Model for the Thermal and Cure-Induced Strain Predicitions
The model for predicting the resin shrinkage is based on a finite element framework [
24] and is a simple 1D thermomechanical model. Earlier work has shown that cure-induced strain can be captured with a material point model [
21]. In this study, a similar approach is applied. However, to predict the thermal behaviour, the through-thickness response of the resin is required. This allows for the additional effect of the exothermal release of heat during the curing. The model is illustrated with
Figure 12. Here, the resin bulk is illustrated in an xy-plane, with y as the principal model direction through the thickness. The region presented in
Figure 12 is a narrow cutout of a vast resin bulk. The length and width of the domain are much larger than the thickness of the observed area, and possible thermal effects from possible edges can be ignored. Then, by only considering the thickness, the model stretches from the surface of the resin, called boundary B, to boundary A, in the middle of the resin. Imposing the conditions listed below:
- A
, (symmetry of heat flow and displacement);
- B
, , .
Boundary A is a symmetry condition for both the thermal and mechanical behaviour. As the resin is unconstrained, the model can contract in the y-direction. The heat flow
h, from the surface of B,
, is the heat transfer coefficient enforced by the air movement possible from inside the oven. The temperature applied in this boundary,
, is the oven temperature measured for each case in
Table 2. The thermal response of the simulation is monitored at boundary A, at the location corresponding to that of the thermocouple and the FBG sensor in the experiment. The strain produced in the simulation is evaluated at Boundary B.
For the prediction of the thermal behaviour, the necessary parameters are tabulated in
Table 6. The total enthalpy of the reaction
has been measured with DSC for the specific resin system [
17]. The densities in
Table 6, as reported in
Section 6, have been determined experimentally. The heat capacity
and conductivity
were taken from [
25,
26], respectively. The convection coefficient for the air inside the oven has been taken from Carson et al. [
27]. The cure-induced strain predicted by the model develops following the theory in
Section 2. The primary components are the chemical and thermal strain, adding to the simulated cure-induced strain. In
Figure 13, the simulation of the cure experiment
is plotted, and the strain and temperature and strain from
Figure 3 are included on top of the predicted strain and temperature by the model. In
Appendix A, figures of the remaining cases for comparison based on the cases in
Table 2 are compiled. To assist the description of the model behaviour, the following notation is used:
1. Isothermal— Pre-cure;
1. Ramp—Pre-cure;
2. Isothermal—Pre-cure;
2. Ramp —Pre-cure;
Isothermal—Post-cure;
Cooldown after Post-cure.
Table 6.
Thermal properties for the simulations.
Table 6.
Thermal properties for the simulations.
[] | [] | [] | [] | [] | [] |
---|
1900 [25] | 0.14 [26] | | 1088 | 1145 | 15 [27] |
The predicted temperature in the simulation results from the temperature load
, the thermal boundary condition and the cure kinetic behaviour. This results in the development of the noticeable exotherm during the two parts of the pre-cure,
and
. The predicted temperature by the simulation matches well with the monitored temperature from the thermocouple inside the resin. This is also observed in the remaining eight cases studied, found in
Appendix A.
The strain predicted depends on the thermal behaviour, as the temperature, corresponding degree of cure
X, and glass transition temperature
are computed for every increment in the simulation. Once the load transfer point is reached, the incremental thermal and chemical strains develop. In
Figure 13, the simulated cure-induced strain is predicted well. Both in terms of the shrinkage occurring during
, which is influenced heavily by the thermal and chemical strain occurring simultaneously. Followed by the heat-up
, then the post-cure
and the cooldown
, these also show good correlation between experiment and simulation. At the end of the cure, both the experimental observed cure-induced strain
and the simulated
are shown in
Figure 13, as well as the final value of the simulated
X. A comparison of the final simulated degree of cure with the final predicted one based on the thermocouple temperature monitored shown in
Figure 3 is relevant. The differences are negligible; thus, the simulated cure development is accurate within the experimentally predicted. In terms of the differences observed between the simulated and measured strain, the deviation relative to the experiment was found to be within 2%. Hence, the simulation is overall satisfactory. The deviations and cure-induced strains observed for all the simulations and corresponding experiments are tabulated in
Table 7. The overall deviation was found to be within 2–6% and the average deviation around 3%. With a simulation that matches the observed experimental behaviour well in all cases. The results from both experiments and simulations are available for download [
23].
