3.1. Model for Preselection of Materials
In order to extrapolate viscoelastic properties of rubber compounds and thus determine the correlation between glass transition frequencies and temperatures, the Williams–Landel–Ferry (WLF) equation [
18] can be used:
where
βT is the frequency shift factor,
T0 is the reference temperature and
C1 and
C2 are empirical material and reference temperature-dependent parameters. They describe the influence of the free volume.
C1 is defined as the reciprocal fractional free volume at the glass transition temperature
Tg, while
C2 is the ratio of the free fractional volume at
Tg, and
αV is the thermal expansion coefficient [
19].
The standard procedure to identify glass transition temperatures using a DMA is to carry out a temperature sweep with a frequency of 1 Hz. The glass transition is at the maximum of the
tan δ peak. The parameter
βT is the quotient of the peak frequencies at both temperature
T and the reference temperature
T0:
An example of an extrapolation of a
tan δ curve is given in
Figure 3.
If a material for a specific damping application has to be selected, the glass transition frequency at the application temperature needs to be in the range of the load frequency. Referencing the VPAM guidelines [
20], common projectiles have impact velocities
v0 between 300 and 1000 m/s. Having armor thicknesses
d of 5–16 mm and using
leads to impact frequencies
fimpact ranging from 0.19 × 10
5 to 2.0 × 10
5 Hz. The glass transition frequency
fg has to be in that range at room temperature in order to maximize the damping during ballistic impact. According to Equation (2) and using a peak frequency of 1 Hz at the reference temperature, shift factors range between 0.19 × 10
5 and 2.0 × 10
5. Incorporating these shift factors into Equation (1) and using the glass transition temperature
Tg as
T yields an equation that can be used to preselect materials by looking at their glass transition temperature:
Equation (4) solved with both impact frequencies is plotted in
Figure 4.
The glass transition temperature of elastomers generally ranges between −100 and 0 °C. By inserting the glass transition temperature of an elastomer into Equation (4), it is possible to calculate the material constants
C1 and
C2 that the material needs in order to show maximized damping in the given frequency range. By comparison of these values with those reported in the literature, it is possible to estimate if a material can be used as a damping material as long as the set of all three values (
Tg,
C1 and
C2) is known. To find out if a material is suitable, it is necessary to mark a point in the diagram that corresponds to the three values of the material. If it is inside the marked range that is bordered by both graphs, then it will have its glass transition frequency in the ideal range. Having a glass transition temperature of −42 °C, butyl rubber is a promising material with
C1 = 4.78 and
C2 = 81.1 K at a reference temperature of
T0 = +10 °C [
21]. The literature does not mention the exact rubber grade and formulation composition, thus these values can only be taken as an indication for the suitability of butyl rubber. To further evaluate this, it is necessary to determine the material properties of a self-manufactured rubber formulation.
3.2. Time–Temperature Superposition of Butyl Rubber
In order to set up a model for extrapolating viscoelastic behavior to higher frequencies, a conventional butyl rubber compound was analyzed through dynamic mechanical analysis.
First, a temperature sweep (
Figure 5a) was carried out to obtain the glass transition temperature of the rubber compound. Afterwards, frequency sweeps above
Tg (
Figure 5b and
Figure 6) were conducted to characterize the temperature dependence of the glass transition. In both experiments the dynamic moduli
E′ and
E″ were measured for a given temperature or frequency range to examine their temperature and frequency dependency. The peak of
tan δ, which is the ratio of
E′ and
E″, was used to determine the glass transition temperature and frequency.
According to the time–temperature superposition principle, the shift factors
βT (
Table 3) can be calculated using Equation (2). The temperature of −50 °C was chosen as the reference temperature
T0. As expected, the peak frequency of
tan δ shifts to higher frequencies for higher temperatures.
Using the WLF Equation (1), an equation system consisting of two equations for the temperatures −40 and −30 °C can be established for determining the material and reference temperature‑dependent constants
C1 and
C2:
Equation (1) shows that the temperature at which the glass transition occurs at 1 Hz can be calculated by using the room temperature as the new reference temperature
T′. Keeping in mind that constants
C1 and
C2 also depend on the reference temperature, it is necessary to extrapolate the new constants
C′1 and
C′2 for the room temperature by using
and
which lead to
It is now possible to determine the temperature range that corresponds to the frequencies induced during ballistic impact. The plot of Equation (1) including the new values for
C′1,
C′2 and
T′ is shown in
Figure 7a. With decreasing armor thickness and increasing projectile velocity, the temperature that corresponds to the impact frequency decreases. By comparison with the temperature sweep of butyl rubber (
Figure 7b), it can be concluded that frequencies induced during ballistic impact are located in the area of maximized damping of the material. Even for high projectile velocities (1000 m/s) and very thin armors (5 mm), the temperature of −49 °C corresponds to a
tan δ of 0.7, which still is 84% of the maximum tan δ of the analyzed rubber compound. A
tan δ of around 0.1 can be seen as the minimum value necessary for a damping material [
22]. The experimental values of the manufactured butyl rubber are much higher, which leads to the assumption that it could be a well-suited damping material for ballistic protection.
