils has been
investigated and presented in a Coffin–Manson plot. For electrodeposited 18 µm thick Cu foils
in bending-fatigue Merchant et al. (1999) report b and c-values (i.e., the slope of the plastic
and the elastic line respectively) to be similar to that of bulk material; the Nft r -value (number of
loading cycles separating predominant plastic and elastic behaviour) is given by typically N =
Fatigue and fracture properties of thin metallic foils 309
104 . In contrast rolled Cu-foils (12 to 35 µm) exhibit abnormal Nft r - and c-values probably
due to hydrogen embrittlement. Additionally c and Df (the fatigue ductility parameter) are
reported as highly anisotropic which may be explained by texture effects caused by a pancake
like grain structure as result of reduction by rolling. For both electrodeposited and rolled
Cu foils (especially for heat treated foils) enhanced Nf - and Df -data with decreasing foil
thickness are presented.
Hong and Weil (1996) proved for electrodeposited and wrought Cu foils (25 µm and 35 µm
thick, respectively) under tension-tension fatigue that the Basquin-equation is obeyed. Almost
no difference between the LCF behaviour of wrought foils and bulk Cu was found; improved
fatigue resistance expressed by a higher fatigue strength coefficient was reported for electrodeposited foils mainly due to extremely fine grained microstructure (grain size in the order of
1 µm). They also compared monotonic and cyclic stress-strain curves of both electrodeposited
and wrought Cu foils. For both types of foils cyclic hardening was reported which was much
more pronounced in wrought than in electrodeposited foils. This was explained by the absence
of primary slip due to the high dislocation density of the deposits (heavily cold-worked metals
also do not exhibit easy-glide). The criteria given by (Dieter, 1988) originally valid for bulk
material could successfully be applied to electrodeposited foils too (cyclic hardening occurred
if the ratio of the tensile to the yield strength exceeded 1.4 and the strain hardening exponent
was greater than 0.15). Cyclic softening may be predicted if the ratio of the tensile to the yield
strength is smaller than 1.2 and the value of the strain hardening exponent is below 0.15. For
all investigated foils cyclic softening was observed only prior to fracture mainly caused by
necking. Merchant et al. (1999) corroborated these findings for a variety of electrodeposited
foils in the thickness range of 12 µm to 35 µm.
Data concerning the cyclic stress-strain behaviour of thin foils are rarely found in literature.
Recently, Hommel et al. (1999) studied the behaviour of very thin copper films (0.5 µm to
2 µm thick layers) on 125 µm thick polyimide substrates. The results indicate a very strong
strain hardening accompanied by an asymmetric deformation behaviour.
Studies of fatigue crack growth are scarce. Alic and Asimow (1974) were the first to
investigate fatigue fracture behaviour of 25.4 µm thick facecentered (Al, Cu and brass) and
bodycentered cubic (Ta) foils reporting the coalescence of voids and holes ahead of the crack
tip as the basic mechanism of crack extension. In the light of investigations performed in the
past decade taking into account the occurrence of plastic strain gradients in the vicinity of
voids this mechanism seems to be questionable at least for foils in the thickness range of
> 1µm up to about several 10 µm (Fleck and Hutchinson, 1997).
More recent studies are reported about three different stages of crack propagation by (Hong
and Weil, 1996). Scanning electron micrographs of electrodeposited Cu foils indicated that
stage I (crack growth initiated by slip and propagated along the shear plane) is the main portion
of fatigue life. Stage II (crack propagates perpendicular to the applied stress and striations
form) is less pronounced since the number of striations is much smaller than the number of
cycles applied. In stage III the crack acts as stress raisers so that failure occurs until the stress
intensity factor equals the fracture toughness. Preliminary fatigue crack growth studies of free
standing Cu foils with different thickness (in the range of 20 µm up to 250 µm) using a special
testing set up were presented by (Hadrboletz et al., 1999).
The difference in crack growth behaviour of foils with varying thickness was interpreted
in terms of plane stress and plane strain in contrast to bulk material if the validity of LEFM
considerations is assumed.
