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On the perforation of aluminum plates by 7.62 mm APM2 projectiles
International Journal of Impact Engineering 97 (2016) 79–86
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International Journal of Impact Engineering j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / i j i m p e n g
On the perforation of aluminum plates by 7.62 mm APM2 projectiles Zvi Rosenberg, Roman Kositski *, Erez Dekel RAFAEL, P.O. Box 2250, Haifa, Israel
A R T I C L E
I N F O
Article history: Received 5 March 2016 Received in revised form 17 May 2016 Accepted 1 June 2016 Available online 22 June 2016 Keywords: Plate perforation Armor piercing projectiles Ballistic limit velocity Aluminum plates
A B S T R A C T
The paper is concerned with the perforation process of aluminum plates by 7.62 mm APM2 projectiles. In particular, we compare experimental values of the ballistic limit velocities for various aluminum alloys with predictions from our numerically-based model, which has been published a few years ago. The data include different thicknesses of aluminum plates of several alloys, ranging in their dynamic compressive strengths between 0.2 and 0.65 GPa We introduce an empirical correction factor to account for the effect of the brass jacket around the hard steel cores of these projectiles. We also discuss the issues of layered targets and inclined impacts, which may complicate the analysis. The good agreements between our predictions and the data for the ballistic limit velocities enhance the validity of the model, and suggest that it could also apply for other armor piercing projectiles, as well as for other metallic plates. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction The protection capability of armor plates made of strong aluminum alloys has been the subject of intense research for many years due to their high strength and low density, resulting in lower weights for armored vehicles as compared with traditional steel structures. Many publications deal with the ballistics properties of various metals, including strong aluminum alloys, against armor-piercing (AP) projectiles through empirical data and their numerical simulations. A comprehensive review of these perforation models is given in Ben-Dor et al. [1]. The data include the ballistic limit velocities (Vbl) of various plates perforated by a certain AP projectile, as well as its residual velocity after perforation. In Rosenberg and Dekel [2,3], we analyzed the interaction of rigid projectiles with metallic plates using numerical simulations. We derived a relatively simple model which accounts for the values of Vbl for a large range of plate thicknesses impacted by rigid sharp-nosed projectiles. The basic assumption behind this model is that the perforation process is achieved by the so-called ductile hole enlargement mechanism. This is the process by which sharp-nosed rigid projectiles perforate ductile metallic plates, which do not suffer from failure mechanisms such as spalling, scabbing or discing. In order to account for the residual velocities after plate perforation, we used the model of Recht and Ipson [4], which is based on energy conservation considerations. This physically-based model was found to account for several sets of data, as summarized by Rosenberg and Dekel [5].
* Corresponding author. RAFAEL, P.O. Box 2250, Haifa, Israel. Tel.: 972-73-335-4444; Fax: 73-335-5858. E-mail address: [email protected] (R. Kositski). http://dx.doi.org/10.1016/j.ijimpeng.2016.06.003 0734-743X/© 2016 Elsevier Ltd. All rights reserved.
The purpose of the present paper is to demonstrate the ability of this approach to account for recently published data concerning the impact of 7.62 mm APM2 projectiles at aluminum plates made of different alloys. The data we analyze concern the value of Vbl for a given aluminum plate, and the values of the residual velocities as a function of impact velocities of the 7.62 mm APM2 projectile. We highlight the difference between the values of Vbl for the full (jacketed) projectiles and for their hard steel cores, through an empirically derived factor. We also discuss the issue of laminated targets, which are often used for ballistics studies, as well as the impact of 7.62 mm APM2 projectiles at inclined plates.
