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Residual strain development in an AS4/PPS thermoplastic composite
measured using fibre Bragg grating sensors
Larissa Sorensen, Thomas Gmur, John Botsis*
Laboratoire de Mecanique Appliquee et d’Analyse de Fiabilite, Sciences et Techniques de l’Ingenieur,
Ecole Polytechnique Federale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Received 10 November 2004; revised 8 February 2005; accepted 9 February 2005
Abstract
This paper demonstrates the use of fibre Bragg grating (FBG) sensors for the measurement of residual strain development during the
consolidation of a thermoplastic composite. During the processing of the carbon fibre-reinforced polyphenylene sulphide (AS4/PPS)
laminate, FBG sensors respond to changes in material state, for example the glass-rubber transition and solid–liquid transition. The sensors
also permit the observation of wavelength shifts and spectral form changes induced by the contraction of the composite during cooling. The
experimental data are compared to a generalized plane strain, thermoelastic numerical model with temperature dependent matrix dominated
properties. The model provides solutions for two limiting cases: one where the specimen contracts freely, and one where the specimen is
restricted by perfect contact with the mould. Strain values calculated in each case are inserted into optomechanical equations, which convert
the strain state in the FBG to a corresponding change in wavelength. In this way, the modelled cases are compared to FBG wavelength shifts
during consolidation.
q 2005 Elsevier Ltd. All rights reserved.
Keywords: A. Thermoplastic resin; B. Residual/internal stress; C. Finite element analysis (FEA)
1. Introduction
The processing of thermoplastic composites may cause
significant residual strains due to their anisotropic and non-
homogeneous nature. Mismatches in coefficients of thermal
expansion of the component materials cause residual strains
on a microscopic level, while thermal mismatch between
plies of different orientations produces a similar effect on a
laminar scale. On a global level, strains may vary
throughout a laminate due to tool–part interaction and due
to thermal gradients that will vary the local material
properties. The total residual strain field in a composite
material is the combination of all of these effects.
Measurements of residual strains are often acquired by
examining the externally visible distortions of a part, such
as curvature [1,2]. Now, with the development of fibre
optic sensors, it should be possible to obtain internal,
1359-835X/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compositesa.2005.02.016
* Corresponding author.
E-mail address: john.botsis@epf1.ch (J. Botsis).
non-destructive measurements that indicate macroscopic
residual strain build-up in a composite laminate. It should
also be possible to vary the position of these sensors within
the composite to provide strain readings in given plies
throughout the thickness of a part. Ultimately, these sensors
could provide crucial information about the initial quality of
processed parts, followed by real-time information relating
to their health during service.
Published studies consider the response of optical fibre
sensors such as a fibre Bragg grating (FBG) to the
accumulation of residual strains in composite materials.
Most often, research focuses on monitoring the curing of
thermosetting composites where residual strains are the
result of matrix shrinkage during polymerization and
thermal shrinkage during cooling [3–7]. Results from
these studies generally show that the FBG spectra translate
towards decreased wavelengths indicating compressive
residual strains [3,7]. Some articles, including one study
on thermoplastic-metal laminates [8], present split spectra
for non-unidirectional lay-ups [4–6]. The explanation of the
split peak differs between the authors, since without
polarization control, it is impossible to verify the origin of
the peak split. Kuang et al. reason that the observed double
Composites: Part A 37 (2006) 270–281
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L. Sorensen et al. / Composites: Part A 37 (2006) 270–281 271
peaks are caused by a non-homogeneous strain field in their
cross-ply thermoplastic-aluminium laminates [8]. In con-
trast, Guemes and Menendez attribute their FBG response to
unequal transverse strains that cause birefringence in the
optical fibre core. They interpret their spectral splits using a
plane stress model of the optical fibre, thereby neglecting
the contribution of a three-dimensional stress field [4,6].
Okabe et al. provide a generalized plane-strain, thermo-
elastic model with constant material properties to show how
the strains in a cross-ply laminate after curing can create
unequal transverse strains in the fibre optic sensor, thus
causing birefringence [5]. This solution assumes that all
residual strains are solely the result of free thermal
contraction with constant material properties.