310 A. Hadrboletz et al.
Nevertheless foils of thickness in that range which has been chosen for the present experiments are presumed to undergo a special plastic deformation. For foils thinner than about 1 µm
plastic deformation may be treated by discrete dislocation mechanics (Tadmore et al., 1998);
on the other hand foils with thickness exceeding ∼200 µm obey conventional plasticity theory,
which is essentially size independent. For the intermediate thickness range the number of
dislocations is too large for a quantitative analysis of plastic deformation based on mechanics
of individual dislocations. The predominant size-effect is mainly attributed to geometrically
necessary dislocations generated by the occurrence of a strain-gradient (Fleck et al., 1994;
Nix and Gao, 1998). The length scale of this gradient is very large compared to atomic lattice
spacing and ranges from 0.25 µm to 5 µm (Hutchinson, 2000).
The tests conducted in this study induce a predominant stretch gradient characterised by a
length scale of approximately 1 µm which may be of some influence in explanation of a size
effect.
An analysis of the fatigue crack growth- and threshold behaviour concerning both conventional plastic deformation theory and dislocation mechanics caused by plastic strain-gradients
should enable to meet the criteria of a ‘size-effect’ from different ways.
A few investigators studied the dislocation-density and -arrangement. Generally the formation of dislocation tangles followed by the development of dislocation cell walls is observed;
however this process requires a minimum value of grain size (equivalent to the foil thickness
if a single grain layer is considered) which is reported to be between 2 µm (Hong and Weil,
1996) and typically 5 µm (Read, 1998). These dislocation cell walls may form new smaller
grains as observed in electrodeposited Cu foils (Merchant et al., 1999). Fractomicrographs
of both tensile and fatigued foil specimens also indicated that a minimum size of dislocation
cells is required.
It may be summarised that from this short review the fatigue crack growth behaviour of
thin foils is rarely investigated. Thus the objective of the present investigation is to study the
crack propagation behaviour of thin foils. In a first attempt the crack growth process has been
investigated mainly of Cu foils in the thickness ranging from 20 µm up to 250 µm, data of Moand of an Al-foil are added. The comparison of the behaviour of the foils with the appropriate
bulk material should allow to introduce criteria of the so called ‘size-effect’.
1.1. M ATERIALS
As materials in most of the investigations Cu has been chosen, because of its practical use
and its well documented fatigue behavior which should allow comparisons with data on corresponding bulk material. Additionally Mo foils of technical purity and Al foils (99.999%)
(each of a single thickness) have been investigated. Cu foils with a purity of 99.95% were
either rolled or electrodeposited (ED), the first have been examined both in the as received
condition and after heat treatment, the latter have been tested in their original condition only.
For comparison 6 mm thick bulk material (commercial E-Cu) has been investigated.
Microstructural parameters and mechanical properties are given in Table 1.
Grain sizes have been determined by the linear intercept method not taking into account
twins yielding values in the range of a few microns for electrodeposited foils and between 20
and about 200 µm for recrystallized foils depending on the heat treatment.
The mechanical properties in table 1 Rm and A are determined with a commercial microtensiletest machine (Fa. Messphysik, Fürstenfeld, Austria) and these values were determined with
an accuracy of about 5%. The strain values in the load-strain curves were measured by use
Fatigue and fracture properties of thin metallic foils 311
of the non-contacting laser-speckle correlation technique which takes care of both rigid body
motions and out-of-plane movements of the specimen (for more details see Anwander et al.,
1999).
As presumed the rolled Cu-foils proved to be highly textured. Investigations by Brokmeier
et al. (1999) indicated that most of the foils showed a cube texture, resulting in Young’s
moduli presented in Table 2. Values depending an the angle between crack growth direction
and rolling direction are listed in Table 2; for Mo and Al single values of E are annotated.
The differences between the calculated and experimentally determined values may be due
to dimensional constraint and indicate that it is necessary to measure the Young’s modulus for
each foil . Most crack growth curves were obtained from specimens oriented in such a way
that the crack path is perpendicular to the rolling direction.