2. A short summary of the model 2.1. The Recht–Ipson model for the residual velocity The residual velocity of sharp-nosed rigid projectile perforating a ductile metallic plate can be calculated by the Recht–Ipson model [3], which is based on energy conservation, through:
MV02 MVr2 = +W 2 2
(1)
where M is the mass of the projectile, V0 and Vr are its impact and residual velocities, respectively, and W is the work done on the target in opening a hole for the passage of the projectile. The value of W can be derived by the condition that at the ballistic limit velocity (V0 = Vbl) the residual velocity is zero. Thus, one gets: MVbl2/2 = W, and Eq (1) results in the well-known relation of the Recht–Ipson model:
Vr = (V02 − Vbl2 )
0.5
(2)
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which should apply for targets perforated by rigid projectiles through the ductile hole enlargement process. In cases where target failure results in back surface debris, the energy conservation equation should include the energy of the debris as well. Such cases are relevant for plates perforated by flat-nosed projectiles, when a cylindrical plug is sheared from the target, or when the target material is relatively brittle and its debris is spalled off from the back surface. As discussed by Rosenberg and Dekel [3], the most important result from Eq. (2) is that the entire physics of the perforation process is manifested by the value of Vbl for the given projectile/plate pair. Once this value is determined, either experimentally or analytically, the residual velocities can be calculated by Eq. (2), as demonstrated in Ref. [5] for various sets of projectile/ plate combinations. In the next section, we summarize the numerically-based model, which was presented in Ref. [2], for the predicted Vbl values, through the relevant properties of the target and the projectile. It is important to note here that several workers use a somewhat different relation than Eq. (2), in order to fit their data for the residual velocities, as follows: 1
Vr = a (V0p − Vblp ) p
(3)
where a and p are empirical constants which are best-fitted to the data. This relation is often related to Recht and Ipson [4] while there is no mention of such a relation in that work. In fact, this is an empirical relation due to Lambert and Jonas [6] who proposed it for the residual velocities of eroding long rod penetrators, after they perforate metallic plates. The application of this empirical relation has been primarily used to better illustrate the relationship between the different data points from experiments or numerical simulations. However, we prefer to use the physically-based relation of Recht and Ipson [4], especially when one deals with pointed projectiles perforating plates of ductile materials. It turns out that most best-fits to the Lambert–Jonas relation for pointed nose projectiles show that the term a in Eq. (3) is close to 1 and p is close to 2, confirming the accuracy of the Recht–Ipson model. As we shall demonstrate here, the use of Eq. (3) for the data of rigid projectiles may obscure important insights, which can be gained by analyzing the data through the physically-based relation in Eq. (2).
2.2. The numerically-based model for Vbl The perforation process of metallic plates by rigid projectiles is dominated by the free surfaces of these plates, resulting in a timevarying resisting stress, which acts on the projectile during perforation. As discussed in Ref. [3], a simple penetration model that is based on this time-varying resisting stress, is difficult to construct. Instead, we defined an effective (constant) resisting stress (σr), which the finite plate exerts on the projectile. The basic idea behind this concept is that the work done by this effective stress, throughout the perforation process, is equal the work done by the actual time-varying resisting stress. Thus, one can write for this work: W = πr2σrH, where r is the radius of the projectile, and H is the thickness of the plate. Using this relation together with the one derived from the Recht–Ipson model: W = MVbl2/2, we get:
⎛ 2σ H ⎞ Vbl = ⎜ r ⎟ ⎝ ρ p Leff ⎠
0.5
(4)
