3.1. Mechanical Testing Analysis
Views of propellant grains after quasi-static compression are shown in
Figure 2a. The samples of the JA-2 propellant remained intact, while cracks were observed in the case of the 12/7 and SC propellants. The JA-2 samples subjected to the dynamic load also remained intact (see
Figure 2b), whereas multi-cracks were observed in the case of 12/7 propellant. The specimens of the SC propellant were completely crushed, as shown in
Figure 2b and
Figure 3, in which three selected frames of the high-speed camera record are presented.
In turn,
Figure 4 presents examples of strain rate curves for each propellant obtained from the SHPB experiments. The strain rate value corresponding to the first maximum is close to the value at which the yield of propellant materials takes place. The second maximum in the SC propellant case corresponds to total crushing of the propellant grains. Therefore, we decided to characterize the tests by values corresponding to the first maximum of the strain rate. These are given in
Table 5. In the quasi-static test, the strain rate value was of the order of 0.001 s
−1.
Figure 5a–c present the true stress–strain plots obtained from the quasi-static and the SHPB tests.
The single-base 12/7 propellant shows the highest values of the yield stress in both the quasi-static and dynamic experiments. In turn, the JA-2 propellant is the softest of the tested propellants, and it reveals the highest strain rate sensitivity. The SC propellant has intermediate values of the yield strength. Both the 12/7 and SC propellants show a modest strain rate sensitivity. Moreover, the JA-2 propellant is characterized by the almost perfectly plastic behavior after the yield (low strain-hardening effect). In turn, the 12/7 and SC propellants reveal mechanical weakness i.e., decreasing flow stress with increasing strain, similar to the strain-softening behavior observed in some metal alloys [
29]. However, in the case of the present work, the mechanical weakness of the 12/7 and SC propellant specimens can be explained by the cracking. It should be noted here that in the case of the 12/7 propellant, the specimens remain intact despite the cracking. This is reflected in the regular shape of the descending parts of the stress‒strain plots. In turn, the descending parts of the stress‒strain curves for the SC propellant show a great scatter. This reflects the random character of the damage process of the grain specimens.
The results obtained for the JA-2 propellant can be compared with the results obtained in [
15], where the SHPB test results for this propellant are reported.
Figure 6a presents a strain rate dependence of the visco-plastic flow stress for the JA-2 propellant. The values of the flow stress determined in [
15] and in this work are shown. Moreover, the results of the drop weight tests from [
8] are also shown. The results of the present work agree well with the results of the SHPB tests obtained in [
15].
As was shown in
Figure 6a, the strain rate dependence of the flow stress can be approximated by the following relation:
In turn, data for the LOVA-type XM39 propellant published in [
8,
15] are shown in
Figure 6b together with data determined in the present work for the SC propellant. Although both propellants are of the same type, their mechanical properties differ considerably. The SC propellant has much greater strength for low strain rates and is much less strain rate sensitive. However, the two propellants show a brittle behavior, and this is manifested by the negative slopes of the visco-plastic parts of the
σ(ε) curves.
3.2. Analysis of Closed Vessel Tests Results
A comparison of the scale of the load rates in the quasi-static, closed vessel, and SHPB tests is given in
Figure 7 for the JA-2 propellant. The SHPB test with the lowest strain rate was chosen. It follows from
Figure 7 that the load rates for the closed vessel tests are intermediate between the characteristic rates of stress for the SHPB and the quasi-static experiments. This means that based on the results of the dynamic and quasi-static strength tests, we can draw inferences concerning the mechanical behavior of the propellants in closed vessel tests.
Table 6 presents the values of the propellant force
f, the covolume
η, and the exponent in the burning rate law
n. Two sets of parameters values are presented: without correction for heat losses and corrected values. Using these values, the experimental shape functions were calculated. The plots of the shape functions are presented in
Figure 8a–c.
There is a clear correspondence between the results of the investigations of the mechanical properties of the tested propellants and their shape functions. The JA-2 propellant, showing plastic behavior and remaining intact under the loading, has a shape function that deviated very little from the theoretical shape function. The observed deviations can be attributed to effects of the ignition. Propellant grains are not ignited at the same time. Therefore, some of them end up burning earlier than others. This causes the transition to the degressive phase of burning to begin earlier than in the case when all the grains are ignited at the same time, as is assumed in calculations of the theoretical shape function.
