The penetration and ricochet of ogive-nosed rigid projectiles obliquely impacting metallic targets

Highlights

The oblique penetration of ogive-nosed rigid projectiles in metallic targets

Introduction

The penetration process of projectiles impacting metallic targets at non-normal incidence has been the focus of vast research, due to its relevance for armor applications. These include oblique impacts and impacts at an attack angle (yaw and pitch). The review article of Goldsmith (1999) covers much of the published information concerning these issues, as far as experimental data and analytical modeling are concerned. One of the interesting phenomena that take place in oblique impacts is that of the projectile’s ricochet at a certain obliquity. These events are very important for armor designers as they can reduce the lethality of long rod penetrators. As discussed by Segletes (2006), the studies concerning ricochet belong to two distinct groups, namely, those which deal with short rigid projectiles such as the armor-piercing (AP) projectiles, and those dealing with non-rigid long rods which are either bending or eroding during their interaction with the target. The ricochet of long rods has been studied by several workers who followed the different behavior of these rods. For example, Rosenberg et al. (1989) focused on the plastic hinge which is formed around the rod/target interaction area, while Warren and Poormon (2001) followed the ricochet of long steel rods that are bending as they penetrate thick aluminum targets. In contrast, the analytical work of Tate (1979) deals with the ricochet of a rigid long rod gouging the surface of the target without penetrating it.

As far as the ricochet of rigid projectiles is concerned, there are very few works, mainly experimental, which deal with this phenomenon and their results will be discussed in this paper. One should note the need for a comprehensive model for the trajectory changes of these projectiles, the ricochet angles of various projectile/target pairs, and the penetration depths of rigid projectiles at inclined impacts. The present study is an attempt toward such a model through the insights gained by the numerical simulations. Other workers followed the trajectory changes of rigid projectiles at oblique impacts while they perforate metallic plates, such as Zener and Peterson (1943), Ipson et al. (1973), Piekutowski et al. (1996), Roisman et al. (1999), and Chen et al. (2006). The asymmetric force on the projectile’s nose upon oblique impact is enhanced by the impact face of the target, leading to its continuous trajectory change. In contrast, the back surface of the plate tends to realign the projectile in the opposite direction, complicating the interaction even further. In order to avoid these difficulties, the present study is focused on the trajectory changes of rigid projectiles during their deep penetration into thick metallic targets (no back surface effects). The impact obliquity (α₀) of the projectile is defined as the angle between its velocity vector and the target’s impact face. The change in trajectory can result in a ricochet of the projectile, if its initial obliquity is below a certain threshold value (α_ric), when the projectile is advancing along a trajectory parallel to the target’s surface. Using 3D numerical simulations we followed the trajectory changes and the threshold ricochet angles, of CRH3 ogive-nosed rigid projectiles, as they penetrate different metallic targets. In addition, the penetration depths of these projectiles were obtained by these simulations, and they were analyzed accordingly. The main results of this study are: 1. simple numerically-based relations are shown to account for the dependence of the trajectory changes in terms of the corresponding ricochet angles, 2. these threshold angles are shown to depend on the properties of the given projectile/target pair, and 3. simple numerically-based relations are shown to account for the normalized penetration depths of these rigid ogive-nosed projectiles at oblique impacts.