Additional progress must be made to understand three-
dimensional residual strain development and the resulting
spectral responses of embedded optical fibres, particularly
those embedded in thermoplastic composites. Conse-
quently, the first goal of this paper is to highlight the
capacities of FBG sensors for process monitoring and
measuring residual strain development in a thermoplastic
composite. By following the FBG spectral response during
the consolidation and demoulding of a composite, insight
can be gained into the ability of an FBG to follow
material and process changes. The sensor will also react
to the development of residual strains on a macroscopic
level.
Secondly, this paper attempts to better interpret the FBG
sensor output and its relationship to the far-field composite
strains by modelling the consolidation process. A numeri-
cal, thermoelastic model with temperature dependent matrix
properties investigates two limiting cases for the process: a
freely contracting composite, and a composite constrained
by perfect contact with the mould lid. These two cases are
considered due to current difficulties in precisely knowing
the mould–specimen contact behaviour. The cases
described in this work provide solutions of the strains in
the fibre sensor, which can be converted into equivalent
spectral changes and compared to experimental findings.
The model also calculates the residual stress and strain fields
throughout the composite. Finally, results from the
modelled cases are used to highlight the potential errors
that can be made when interpreting FBG spectral responses.
Fig. 1. Wavelength response of a uniform FBG to the application of a
homogeneous strain field.
2. Working principles of FBG sensors
FBG sensors are created by modulating the index of
refraction along a length of the core of an optical fibre.
When illuminated by a broadband source, it will strongly
reflect light at the Bragg wavelength according to:
lB Z 2n0L (1)
where lB is the Bragg wavelength, n0 is the effective index
of refraction and L is the grating period. As a result, sensors
with constant periods and index modulations will reflect
light at a single Bragg wavelength. Changes in the index of
refraction and the period of the grating due to strains and
temperature variations will perturb the Bragg wavelength of
an FBG. For strain and temperature fields that are uniform
along the length of the sensor, the relative shift in Bragg
wavelength is described by Sirkis [9] using the following
equations:
Dlbx
lB
Z 3z Kn2
0
2½p113x Cp12ð3z C3yÞ�CxDT (2a)
Dlby
lB
Z 3z Kn2
0
2½p113y Cp12ð3x C3yÞ�CxDT (2b)
where Dlbx,y are the shifts in Bragg wavelength for both of
the major polarization axes in the fibre, 3x,y,z are the total
principal strain components in the fibre core (3mechanicalCaDT), p11 and p12 are the strain-optic Pockel’s constants, x is
the thermo-optic constant and DT is the change in
temperature.
When the transverse strains are equal in low-birefringent
fibres (3xZ3y) the equations (2a) and (2b) become
equivalent, thus representing a single Bragg wavelength
shift. This corresponds to the case most often considered in
sensor applications and is shown in Fig. 1. The single
equation can be further reduced by assuming that the
transverse strains are related to the longitudinal strains by
the Poisson’s ratio of the glass fibre (3xZ3yZKn3z) and that
there is no temperature change:
Dlb
lB
Z ð1 KpeÞ3z (3)
where pe is the effective photoelastic constant that
incorporates the photoelastic constants pij and effective
refractive index n0. Using this equation, it is possible to
relate the measured wavelength shift to longitudinal strains
in the sensor. Although this approximation works well
in situations where uniaxial strains dominate, there are cases
when Eq. (3) is not realistic [10]. This is the case when the
transverse strains in the fibre core are not related to the
longitudinal strain by Poisson’s ratio. If a sensor is
embedded in a material that undergoes thermal contraction
L. Sorensen et al. / Composites: Part A 37 (2006) 270–281272
or chemical shrinkage, the exact state of strains transferred
to the fibre core must be considered. For an isotropic
material with a low elastic modulus, little error will be
incurred by considering the above simplification. However,
in an orthotropic composite such as a carbon-reinforced
polymer, thermal contractions in the fibre direction are
extremely small (approaching 3zZ0) compared to those in
the transverse direction. This may result in significant errors
if the simplified pe assumption (Eq. (3)) is used to relate
strains to wavelength shifts.
Unequal transverse strains will also create a response
from the FBG that cannot be interpreted using Eq. (3).