1.2. E XPERIMENTAL
SET- UP AND PROCEDURE
The experimental set-up for fatigue testing is shown in Figures 1a and 1b. The foil-specimens
with dimensions of 20 × 20 mm are glued with a commercial strain gauge adhesive to a bar
shaped holder manufactured of a high strength Al-alloy. The specimen holder is part of a
resonance testing equipment which is excited to longitudinal push-pull vibrations with zero
mean stress (i.e., R = −1) at a testing frequency of 20 kHz. A standing wave is formed in
such a way that the length of the specimen holder corresponds to half of the wavelength. The
distribution of displacement and strain varies sinusoidally along the specimen-holder with the
maximum of the strain in the midsection. In this area a slot of 1.2 mm width and 0.5 mm
depth is machined into the one side of the holder . The foil is glued across this slot and is
free standing in this area. The width of 1.2 mm assures that the strain value is kept almost
constant at its maximum value; the depth of 0.5 mm is chosen so that there is enough space
for eventually penetrating adhesive and is otherwise not too large to guarantee a proper mode
of loading. This is also the reason that an identical slot is machined on the other side of the
holder (see Figure 1b). The holder of a high strength Al-alloy dissipates only a small amount
of vibration energy and due to the foil thickness being very small compared to the thickness
of the holder (t = 8 mm) detrimental effects such as heating can be neglected.
For monitoring of the strain of the loaded foil miniature strain gauges of the type LY 11/
(Hottinger–Baldwin) with an active length of 0.6 mm have been positioned to the locations
SG1, SG2 and SG3. From calibration measurements it is feasible to use each of these positions
for controlling the test. Identical waveforms of the strain signals could be observed for all three
strain gauges. This implies loading of the foil without buckling. This is not surprising because
the critical force for buckling is proportional to B 3 /b (B = 20 mm and b = 1.2 mm). For
fatigue crack growth testing a lancet-shaped notch (1.5 mm long, ∼0.5 mm wide) is introduced
in the center of the free-standing zone of the foil by electro-discharge machining. Crack length
can be monitored by a travelling light microscope with a resolution of approximately 1 µm.
From these curves the derivatives were determined to obtain crack growth rates. The strain
value may be recorded with three strain gauges alternatively. Crack growth was recorded
continuously with a computer-aided data acquisition system. Crack growth rates were in
the range of 10−9 to 10−13 m cycle−1 . The crack length as function of loading cycles was
obtained for constant strain amplitude loading. Intermittently the specimens were investigated
in a scanning electron microscope using the electron-channelling contrast imaging (ECCI)
technique to reveal changes in the global dislocation arrangement (see Chen et al., 1997). 1Kvalues have been calculated using the commonly known equation for through cracks according
Material
Cu rolled
Cu (ED)
Thickness
(µm)
Grainsize (µm)
As received Recryst.
600 ◦ C/ 2 h
9
20
35
50
78
100
125
200
250
35
105
25
10
20
15
15
15–25
15–25
15–25
15–25
2–5
7–10
Al rolled
125
Mo
30
25
15
20
30
35
40
40
45
10
15
Recryst.
550 ◦ C/ 2 h
200
Recrsyt.
50
Recryst.
900 ◦ C/ 2 h
30
40
60
80
120
130
Rm (MPa)
As received
130
172
179
165
188
210
266
Recryst.
600 ◦ C/ 2 h
Recryst.
900 ◦ C/ 2 h
92
124
157
120
142
155
180
192
178
142
165
180
A (%)
As received
4.5
2.8
11.3
11
15
15.5
2
Recryst.
600 ◦ C/ 2 h
Recryst.
900 ◦ C/ 2 h
7.5
10.5
12
13
17
17.5
23
22.5
26
17
20
151
247
238
255
204
124
157
Recryst.
550 ◦ C/ 2 h
23
17
15
2.6
8
27
17
35.5
Recryst.
550 ◦ C/ 2 h
17,5
312 A. Hadrboletz et al.
Table 1. Microstructural parameters and some mechanical properties of the materials investigated.
Fatigue and fracture properties of thin metallic foils 313
Table 2. Young’s modulus in as received and heat treated rolled
Cu-foils depending on the angle between crack growth direction
and rolling direction.