Simulation results

General observations

The 3D Lagrangian explicit non-linear finite element LS-DYNA code was used in this study for the trajectory changes of rigid projectiles impacting obliquely at thick metallic targets. The CRH3 ogive-nosed projectiles, with diameters of 7.1 mm and various aspect ratios, were treated as rigid bodies and for the targets the von-Mises yield criterion was used, in order to simplify the analysis. The mesh size in the target was 0.3 mm, namely, 11 elements on the radius of the projectile in order to assure convergence of the results, as recommended by Rosenberg and Dekel (2012). The erosion parameter, which is the maximal effective plastic strain allowable in a given element, was set at 1.0. This value was shown to be large enough as far as the accuracy of these 3D simulations is concerned. In fact, one set of simulations was compared with the results of another set, with an erosion parameter of 1.5, and an agreement to within 1° was found in the obliquity changes for the two sets. Considering the fact that simulations with higher erosion parameters take more time and they often fail to be completed, it was decided to use an erosion parameter of 1.0 in all simulations presented here. Most of these were performed with aluminum targets having von-Mises strength of 0.43 GPa, since this is the dynamic flow stress of the 6061-T6 alloy, at a plastic strain of about 0.3 and at a strain rate of 10⁴ s⁻¹. As was shown by Rosenberg et al. (2018), these are the average values of the strain and the strain-rate in a metallic plate during a perforation process. Using these values for the average dynamic flow stresses simplifies the analysis without impeding its relevance. Thus, the von-Mises yield criterion was applied to all the targets in this study, with no hardening or rate effects. The AUTODYN code’s library was used for the parameters in the Gruneisen equation of state, of this aluminum alloy, and for its elastic constants (E = 69 GPa and ν = 0.33). No friction was used in these simulations since it was shown by Rosenberg and Vayig (2021) that friction plays a negligible role in the penetration process at ordnance velocities (0.5–1.5 km/s). The aluminum targets were large enough to be considered as semi-infinite, with a diameter of 254 mm and thicknesses which were at least twice the penetration depths of the corresponding projectiles. The simulations were performed for a half-space in order to reduce computing time. More details on the simulation scheme are given in the Appendix, at the end of this paper. As will be shown later, steel targets were also used in several simulations. However, it was found that target density does not play a role in these oblique interactions, as in the case of normal impacts, as shown by Rosenberg and Dekel (2009). Thus, most of the simulations were performed with aluminum targets.

The basic features of the interaction between a rigid ogive-nosed projectile and an inclined target are shown in Figure 1, through a series of snapshots depicting the change in the projectile’s trajectory due to the asymmetric force on its nose. The steel projectile, with an aspect ratio of L/D = 5, impacts the aluminum target at V₀ = 1000 m/s and obliquity of α₀ = 40°. Note that the targets in our simulations were much larger than those shown in the figure.

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Figure 1.

The trajectory change from α₀ = 40° at impact to the final angle of α_f = 23°.

As seen in this figure, by a time of 75 μs the projectile completed its trajectory change from the initial obliquity of α₀ = 40° to the final angle of about α_f = 23°. A more detailed presentation of the simulation results show that the trajectory change starts upon impact and is over by about 50 μs after impact. In order to have a better insight into this interaction one can follow the deceleration history of this projectile, as shown in Figure 2, which also shows its deceleration history at normal impact. One can see a high peak in the projectile’s deceleration, starting at t = 20 μs after impact and lasting up to about t = 45 μs, after which the deceleration history of the oblique impact follows that of the normal impact. By closely following the trajectory changes one finds that at about 20 μs the projectile has penetrated to about half its length and that by 45 μs it is fully embedded in the target. In fact, this is the case for all the impacts simulated here, as far as the timing of the trajectory changes are concerned. Moreover, the simulations show that the timing of the peak in the deceleration history coincides with the wake separation and reattachment processes during penetration. As explained by Kong et al. (2014) and by Wei et al. (2021), the wake separation takes place when the projectile’s shank detaches from the target, and reattachment is referred to the closing of the gap between them. Note also the second (lower) peak, at about t = 65 μs, which indicates that another separation/reattachment event takes place, to a lesser extent, at this time. A filtering scheme was used for these plots in order to reduce the noise on these records.

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Figure 2.

Deceleration histories of the projectiles at α₀ = 40° (black) and at normal impact (red).

Figure 3 presents the simulation results for the change in the obliquity angles, δ = α_f − α0 as a function of their initial obliquities (α₀), for L/D = 5 projectiles impacting the aluminum target (Y = 0.43 GPa) at a velocity of 1000 m/s. A guideline, connecting these results, was added in the figure for presentation purpose. The intersection of this guideline with the dotted straight line (δ = α₀) means that α_f = 0 at this point. This is the threshold ricochet angle (α_ric) where the trajectory of the projectile is parallel to the target’s surface, as defined above. For the set of simulations presented in Figure 3 one finds that α_ric = 32°.

