Acta Materialia 53 (2005) 647–657
www.actamat-journals.com
Determination of interface integrity in high volume fraction
polymer composites at all strain levels
G. Anagnostopoulos a, D. Bollas a, J. Parthenios
a
a,b
, G.C. Psarras b, C. Galiotis
a,b,*
Institute of Chemical Engineering and High Temperature Chemical Processes, Foundation for Research and Technology –
Hellas (FORTH), P.O. BOX 1414, Patras 26504, Greece
b
Department of Materials Science, School of Natural Sciences, University of Patras, Patras 26504, Greece
Received 9 February 2004; received in revised form 5 October 2004; accepted 12 October 2004
Abstract
In fiber-reinforced composites, the mechanical performance is strongly dependent upon the quality of fiber/matrix interface. In
the present work, a novel technique is presented whereby all important interface parameters of high volume fraction composites are
determined in situ on standard tensile coupons. The method involves the introduction of a small surgical cut to the upper ply of a
unidirectional aramid/epoxy laminate prior to vacuum bagging. During the curing procedure and subsequent consolidation in an
autoclave, the resin penetrates the area of the cut and thus no effect to the integrity of the composite is observed. The stress transfer
profiles during mechanical loading emanating from the fiber break(s) can be obtained by means of the technique of laser Raman
microscopy. The technique allows the determination of the maximum interfacial shear stress at all far-field strain levels and therefore
presents the only alternative for the accurate estimation of interface integrity in high volume fraction polymer composites.
Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Raman spectroscopy; Tension test; Fiber reinforced composites; Interface; Aramid fibers
1. Introduction
It is widely acknowledged that the fiber/matrix interface influences the properties of fiber-reinforced polymer
composites. The stress transfer efficiency from the matrix to the fibers, the stress build-up in broken fibers
and the redistribution of the stresses in the neighboring
intact fibers are all governed by the interfacial strength
and integrity [1–4]. Over the years a lot of research has
taken place in an attempt to evaluate the fiber/matrix
integrity by using single-fiber (short or long) model
geometries [5–11] and a variety of experimental techniques. Useful reviews of these micromechanical tech*
Corresponding author. Tel.: +30 261 096 525 5; fax: +30 261 096
522 3.
E-mail addresses: [email protected], [email protected]
(C. Galiotis).
niques can be found in [12]. Attempts to assess the
interface strength in high volume fraction composites
have also been made by employing microindentation
methods but the results are affected by the specimen fabrication procedures and the data reduction schemes
[5,9].
Direct non-destructive measurements of stress/strain
in fibers can be made using the technique of laser
Raman microscopy (LRM). This technique has been
proved to be very effective in obtaining stress and strain
profiles at the vicinity of fiber discontinuities in both single fiber model composites and high volume fraction
coupons [13–16]. In discontinuous fiber composites of
any volume fraction, the LRM can be readily applied
and produces valuable data. However in the case of continuous fibers, the stress transfer is only activated at a
fiber break. Since composites are designed to operate
at low deformations, it is questionable whether the
1359-6454/$30.00 Ó 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.actamat.2004.10.018
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G. Anagnostopoulos et al. / Acta Materialia 53 (2005) 647–657
inherent strength of the interface at low deformation can
be assessed due to the lack of available fiber breaks for
stress transfer measurements. Furthermore, reinforcing
polymer fibers such as aramid, do not exhibit a clear
break at fracture and hence stress transfer profiles are
difficult to obtain from the incurred discontinuity. In
[15] the unification of all interfacial measurements
through the technique of LRM has been presented.
In this paper, a new experimental procedure is presented for the in situ measurement of the interfacial
strength in standard composite coupons that incorporate continuous fibers. A small surgical fiber-cut is introduced to the upper layer of unidirectional aramid/epoxy
specimens prior to vacuum bagging (Fig. 1). After the
cutting, and due to the curing and the post curing, the
resin consolidates this discontinuity. Such a minute fiber
severance is not expected to affect the mechanical properties of the composite. Standard tensile coupons are
subsequently prepared and their integrity is assessed
from the induced discontinuity at various levels of applied load by means of LRM.
