INVESTIGATION OF RESIDUAL STRESSES DEVELOPMENT IN A SINGLE
FIBRE COMPOSITE WITH FBG SENSOR
F. Colpo, D. Karalekas 1 , J. Botsis
Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015, Switzerland
[email protected], [email protected], [email protected]
ABSTRACT
A combined experimental and numerical study was undertaken to improve understanding of the development of residual
strains in a model thermoset polymer matrix composite due to subsequent thermal post-curing treatment. In the present work,
an embedded long fiber Bragg grating (FBG) sensor was used to measure the complete strain distribution built-up in a partially
cured epoxy cylinder which was then post-cured at two different temperature profiles. The optical glass fibre was centrally
located within the epoxy, thus acting as reinforcement and as a sensor providing relatively non-invasive strain measurements.
Such residual strains are due to volume shrinkage of the epoxy resin during curing and post-curing treatment and by the
mismatch between the elastic and thermal properties of the two constituents. The local Bragg wavelength shift due to the
residual strains along the FBG and changes in temperature were obtained using a technique based on optical low coherence
refletometry and inverse scattering. These strains are in good agreement with numerical simulations in which the residual
strains in the matrix were treated as an equivalent thermoelastic problem.
1. Introduction
One of the most critical reliability issues in polymer-based composites is the magnitude of residual stresses generated during
the consolidation process. These process-induced residual stresses and strains lead to significant problems such as shape
distortion, warpage, matrix cracks and delaminations, and can have significant effects on the mechanical properties of a
composite product. They are mainly caused by volumetric change of the epoxy resin, due to thermal expansion and chemical
shrinkage, during curing and post-curing treatment and by the mismatch between the elastic and thermal properties of the
composite constituents. Therefore, the capability to predict and measure the processing residual stresses and strains is of
great importance in improving the performance of composite structures.
During the cure cycle of a polymer composite the matrix changes from a liquid-like uncrosslinked material in the early stages
of cure to a viscoelastic solid at the end of curing. The residual stresses that arise during cure are influenced by this complex
constitutive behaviour. Several studies have investigated and reported on the optimization of cure conditions (cure rate, cure
cycle, pressure and cooling rate) to reduce the process-induced interfacial residual stresses and matrix residual stresses [1-6].
Additionally, numerous works have been performed to predict both analytically and numerically the residual stresses by using
elastic and viscoelastic approaches [7-9]. A number of real time cure monitoring techniques have been used in the past,
including dielectric analysis, ultrasonic scanning and nuclear magnetic resonance [10-12]. Recently, various investigators have
been using fiber optic sensors for curing of composite materials and monitor the induced residual stresses [13]. Some of the
main advantages of these sensors are that they are small in size and lightweight, and can be embedded in composite
materials in a minimally invasive manner, without significant detrimental effects on their mechanical properties. For instance,
extrinsic Fabry Perot interferometric (EFPI) and Bragg fiber optic sensors have been implemented to get local strain fields
during the whole cure stage [14]. Further, fiber Bragg sensors (FBG) have been used for simultaneous monitoring of the
fabrication strain and the cure temperature [15]. A single embedded FBG sensor within an epoxy resin, placed into a
rectangular mold, was implemented to detect on-line the resin gelification onset [16] and the strain build evolution during the
whole non-isothermal cure cycle [17]. Nevertheless, it is important to point out that in these FBG-based investigations the
information regarding the build-up of process induced strains was extracted by recording the change of FBG wavelength (shift
of the Bragg wavelength peak), to obtain the maximum value of the deformation induced to the grating. However, during curing
and post-curing, the strain distribution along the FBG is non-uniform and the recorded FBG spectra either broaden and/or
consist of several peaks. Thus, it is difficult to extract the actual strain distribution along the embedded fiber. Consequently, the
use of FBG is limited to giving some qualitative trends in material processing and residual stress.
In this work, a technique based on Optical Low–coherence Reflectometry (OLCR) and inverse scattering [18, 19] was
implemented to interrogate the FBG in real time measurements of the generated residual strains distribution, along the entire
length of an optical fiber embedded into an epoxy cylinder, at the end of two consecutive post-curing cycles. The experimental
work was accompanied by numerical analysis in which the optical fiber-host system was considered and modeled as a two
1
On sabbatical leave from the University of Piraeus, Greece
phase, single fiber composite to obtain the process-induced residual stresses when the specimen was brought to room
temperature. The results demonstrate that the used fiber optic sensor system can provide the complete strain profiles inside
the model composite during subsequent post curing processes.