Thickness (µm)
9
20
35
50
78
100
200
35 (ED)
105(ED)
Mo
Al
Young’s modulus (GPa)
0◦
45◦ 90◦
Specifications
122
136
112
104
131
68
128
129
115
111
104
151
116
A, C
R, C
E
A, C
R, C
E
A, C
R, C
E
R, C
R, C
R, C
E
E
A, E
A, E
E
E
320
70.5
127
124
104
131
126
108
126
125
125
133
131
110
87
125
128
112
104
123
82
128
130
132
111
106
141
112
109
92
102
Specification: A, as received.
C, calculated values according Hill’s approach (Stüwe, 1974).
E, experimentally determined values.
R, heat treated (recrystallized).
linear elastic fracture mechanics (LEFM) considerations. These have been determined using
the appropriate value of the Young’s modulus (see Table 2). The values of the geometric
correction function have been taken from literature (Murakami, 1987). To determine 1Kt h
the applied load was decreased stepwise in increments of 5% of the preceding strain value
until the crack growth rate was smaller than 10−12 m cycle−1 .
2. Results and discussion
2.1. FATIGUE
CRACK GROWTH CURVES
The crack growth behaviour at a given constant strain value of a 30 µm thick Mo-foil is
presented in Figure 2. Figure 2a depicts the crack length a as function of the number of loading
cycles N, whereas in Figure 2b the crack propagation rate da/dN depending on the crack length
is shown. In Figure 2c a SEM micrograph of the crack path is added.
Figure 3a presents a crack growth curve of rolled copper with a thickness of 20 µm indicating intermediate crack arrest. This is also illustrated more clearly in Figure 3b indicating
the dependence of the crack propagation rate on crack length. In Figure 3c several of crack
growth curves at constant strain ranges for rolled Cu foil specimens of different thickness
314 A. Hadrboletz et al.
Figure 1a. Computer controlled resonance fatigue test system at 20 kHz for crack growth measurements.
Figure 1b. Detailed view of the foil attached to the specimen holder.
and heat-treatments are given. For the purpose of comparison a crack growth curve of bulk
material (6 mm thick strip) is added.
In addition a crack growth curve and the SEM-micrograph of the corresponding crack path
of a 125 µm thick recrystallized Al foil are presented in Figures 4a and 4b, respectively.
Crack growth curves of foils with thickness < 200 µm presented above are characterised
by the occurrence of intermediate saturation values and negative curvature , this indicates a
decrease of compliance with increasing crack length. Contrary the behaviour of bulk material
is opposite. This is due to an increase of the compliance with crack length.
Crack paths in the Mo- and in the Al-foils appear serrated; especially for Mo the crack
propagates mainly along grain boundaries, additionally bifurcation occurs.
Fatigue and fracture properties of thin metallic foils 315
Figure 2. (a) Crack length versus number of loading cycles of a Mo foil (recrystallized) with a thickness of 30 µm.
(b) Crack propagation rate versus crack length of a Mo foil (recrystallized) with a thickness of 30 µm.
This anomalous crack growth behaviour is investigated in more details for a 100 µm thick
annealed Cu-foil as an example.
In Figure 5a three positions A, B and C are distinguished. Region A is closer to the startingnotch, B belongs to an area of intermediate crack-arrest and C corresponds to the crack tip.
The inserts show three ECCI-micrographs of the areas A, B and C .The micrograph taken
from region A indicates the crack path influenced by microstructural features resulting in
intermediate crack arrest. From the micrograph of location B it may be deduced that crack
arrest occurs if the growing crack interacts with grain boundaries, impurities or any other
microstructural barriers. This arrest is always accompanied with a plastically deformed region,
the extent of which may be deduced from the presented micrograph.
316 A. Hadrboletz et al.
Figure 2. SEM micrograph (ECCI technique) of the crack path in a Mo foil (recrystallized) with a thickness of
30 µm.
Figure 3a. Crack length as function of number of loading cycles of a Cu foil (as received) with a thickness of
20 µm.
Figure 3b. Crack propagation rate as function of crack length of a Cu foil (as received) with a thickness of 20 µm.
Fatigue and fracture properties of thin metallic foils 317
Figure 3c. Crack length as function of number of loading cycles of Cu foils (recrystallized) with varying thickness.
Figure 4. (a) Crack length as function of number of loading cycles of an Al foil (recrystallized) with a thickness
of 125 µm. (b) SEM micrograph of the crack path of an Al foil.
