HVI experiments of MRE fragments impacting armour steel plates were performed using a 7.62 mm calibre powder gun. The set-up allows the fragments to reach velocities up to 1600 m/s. A double infrared (IR) light barrier LS 260 was employed as impact velocity measurement system for the fragments before the impact. The IR were 0.5 m apart and can capture velocities up to 2000 m/s. A Shimadzu HPV-1 high-speed camera, triggered by the IR system, records the shots. The target is installed inside a closed ballistic chamber, equipped with a window allowing high-speed recording.
Among the parameters affecting the fragment/plate interaction, the impact pitch angle (α), defined as the angle formed between the longitudinal axis of the fragment and the horizontal flight direction, plays a crucial role, as discussed by Zukas [
9] and Rosenberg et al. [
10]. The experimental set-up, shown schematically in
Figure 2, enables measuring α but does not allow the measurement of the yaw angle, i.e., the angle between the longitudinal axis of the fragment and the vertical reference plane, nor any rolling rotation.
Five shots were performed without lighting to focus on the IIER. This set-up does not allow evaluating the angle of impact α but is necessary to focus on reaction initiation and evolution.
2.1. Materials
Mixed Rare Earths (MRE) are commercially available pyrophoric mixtures. The compound used in the experiments is composed mainly of cerium (Ce, 49%) and lanthanum (La, 23%). A detailed list of the components of commercially available MRE from different suppliers is available [
11].
Table 1 lists the mechanical properties of MRE and armour steel used in this paper.
A total of 94 cylindrical fragments were used to perform the analyses. The fragments can be divided into three categories: MRE samples with a diameter of 5 mm, MRE samples with a diameter of 3.5 mm, and steel fragments with a diameter of 4.6 mm. All fragments had a length (L) over diameter (d) ratio (L/d) of one. The masses of the MRE fragments are 0.6 and 0.2 g, respectively, while the steel fragments weigh 0.6 g. The fragments were encapsulated in a plastic sabot, which in turn was fixed on a 7.62 mm cartridge. The shooting velocities were controlled by adjusting the amount of gun powder used to fill the cartridge. During the flight, the fragment separates from the sabot due to the different kinetic energies.
Similarly to what described by Waite et al. [
6] and Hillstrom [
7], steel fragments were used in the study. Different solutions were evaluated to achieve high comparability between shots involving fragments made from different materials. In the first approach, the geometrical features and masses were kept constant. In this manner, the experiments had the same initial kinetic energy and the same impact surface. In order to achieve this, considering the different densities of the materials, a hole was drilled on the back end of steel cylinders with diameters of 5 mm and L/d ratio of one to reduce the mass to the desired value. However, this solution was discarded as it influenced the deformation and failure of the fragments significantly. Consequently, an agreement was made to keep the mass of the fragments constant, keeping the
L/d ratio fixed to one. Therefore, the diameter of the steel fragments was decreased to 4.6 mm steel fragments obtaining a mass of 0.6. Armour steel plates with three different thicknesses (2 mm, 2.5 mm and 3 mm) were used in the experiments.
Figure 4 shows a 10 mm × 10 mm × 2 mm plate and the different fragments employed.
2.2. Influence of Impact Pitch Angle (α) on HVIs
The impact pitch angle (α) plays a crucial role on impacts, as discussed by Zukas [
9] and Rosenberg et al. [
10]. However, Zukas [
9] highlighted that the influence of α is inhibited when materials with significantly different hardness values interact, as in the case of rigid penetrators impacting a relatively soft target. Similarly, the hardness of the plates used in the experiments described is up to three times (270 ÷ 380 HB) the hardness of the MRE fragments (120 HB). Therefore, the influence of α on the experimental outcomes needed to be quantified. An analysis of the
of the plugs recorded for 5 mm MRE fragments impacting 2 mm plates was performed to quantify the threshold value of α.
The Recht-Ipson (RI) [
12] formula, shown in Equation (1), was used to fit the experimental data
where
a and
b are fitting parameters.
The
CIV indicates the perforation capabilities of a fragment.
CIV was defined as the statistically determined minimum velocity necessary to perforate a particular target with a specific fragment, with no residual velocity detected. The definition of perforation is central in this context: for our purpose, a complete perforation is considered when a plug is entirely detached from the plate, and IIER is clearly visible in the form of a reaction spreading behind the target. The
CIV is evaluated summing the four highest impact velocities for shots that result in no complete perforation to the four lowest impact velocities resulting in complete perforation of the target, dividing the result by the total number of shots considered, as expressed in Equation (2)
where
represents the
shot resulting in perforation and
represents the
shot resulting in no perforation of the plate;
represents the total number of shots used to evaluate the
CIV, which in this work is 8.
The Least Square Method was used to perform the curve fitting. It was observed that, for values of α lower than 30° degrees, the effects on residual velocities do not influence the RI curve fit, as visible in
Figure 5. It was also observed that higher values of α resulted in lower
, indicating an grater amount of energy dissipated by the target.