2. FBG principles
A Bragg grating is a permanent periodic change of the refractive index in the fibre core, whose fundamental purpose is to
reflect a narrow spectral component of a broadband light source. The reflected light is centred on the Bragg wavelength λ B ,
which is directly related to the grating period Λ, and the mean effective refractive index neff , through the resonance or Bragg
condition [20]
λ B = 2neff Λ
(1)
When the FBG is homogeneous, its parameters are constant along the grating length (z axis) and the Bragg condition (1) is
independent of the position along z. When uniform changes in strain and/or temperature occur, along the FBG, its spectral
response exhibits a simple shift of the Bragg peak without modification of its shape. This is the simplest loading case on an
FBG and has been used extensively as a measure of applied strain in conventional sensing applications with FBGs.
In most cases, the loading conditions result in non-homogeneous strains along the fibre. In such cases, the grating parameters
are position dependent. The coupled-mode formalism then provides a direct mathematical tool to describe the interaction of
light propagating in the grating. This analysis uses a complex coupling coefficient q ( z ) whose amplitude and phase along the
FBG’s length need to be determined [18]. The Bragg wavelength becomes a function of z and a local Bragg condition
λB ( z ) = 2 neff Λ eff ( z )
replaces (1). Certain experimental configurations (as in the present case) preserve cylindrical symmetry
and the transversal strains
of the fibre and
εz
ε x , ε y (or ε r , εθ ) applied to the fibre are such that ε x = ε y = −ν f ε z ( ν f is the Poisson’s ratio
the applied axial strain) [21]. Thus, assuming constant temperature, the local Bragg wavelength shift
Δλ B ( z ) = λ B ( z ) − λ B 0 ( z ) is then related to ε z ( z ) by [19]
Δλ B ( z )
λ B0 ( z )
The photoelastic coefficient
= (1 − pe ) ε z ( z )
pe is given by pe =
neff 2
2
(2)
( (1 − ν ) p
f
12
)
− ν f p11 which can be experimentally measured and
p11 , p12 are the strain-optic constants for the optical fibre. In a typical experiment, λ B ( z ) is deduced from the OLCR-based
method and the strains along the fiber from (2).
3. Materials, sample preparation and experimental procedure
The epoxy system used in this work was a 70:30:10 per weight mixture of DER330, DER 732 Dow Epoxy resins and of a DEH
26 hardener. Cylindrical epoxy specimens having an outer diameter of 8 mm and length of 40mm were fabricated. The
cylindrical configuration was selected because it is simpler to manufacture and to use in numerical simulations due to
symmetry conditions. Further, it allows the results to be compared with existing analytical models since such a configuration
minimizes free surface effects and can lead to a sufficient level of residual stresses on the fibre, independent of the zcoordinate in a large portion of the fibre. Each specimen contained a standard embedded fiber of 0.125 mm diameter, centrally
located along the cylinders axial direction. The optical fiber-host epoxy system is considered as a two phase composite with a
very small fiber volume ratio. The room temperature elastic properties considered for the matrix epoxy are, Young’s modulus
Em = 2.35 GPa and Poisson’s ratio νm = 0.38, as obtained by testing of dog-bone epoxy specimens. The corresponding ones
for the embedded E-glass fibre were, Ef = 70 GPa and νf = 0.19. The optical fiber used in this thermal study was equipped with
a FBG of 24mm in length having a wavelength of 1300 nm. The FBG grating was positioned in such a way that 17mm of its
grating length was located within the cylindrical specimen while the rest 7mm outside. As a result, the embedded part of the
Bragg grating would respond to both temperature changes and curing induced strain built-up while the part which is outside
the cylindrical specimen would be sensitive only to thermal effects due to temperature increase during post-curing.