In position C the interaction of the strain field at the tip of the crack with a twin boundary
can be seen from the change in channelling-contrast. The plastic zone around the tip of the
fatigue crack is small, the adjacent grain contains a typical vein-structure with a PSB .
The special dependence of the crack propagation rate on the number of loading cycles for
foils with thickness < 200 µm is very similar to the characteristic behaviour of short cracks
(Miller and de los Rios 1986). This is not surprising since cracks in foils may be regarded as
microstructurally small at least in one dimension although they are not necessarily physically
short. Taking into account that especially rolled and heat treated foils often consist of a single
or double layer of grains through the thickness the crack propagation behaviour is like that in
a single crystal which is known to be almost unaffected by closure. Therefore corresponding
stress intensity values may be regarded as effective values.
As a typical example revealing the dislocation arrangement surrounding a crack propagating at a rate of 10 −9 m cycle−1 in Figure 5b a fine cell structure (cells < 1 µm) in a region
of 20 µm width around the crack is shown. In a certain distance from the crack most of the
318 A. Hadrboletz et al.
Figure 5a. Crack length versus number of loading cycles of a Cu foil (recrystallized) with a thickness of 100 µm
and SEM micrographs at different locations of the crack path.
Figure 5b. Dislocation arrangement surrounding a fatigue crack (propagation rate 10−9 m cycle−1 ) of a 100 µm
thick recrystallized copper foil obtained with the ECCI technique.
Fatigue and fracture properties of thin metallic foils 319
grains exhibit a vein structure with persistent slip bands (PSBs), frequently ceasing within
the grain. From the observed microstructures the local cyclic plastic strain values may be
deduced by a comparison of the prevailing microstructure with the results of the corresponding
stress-cyclic plastic strain curve as presented in (Chen et al., 1997). Thus at the border of the
region containing cells the plastic strain εpl exceeds 6 × 10−4 , the region containing PSBs
corresponds to plastic strain values of 6 × 10−4 > εpl > 10−5 .
From Figure 3c a change in the crack growth behaviour is obvious if the foil-thickness is
larger than approximately 150 µm. This behaviour may be explained by the assumption of a
state of plane stress for crack growth in foils thinner than 150 µm and a state of plane strain
for foils with thickness exceeding 100 µm. Crack resistance in plane stress is increasing with
increments of crack length so that a larger stress is required to maintain crack growth (see
Broek, 1988), otherwise for constant strain (as it is the case for the present experiments) a
diminishing propagation rate with increasing number of loading cycles results, which has
been actually observed.
These assumptions have been made firm by use of the rp /t-criteria (ratio of the prevalent
radius of plastic zone to the foil-thickness). Due to LEFM-considerations (Broek, 1988) they
are as follows:
rp /t < 0, 025 for cracking in plane strain,
rp /t > 1 for cracking in plane stress
The radius of the plastic zone may be either calculated from Equation 1 or may determined
rp = 1/2π(Kmax /σy )2
(1)
Kmax maximum value of stress intensity factor during loading cycle; σy yield stress after
saturated cyclic hardening,
from the ECCI-micrographs directly defined by the extent of characteristic dislocation
structures. Values of rp and rp /t at the tip of the growing crack for Cu-foils of different
thickness and different crack lengths are summarised in Table 3.
The given rp /t-values really indicate that for crack growth in foils a transition from a state
of plane stress to a state of plane strain with increasing foil thickness occurs.
If the grain size of the foils investigated are taken into consideration the criteria mentioned
above may be replaced by the following: size effects may occur if the ratio of the grain size to
the foil-thickness is close or larger than unity.
To check these criteria electrodeposited Cu foils, known to be extremely fine grained, have
been investigated additionally.
In Figure 6 a-N curves of electrodeposited Cu foils of two different thickness are shown.
Unlike the behaviour of rolled and annealed foils (Figure 3) the crack growth behaviour is
equal to that known from bulk material. For both ED Cu foils the behaviour of crack propagation rate does not show intermediate crack arrests. A typical micrograph of the crack path
in a foill with 105 µm is also presented, cracking along grain boundaries can be seen.