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Figure 3.

Numerical results for trajectory changes and onset of ricochet of the L/D = 5 projectile.

One can see that the trajectory changes are very high for impact angles beyond the threshold ricochet (α_ric), since at these angles the projectile moves toward the impact face of the target, where it experiences a much lower resisting stress. This is seen in Figure 4 which shows this projectile impacting the aluminum target at an obliquity of α₀ = 30° (α₀ < α_ric), at a velocity of 1000 m/s. One can see that the target’s face above the penetration channel is pushed upwards (as indicated by the arrow in the figure) offering less resistance to penetration and, as a result, a complete ricochet.

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Figure 4.

The full ricochet of the L/D= 5 projectile at α₀ = 30°.

The deceleration history of the projectile at this obliquity (α₀ = 30°) is shown in Figure 5. One can see that the first peak in the deceleration history is similar to that shown in Figure 2 for α₀ = 40°, as far as its magnitude and timing are concerned. However, the second peak is much higher than the corresponding one in Figure 2, and there is a third peak which enhances the change in the projectile’s trajectory. Note the decreasing trend in the deceleration history, as the projectile is headed toward the free face of the target.

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Figure 5.

Deceleration history for the ricocheting projectile shown in Figure 4.

In order to determine the role of the target’s properties in these interactions we followed the L/D = 5 steel projectiles, impacting aluminum and steel target having the same strength (Y = 0.43 GPa) using the von-Mises yield condition, as explained above. The elastic constant of the steel, as well as its equation of state parameters, were taken from the same source as for the aluminum target. Figure 6 compares the simulation results for the projectiles impacting these targets at 1000 m/s. Since the two sets of results are very close, one can conclude that the density of the target does not play a significant role in the interaction. The small difference between the two sets of simulations is due to the difference in the penetration resistance (R_t) of aluminum and steel targets having the same strength. As will be shown below, this resisting stress plays a crucial role on the value of α_ric. Thus, the aluminum targets were used for all the simulations described below and their results apply for all target materials.

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Figure 6.

Simulation results for aluminum targets (circles) and steel targets (squares).

The parametric study

The next sets of simulations were performed as a parametric study where one parameter of the projectile, or the target, was changed in each set. This is the best way to follow the dependence of a given process on each of the relevant physical/geometrical properties of the materials involved. The main interest here is in the trajectory changes of the projectile up to its threshold ricochet angle (α_ric), where the projectile is moving along a line parallel to the target’s face. For lower values of α₀ the projectile is heading toward the impact face of the target, as seen in Figure 4, and these impacts will not be discussed henceforth. Thus, the results for α_f (α₀), where α₀ ⩾ α_ric, are presented here, and for these obliquities the following functional dependence of α_f (α₀), applies for all of the simulation results:

a f = 90 [ ( α 0 − α ric ) / ( 90 − α ric ) ] 0 . 75

(1)

which results in the correct values of α_f = 0 for α₀ = α_ric, and α_f = 90° for α₀ = 90°.

Figure 7 shows the results for steel projectiles with different aspect ratios of L/D = 3, 5, 7.5, and 10, impacting the 0.43 GPa aluminum target at 1000 m/s. One can clearly see the significant difference between the results for these projectiles, with the expected result that longer projectiles are less sensitive to the impact angle (lower deflections), due to their higher moments of inertia. One can also see the good agreement between the functional dependence of α_f (α₀), as given by equation (1), and the simulation results. This agreement will be demonstrated in all the other cases presented below.

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Figure 7.

Simulation results for projectiles with different aspect ratios and their agreement with equation (1).