After pouring the mixture into the mould, the system was left to cure at room temperature for 24 hours. When the specimen
fabrication was completed, a precise measurement of both the length and the position of the grating carried out using the
OLCR apparatus. Then, the cylindrical specimen with the embedded FBG sensor was removed from the mould and placed in
an air conventional oven where it was thermally treated following a pre-selected temperature cycle. Each followed post-curing
process was consisted of three stages. In the first stage, the ramp-up, the temperature was increased to the desired one within
two hours. The second stage which was the cure post-cured phase, where the specimens was left for 9 hours. After postcuring the composite specimen was cooled down to room temperature after opening the oven door at the end of the postcuring plateau. Two separate, but consecutive, post-curing cycles were applied to the same specimen. In the first one the
post-curing temperature remained the same, namely at 70 0C, while in the second one it was raised to 110 0C. During the
post-curing process a J-type thermocouple was placed in the oven to record the applied temperature cycle using a data
acquisition system. Figure 1 shows the applied temperature profiles as measured by the thermocouple inside the oven during
0
0
the 70 C and 110 C thermal cycles. The small recorded temperature drifts at the curing plateau are due to the oven operation
to maintain the set cure temperature. Symbols 1, 2, 3A, 3B, 3C, 3D and 4 designate the points were Bragg response was
measured.
120
100
Temperature (°C)
3A
3B
3D
3C
80
2
60
70 °C Thermal Cycle
110 °C Thermal Cycle
40
4
1
20
After demolding
End of thermal
0
0
5
10
15
20
25
Time (hours)
Figure 1. Post-curing temperature profiles at 70 0C and 110 oC, respectively. Insert indicates the specimen geometry (not on
scale) the FBG and the coordinate system.
4. Numerical modeling
The specimen is considered as a cylindrical fibre-reinforced composite with two concentric material sub-domains described by
the cylindrical coordinate system (r, θ, z) (Figure 1). The fibre is considered as a central cylinder of radius rf = 0.0625 mm and
the surrounding matrix domain corresponds to the annulus of inner rf and outer radii rm = 4 mm. The glass and epoxy are
assumed to be linear elastic and isotropic materials with perfect interface conditions between them. To model the matrix
shrinkage effect, the problem is considered analogous to a thermo-elastic one. We also consider the elastic properties of the
materials to be independent of temperature and not to change as the degree of cure advances during post-curing. As a
consequence, the only residual stresses are those associated with curing shrinkage of the epoxy when the system returns to
room temperature. To simulate the residual strains at room temperature, a matrix shrinkage function Sm (r, z) is introduced in
the general strain-stress relations as described in detail in refs. [19, 22]. The form of the Sm function is obtained from the
strain measurements along the FBG. Due to the cylindrical symmetry of the specimen, an axisymmetric FE model is used to
TM
determine the residual stress state in the specimen. Numerical simulations are performed with the commercial ABAQUS
code by meshing only one half of the rz-plane. Along the longitudinal and transverse directions, the matrix domain is
discretized into 300×300 elements and for the fibre, 300×30 elements are used. The mesh is constructed with 8-node biquadratic axisymmetric quadrilateral elements and is refined towards the ends and at the fibre–matrix interface to
accommodate strong variations of the field quantities.
5. Results and discussion
From the obtained measurements, the cylindrical specimen experienced substantial non-uniform compressive strains due to
epoxy’s consolidation during curing and further thermal post-curing treatment. It has been seen that a maximum strain of 2000
micro-strains is recorded close to the central region of the specimen right after curing at room temperature [19]. However, the
magnitude of the obtained strains increases considerably when the specimen is thermally post-cured at higher temperatures,
0
at 70 and 110 C respectively.
A 3-D representation of the FBG spectra recorded by the same FBG during the first thermal cycle is shown in Figure 2, leading
to a more evident picture of the complete spectra evolution with temperature change at different stages of the heating
up/hold/cooling down cycle. It is seen that the obtained peaks, that relate to different reflected spectra recorded from the freeend of FBG and correspond to the same temperature of interest (e.g. room temperature, cure temperature), have the same
wavelength peak value (nm). However, such spectra are of limited value since they do not give any details on the local
wavelength evolution and correspondingly on the strains on the embedded fibre. In Figure 3 the corresponding local Bragg
wavelength evolutions along the embedded fiber are plotted. The ones corresponding to the applied post-curing plateau
(namely 3A, 3B, 3C and 3D) reflect the material changes due to further cure reactions that take place and the thermal
expansion mismatch between the fibre sensor and the host-matrix. In addition, they account for temperature gradients inside
the epoxy specimen, especially at the beginning of the heating plateau (point 3A), where the epoxy material around the fibre is
exposed to a lower temperature compared to the ones recorded by the oven thermocouple or experienced by the outer
surfaces of the cylindrical specimen. These thermal differences are expected to be more pronounced during cooling stage
since the outside surfaces of the cylindrical specimen would contract first. The corresponding plots for the second post-curing
cycle are presented in Figures 4-5. As can been seen from the local Bragg wavelength evolution, a lower applied temperature
profile leads to less residual strains since they are induced by the thermal expansion mismatch between the reinforcing fibre
sensor and the epoxy matrix and the matrix shrinkage during the cool-down period.