These strains deform the sensor so that its cross-section
becomes non-circular (elliptic if strains are symmetric),
and birefringence is induced in the fibre. This causes the
light to follow either the fast or slow axis, making the
sensor response polarization dependent. For each polariz-
ation axis, a different wavelength will be reflected as
predicted by the set of two equations in Eqs. (2a) and
(2b). If polarized light is sent separately along each of the
polarization axis, then two distinct peaks are distinguish-
able (Fig. 2); however, if light is sent along an arbitrary
path, then the resulting wavelength spectrum will be a
mixture of the light reflected along both axes. This dual
peak spectrum could be misinterpreted by attributing its
source to a non-uniform longitudinal strain field.
Consequently, the polarization dependence of a reflected
spectrum should be tested in order to determine the origin
of its multiple peaks.
In general, the overall strain state in the FBG sensor core
should be considered before simplifying Eqs. (2a) and (2b).
For those cases where simplification is not realistic, one is
left with a set of equations that includes three mechanical
strain components and a temperature component. The
temperature component may be removed via calibrated
temperature correlation and various other methods [11].
However, since three strain unknowns remain in the set of
two equations, additional input is required to solve the
residual strain state. This can be accomplished by creating
an appropriate model to describe the strain state in the fibre
core and then comparing it to experimental output from the
FBG sensor.
Fig. 2. Influence of unequal transverse strains on the spectral response of an
FBG.
3. Experimental method
3.1. Characterization and preparation of the fibre Bragg
grating sensors
This study used polyimide-coated, low-birefringent,
single mode FBG sensors with a cladding diameter of
125 mm. All fibres were annealed at 320 8C for 2 h in order
to stabilize their optical response during the high-
temperature processing cycle [12]. Both 3 and 22 mm
gauge length sensors were used during testing, all operating
at a wavelength of approximately 1300 nm. The 3 mm
sensor provided strain measurements after specimen
consolidation. The 22 mm sensors were used to follow
wavelength evolution during processing. These long gauge
length sensors also had the benefit of narrow bandwidths:
less than 0.03 nm for the full width, half maximum of their
spectra. Such narrow bandwidths provided a clear distinc-
tion of any splits in the spectral form that may have been
caused by load-induced birefringence both during and after
the consolidation process.
Before embedding, the region around the FBG grating
was stripped of its polyimide coating using hot sulphuric
acid. The sensors were rinsed with alcohol and then treated
with Silquest RC-2 silane that generates a strong interface
between glass fibres and polyphenylene sulphide (PPS)
polymer [13]. The fibres were dried for a minimum of 1 h at
120 8C after the silane treatment.
Reference spectra were measured for all FBG sensors
before embedding and their equivalent photoelastic constant
pe was measured by hanging calibrated weights on the ends
of the optical fibres. The measured peZ0.29 corresponded
well to that calculated using p11 and p12 provided by
Springer and van Steenkiste [14]. All other fibre optic
properties can be found in Table 1. Fibre Bragg gratings
were also characterized with respect to their temperature
sensitivity by matching their wavelength shifts to that of
250 mm diameter type K thermocouples. This allowed their
temperature-induced response to be separated from their
mechanical response.
3.2. Specimen preparation
Four specimens were fabricated from Cytec’s AS4/PPS
(carbon fibre—polyphenylene sulphide) Fiberite composite
prepreg. The prepreg was cut into 200 mm by 50 mm strips,
with a consolidated ply thickness of approximately 130 mm.
Table 1
Properties for the optical fibre and FBG
Mechanical properties Optical properties
Ef 70 GPa p11 0.17 [14]
nf 0.16 p12 0.36 [14]
af 0.5!10K6 1/8C n0 1.45
Fig. 3. (a) Composite laminate schematic, showing position of FBG sensor and thermocouple. (b) Micrograph showing the distribution of consolidated
composite around the embedded 125 mm diameter FBG sensor.
L. Sorensen et al. / Composites: Part A 37 (2006) 270–281 273
The strips were cleaned with a damp alcohol rag, air dried
and stacked into a matched-metal mould in a unidirectional
configuration [0]28. Each specimen included an FBG sensor
that was centrally located either between the middle plies, or
between the 3rd and 4th outer plies. Each sensor was
correctly positioned in the x-direction by passing it through
slits centred in either end of the mould and the exiting
portions of the fibre were protected by PTFE tubing to
prevent fracture. Long gauge length FBG specimens also
included thermocouples that were embedded in an adjacent
ply to provide a means for temperature compensation. A
schematic of the specimen configuration and a micrograph
of an embedded FBG are shown in Fig. 3(a) and (b) and a
list of the four specimen configurations is provided in
Table 2.