2. Experimental
In the present study two different laminate specimens,
incorporating KevlarÒ29 (Du Pont) aramid fibers, were
prepared and tested. The detailed preparation procedure
for lamination and curing is given below. As mentioned
above, in both cases prior to vacuum bagging small cuts
were made on the surface of the top prepreg layer using
a surgical scalpel (Fig. 1). As explained later, with this
Fig. 1. Schematic representation of the procedure employed in order
to induce a fiber discontinuity into the composite coupons. Indicative
microphotographs (after surface polishing) from corresponding
aramid fiber epoxy resin laminates are also shown.
cutting method, some of the fibers are bending over
and this may affect the stress mapping near the fiber tip.
2.1. Preparation of aramid/epoxy specimens
The test specimens were made using unidirectional
KevlarÒ29/LTM217 resin pre-impregnated tapes
(Table 1). The LTM217 one-part epoxy resin was supplied by the Advanced Composites Group, UK. The
selection of the resin was based on its suitability for prepreg manufacturing and on its high Tg (P150 °C). The
composite laminates consisted of 4-ply unidirectional
KevlarÒ29/LTM217 prepreg tapes of 53.5% in fiber volume fraction. The curing procedure involved initial heating to 70 °C for 12 h followed by post-curing at 140 °C for
1 h in an autoclave system. The diameters of individual
KevlarÒ29 filaments, supplied by Du Pont de Nemours
Inc., were measured by employing a laser diffraction unit
and their mean value was found to be 15.9 ± 0.5 lm.
The same aramid fibers were used for the fabrication of the model composites and were embedded in
an Araldite epoxy matrix [17]. Short individual filaments of 1–3 mm in length were carefully chopped
from the continuous yarn by means of ceramic scissors. Rubber dog-bone moulds were half-filled with
a two-part, solvent free epoxy resin supplied by
Ciba-Geigy. The resin was prepared following the
manufacturerÕs specifications by mixing 100 parts of
a modified epoxy novolac resin (Ciba LY 1927) to
36 parts of amine hardener (Ciba HY 1927 GB) by
weight. The resin was allowed to set partially for
about half hour. A single short filament was very
carefully lifted by means of a very fine pair of tweezers and positioned at the surface of the epoxy layer.
Extreme care was taken to align the filament parallel
to the specimenÕs symmetry axis. A second batch of
epoxy resin was prepared and poured into the mould.
The whole procedure was completed by transferring
the mould into an environmental chamber and allowing the specimens to cure for seven days at room temperature with temperature fluctuation at a minimum
of ±1 °C.
Specimens that contained misaligned fibers or other
defects, such as air bubbles, were discarded. The selected
specimens were ground uniformly in order to bring the
embedded fiber to about 400–500 lm from the resin surface. At the final stage, they were extensively polished to
a high level in order to prepare a perfectly smooth
surface.
Table 1
The material properties of the aramid/epoxy resin composite
Material
Ò
Aramid fibers Kevlar 29, DuPont USA
Epoxy resin matrix LTM 217ACG, UK, cured at 140 °C
YoungÕs modulus E (GPa)
Tensile strength r (MPa)
Fracture strain e (%)
70.0
2.0
2500
55
3.5
3.0
G. Anagnostopoulos et al. / Acta Materialia 53 (2005) 647–657
2.2. Raman microscopy–experimental set-up
A new remote Raman microprobe was used for all
Raman measurements in full composite specimens
(Fig. 2). This microprobe was designed in-house and
developed by Jobin/Yvon DILOR. The microprobe
uses an optical fiber for Raman light collection, while
the laser source has been incorporated into the main
body of the microprobe [18]. Laser source is a miniature diode pumped solid state laser (lGreen SLM, Uniphase) excited at 532 nm with 50 mW maximum
output power. The use of a set of attenuated filters
in front of the laser output mirror provides the capability of changing the incident laser power to the desired
level. The laser beam is first filtered through a microscope objective and a spatial filter then is directed to
the microscope objective by reflection on the notch filter (O.D. = 7). The 180° scattered light passes through
the notch filter and guided through the optical fiber to
a SPEX 1000M single spectrometer. A main feature of
the new probe is the capability of controlling the polarization of both laser and collected Raman light. Moreover, the use of an adjustable pinhole placed in the
optical path of the collected Raman light gives the
advantage of confocality to the microprobe. Detailed
description of the new Raman microprobe will be given
elsewhere [19].