Regarding both the rp /t-values (added in Figure 6) being well below unity and the ratio of
grain size to foil-thickness which ranges from approximately 20 for the 35 µm thick foil to
approximately 50 for the 105 µm thick foil the observed behaviour is not surprising.
320 A. Hadrboletz et al.
Figure 6. Crack length as function of number of loading cycles of electrodeposited Cu foils of two different
thickness, the micrograph indicates typical features of a crack path in the 105 µm foil.
Table 3. Radii of plastic zone along the crack path for foils of
different thickness; calculated from (1).
Foil thickness (µm)
Crack length (µm)
rp (µm)
rp /t
20 (σy = 112 MPa)
400
1200
1900
3400
3800
4150
900
1350
1500
250
700
1200
400
800
1200
8
23
38
82
77
40
27
43
47
16
48
82
29
57
87
0.40
1.15
1.90
1.64
1.54
0.80
0.27
0.43
0.47
0.08
0.24
0.42
0.006
0.013
0.019
50 (σy = 70 MPa)
100 (σy = 77 MPa)
200 (σy = 60 MPa)
Bulk (σy = 60 MPa)
Fatigue and fracture properties of thin metallic foils 321
Figure 7. da/dN - 1K curves of rolled (recrystallized) and electrodeposited Cu foils.
2.2. FATIGUE
THRESHOLD VALUES
As an example in Figure 7 two curves – crack propagation rate versus stress-intensity factor
range – for a rolled and electrodeposited Cu foil (of comparable thickness) are given. These
graphs also illustrate the way to determine threshold-values; an almost identical behaviour in
the slow crack-growth regime and threshold values is observed.
In Table 4 a selection of ranges of threshold stress intensity factor values for foils of
different materials and thickness is given.
A brief summary of the given results of Cu-foils reveals the following facts:
• The scatter of 1Kt h -values for a given thickness is considerably large
• The angle between the direction of crack growth and the rolling direction influences the
1Kt h -values severely.
• A trend of increasing 1Kt h -values with increasing foil thickness can be observed.
• The threshold values are comparable with the effective fatigue threshold of bulk material
The large scatter of the 1Kt h data may be related to the pronounced interaction of the crack
path with grain boundaries. This is especially true for the case of foils which thickness is in the
order of the average grain size, so that the whole foil may be regarded as a two-dimensional
array of grains.
Copper foils are often highly textured. As it is known and may deduced from Table 2
texture is the reason of the variation of the value of the experimentally determined Young’s
modulus from 68 GPa up to 132 GPa. For bulk material commonly a value of 126 GPa is
assumed, so that neglecting texture effects the error of 1Kt h is considerable.
The trend of the increasing threshold-values with increasing foil thickness may be explained by a more detailed evaluation of the ratio of grain size to thickness and the grain
size distribution (mostly influenced by predeformation). Higher threshold values in thicker
foils (consisting of a few layers of grains) are mainly attributed to an irregular distribution of
grains of different shapes, sizes and amounts of deformation.
da/dN versus N curves of foils consisting of a single or double layer of grains through the
thickness resemble the behavior of short cracks. These are essentially closurefree and their
threshold-values are close to effective values. The same may be stated for single grain lay-
322 A. Hadrboletz et al.
Table 4. Fatigue thresholds of foils of different thickness.
√
Range of 1Kt h (MPa m)
Material
Thickness
(µm)
Heat
treatment
Mo
Cu (rolled)
30
20
Recryst.
As received
850 ◦ C/ 6 h
9.5–10
2.6–2.8
1.8–2.6
As received
As received
As received
600 ◦ C/ 4 h
900 ◦ C/ 2 h
1.9–2.0
2.1–2.9
3.0–3.9
2.8–2.9
2.8–3.2
900 ◦ C/ 2 h
2.4–2.7
600 ◦ C/ 4 h
3.3–3.7
2.8–3.2
2.2–2.4
2.9
50
78
100
Cu (ED)
∗ 1K
200
Bulk∗
35
105
As rec.
As rec.
Remarks
Variation of Young’s
modulus due to texture
One specimen
Crack direction normal to
rolling direction
Crack direction 45◦
to rolling direction
t h,eff (see Chen et al., 1991).
ered foils so that the determination of threshold-values of such foils may be regarded as an
attractive way of getting 1Kt h,eff -values. Almost comparable threshold values of the ED foils
with rolled material is observed. From the prevalent grain boundary cracking (see Figure 11)
resulting in a small amount of plastic deformation it may be deduced that closure is less
pronounced.