In the graph, the blue markers indicate the residual velocities measured for shots impacting the target with 0 degrees, the black markers indicates the residual velocities for impacts at α up to 30° degrees, and the red markers indicate impacts at values of α higher than 30° degrees. A significant effect in the outcomes is evident on the red markers.
The RI curves depicted in
Figure 5 were defined by fitting the different dataset represented in the graph. In particular, the blue curve was obtained by fitting the residual velocities from normal impacts, the black curve from all the shots up to 30° degrees, and the red curve by considering the entire data set, including shots having impact angles higher than 30° degrees. The curves are sufficiently similar. However, the residual velocities indicated by the red markers are evidently lower than the value predicted by the RI curves.
Therefore, it is reasonable to use α higher than 30° degrees as exclusion criterion in the analytical analysis of the problem. Bratton et al. [
13] described a similar threshold value for IRM impacting 4130 steel targets
2.3. Analysis of Critical Impact Energy
The
critical impact velocity (CIV),
critical kinetic energy (
) and residual velocity quantify the ballistic properties of HVIs of the fragments assessed. The
represents the minimum kinetic energy necessary for the fragment to perforate a specific target. The
is obtained by imposing the
CIV value as initial velocity, as described by Equation (3).
The analysis of the experiments described in this paper starts from the energy balance. Equation (4) expresses the energy balance formulated by Grady et al. [
14], which described the HVI of cylindrical brittle fragments on steel targets.
where
is the initial mass of the fragment,
and
are, respectively, the residual mass of the fragment and the mass of the plug ejected from the plate;
indicates the initial velocity and
was defined by Grady et al. [
14] as “excess energy”, which was expressed as the following sum
where the value
is the residual kinetic energy associated with the expansion of the shattered fragment;
Wp is the energy dissipated by the plate in the perforation process, and
Ef is the energy absorbed by the fragment for shattering. The balance can be simplified by imposing the
CIV as the initial velocity in Equation (4): as result, the residual kinetic energy becomes null. Therefore, by combining Equations (3)–(5), the balance can be rewritten as follows
Further simplification of the balance can be made following the observations of Grady et al. [
14], which remarked that the term
Ef is negligible for brittle materials, as they experience fracture without any significant plastic deformation and, therefore, without or with minor energy dissipation. Consequently,
EKin Crit can be approximated to
Wp, as reported in Equation (7).
The quantitative evaluation of the term
Wp has been the subject of intensive study. Empirical equations for the evaluation of the term
Wp were defined in the forties by Bethe [
15] and later by Taylor [
16] and have been continiously improved over the years as summarised by Rosenberg et al. [
10]. In this work an empirical formula, shown in Equation (8), is used for the evaluation of the term
Wp
where
d is the diameter of the fragment,
h is the thickness of the plate, and
is the ultimate stress of the plate, multiplied by the constant
k, dependent on the strain rate. Historically, the term
represented a multiplier of the strength of the plate obtained experimentally, dependent on the mechanical and geometrical features of fragments and plates involved. Recent studies such as the investigations discussed by Meyer et al. [
17] and Stepanov [
18] provide a physical explanation to the need of a multiplier for the ultimate strength
during a dynamic impact. In particular, the study performed by Meyer et al. [
17] discusses the increase of yield strength observed in metals during very high strain rate loadings (
). Stepanov [
18] also investigated the influence of strain rate of impact and explosive loading conditions on the mechanical properties of high strain steels, observing that for
the strength values exceed the static value several times. The strain rates characterising the HVI discussed in this work are estimated to be in the ranges of 4 ÷ 5 * 10
5 s
−1, and, therefore, the term
k in Dquation (8) is associated with the significant increase of yield strength experienced by the armour steel plates during the impacts. The estimated strain rate values were obtained by dividing the impact velocities by the thickness of the plate, as discussed by Cagle et al. [
8]. The values obtained align with indications by Zukas [
9].
By combining Equation (7) with the formula in Equation (8), the balance could be expressed as follows
This last form of the energy balance could be rearranged through algebraic manipulation to make it adimensional, similarly to what was done by Aly et al. [
19]. In the paper, seven different empirical equations valid for predicting
for cylindrical and hemispherical fragments impacting metallic plates were compared. The different equations were expressed in adimensional form to identify the non-dimensional parameters affecting the normalised
. It was observed that the parameter (
h/d), where
h is the plate thickness and
d is the diameter of the fragment, plays a critical role in evaluating the energy necessary for perforation.
Similarly to what was described by Aly et al. [
19], the energy balance in Equation (9) was rewritten in adimensional form as a function of the parameter
where
represents the normalised
. It can be observed that the parameter
represents the following expression
which allows estimating the value of the multiplier
k.
The analytical model in Equation (10) was used to fit the
calculated from the experimental data. The expression shows that
is equal to a two-term quadratic equation. However, according to Grady et al. [
14], the linear term is negligible in first approximation. An assessment of the validity of the approximation, the estimation of the parameters
and
from Equation (10) and
k from Equation (11) are described in
Section 4.