The dispersed Raman signal was then converted to an
electrical signal and stored in a PC. The obtained spectra were analyzed by fitting the raw data with Gaussian
and/or Lorentzian distributions [20]. This remote set-up
is particularly useful for composite materials subjected
to mechanical loading, but it can, also, be used in a
whole variety of technological applications where con-
649
ventional micro-Raman arrangements impose space
restrictions on the actual measurements.
2.3. Mechanical testing
The full composite specimens were subjected on tensile test with an MTSÒ 858 servo-hydraulic tabletop
mechanical frame (maximum load of 25 kN and of a
testing frequency of up to 300 Hz). This frame has a
whole range of load cells and gripping assemblies, which
make this instrument capable of testing fibers, plastics,
as well as, stiff materials such as ceramics, metals and
composites (Fig. 2). The laser power on the specimen
was kept low (2 mW) to avoid material damage due to
local overheating. The gauge length of the specimen
was 30 mm and the specimen width was 10 mm. In order
to determine the residual thermal fiber stress prior to the
application of an external load, the Raman response of a
large number of KevlarÒ29 fibers located at random
positions within the specimens was recorded. The location of the measurements was chosen to be far from
the original fiber-cut to ensure that any local damage induced by the cutting procedure does not affect the results. At different levels of external load, Raman data
were collected by scanning the Raman probe point-bypoint along a broken fiber on either side of the induced
discontinuity. The applied strain was measured via conventional strain gauges. Two aramid/epoxy coupons
with induced discontinuity were prepared and tested under identical conditions.
As for the model composites, a microtensometer was
employed for specimen deformation [17]. It was designed and manufactured for tensile testing of this particular coupon geometry and can be accommodated
Fig. 2. Experimental set-up showing the remote Raman probe coupled to the mechanical frame (MTS). The excitation source (solid laser at 532 nm)
is integrated into the body of the microscope.
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G. Anagnostopoulos et al. / Acta Materialia 53 (2005) 647–657
under the Raman microscope. The candidate specimen
was carefully mounted on it by clamping its ends between its jaws and secured by tightening the screws.
The local strains were continually monitored during
tightening to avoid imparting undesirable bending tensile or compressive stresses to the specimen. Finally,
the microtensometer was positioned on the experimental
stage under the Raman microscope. The stage is translated in the xy-direction (specimenÕs plane) and the
z-direction (laser beam direction) by means of three
heavy-duty micrometers with a resolution of 10 lm.
3. Results
3.1. Aramid fibers as stress and strain sensors
Aramid fibers, such as the well-known KevlarÒ family (Du Pont) are excellent Raman scatterers [17].
Their spectrum includes a strong band at 1611 cm1,
which corresponds mainly to the phenyl ring/C–C
stretching. Therefore, well-defined relationships between Raman wavenumber shift and the applied stress
or strain can be obtained by these fibers. At this point
it is worth noting that all the band shifts are referred
to the change of band positions as compared to the
corresponding values of the freestanding fibers in air
(Fig. 3). The observed stress or strain dependence of
the specific vibrational mode allows the conversion of
the Raman wavenumber to values of stress or strain
in any application that involves the aramid as reinforcing fibers.
3.2. Stress and strain mapping in aramid/epoxy
composites
In order to investigate the stress transfer mechanisms,
at constant temperature, in aramid/epoxy composites, a
methodology has been developed. In particular, the total
wavenumber shift for the 1611 cm1 band depends on
two terms, according to the following equations:
Dm ¼ DmM þ DmR ;
ð1Þ
DmM ¼ k 1611 rf ;
ð2Þ
where DmM is the shift due to the mechanical field induced by the applied stress or/and strain and DmR is a
constant term, which represents the shift due to the
residual thermal stresses, caused by the mismatch of
the thermal expansion coefficients of the aramid fibers
and the epoxy resin. The stress calibration factor,
k1611, represents the sensitivity of the 1611 cm1 to the
applied mechanical stress (Table 2). The axial stress in
the fiber, rf, at any point along the embedded fiber
can be obtained easily from Eqs. (1) and (2)
rf ¼
Dm DmR
:
k 1611
ð3Þ
3.3. Axial stress distribution in aramid/epoxy composites
The stress transfer profiles at certain levels of applied
tensile strain were obtained for both specimens. The obtained results are presented in Figs. 4(a), 5(a) and 6(a);
the position of the fiber break in these figures is considered
to be the origin of the X-axis. The measurements took
place along one particular fiber, with a 2 lm step. As expected, the stresses build up from the discontinuity and
reach a maximum value (far field stress) at a certain distance away from it. At 0.8% of applied strain (Fig. 4(a)),
the fiber stress builds to a maximum plateau value of
500 MPa at a distance of 250 lm. By increasing the
applied strain to 1.5% and 2.0% (Figs. 5(a) and 6(a)),
the axial stress now builds to maximum plateau values
of about 900 and 1250 MPa at distances of 300 and
400 lm, respectively, away from the discontinuity.