where ρp and Leff are the projectile’s density and its effective length, respectively, and their product is determined through: M = πr2ρpLeff. Through a series of numerical simulations, Rosenberg and Dekel [3] derived the dependence of the effective resisting stress (σr) on
the plate’s properties and on the projectile’s diameter (D). They found that σr is dependent on the target’s dynamic compressive strength (Y) and on its normalized thickness (H/D), through the following relations:
σr ⎛ H⎞ = 2 + 0.8ln ⎜ ⎟ ⎝ D⎠ Y σr =2 Y
1 H < <1 3 D
H σr 2 = +4 Y 3 D
H 1 ≤ D 3
H ≥1 D
(5a)
(5b)
(5c)
These relations are due to the different states of stress which the plate experiences during its perforation, as discussed in Ref. [5]. In particular, Eq. (5b) applies for plate thicknesses where the state in the plate is that of plane stress, while Eq. (5a) describes the transition to the more complex three-dimensional stress state. In the present paper, we shall deal with projectile/plate combinations for which H/D > 1.0, and we use Eq. (5a) throughout the paper. Rosenberg and Dekel [5] pointed out that the value for Y should be the dynamic compressive flow stress of the target material, at a strain of about 0.3 and strain rates of the order of 103 s−1. These are the typical values of the rates and strains in the target elements adjacent to the perforating projectile, which experience large compression during the perforation process. Dynamic compressive flow stresses are easily obtained by Kolsky bar tests, which result in compressive stress–strain curves for large strains at the relevant rates. Moreover, these tests should be performed on specimens taken from the plates that are used for the ballistics experiments, in order to avoid possible variations in the properties of plates taken from different lots. In order to understand why the relevant dynamic flow stress (Y) has to be determined at a strain of about 0.3, consider the numerical simulations of Rosenberg and Dekel [3] for rigid projectiles perforating plates of stainless steel 304L. This material has a very large hardening coefficient, which increases its flow stress from a value of 0.34 GPa at yield, to more than 2 GPa at its ultimate value. These simulations showed that the SS304L plates behave in the same way as plates having a constant (von-Mises) strength of Y = 0.82 GPa. Using the constitutive relation for SS304L, with its strong hardening property, one finds that this intermediate value of flow stress corresponds to a strain of 0.3. This result indicates that the average strain of the target elements around the projectile is about 0.3, which is the reason for choosing the value for Y at this strain from dynamic compressive stress–strain curve. In the same way, one may consider the rate of 10−3 s−1 as representing an average strain rate for these elements. Note that the value of 0.3 for the average strain does not mean that the material fails at this strain. In fact, the proposed model applies for ductile materials for which the strain to failure is usually much larger. In order to demonstrate the validity of our approach, consider the data of Roisman et al. [7], who shot tungsten-alloy ogivenosed projectiles, with M = 88 g and D = 11.3 mm, at stacks of three 6061-T651 aluminum plates with a total thickness of 120 mm. The experiments included normal impacts at stacks of three 40 mm thick plates, as well as inclined impacts at an obliquity of 30° with two plates of 40 mm and one of 25 mm. From Eq. (5a), we find that that σr = 3.89Y for this case and with Y = 0.42 GPa, as measured for the 6061-T651 alloy in our Kolsly bar system, we get from Eq. (4): Vbl = 668 m/s for this case. With this value of Vbl we can calculate the Vr(V0) curve from Eq. (2). Fig. 1 shows the excellent agreement between the model’s predictions and the data from Ref. [7], as far as Vbl and Vr(V0) are concerned.
Z. Rosenberg et al. / International Journal of Impact Engineering 97 (2016) 79–86
800
1000 900
81
data model
700
data model
800
600 700
500 Vr (m/s)
Vr (m/s)
600 500 400
400 300
300
200
200
100
100 0 600
700
800
900 V0 (m/s)
1000
1100
1200
Fig. 1. The agreement between our model (Vbl = 668 m/s) and the data from Ref. [7].
0 500
600
700 V0 (m/s)
800
900
Fig. 2. Comparing the model predictions (Vbl = 517.4 m/s) with the data for 20 mm plates of 5083-H116 aluminum perforated by the steel cores.