In order to have a better insight into the asymmetric force which acts on a longer projectile (L/D = 10), consider its deceleration history, as shown in Figure 8. The figure compares the deceleration histories for the L/D = 5 and 10 projectiles, impacting the Y = 0.43 GPa aluminum targets at 1000 m/s, at an obliquity of α₀ = 40°. One can see that the two curves are very similar qualitatively and that that the asymptotic (constant) levels are related by a factor of 2.0, since the deceleration of a rigid projectile is inversely proportional to its length, as shown by Rosenberg and Dekel (2009). Even the peaks in these deceleration histories are related by a factor of 1.75. The main difference between these curves is in the timing of the peaks. In both cases the onset of these peaks correspond to the embedment of half the projectile’s length in the targets, and they last for about 20 µs which correspond to the full embedment of these projectiles. As described above, these peaks are related to the process of wake separation and its reattachment.

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Figure 8.

Deceleration histories of the L/D = 10 projectile (black) and the L/D = 5 projectile (red), at α₀ = 40°.

The next set of simulations followed the influence of the target’s strength on the trajectory changes, for L/D = 5 steel projectiles at a velocity of 1000 m/s. Figure 9 shows the results in terms of α_f (α₀) for aluminum targets with Y = 0.43, 1.0, and 1.5 GPa. Note the good agreement between the simulation results and the relation in equation (1).

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Figure 9.

Results for different target strengths: 0.43 GPa (triangles), 1.0 GPa (circles), and 1.5 GPa (squares).

The dependence of the interaction on the projectile’s impact velocity was determined by sets of simulations with V₀ = 750, 1000, and 1500 m/s, for the L/D = 5 projectile impacting the Y = 0.43 GPa target. Figure 10 shows these simulation results and one can see that the higher impact velocities resulted in lower threshold ricochet angles.

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Figure 10.

Simulation results for different impact velocities; 750 m/s (squares), 1000 m/s (open circles), and 1500 m/s (full circles).

Finally, the sensitivity of the interaction to the density of the projectile was checked in a specific set of simulations for a WHA projectile (ρ_p = 17.5 g/cm³), with an aspect ratio of L/D = 5, impacting the aluminum target at 1000 m/s. Figure 11 compares the results for the two projectiles (steel and WHA), and one can see that the higher density projectile is less deviating, as expected.

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Figure 11.

Simulation results for different projectile densities.

At this point our simulation results were compared with two sets of experimental results which were published on this issue. Note that there are very few experimental works from which one can gain relevant information on the trajectory changes of rigid projectiles, impacting thick targets at different obliquities. Senthil et al. (2017) impacted 7.62 mm APM2 projectiles at mild steel plates of various thicknesses, at impact velocities of about 820 m/s. Most of these impacts resulted in perforation of the plates, but for thicker plates (20–25 mm) they found that the onset of ricochet was at α_ric = 39°–40°. Thus, a set of simulations was performed for an L/D = 4.3 ogive-nosed steel projectile, resembling the hard steel cores of their projectiles, impacting steel targets with an effective dynamic strength of Y = 0.8 GPa. This value represents the flow stress of mild steel at high strain rates, as measured by Vural et al. (2003). Figure 12 shows our simulation results, in terms of δ(α₀), for this projectile/target pair and one can see that they resulted in a ricochet angle of 40°, in excellent agreement with the results of Senthil et al. (2017). It is important to note that the values of α_ric for thinner plates (10–12 mm), in that work, were lower by about 10° due to the effect of the back surfaces.

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Figure 12.

Simulation results for the 7.62 APM2 projectiles impacting mild steel targets at 820 m/s (circles), and 6082-T4 targets at 830 m/s (squares).

Another series of tests was performed by Børvik et al. (2011) who shot similar projectiles at 20 mm thick 6082-T4 aluminum plates, at 830 m/s. These experiments resulted in a threshold ricochet of about α_ric = 32°. Our simulation results for this projectile impacting the aluminum target (with Y = 0.4 GPa) are also shown in Figure 12. The predicted threshold ricochet angle is α_ric=35°, which is somewhat higher than the experimental result of Børvik et al. (2011) A probable reason for this difference is that the 20 mm plates in these experiments were not thick enough to be considered as semi-infinite, and their back surface influenced the projectile’s trajectory.