3.4. Interfacial shear stress distribution in aramid/epoxy
composites
The stress transfer profiles can be converted into
interfacial shear stress profiles, srz, along the length of
Table 2
Stress/strain sensitivity of KevlarÒ29
Fig. 3. Raman wavenumber shift of the 1611 cm1 band as a function
of stress and strain for KevlarÒ29 fibers. The solid lines correspond to
least-squares fits to experimental data.
KevlarÒ29 (1611 cm1)
Stress sensitivity
(cm1/GPa)
Strain sensitivity
(cm1/%)
4.0 ± 0.5
2.38 ± 0.13
G. Anagnostopoulos et al. / Acta Materialia 53 (2005) 647–657
Fig. 4. (a) Stress profile and spline fit of the examined Kevlar fiber at
0.8% applied composite strain; (b) the corresponding interfacial shear
stress profile.
the fiber by means of a straightforward balance of forces
argument, which leads to the following relationship [16]:
r drf ðzÞ
srz ¼ ;
ð4Þ
2
dz
where r the radius and z the distance along the fiber
length. The ISS profiles (Figs. 4(b), 5(b) and 6(b)), srz,
were derived using cubic spline fitting to the raw data
at 0.8%, 1.5% and 2.0% of applied strain calculating
the derivatives, drf/dz, from the fitted functions and
employing Eq. (4) (see also Appendix A). As has been
stated elsewhere [16], the balance of forces argument is
of general validity and can also be applied to the stress
field acting on fibers adjacent to the fiber break or discontinuity. The resulting ISS profiles at 0.8%, 1.5%
and 2.0% of applied strain are shown in Figs. 4(b),
5(b) and 6(b), respectively. At 0.8% of applied strain,
the ISS maximum (ISSmax) of about 40 MPa appears
at the fiber discontinuity and decays to zero at a distance
651
Fig. 5. (a) Stress profile and spline fit of the examined Kevlar fiber at
1.5% applied composite strain; (b) the corresponding interfacial shear
stress profile.
of about 250 lm from it (Fig. 4(b)). The corresponding
ISS profile at 1.5% applied strain exhibits the characteristic ‘‘knee’’ at the fiber cut and reaches a maximum value of 22 MPa at a distance of about 70 lm (Fig. 5(b)).
Moving to higher strain levels, at 2.0%, the ISSmax is
about 18 MPa and remains almost constant for
200 lm and then decays to zero (Fig. 6(b)). In Fig. 7
the resulting ISSmax as a function of the applied tensile
strain from both specimens are presented. It is clearly
seen that ISSmax increases with strain, reaches a maximum of almost 40 MPa at an applied strain of 1.2%
and then decreases reaching a plateau value of 18
MPa at high strains. The various methods of experimental determination of the upper ceiling of ISS are compared with existing analytical models in a subsequent
publication [21].
A parameter that characterizes the integrity of the
interface is the transfer length or ‘‘ineffective length’’,
Lt, defined as the distance from the discontinuity where
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G. Anagnostopoulos et al. / Acta Materialia 53 (2005) 647–657
Fig. 8. Transfer length in the full aramid/epoxy composites as a
function of the applied tensile strain.