3.3. Composite mechanical properties
Mechanical properties of the AS4/PPS composite were
obtained using a mixed numerical–experimental identifi-
cation method based on modal analysis [15] and key values
were confirmed with both manufacturers data [16] and
independent tensile tests. Temperature dependent transverse
moduli trends were measured using a Rheometrics Solids
Analyzer (RSAII) which performed dynamic mechanical
thermal analysis (DMTA) in three point bending at 1 Hz.
Table 2
List of specimen characteristics
Specimen FBG gauge
length (mm)
FBG
location
Compensating thermo-
couple location
U1 3 Middle None
U2 22 Middle Adjacent ply
U3 22 Outer Adjacent ply
U4 22 Middle Adjacent ply
Table 3
Room temperature properties for AS4/PPS composite
Longitudinal modulus, E11 128 GPa
Transverse modulus, E22 10 GPa
Shear modulus, G12 5.7 GPa
Longitudinal Poisson’s ratio, n12 0.30
Transverse Poisson’s ration, n23 0.49
Longitudinal coefficient of thermal expansion, a11 1!10K6 1/8C
Transverse coefficient of thermal expansion, a22 28!10K6 1/8C
These transverse moduli were then multiplied by a factor of
1.18 to bring the room temperature modulus in line with
accepted values since the DMTA can provide accurate
trends but not necessarily accurate absolute values [2].
Where required, shear moduli were assumed to follow the
same temperature trend as the transverse modulus.
Coefficients of thermal expansion (CTE) in the
transverse direction were measured up to 270 8C using a
Perkin Elmer thermomechanical analyser (TMA7). This
technique was not sensitive enough to measure the
extremely small CTE in the fibre direction, therefore, the
embedded FBG’s were used to provide this information.
Specimens were placed in a freezer and the temperature
drops read by the embedded thermocouples were compared
to the wavelength shifts. Wavelengths were then related to
the strain state using Eqs. (2a) and (2b). The thermally
induced strains in the fibre were also calculated analytically
by assuming the generalized plane strain case of a
cylindrical inclusion in an infinite host used by Sirkis
[10]. By combining this solution with Eqs. (2a) and (2b) and
assuming room temperature material properties and the
previously measured transverse CTE, it was possible to
extract the value of the longitudinal CTE. This value was
assumed to remain constant for all temperatures due to the
domination of the carbon fibres. Room temperature values
of all material parameters are provided in Table 3, given a
fibre volume fraction of 60%.
3.4. Specimen consolidation and data acquisition
Specimens were consolidated in a matched-metal mould
placed in a Fontijne hot-press under pressure and
temperature control, as shown in Fig. 4. The processing
profile represented by Fig. 5 was chosen to coincide with
those suggested by the prepreg manufacturer [17]. They
Fig. 4. Cross-section of the specimen in the matched-metal mould.
Fig. 5. Processing cycle parameters, including the mould temperature and the platen pressure applied on the mould.
L. Sorensen et al. / Composites: Part A 37 (2006) 270–281274
allowed the thermoplastic to completely melt and then
solidify into the mould form upon cooling.
Each FBG sensor was characterized before and after
consolidation with a tunable laser-based system capable of
polarization control. Additionally, specimens with 22 mm
long gauge lengths were monitored throughout the
consolidation process. During processing, a tunable laser
and photodetector were coupled to the FBG sensor fibre to
provide continuous measurements of the reflected spectra.
The temperatures of the embedded thermocouple and of a
thermocouple inserted into the mould (Fig. 3) were
measured concurrently via a temperature data acquisition
system.
The available optical monitoring system used during
fabrication did not posses sufficiently fast polarization
control, thus spectral measurements were taken with
arbitrary and possibly changing polarization states. The
monitored Bragg wavelength was consequently defined by
either the maximum of a single or multiple-peak spectrum,
or by the maximum of the rightmost peak in a double-peak
spectrum. The requirement for this arbitrary standardization
stemmed from the tendency of the spectra to change form
and centre of gravity due to load-induced birefringence.
When possible, peak separation due to birefringence was
measured during processing in order to complete the
spectral data from these tests.