The proposed model

Thus far it has been shown that the trajectory changes of rigid projectiles, obliquely impacting metallic targets, can be accounted for by equation (1) through a threshold ricochet angle (α_ric) which characterizes the projectile/target pair. In order to find a physically-based relation between α_ric and the properties of the projectile and the target one has to consider the work of Zener and Peterson (1943). Their model for the trajectory change during oblique penetration is worth repeating here, due to its insight into the physics of this interaction. They start with the notion that if torque (N) acts for a time (T) on a projectile with a moment of inertia (I_m), the change in obliquity during this time (δ) is given by:

δ = NT 2 / 2 I m

(2)

where δ is given in radians. This equation is derived by integrating (twice) the equation for the angular acceleration of a rotating body, namely, d²δ/dt² = N/I_m. According to Zener and Peterson (1943) “the torque N is proportional to the transverse force on the projectile which, in turn, is proportional to some measure of the target’s resistance to plastic deformation (R).” They also assume that the duration of the force (T) is proportional to 1/V (where V is the impact velocity), and that the moment of inertia is proportional to the projectile’s density (ρ_p). These assumptions lead to the following relation for the trajectory change:

δ = kR / ρ p V 2

(3)

where k is a constant which accounts for the approximations made in this analysis, such as the fact that the value of V is changing during the process.

With these insights one may conclude that the non-dimensional parameter, which combines the relevant entities of the oblique impact, is A = R_t/ρ_pV₀², where ρ_pV₀² is related to the projectile’s impact pressure and R_t is the target’s resistance to penetration. This resisting stress is given by the numerically-derived relation of Rosenberg and Dekel (2009):

R t = Y [ 1 . 1 ln ( E / Y ) − b ]

(4)

where E is the target’s Young modulus and Y is its dynamic flow stress. The constant b in this relation is equal to 1.15 for CRH3 ogive-nosed projectiles, and it has different values for other nose shapes. Since Young’s modulus (E) has different values for aluminum and steel, equation (4) results in different values of R_t for these targets even when they have the same strength (Y). This may be the reason for the small difference which was found above for the trajectory changes in these targets (see Figure 6). One should also note that the same non-dimensional parameter (A) as defined above, was found by Vayig and Rosenberg (2021) to be the normalizing parameter for the yawed impact of rigid rods at metallic targets.

In order to demonstrate the validity of this parameter as a normalizing entity, one can compare the simulation results for three projectile/target pairs having the same value of this parameter (A = 0.243). These pairs are: (1) steel projectiles impacting the Y = 0.43 GPa targets at V₀ = 1000 m/s, (2) steel projectiles impacting the Y = 1.0 GPa targets at V₀ = 1360 m/s, and (3) WHA projectiles (ρ_p = 17.5 g/cm³) impacting the Y = 1.0 GPa targets at 907 m/s. Note that these are aluminum targets and their R_t values, as calculated by equation (4), are: 1.91 and 3.52 GPa for Y = 0.43 and 1.0 GPa, respectively. Figure 13 shows the simulation results for the three pairs and one can see that they fall on a single curve. Note that this figure presents the obliquity changes (δ) for impact obliquities even beyond the threshold ricochet angle (α_ric).

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Figure 13.

Simulation results for the three projectile/target pairs having the same value of A; steel projectile on 0.43 GPa targets at 1000 m/s (squares), steel projectile on 1.0 GPa targets at 1360 m/s (circles), and WHA projectiles on 1.0 GPa targets at 907 m/s (triangles).

It is interesting to note that the non-dimensional parameter (A) also appears in the model of Rosenberg and Dekel (2009), for the normalized penetration (P/L) of a rigid rod impacting metallic target at normal incidence, namely:

P / L = ρ p V 0 2 / 2 R t

(5)

from which one gets P/L = 1/2A, where L is the projectile’s length. This relation means that high values of A correspond to low penetration depth. In particular, for A = 1.0 one gets a normalized penetration of P/L = 1/2 which means that the projectile in this case will not be fully embedded in the target. Consequently, this projectile will not experience a full trajectory change, as discussed above. In order to have a full projectile embedment, the value for this non-dimensional parameter should be less than A = 0.5.