Fig. 6. (a) Stress profile and spline fit of the examined Kevlar fiber at
2.0% applied composite strain; (b) the corresponding interfacial shear
stress profile.
the ISS tends to zero (for practical purposes 1 MPa is taken here as ‘‘zero’’). In Fig. 8 the dependence of transfer
length on the applied strain is presented. As can be seen,
the Lt increases with applied strain and reaches a maximum of 300 lm just prior to the onset of interface failure. At higher strains the propagation of interface
failure brings about a further increase of the transfer
length to a maximum value of 900 lm at the maximum
attained strain of 2.5%.
Finally, for comparison purposes in Fig. 9(a) and (b),
the axial stress and the corresponding ISS distributions
for the single KevlarÒ29 fiber model composites are presented [17]. The obtained values of maximum ISS per
strain level have been added to the full composite results
of Fig. 7. In spite of the differences in the type of epoxy
used, the RT curing of the single fiber coupons ensured
that there were no significant residual stresses that may
have changed the stress state at the interface. Hence, it is
not surprising that the overall trend of the maximum
ISS as a function of tensile strain is strikingly similar.
4. Discussion
Fig. 7. Maximum Interfacial Shear Stress as a function of the applied
tensile strain for both model and full aramid/epoxy composites.
The overall picture of the interface integrity as a function of applied strain is very similar to what has been obtained previously in single aramid fiber/epoxy
composites, albeit under different matrix/curing conditions [17]. As seen in Fig. 7, there seem to be two distinct
stress transfer regions: (a) a perfectly elastic one in
which the ISS decays from a maximum value at the induced discontinuity to almost zero at some distance
away, and (b) a post-interface-failure region in which
the ISS maximum has been shifted away from the induced cut and the characteristic ISS ‘‘knee’’ appears
G. Anagnostopoulos et al. / Acta Materialia 53 (2005) 647–657
653
Fig. 9. (a) Fiber stress profiles and (b) ISS profiles for KevlarÒ29/epoxy model composites.
(Figs. 5(b) and 6(b)) [17,22]. It is indeed remarkable that
the presence of neighboring fibers and the existing differences in the local stress state as one moves from single
fiber to practical high volume fraction composites has
not altered the way the aramid/epoxy interface performs
as a function of an increasing tensile strain. The maximum ceiling of ISSmax varies between 40 and 45 MPa,
being of the same order of magnitude with the shear
yield strength of the resin [23]. Obviously, the maximum
attained ISS values between the two classes of specimens
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G. Anagnostopoulos et al. / Acta Materialia 53 (2005) 647–657
differ as they should do due to differences in matrix
chemistry and curing conditions but for such types of
composites the results presented here and elsewhere
[24], indicate that the upper ISS ceiling of 40–45 MPa
is governed by local matrix yielding. For KevlarÒ49/
epoxy systems, this assertion has also been confirmed
by finite element analysis of the axial stress transfer
and corresponding ISS distributions [25].The presence
of matrix plasticity near the fiber end in tandem with
the viscoelastic properties of both resin and fibers leads
to a drop in the ISSmax that the system can sustain at
higher levels of strain (Fig. 7). Similarly, the values of
Lt (Fig. 8) increase approximately linearly with applied
strain in the elastic region. However, considerable
fluctuations are observed at higher strains. Since the Lt
is defined as the length over which the ISS reaches
1 MPa, these fluctuations are attributed to the corresponding ISS fluctuations, as a result of the presence
of interface damage beyond 1.4%.
A schematic of the post-interface-failure picture for
these types of composites is given in Fig. 10. Fiber–fiber
interaction can induce changes to the axial stress profiles
as long as the neighboring fiber breaks can induce perturbations in the local shear field [26,27]. In high volume
fraction composites fiber breaks are registered on adjacent fibers by the characteristic ‘‘spikes’’ observed in
Figs. 5(a) and 6(a). The stress concentration factors
for these perturbations can be calculated from the equation [28]
rz
k¼ ;
ð5Þ
r0
where rz is the peak stress at the middle of the stress
concentration distribution and r0 is the applied far-
Fig. 10. Schematic representation of the axial stress distribution on a
fractured fiber and one nearest neighbor. The transfer length in the
broken fiber and the corresponding positively-affected-length (PAL) in
the neighboring fiber is shown.