4. Experimental results
During the heating portion of the consolidation process,
Bragg wavelength shifts and spectral forms provided
qualitative indications of changes in the surrounding
composite prepreg. The magnitude of spectral shifts and
their forms depended on the contact between the optical
fibre and the composite. For example, when the sensor was
placed between the prepreg sheets and pressure applied on
the mould, the measured spectra transformed from a single
peak into multiple peaks due to the non-uniform contact
stress between the rigid, curved prepreg plies and the optical
fibre [18]. Later in the process, around the glass-transition
temperature (Tg) of 90 8C, the polymer matrix softened
allowing for better contact with the FBG. As a result, the
spectra presented a double peak after this transition. They
did not return to a single peak form because of the mould
pressure which was pressing the fibre into an elliptic shape
causing birefringence as described by Eqs. (2a) and (2b)
when 3xs3y. The deformed shape can be assumed due to
geometric and loading symmetry.
Another indication of the passage through Tg was
given by the magnitude of the Bragg wavelength. In
Fig. 6, the temperature corrected wavelength shifts showed
a sharp jump around Tg. A similarly marked step in Bragg
wavelength was observed at the melting temperature (Tm),
of 280 8C. This jump was also followed by a change in
the spectral forms, from double peaks to single peaks.
This transformation was explained by the liquid state of the
polymer matrix that permitted viscous flow of the composite
around the fibre optic sensor creating equal transverse
strains.
Later in the melt state of the process, two step pressure
increases were applied. During each of these pressure steps,
there were corresponding increases in Bragg wavelength
due to the increased pressure in the mould. If one considered
that the FBG and polymer were now in perfect contact, then
it would be possible to model these instantaneous
wavelength shifts using a low transverse modulus for the
composite as described later in this paper.
After solidification, the FBG sensor was considered to have
perfect adhesion with the matrix, and consequently wave-
length shifts and spectral forms then represented strains
transferred from the surrounding composite specimen into the
core of the optical fibre. During cooling, the Bragg wavelength
followed the contraction of the composite plate (Fig. 6).
Fig. 6. Typical changes in the wavelength peak (right peak if split) during processing.
L. Sorensen et al. / Composites: Part A 37 (2006) 270–281 275
Some fluctuations were observed in the wavelength measure-
ments corresponding to the on/off water cooling process. After
viewing video recordings of the specimen and mould during
processing, it was evident that this on/off sequence caused
significant movement and pressure changes that would explain
the periodic wavelength fluctuations. The video recording also
indicated that the matrix retained the capacity for viscous flow
until temperatures around the recrystallization temperature
(TcZ235 8C) published for PPS [19].
Itwasalsonoticed that thespectralpeaksplit returnedinthe
vicinity of this Tc; however, it was difficult to pinpoint the
exact temperature where this occurred due to the lack of
polarization control during processing. It is likely that the
mould–composite interaction during thermal contraction
caused the birefringence leading to a peak split. At the end
of cooling, the mould was released from the press, resulting in
a wavelength step increase but no peak split change. The same
type of wavelength jump was again observed when the
specimen was removed from the mould. These step increases
in wavelength were related to the release of constraints caused
by mould contact; however, since the peak split did not
Fig. 7. Spectral measurements of U3 before and after consolidation. The
two right-hand peaks represent the two major polarization axes.
change, the ratio of transverse strains was considered
constant.
After demoulding, it was possible to measure the complete
spectral response of the embedded FBG sensors with a
polarization controlled laser using 1 pm scan steps. Fig. 7
shows a typical wavelength response before embedding (left)
compared to the spectra obtained after consolidation (right).
All spectra exhibited uniform single-peak forms, indicating
uniform longitudinal strain fields. However, since the two
right-hand spectra were obtained from the same specimen
simply by adjusting the polarization angle of the light by 908,
this indicated that the dual peaks were caused by strain-
induced birefringence in the embedded fibre. By combining
Eqs. (2a) and (2b), transverse strain differences in the core of
the optical fibre were calculated using the wavelength
separation between the two peaks:
3x K3y Z2
n20ðp12 Kp11Þ
� �lbx Klby
lB
(4)
The separations between lbx and lby were measured with
the polarization controlled system, and then inserted into
Eq. (4) toobtaincorresponding transversestraindifferencesas
listed in Table 4. Measurements taken during processing are
provided later in the paper; however, it is interesting to note
that the removal of the composite specimen from the mould
and from the press caused no change in the transverse strain
difference (peak split), even though significant jumps were
observed in the overall wavelength. This led to the conclusion
Table 4
Differences between the wavelengths of the two major polarization axes
caused by transverse strain differences
Specimen Peak split (nm) Strain difference (mm/m)
U1 0.040 154
U2 0.071 273
U3 0.038 146
U4 0.030 116
L. Sorensen et al. / Composites: Part A 37 (2006) 270–281276
that the forms of strain distributions were ‘frozen’ into place
during processing, and then proportionally released upon
removal from the mould. For measurements taken during
processing, the lack of polarization control would not permit
the visualization of peak splits less than approximately 30 pm
due to the bandwidth of the original spectra. The resolution
was also limited to G5 pm due to the laser step spacing.