The trend of decreasing 1Kt h -values with decreasing foil-thickness may be explained by
the influence of plastic strain gradients in the foils too.
According (Wei and Hutchinson, 1997) the effect of strain gradient hardening on crack
growth results in a lowering of the total work of fracture, which is otherwise known to be proportional to the threshold value (Taylor, 1981). For a more precise evaluation the characteristic
length l of the plastic strain gradient is to be compared with R0 defined as:
R0 = E00 /3π(1 − ν)2 σy2 ,
E, Young’s modulus; 00 , work of separation in the fracture process; ν, Poisson’s ratio; σy ,
tensile yield strength,
which is a length quantity that scales the plastic zone size.
To demonstrate the influence of strain gradient hardening on crack growth the following
example is given. Considering a material with a strain hardening exponent of 0.2 and constitutive length parameters to be equal in stretch and rotation, the peak separation stress being
four times the yield stress, an increase of l/R0 from 0.2 to 1.0 yields a lowering of the ratio
of the total steady state work of fracture to the work of separation of the fracture process to
about half of its origin value. This ratio is decreased to even less than a third if no plastic strain
gradient is taken into account. Since the threshold value is proportional to the square root of
the work of separation this means a reduction of threshold values from about 30% to 50%.
Fatigue and fracture properties of thin metallic foils 323
Figure 8. SEM fractomicrographs of Cu foils tensile tested (a) 20 µm (recrystallized). (b) 78 µm (recrystallized).
Therefore a quantitative evaluation of the lowering of threshold values in thin foils requires
the knowledge of several parameters such as the strain hardening exponent and the constitutive
lengths parameters of strain gradients.
3. Aspects of the fracture process
To reveal a size effect of the fracture behaviour of thin foils in a first attempt fractomicrographs
of tensile loaded specimens of different thickness have been investigated. As examples the
fractomicrographs of a 20 µm thick foil (consisting of a single layers of parts of grains) is
presented in Figure 8a and that of a about 78 µm thick foil (several grains across the thickness)
is given in Figure 8b. For foils below a thickness of about 70 µm ductile failure with typical
knife edge rupture without voids and dimples appears. Similar to the findings of Hong and
Weil, (1996) for Cu foils in this regime of thickness the fracture is of a transgranular type. For
foils with larger thickness voids and dimples can be observed; these dimples are very small
(about 5 µm) and comparable with the constitutive length scale of plastic strain gradients,
324 A. Hadrboletz et al.
Figure 9. Fracture strain of Cu foils dependent on thickness.
indicating that the fracture process is severely influenced by the occurrence of strain gradients.
A comparison with the observed cell sizes in the corresponding ECCI-micrographs yields
diameter values in a similar range. Some theoretical derivations (by Laird et al., 1986) give
5 µm as the lower bound of the size of dimples, corroborating the experimental findings
presented here. For thin foils in the thickness range of about 1 µm Arzt (1998) reports the
lack of dimples on the fracture surfaces. This minimum size of approximately 5 µm makes it
impossible to fit into a foil thinner than this size.
In a different approach the fracture strain as function of the foil thickness has been investigated. Results are depicted in Figure 9 for foils as received and after two different heattreatment procedures. For all three types a markedly decrease of fracture strain with decreasing
foil-thickness is obvious, which may be explained that the length parameter of the plastic
strain gradient is comparable to the foil thickness. The rate of decrease being pronounced with
higher annealing temperature. Although it is not illustrated it should be taken into account that
bulk material exhibits values of fracture strain exceeding 50%. For comparison in single or
double grain layered structures (i.e. foil thickness up to about 100 µm) the fracture strain
never exceeds a value of 20%. From this and the fact of crystallographic crack planes well
documented in the typical knife edge rupture it may be concluded that the number of activated
gliding systems is reduced if the ratio of the foil-thickness to the grain size approaches unity.
The microstructure of such a thin foil resembles an arrangement of neighbouring single
crystals in one plane. This means for a prevalent cube texture a concentration of slip activity
in one or two systems.