field stress which coincides with the fiber axial stress
far away from the fiber discontinuity. The ‘‘spikes’’ observed in Figs. 5(a) and 6(a), correspond to values of
stress concentration of the order of 1.10 at 1.5% which
then increases to 1.24 at 2.0% applied strain for the
adjacent fiber fracture appearing at distance of 600
lm from the fiber discontinuity. The second adjacent
fiber fracture event, which is evident also at 2.0%, appears at a distance of 100 lm from the fiber discontinuity. At that position (Fig. 6(a)), the fiber stress has
not reached its maximum value r0 due to its proximity
to the fiber cut. Since the primary source for the stress
concentration here is the fracture of an adjacent fiber
at a far field stress of r0, Eq. (5) cannot be readily
used. An equivalent stress concentration can be estimated from
k¼
rz Drz þ r0
¼
;
r0
r0
ð6Þ
where rz is the equivalent stress rise on the top the far
field stress r0 which is the sum of its magnitude, Drz plus
r0. Application of Eq. (6) yields a concentration value of
1.24 which is identical (as it should be) with that found
at a distance of 600 lm. This relatively high value cannot lead to fracture of the fiber under examination since
at that position the axial fiber stress is only 400 MPa
(Fig. 6(a)). This ‘‘staggering or shielding’’ effect, which
reduces the severity of stress concentration in a neighboring fiber, has been postulated analytically by Sastry
and Phoenix [26,27] and it is verified experimentally
for the first time here.
In order to assess the overall integrity of the composites to the application of a tensile stress field, Raman
sampling has been performed along the fiber direction,
at random points far away from the induced discontinuity. The resulted distribution was derived from a set of
200 measurements. The same procedure is repeated
within a batch of freestanding fibers in air, just prior
and just after each experiment. The results are fitted
with Gaussian functions as described elsewhere [29]. In
order to derive the net Raman wavenumber shift due
to curing and post-curing process the resulting distributions of fibers in air and of fibers embedded in the composite must be statistically subtracted. The probability
density function of the difference of two sets of measurements described by Gaussian distributions is given by
[30]
ðzlÞ2
2ðs
2 þs2 Þ2
1
2
pffiffiffiffiffiffi ;
P ðzÞ ¼ 2
ð7Þ
2
ðs1 þ s2 Þ 2p
where z = x2 x1 is the difference of the two variables,
l ¼ x2 x1 the difference of the corresponding arithmetic means with x1 being the smaller value and s1, s2 the
standard deviations of the initial distributions. Finally,
the resulting Raman shift distribution is converted into
G. Anagnostopoulos et al. / Acta Materialia 53 (2005) 647–657
strain through the predetermined Raman wavenumber
calibration factor (Fig. 3).
By employing the same mathematical procedure the
far field strain, for the loaded specimens, was assessed
for all levels of the applied strain. The obtained distribution functions are presented in Fig. 11 and the corresponding average values of far field strain and
standard deviation are given in Table 3. As can be seen,
at the onset of the experiment the average fiber strain is
negative due to the presence of compressive residual
stresses in the fiber [31,32]. The gripping of the specimen
alleviates some of these stresses and the subsequent
application of tension leads to the increase of the average fiber strain. However, as is shown in Fig. 11, the fiber strain distributions become significantly broader as
the external strain increases, which indicates the presence of a gradient of stress in the embedded fibers. This
gradient can be attributed to a combination of: (a)
occurrence of premature natural fiber breaks; (b) resulting stress concentrations in neighboring fibers; (c) interface failure and subsequent increase of transfer length. It
is indeed indicative of the stochastic nature of these
composites that at strains as high as 2% the full width
of the distribution as expressed by twice the standard
deviation, can be as high as 1% strain (Table 3). Such
important considerations must be introduced to all analytical and/or design calculations that deal with fibrous
materials.