Although the experimental configuration gave overall
wavelength shift and peak split data, they were insufficient
to completely describe the three dimensional strain state if
longitudinal strains are small compared to transverse strains.
For this reason, a model of the consolidation process must
be used to solve for the strain field in the FBG.
5. Numerical modelling
5.1. Description of the numerical modelling
A generalized plane strain thermoelastic model is used to
calculate the strain field both in the centre of the optical fibre
and in the surrounding composite. This incremental model
follows the cooling portion of the consolidation cycle so that
residual strain development can be calculated starting from
the solidification of the composite laminate. The strain data
are then inserted into Eqs. (2a) and (2b) and the resulting
wavelength shifts compared to experimental data.
Considering the symmetry of the specimen geometry and
the applied loads, only one quarter of the specimen is
analyzed. The specimen is subjected to two sets of boundary
conditions in order to show the effect of the loading pressure
and mould–specimen contact. In the first case, the model is
described as ‘unconstrained’ since it represents a freely
contracting specimen without mould contact. Although, this
situation does not match reality, it is chosen as a reference
state. A second ‘constrained’ boundary condition set is
chosen to provide the limiting solution where the contact
between the specimen and the mould lid is assumed perfect.
These two sets of boundary conditions, subsequently
referred to as the ‘unconstrained’ model and the ‘con-
strained’ model, are shown in Fig. 8.
The numerical analysis of these two cases is performed in
ABAQUS 6.4 using an incremental thermoelastic model
Fig. 8. (a) Unconstrained model boundary conditions, (b) constrained model bound
scale).
with quadratic, generalized plane strain elements
(CPEG8R). The radius of the optical fibre is 0.065 mm
compared to the specimen half thickness of 1.82 mm and in
the constrained case, the mould lid thickness of 15 mm.
Since the fibre optic region is the area of interest, the mesh is
refined to provide appropriately small elements around the
FBG sensor. For the unconstrained model this includes 14,
599 elements and 44,439 nodes. The constrained model uses
14,974 elements and 45,646 nodes.
In developing these numerical models various factors
should be considered. For the AS4/PPS composite system,
with a high melting point of 280 8C, the laminate thermal
contraction is considered to have the most influence on
residual strain development. Mould contact and pressure are
also considered significant due to the development of
birefringence in the experimental results, which implies
unequal transverse strains. Global strains caused by thermal
gradients are neglected due to the small temperature
differences recorded between the inner, outer and mould
thermocouples during cooling. Crystallization development
is also omitted from the model due to the difficulty of
performing tests to accurately quantify its evolution and due
to an uncertainty in its importance as discussed by Sonmez
and Eyol [20]. The recrystallization temperature, is
however, considered an important reference point, since
above Tc the matrix is so soft that no significant residual
strains are expected to accumulate.
The simulations of this consolidation process use
constant elastic, isotropic material properties in the fibre
optic (Table 1) and in the steel mould. The modulus of steel
is 200 GPa, its Poisson’s ratio 0.3 and its coefficient of
thermal expansion 12 mm/8C. In the composite region the
material is elastic, transversely isotropic, and the fibre
dominated properties (E11, a11) and Poisson’s ratios are
considered constant with temperature (Table 3). In order to
better model the stress and strain development due to
consolidation, the temperature dependence of the matrix
dominated composite properties (E22, G12, a22) is taken into
account as shown in Fig. 9. By applying temperature
dependence, it is assumed that such an incremental elastic
model sufficiently describes the process without the need for
an experimentally and numerically expensive viscoelastic
model [1,2,21]. The use of temperature dependent moduli
ary conditions where the mould is perfectly attached to the composite (not to
Fig. 9. Temperature dependent transverse modulus E22 and CTE a22, with piecewise constant steps shown only on the modulus curve. (G12 follows the same
trend as E22.)