In Figure 10 a SEM-micrograph of a foil with a thickness of 100 µm ruptured after cyclic
loading is presented. In comparison to the fracture topography in tensile loading the fracture
surfaces in fatigue appear more flattened and occasionally striations with a typical spacing of
about 1 µm are observed. This indicates that the crack propagation rate has been in the order
of 10−8 m cycle−1 (regime II). Similar findings in a 35 µm thick rolled Cu-foil have been
reported by (Hong and Weil, 1996). Furthermore dimples with a characteristic size of about
5 µm are observed in the tensile ruptured area.
In Figure 11 the fracture surface of a 35 µm thick Cu foil parallel to the plane of fracture
is presented. An almost entirely crystallographic mode of fracture is observed. Taking into
account the extremely fine grained structure of this electrodeposited foil this behaviour is not
Fatigue and fracture properties of thin metallic foils 325
Figure 10. SEM fractomicrograph of a Cu foil (recrystallized) with a thickness of 100 µm after fatigue testing.
Figure 11. SEM fractomicrographs of an electrodeposited Cu foil (35 µm) after fatigue testing.
surprising. Since the foil consists of multiple grain layers across its thickness the crack growth
behaviour is comparable to bulk material which ratio of thickness to grain-size is in the same
magnitude.
4. Summary and conclusions
In the present study the fatigue and fracture behaviour of foils with thickness ranging from
20 µm to 250 µm have been investigated and compared to the corresponding bulk material.
Investigations of rolled, annealed and electrodeposited Cu foils of different grain sizes
revealed that the so called size effect may be observed if the ratio of the grain size to the
foil thickness is close to or larger than unity and the constitutive length scale of the apparent
plastic strain gradient is a considerable portion of the foil thickness.
This results in a reduction of the number of the activated slip systems corroborated by
the decrease of the fracture strain. Similar considerations explain the size effect of the fracture
process characterised by the complete lack of dimples or the occurrence of very small dimples
in size comparable to the characteristic length of the apparent plastic strain gradient.
326 A. Hadrboletz et al.
A size-effect was detected for fatigue crack growth data of rolled foils. For foils of thickness up to about 150 µm intermittent crack arrest (interaction with microstructural barriers)
and a negative curvature of the crack growth curve was observed. This was explained in
terms of LEFM based on the crack resistance and the rp /t criteria resulting in a transition
from plane stress to plane strain. The observation of the extent of the plastic zone in the
vicinity of the crack is performed by application of the ECCI technique. Due to the fact that
the electrodeposited foils consist of a multilayered fine grain structure (with a plastic strain
gradient developed differently to that in a coarse grain structure) no size effect is observed and
the fatigue crack growth behaviour is similar to bulk material. The results of Mo and Al foils
indicate that similar considerations may be applied to explain the size effect of metallic foils.
Values of the fatigue threshold stress intensity factor of thin foils are almost unaffected by
closure and thus comparable to effective values of bulk material. Therefore the determination
of fatigue thresholds in foils may occasionally replace testing of conventional macrospecimens.
The presented resonance testing system operating at a frequency of 20 kHz allows fatigue
loading of free-standing foils with different thickness at R = −1. For positive R-values the
special designed specimen holder can be inserted in a conventional servohydraulic testing
machine.
The observed changes in the global dislocation structure by use of the ECCI method
allowed to understand the interaction of the growing crack with the prevailing dislocation
structure in foils with varying thickness. Thus a combination of the fatigue testing device with
ECCI-techniques allows a more basic approach to the size effect.
A further development of the presented testing equipment allows to investigate the size effect of thin wires on the fatigue life approach. Such investigations permit deeper understanding
of the fatigue in quasi-onedimensional arrangement of grains.
This investigation results in new length parameters in comparison to the apparent grain size,
which could support the plastic strain gradient theory for the explanation of the size effect.
Acknowledgements
The authors thank the Austrian national science foundation (P 12311-TEC, the Bundesministerium für Wissenschaft und Verkehr (COST action 510) and the Jubiläumsfonds der österr.
Nationalbank for financial support. An additionally support of Prof. Dr R. Stickler with the
ECCI technique is greatly appreciated.
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