To sum up, by introducing a fiber discontinuity to
real life polymer composites, a detailed view of the interface integrity is obtained. However, the geometry of the
cut end is dependent on the cutting methodology and,
subsequently, this will strongly affect any stress concentrations generated at the fiber end. In particular, the cut
fibers had often tapered ends and therefore it was diffi-
655
Table 3
Statistically average values of axial fiber strain and corresponding
standard deviations taken from 200 random measurements
Applied strain (%)
Far field strain (%)
2s
0.0 (Ungripped)
0.0 (Gripped)
0.2
0.5
0.8
1.0
1.5
2.0
0.06
0.05
0.28
0.60
0.91
1.24
1.73
2.28
0.32
0.42
0.53
0.53
0.65
0.61
0.66
0.94
cult to obtain the stress transfer profiles at the locality
of the fiber tip. In current work, this problem is eliminated by introducing a fiber discontinuity by controlled
burning of the fiber via an Ar+ laser beam emitting at
514.5 nm [21,33].
5. Conclusions
In this paper a new experimental procedure has been
presented for the in situ assessment of the interfacial
characteristics of aramid/epoxy unidirectional composite coupons. The technique involves the introduction
of a small surgical fiber cut to the upper prepreg layer
prior to vacuum bagging. The obtained resultant data
have proven the validity of this methodology for deriving the maximum interfacial shear strength as well as
the transfer length and the extent of interfacial damage
at all strain levels. The mode of interface failure appeared to be local matrix plasticity and the upper ceiling
of ISS obtained (45 MPa at 1.4%) is comparable with
the shear yield strength of the resin.
Acknowledgements
The authors acknowledge the European Commission
through the Dampblade project (No: ENK6-CT-200000320) funded by the ENERGIE program and the
Adapt project (No: BRPR-CT97-468) funded by the
Brite/EuRam program for supporting the work presented here. Finally special thanks must be given to
Dr. C. Vlattas for conducting the single fiber model
composites experiments presented here.
Appendix A
Fig. 11. The resulting normalized distributions of fiber strain at
different levels of the external applied strain. Each distribution
corresponds to 200 data points.
The presence of errors and uncertainties in a set of
experimental data is an unavoidable fact, which can be
tackled in two ways: (a) by reducing the systematic errors through an improved experimental procedure, and
(b) by filtering the experimental/undesirable noise using
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G. Anagnostopoulos et al. / Acta Materialia 53 (2005) 647–657
first and second derivatives. The x-values of the joints
are called knots, or more precisely, interior knots. The
number of the knots determines the number of regions
of the experimental data, where a cubic polynomial fit
is applied.
The choice of the number and position of the knots is
determined by optimizing the following criteria: (a) The
R2-value, or the coefficient of correlation, which can be
translated into a statement about the expected residuals
(root mean square deviations) of the particular fitting.
Therefore, a fitting is valid when the R2 approaches to
unity. (b) The v2-value has the following formula
P ðObserved valueExpected valueÞ2
X2 ¼
and comprises a miniðExpected valueÞ
mization criterion for the difference between the experimental value and the calculated value. A satisfactory
approximation of the experimental data is chosen when
the v2 approaches zero [30].
In Fig. 12, three different attempts of cubic spline
interpolation to the same set of experimental data are
shown. A different number of interior knots is used in
each case. Keeping the number of interior knots at
low level, smoothness can be achieved, but the fit may
be poor (low closeness, Fig. 12(a)). Experimental noise
is underestimated resulting to insufficient filtering and
inadequate treatment of data. On the other hand, if
the number of knots is too high, the noise is overestimated, and the results suffer from low smoothness.
The resulting fit is too close to the data, tending to have
unwanted fluctuations between the data points (poor filtering, Fig. 12(c)). The best cubic spline attempted here
is presented in Fig. 12(b); the number and position of
knots are appropriate, reflecting the different regions
of the data set and thus their physical meaning. Experimental noise is sufficient filtered, since high smoothness
and closeness are achieved.
References
Fig. 12. Cubic spline interpolation with uniform knot sequence for the
KevlarÒ29/epoxy full composites.
suitable statistical tools. With respect to (b), any statistical interpretation cannot be defined monotonically as
it will depend on the nature of the elements of the data
set, their number and previous experience. In any case,
in order to apply a satisfactory fit to the experimental
data, smoothness and closeness of the chosen approximation are of paramount importance.
In this paper, a cubic spline function was chosen in
order to fit the fiber stress versus the distance along
the fiber (Figs. 4(a), 5(a), 6(a)). This cubic spline function consists of a number of cubic polynomial segments joined smoothly end to end with continuity in
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