In the case of the 12/7 and SC propellants, the relative burning surface exceeds the nominal relative burning surface by 20–30%. This means that during the burning process, an additional surface is created due to cracking of the propellant grains. The fragments created have shapes that cause the burning surface to decrease with the increasing volume of burned propellant. This means that the burning process has a degressive character. In the geometric law, the degressive phase of burning of seven-perforated propellants begins for z = 0.87. In the closed vessel tests, this phase begins at z = 0.7–0.8 for the JA-2 propellant, at z = 0.4–0.6 for the 12/7 propellant, and at z = 0.2–0.3 for the SC propellant.
The physical burning law, Equation (6), assumes that there is no dependence of the burning surface on the pressure value. From this assumption, it follows that no differences between the shape functions determined for different values of the loading density occur. In the case of the JA-2 propellant, there are only slight differences. In the case of the 12/7 and the SC propellants, these differences are pronounced.
Is it justified to use the results of uniaxial compression tests to explain the behavior of propellants in closed vessel tests? Theoretically, the mechanical loading of propellant grains in the closed vessel tests is isostatic, because the grains are loaded by the pressure of the surrounding hot gases. However, there are some deviations from loading of this character. When the propellant bed is ignited, the pressure acts on a bed of unignited grains, so the load is not isostatic. The behavior of propellant beds under quasi-static uniaxial loading was investigated in [
30]. Beds of the single-base propellants M1 and M14, double-base propellant JA-2, triple-base propellant M30, and composite propellant M43 were loaded.
Figure 9 presents plots of the dependence between the density of a propellant bed and the stress. The behavior of the beds of propellants is just as can be expected based on the results of uniaxial loading of propellant grains presented in
Section 3.1. The single-base propellants show the highest resistance to the load, while the JA-2 propellant shows very low resistance. The composite propellant M43 shows an intermediate resistance.
When the propellant grains are fully ignited, the value of the pressure inside the perforation may exceed the pressure acting on the outer surface. This is caused by a delay in evacuation of the gases produced from the perforation. The higher value of the pressure inside the perforation accelerates the burning rate. As a result, the propellant burns faster, as in the case when the burning rates on the outer and inner surfaces are the same. Apparently, this is equivalent to an increase of the burning surface. This effect was pointed out in [
1] as a reason for deviations from the geometric burning law. However, this effect is not observed in the case of the JA-2 propellant. We can attribute this to the mechanical properties of this propellant. The higher value of the pressure inside the perforation may cause an increase in the diameter of perforation, which facilitates the evacuation of gases. So, the deformation of grains compensates for the effect of accelerated burning of the inner surfaces. However, despite the deformation, the grains remain intact, and no additional surface is created. For the 12/7 and SC propellants, the brittle behavior was observed. This means that the deformation may be accompanied by cracking.
Two other factors should be taken into account in explaining the results of the closed vessel tests. The magnitude of the stresses inside the propellants grains depends on the deviations from the symmetry of the grains: the larger the deviations, the higher the stresses. The JA-2 propellant grains have a perfect shape, while the shapes of the laboratory-made SC propellant grains show relatively large deviations from the ideal shape. This applies to a lesser extent to the 12/7 propellant. Higher inner stresses in materials prone to brittle behavior may be the reason for the observed increase in the burning surface.
The ignition temperature is the second factor that may have an influence on the behavior of propellants in closed vessel tests. It is the lowest for the JA-2 propellant, higher for the 12/7 propellant, and the highest for the SC propellant. The higher the ignition temperature, the longer the ignition time. This can be illustrated by the initial fragments of pressure records shown in
Figure 10. The moment when the value of 0.5 MPa is attained is chosen as a starting time. The ignition time of the JA-2 propellant is the shortest, while the ignition time of the SC propellant is the longest. If a propellant is difficult to ignite, a situation is possible when only a part of the perforation of the grains is ignited. In this case, the loading scheme of propellant grains deviates considerably from the isostatic one and may cause the cracking of grains. This factor does not act in favor of the SC propellant.
The prolonged ignition process may cause heating of the outer parts of grains. This causes an accelerated burning of these parts. This effect was considered in [
31,
32] as a reason for deviations from the geometric burning law. The initial peaks of the
ϕ values in
Figure 8a–c may be attributed to this effect. However, this effect is important for propellant grains with low web values. The grains of the tested propellants had relatively high web values. In the course of the burning process, the initially heated layers are burned. So, after a transient faster burning, the burning process should obey the geometric law of burning. Instead of this, degressive burning was observed for the 12/7 and SC propellants. This can be explained only by the crushing of propellant grains.