3.3. Impact Behavior of Rubber-Coated Plates
In
Section 3.2, it is shown that induced frequencies during ballistic impact are located in the area of maximized damping for the butyl rubber compound used, as the material undergoes the glass transition in this frequency range. To provide proof to the claim that this phenomenon positively effects ballistic protection, ballistic experiments were conducted with three different rubber formulations with different glass transition temperatures to cover three cases: an ideal glass transition temperature (in agreement with the results from
Section 3.2), a glass transition temperature that is higher than the ideal one, and one that is lower than the ideal one. If an optimized glass transition temperature and thus the maximized damping in this area has a positive effect on ballistic protection, then the rubber compound with the ideal
Tg should show the best results.
As the influence of fillers and crosslinking system on the glass transition temperature is limited, it has to be varied by choice of different polymers. Butyl rubber (IIR), styrene–butadiene rubber (SBR), and a butadiene rubber (BR)/natural rubber (NR) blend (ratio of 9:1) were used. The elastomers and their respective glass transition temperatures are listed in
Table 4. In attempt to reduce other possible influence factors on the results, e.g., hardness, all three formulations were prepared with different carbon black contents to ensure that all three rubber compounds possess the same hardness (~50 °ShA)
In the first experimental series, the impact behavior of the rubber-coated aluminum plates was observed with a high-speed camera. The images of the impact experiments are shown in
Figure 8.
The impact responses of the materials with different glass transition temperatures vary strongly. BR/NR (
Figure 8a) shows a ductile behavior whereas the rubber bulges away from the projectile. SBR (
Figure 8c) responds in a brittle manner and the rubber splinters away. This behavior is expected since the glass transition temperature of SBR is too high, meaning the impact frequency is higher than the glass transition frequency, and thus the material undergoes the transition while the glass transition temperature of BR/NR is too low, leading to the impact not triggering the glass transition.
For the IIR specimen (
Figure 8b), the material shows a mix of both responses. The rubber splintered away like in the case of SBR, but bulging can also be observed. This can be interpreted as the glass transition being in the correct magnitude. As the mean molecular mass of elastomers and polymers, in general, has a broad distribution, its properties also show a larger variety. Some of the polymer chains already underwent the phase transition from ductile to brittle, others still behave in a ductile manner.
The results of the DMA experiments in
Section 3.2, stating that impact frequencies are in the range of the glass transition of IIR, can be validated by these impact behavior experiments. By using an ultra-high-speed camera, it is clearly possible to differentiate between IIR and other rubbers with non-optimized glass transition temperatures.
3.4. Energy of Absorption of Armor Plates
To provide another proof of the proposed rubber compounds, the same rubber formulations were examined as part of steel–rubber–aluminum sandwich plates. Using flash X-ray images, the residual velocity
vr of the projectile was determined. The kinetic energy
Ekin before and after the impact can be calculated using
with
m being the mass of the projectile (4.6 g) and
v being either the impact velocity
v0 or the residual velocity
vr. The absorption parameter
A describes the percentual amount of absorbed kinetic energy and can be calculated as follows:
The results show that the kinetic energy absorption is highly dependent on the impact velocity
v0. To compare the obtained results, it is necessary to either limit the results used to shots where the impact velocity
v0 is in a narrow range around a specific velocity (e.g., 600 m/s) or to take into account the standardization of the absorption values. The dependency between the absorption parameter
A and the impact velocity
v0 for the different armor configurations can be seen in
Figure 9. As a reference, the same configuration without the rubber interlayer (6 mm steel + 4 mm aluminum) is used.
At every impact velocity
v0, butyl rubber shows better energy absorption than the other configurations. To better compare the actual energy absorption, the values of the absorption parameter
A are standardized to an impact velocity
v0 of 600 m/s according to
The standardized mean values for each configuration are given in
Table 5 and plotted in
Figure 10.
A600/A600,ref is used as a measurement for the improvement over the reference configuration without rubber.
While all rubbers show at least a small improvement over the reference configuration, butyl rubber shows a significant increase of the energy absorption of 8.3%.
Another set of ballistic experiments was carried out to investigate the influence of both hardness and damping of the rubber layer on ballistic protection, in addition to the previously analyzed influence of the glass transition temperature
Tg. Experiments were carried out with the rubber compound showing the most promising results in the previous experiment (IIR). In principle, the same formulation for IIR like in the previous experiments was used, but hardness and damping were varied by changing the carbon black content of the formulation. It has to be noted that it is not possible to vary both properties individually, as the carbon black content (or any other formulation parameter for that matter) will always have an influence on them simultaneously. With an increase in carbon black content the hardness will increase, and the damping (
tan δ) will decrease. The standardized mean values for each configuration are given in
Table 6 and plotted in
Figure 11. The IIR compound with 50 phr carbon black and a hardness of 50 °ShA is the same rubber compound that was used for the experiments shown in
Figure 10.
It is assumed that both high hardness and high damping of the rubber have a positive impact the effectiveness of the armor configuration. As both properties are conversely dependent on the varied carbon black content, it is expected that there is an overlapping of both effects. In that case, the best energy absorption should be observed for intermediate carbon black contents. As can be seen in the results, there seems to be a maximum in energy absorption around the configurations with a carbon black content of 50 and 65 phr. For very low hardnesses, the effectiveness of the armor is not better than the reference, while for a very high hardness but a low damping, the energy absorption gets smaller again.
From these results it can be concluded that while both hardness and damping of the rubber interlayer have an influence on ballistic protection, the influence of hardness is more important and the best results are achieved with medium carbon black contents of around 50–65 phr. It might be possible to further increase the energy absorption by finding the optimum in carbon black content, but given the small overall improvement, it is not expected that this will yield much better results.