To better clarify how the thermal strain prediction affects the model behaviour during the curing, the experimental and simulated cure-induced strain is plotted in
Figure 14 as a function of temperature. The figure demonstrates that the model predicts the cooldown during
well, although it underestimates the shrinkage somewhat in magnitude. During the following heat-up
, the expansion observed in the experiment is parallel with the expansion simulated. Therefore, the simulation can capture the expansion and contraction observed experimentally while curing progresses. This is important as the contraction and expansion occurring during
and
both occur well above
. This means that the expansion and contraction should be influenced by curing as per the thermal expansion model applied in
Section 2.4.
The final cooldown
that occurs from
and down to room temperature is unaffected by any significant changes in
X and demonstrates that the model can also capture the cured contraction well from just below
and until far away from
. The simulated strain is plotted as a function of the degree of cure,
X, against the experiment in
Figure 15, where
Figure 15a demonstrates the temperature development of the experiment
, simulation
and the oven temperature
as a function of
X. The temperatures of the experiment and simulation agree. There is a slight variation between the oven temperature and the resin temperatures.
This lag appears due to the heat flow through the thickness of the sample.
Figure 15b demonstrates the experimental and simulation strains as a function of
X. This makes it easy to distinguish the thermal strain from the chemical strain observed in the simulation. As the temperature drops during the pre-cure
, the simulated thermal strain is also observed to drop. The simulated chemical strain also decreases continuously as the degree of cure increases. It should be possible to check whether the chosen volumetric shrinkage model (
4) adapted for the chemical strain matches the experimentally observed behaviour. The simulated strain
is seen to under-predict the shrinkage occurring during slightly
, but follows in parallel with the experimental strain for the duration of
. After that, the curing ends with the cooldown
. Even though there generally is this slight offset between experimental and simulated, the offset does not increase or decrease slightly. Indicating that the proposed shrinkage behaviour follows the experimental behaviour well. The simulations are, therefore, quite capable of determining the effects observed experimentally.
To demonstrate this graphically across the range of cases,
Figure 16 shows an extended version of
Figure 5. The final simulated and measured values of cure-induced strains are plotted together, and the possible differences are shown. The trendline adapted in
Figure 5 is not applied here as the pre-cure has shown a high dependency on cure-induced strains. The fact that a very low achieved
for case
and
results in high cure-induced strain signifies the dominating effect due to pre-cure and, more precisely,
. This is attributed mainly to the thermal expansion model applied. Stressing that even though
influences the level of cure-induced strain observed it is necessary to consider the complex thermal expansion to determine the cure-induced strain accurately.
Adapted in the simulation, this cure-dependent thermal expansion results in a similar low expansion during the heat-up for the cases
and
, as observed in
Figure 17a. Demonstrating that the simulation can capture the complex thermal expansion behaviour observed. The low thermal expansion at a lower degree of cure results in more shrinkage being transferred at the cooldown after post-curing. As simulated for the two cases,
and
agreed well with the measured strain. This limits the ability to reduce the cure-induced strain for
from −20 K to 0 K for this specific resin system. At the other end of the
axis in
Figure 16, the cases
and
as well as
and
show that an increase in cure-induced strain is present. However, the difference in
between
and
is approximately 5% which, even though it is a more considerable difference than the 3.4%
and
, the effect of
is more dominating for higher values of
. This is because the resin has cured significantly more; thus, the cure-dependent thermal expansion has developed much more, making the temperature at load transfer much more critical. This effect is reflected in the simulation and is due primarily to the implemented thermal expansion model. Therefore, the developed simulation can accurately predict the complex shrinkage observed over various experimental cases.