o
Figure 2. 3-D representation of the FBG spectra measured during the entire first thermal cycle at 70 C.
o
Figure 3. Local Bragg evolution, reconstructed with the OLCR technique, during the entire first thermal cycle at 70 C.
Figure 4. 3-D representation of the FBG spectra measured during the entire third thermal cycle at 110 oC.
o
Figure 5. Local Bragg evolution, reconstructed with the OLCR technique, during the entire third thermal cycle at 110 C.
In Figure 6 the strain evolutions, experimental and simulated, along the embedded fibre at the end of the thermal cycles at 70
o
and 110 C are presented, as obtained by using Eq. 2 where pe = 0.2148. It is seen that the resulted strain profiles at 70 and
o
110 C have maximum compressive values of 5900 με, and 6600 με in the central region, respectively. The corresponding
simulated strain distributions along the fibre, correctly follows the experimental measurements. It is evident from the shape of
the graphs that the strain profile is of parabolic shape, with the maximum strain value occurring at the center of the specimen.
It is noted that in Figure 6 the measured and calculated residual strains do not reach a zero value at the FBG entry point to the
specimen. This is due to the fact that there are compressive strains sensed by the free end of the FBG which has been
covered by a small quantity of resin material that extends beyond the specimen main body.
0
0
2
4
6
8
10
12
14
-1000
εz (r=0, z) [με]
-2000
-3000
Experimental
Numerical
-4000
-5000
T = 70 C
-6000
T = 110 C
-7000
Embedded part of the FBG [mm]
Figure 6. Comparison between measured and simulated strain distributions along the grating length at the end of the 70 and
o
110 C cycles, respectively.
In Figures 7a and 7b the axial and radial stresses evolution in the plane z=0 and at the end of the thermal cycles at 70 and 110
C are presented. As expected, σz, σr, σθ are significantly compressive at the reinforcing optical fibre while σr, and σθ in the
matrix retain their maximum values at the fibre/matrix interface (note that the shear stress is zero at z=0). Interestingly, the
influence of the embedded fibre on the residual stresses expends up to eight fibre radii.
o
10
100
0
σθ
-100
σz
0
-200
-300
σr
-5
σz
-400
σr & σθ
-500
σz(r,z=0) [MPa]
σr(r, z=0) & σθ(r,z=0) [MPa]
5
-10
-15
-600
0
0,5
1
1,5
2
2,5
3
3,5
4
r [mm]
Figure 7a. Axial and transversal stresses evolution along the plane z=0, at the end of the second thermal treatment cycle at
o
70 C.
10
100
0
σθ
-100
σz
0
-200
σr
-300
-5
σz(r,z=0) [MPa]
σr(r, z=0) & σθ(r,z=0) [MPa]
5
-400
σz
-10
-500
σr & σθ
-15
-600
0
0,5
1
1,5
2
2,5
3
3,5
4
r [mm]
Figure 7b. Axial and transversal stresses evolution along the plane z=0, at the end of the third thermal treatment cycle at
o
110 C.
6. Conclusions
In this work, a single FBG fibre centrally located inside an epoxy cylinder has been used to monitor the complete strain build
up and evolution that occur, along its sensing grating, when a partially cured test sample undergoes successive thermal cycles
of post-curing. The reconstructed local Bragg results, using the OLCR technique, do provide significant information regarding
strain evolution at different time intervals during the heating plateaus and as well as at the end of their corresponding cooling
down stage. An axisymmetric finite element (FE) model, based on an equivalent thermo-elastic approach, was used to
determine the residual stresses in the thermally treated cylindrical specimens. In the performed analysis the specimen was
modelled as a cylindrical fibre-reinforced composite consisted of two concentric cylinders. The numerical results were in good
agreement with the ones obtained experimentally.
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