Once the validity of the parameter A = R_t/ρ_pV₀² was established as the normalizing entity for the trajectory changes, one has to relate it to the values of α_ric which were obtained in the simulations. This relation is shown in Figure 14 in terms of α_ric as a function of A, for all the L/D = 5 projectiles impacting the various targets at different velocities. These results can be expressed by the following relation:

α ric = 90 − 47 A − 0 . 15 for L / D = 5 projectiles

(6)

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Figure 14.

The relation between the threshold ricochet angles (α_ric) and the normalizing parameter (A) for the L/D = 5 projectiles.

which is shown in Figure 14 as a curve through the simulation results.

At this point it is interesting to compare our simulation results with test results from Roisman et al. (1999). They shot CRH3 ogive-nosed WHA projectiles (ρ_p = 17.67 g/cm³) with L/D = 5, at stacks of two or three plates of aluminum 6061-T651. Most of their tests resulted in complete penetration, but one can check the results from two shots, at α₀ = 45° and at impact velocities around 580 m/s, for which the projectiles were embedded in the targets. By measuring the orientation of the embedded projectiles in Figures 8 and 11 of Roisman et al. (1999), one finds that the final obliquities in these tests were about α_f = 25°. Using the relevant data for these shots one finds that A = 0.325, and through equation (6) one gets an expected value of α_ric = 34.5° for this projectile/target pair. Inserting this value in equation (1) an expected value of α_f = 26° is obtained for the final obliquity in these tests, which is in excellent agreement with the experimental results of Roisman et al. (1999).

Finally, consider the simulation results for projectiles of different aspect ratios, which were presented in Figure 7. Figure 15 shows the relation between α_ric and L/D for projectiles with L/D = 3, 5, 7.5, and 10, impacting aluminum targets (Y = 0.43 GPa) at 1000 m/s. Clearly, α_ric decreases with increasing L/D as expected by the analysis of Zener and Peterson (1943), since the projectile’s moment of inertia increases with its length. The curve in Figure 15 is drawn to illustrate that the values of α_ric follow an (L/D)^−0.5 dependence. Thus, in order to obtain the value for α_ric for a projectile with a given aspect ratio, L/D = θ, one has to multiply the value of α_ric for L/D = 5 by a factor of (5/θ)^0.5, namely:

α ric ( L / D ) = α ric ( 5 ) ( 5 / θ ) 0 . 5

(7)

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Figure 15.

The dependence of α_ric on the projectile’s aspect ratio (L/D).

where α_ric (5) is given by equation (6). This relation results in α_ric values which are within ±1° of those obtained by the simulations.

The penetration depths at oblique angles

One can also analyze the penetration depths (P) of these rigid ogive-nosed projectiles, as a function of their obliquity angle. In the following discussions the focus is on their normalized penetration depths (P/P₀) where P₀ is the penetration depth at normal impact. Our first step was to check whether the normalized penetration depths (P/P₀) are related to the parameter A, which was defined above. Figure 16 shows these normalized penetrations as a function of obliquity (α₀) for the three projectile/target pairs discussed in Figure 13, all of them having the same value for the parameter A. It is clear that for these different cases one gets the same dependence of P/P₀ on the obliquity angle.

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Figure 16.

The normalized penetrations of the three projectile/target pairs from Figure 13.

The next step was to check the P/P₀ results for L/D = 5 and 10 projectiles impacting the 0.43 GPa aluminum target at 1000 m/s. These projectile/target pairs have the same value of A = 0.245 and their P/P₀ results are shown in Figure 17. Obviously, they do not fall on a single curve, which means that the parameter A cannot be used as normalizing entity for P/P₀.

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Figure 17.

Results for L/D = 5 and 10 projectiles with the same value for A.