L. Sorensen et al. / Composites: Part A 37 (2006) 270–281 277
that are very low, close to Tm, also means that no significant
residual stresses will accumulate before Tc.
In the numerical model, transverse modulus, shear
modulus and transverse CTE are considered piecewise
constant by step, corresponding to thirteen temperature
increments calculated in both models. As shown in Fig. 9,
temperature increments have different magnitudes depend-
ing on the rate of change of the transverse modulus in a
given temperature range. In temperature ranges where the
modulus varies quickly, the steps are smaller, whereas, the
entire process below Tg is considered in one step.
Calculations are performed at each temperature incre-
ment using the corresponding material properties to solve
for the incremental stress–strain state in the specimen. At
any point in the cooling process, the total stresses and strains
are the sums of the temperature-induced stresses and strains.
In the constrained case, the moulding pressure is also
applied at each temperature step; thus the total accumulated
strains include the current pressure-induced strains in
addition to the sum of the temperature-induced strains.
The strains calculated at each increment of the
unconstrained model are verified using an analytical
generalized plane strain model for a fibre in an infinite
orthotropic matrix subjected to thermal contraction [9].
Both the numerical and analytical calculations provide
identical results for strains at the fibre core.
6. Results of numerical analysis
6.1. Optical fibre core
In Fig. 10, results from the two numerical models are
compared with experimental measurements taken during the
cooling portion of the consolidation process. This graph
shows the experimental evolutions of the rightmost peak
shift compared to the modelled Bragg wavelengths for the x-
axis of polarization. The experimental results are set to a
reference point corresponding to zero strain at 235 8C
following video observations of matrix behaviour (Section
4). It is assumed that the viscous composite will not retain
any significant residual strain above Tc. At solidification,
(TmZ280 8C) the results from the constrained and uncon-
strained models differ due to the addition of the moulding
pressure as a boundary condition in the constrained model.
While the specimens cool in the mould, the peak shifts
closely follow the constrained model results. When they are
released from the mould, the experimental wavelengths
jump towards the results predicted by the unconstrained
model. This implies that the specimen–tool interaction
contributes significantly to strain development during
cooling.
This specimen–tool interaction can be further investi-
gated by looking at the birefringence induced in the
experiments compared to that predicted by the models. In
the fibre core, transverse strains in the unconstrained model
develop to be equally compressive, thus this model does not
explain the peak splits observed during and after consolida-
tion (Fig. 11). In contrast, due to tool–part contact in the
constrained model, the transverse strains diverge signifi-
cantly during cooling. The small difference in strains upon
application of moulding pressure does not cause any
measurable peak split, supporting the single peak observed
in FBG measurements.
In Fig. 12, the separations of the transverse strains (3xK3y) are compared to the experimental strain differences
measured via Eq. (4). While the unconstrained model does
not predict any birefringence, the perfect mould–specimen
Fig. 10. Evolution of wavelength shift (rightmost peak when required) during cooling: experimental and modelled.
L. Sorensen et al. / Composites: Part A 37 (2006) 270–281278
contact assumed in the constrained model clearly introduces
excessive birefringence. It does, however, provide a
plausible explanation for the origin of the experimentally
observed peak splits. A solution considering realistic
frictional contact between the mould and the composite
would be required to provide the intermediate solution that
better follows the experimental results.
6.2. Strains in the composite
Next, one must consider that the strains measured by an
FBG represent the strains in the fibre core and not
necessarily those in the surrounding composite material.
The numerical analysis used in this section provides insight
into the stress and strain distribution throughout the
composite specimen, by considering the FBG to be
embedded in a homogeneous laminate. Fig. 13 shows that
all residual stresses remain low or zero in the far-field
composite of both models, except for the transverse stresses
(x-direction, constrained model) which surpass half of
Fig. 11. Evolution of strains in the optical fibre core during cooling as
calculated by the unconstrained and constrained models.
the matrix fracture strength [22]. The origin of these high
stresses (sx) is the thermal expansion mismatch between the
steel mould and the 908 direction of the composite. During
cooling, the steel mould restricts the contraction of the
composite, thus causing tensile stresses in the x-direction.
These tensile stresses would significantly reduce the
resistance of the composite to transverse matrix cracking.
Examining the plot of strain distribution in Fig. 14 one
observes large compressive through-thickness strains both
around the optical fibre and in the far-field composite. At the
fibre interface these compressive strains reach 3%, which in
tension represents the failure strain reported for the PPS
matrix. Interestingly, the optical fibre disturbance of the
strain field becomes negligible at a distance of approxi-
mately three fibre radii from the FBG. Although, the cases
investigated herein represent two extreme situations of the
consolidation process (unconstrained and constrained), the
actual residual strain distribution should fall between
the results presented in Fig. 14. Consequently, the strains
Fig. 12. Evolution of transverse strain difference in the fibre core during
cooling: modelled and experimental.
Fig. 14. Strains along the y-axis after cooling as calculated by the unconstrained and constrained models.
Fig. 13. Stresses along the y-axis after cooling as calculated by the unconstrained and constrained models.
L. Sorensen et al. / Composites: Part A 37 (2006) 270–281 279
in the far-field composite can be estimated to be
approximately K2%.
Fig. 15. Evolution of relative strain error derived from wavelengths
calculated using the unconstrained model results, and then worked
backwards into strains using the pe assumption.
6.3. Potential interpretation error due to pe assumption
Although, it may seem obvious that one should avoid
simplifying the optomechical equations when considering a
birefringent signal, single-peak responses may sometimes
be misinterpreted. When the transverse strains are large
relative to the longitudinal strain, and thus do not follow the
Poisson’s ratio relationship (3xs3ysKn3z), Eq. (3) will
produce significant error in the measured 3z. In the case of
the freely contracting composite modelled in this investi-
gation, the pe simplification causes 180% relative error in
the measured longitudinal strains and about 110% error in
the transverse strains (Fig. 15).
L. Sorensen et al. / Composites: Part A 37 (2006) 270–281280
By studying the residual strain cases modelled in this
paper, it is clear that one must carefully consider the type of
strain field in the core of an FBG sensor before interpreting
its signal. In cases where uniaxial loads are applied to a
specimen, Eq. (3) should provide adequate correlations.
When materials like carbon fibre-reinforced polymers
undergo thermal shrinkage, the fibre-direction strains are
close to zero due to the negative coefficient of thermal
expansion of the carbon fibres. This creates a strain state that
may be difficult to interpret without additional information
or modelling.
7. Conclusions
The FBG sensors used in this study demonstrate their
ability to follow pressure and material changes in a
thermoplastic composite during the consolidation process.
After consolidation, the birefringence of the FBG sensors is
verified using a polarization controlled system, and the dual
peak spectra are attributed to unequal transverse residual
strains. Since the interrogation of a single polarization axis
produces only one peak, the residual stress and strain field
must be constant along the length of the sensor.
The combination of FBG measurement with numerical
modelling can serve as a good tool for measuring residual
strain accumulation. Further comparison of the FBG
response and the modelling indicates that the specimen–tool
interaction plays a significant role in the development of
residual strains in the unidirectional composite. In the case
of the constrained model, the calculated transverse stress
(sx) is about 50% of the matrix fracture strength.
Modelling the consolidation process is a challenging
problem because of the need to realistically describe the
material behaviour, to accurately measure material proper-
ties, and to define appropriate boundary conditions. This
investigation uses a model with temperature dependent
material properties and two types of boundary conditions
that provide limiting solutions to the consolidation process.
In order to improve the accuracy of the calculated residual
strain fields, it is very important to identify appropriate
contact conditions between the mould and the specimen;
however, such an experimental task is very difficult at
present. Thus, in conjunction with accurate and precise
experimental data from FBG, various specimen–mould
contact conditions should be simulated to find a narrow
range of realistic solutions for particular material/processing
conditions.
Acknowledgements
The authors would like to acknowledge the financial
support of the Swiss National Science Foundation, Grant no.
2000-068279. They also thank their partners in this grant,
the Advanced Photonics Laboratory, BIOE, EPFL, for their
continued support on optics issues. Finally, they wish to
thank Cytec Industries Inc. for the composite material used
in this investigation and the Laboratoine de technologie des
composites et polymers, IMX, EPFL for the use of their hot
press.
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