Due to the lack of an analytical model an empirical relation was sought to connect the values of P/P₀ with both the initial obliquity and the ricochet angle (α_ric). This numerically-derived relation is given in terms of the obliquity measured from the normal to the target, β₀ = 90 − α₀, as follows:

P / P 0 = cos ( B β 0 )

(8)

and B was found to be given by:

B = 1 . 0 + 0 . 158 α ric + 0 . 76 ( α ric ) 2

(9)

where α_ric is given in radians. The values for B increase from 1.16 to 1.4, for ricochet angles ranging between 21° (0.367 radians) and 36° (0.628 radians). The fact that α_ric depends on the parameter A, through equations (6) and (7), means that the values for P/P₀ depend, implicitly, on this entity.

In order to demonstrate the validity of the above relations consider Figure 18 which shows several examples for the excellent agreements between their predictions and the numerical results. Note the deviation at the lower parts, near the ricochet angles, where the model somewhat over-predicts the numerical results.

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Figure 18.

Comparing our model and simulation results for (a) L/D = 5 and (b) L/D = 10 projectiles.

A few words about bending rods

In order to demonstrate the different behavior of non-rigid rods obliquely impacting metallic targets, one should consider the work of Warren and Poormon (2001). Their experiments and numerical simulations followed the trajectories of non-rigid rods obliquely impacting thick 6061-T6511 aluminum targets. We simulated their test #1-0468 in which the L/D = 10 ogive-nosed steel rod impacted the target at a velocity of 1184 m/s and obliquity of 45°. The von-Mises strength of 0.43 GPa was used for this aluminum alloy, as explained above. The Johnson-Cook constitutive equation was applied for the Rc44.5 VAR 4340 steel rod, through which the flow strength varies with the plastic strain and strain rate according to:

Y = ( C 1 + C 2 ε p n ) [ 1 + C 3 ln ( ε . / ε . 0 ) ]

(10)

with C₁ = 1.14 GPa, C₂ = 510 MPa, n = 0.19, and C₃ = 0.01.

A few snapshots from the corresponding simulation are shown in Figure 19, and one can clearly see the significant bending of the rod and its tendency to ricochet from the target. The corresponding shadowgraph in Warren and Poormon (2001) of the rod embedded in the target is very similar to the latest snapshot in Figure 19. It is interesting to note the difference between simulations with and without the effect of the impact face, in Warren and Poormon (2001). Their simulations show that the bending rods do not turn around enough, if the effect of the impact face is not taken into account. Obviously, the effect of the impact face is inherent to our simulation, which is the reason that it follows the experimental result. One should also note that according to our model the threshold ricochet angle for a rigid rod under these conditions is α_ric = 20.4°, which is much lower than the α = 45° obliquity in the test with the non-rigid rod.

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Figure 19.

Simulation results for the non-rigid rod impacting the aluminum target at 45°.

Conclusions

This study focused on the trajectory changes of CRH3 ogive-nosed rigid projectiles impacting semi-infinite metallic targets at various obliquities. The main conclusions from this work are: 1. the trajectory changes are related, through a simple relation, to the corresponding threshold ricochet angles (α_ric) of the projectile/target pair. 2. These threshold angles were found to depend on a non-dimensional entity, which is equal to the ratio between the target’s resistance to penetration and the dynamic pressure exerted by the projectile upon impact, as suggested by Zener and Peterson (1943). 3. Simple relations were found to account for the normalized penetration depths of these projectiles as a function of their obliquity and their threshold ricochet angles. This work concerns ogive-nosed CRH3 rigid projectiles of aspect ratios (L/D) between 3 and 10, impacting metallic targets with strengths of 0.43–1.5 GPa, at ordnance velocities (0.5–1.5 km/s). Similar work should be performed for rigid projectiles having spherical and flat noses. Thick metallic targets were considered here, for which the influence of their back surface can be ignored, and similar work has to be done on finite-thickness plates. Finally, the numerically-based relations derived in the present work need to be accounted for by proper analytical treatments.