3. Comparing the model’s predictions with experimental data In this section we compare the model’s predictions with several sets of data, which were published recently, for the perforation of aluminum plates by 7.62 mm APM2 projectiles. We start with the data for the hard steel cores impacting aluminum plates of different alloys, since this is the case where the projectile is a truly rigid one with well-defined dimensions. The case of a full projectile, which includes the soft jacket surrounding the hard steel core, is discussed later where we account for the effect of the jacket on the value of Vbl. We shall also address the issue of target lamination since, often, the data are given for targets that consist of two or three plates that are held together. 3.1. Perforations by the hard steel cores Data for the perforation of Al 5083-H116 plates by both the hard steel cores and the full (jacketed) 7.62 mm APM2 projectile is presented in Ref. [8]. The diameter of the core is 6.17 mm and its weight is 5.25 g from which we get: ρpLeff = 17.57 g/cm2, which should be used in Eq. (4) for the derivation of the corresponding Vbl values. For consistency of units in Eq. (4), one should use GPa for the strength term (σr) and H should be given in centimeters. The values for Vbl are then obtained in units of km/s. The compressive stress–strain curve for this alloy, as given in Ref. [8], shows that at a strain of about 0.3, the flow stress is about Y = 0.4 GPa. Using this value in Eq. (5a), together with H/D = 3.24 for the 20 mm plates, results in σr = 1.176 GPa for this plate/projectile combination. Inserting this value in Eq. (4), together with the value of ρpLeff, results in a predicted value of Vbl = 517.4 m/s for this case, which is in excellent agreement with Vbl = 513 m/s, as deduced in Ref. [8] from the residual velocity measurements. Fig. 2 shows that Eq. (2) for Vr, together with our predicted value for Vbl, accounts for the residual velocity data in a very satisfactory way. This agreement enhances the claim of Rosenberg and Dekel [5], that the whole Vr(V0) curve, for a given projectile/plate configuration, can be calculated through Eqs. (4) and (5) and the Recht–Ipson relation for Vr, as given by Eq. (2). Our next step is to consider the data in Ref. [8] for targets that were made of two adjacent 20 mm plates. Obviously, if the plates are strongly fastened to each other, they can be treated as a single plate which is 40 mm thick. However, as we do not know how well
these plates were held together, before and during the perforation process, we assume that a small interval may have existed between the two plates. In order to assess the effect of such intervals on the ballistic limit velocity, we performed two sets of simulations, one with a single aluminum plate, 40 mm thick and another with two 20 mm plates, separated by an interval of 0.5 mm. The targets were modeled as elastic perfectly plastic, with yield strength of 0.4 GPa. The ogive nosed steel projectile in these simulations had a diameter of 6.2 mm and a mass of 5.3 g, which are very close to those for the hard steel core of a 7.62 mm APM2 projectile. We should note that these simulations were performed in order to check the influence of layering, and the hard steel core steel was chosen in order to avoid the complications arising from the jacket of the full AP projectile. The impact velocity in these simulations was 1000 m/s and the resulting residual velocities were 603 m/s and 629 m/s, for the single and the double-plate targets, respectively. Using the Recht– Ipson model, as depicted by Eq. (2), we get a difference of about 2.6% for the inferred ballistic limits of the single and double-plate configurations. We shall take this difference into account for the doubleplate targets in Ref. [8]. Namely, we shall multiply the calculated values of Vbl for the single plate by a factor of 0.974, in order to predict the value for the double-plate configuration. For the tripleplate targets, we shall use a correction factor of (0.974)2 = 0.95. Note, that these corrections can be smaller if the interval between the plates is smaller, and we may consider the 2.6% difference as an upper bound for such a correction. Another set of simulations for the two arrangements discussed here, was performed at an impact velocity of 800 m/s, which is close to the ballistic limit velocity of this configuration. The inferred values of Vbl for the single and doubleplate targets differed by only 1.8%, which shows that an interval of 0.5 mm between the plates may have a smaller effect at lower impact velocities. The first step in our calculation is to find the value of Vbl for a single plate 40 mm thick, which corresponds to H/D = 6.48. Inserting this value and Y = 0.4 GPa in Eq. (5a) results in: σr = 1.4 GPa for this plate/core combination. With this value for σr, Eq. (4) results in a predicted value of Vbl = 797.8 m/s for the 40 mm plate. Multiplying this value by the correction factor of 0.974 from our simulations, we get a predicted value of Vbl = 777 m/s for the doubleplate target. Note that Børvik et al. [8] deduced a value of Vbl = 767 m/s
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from their data, which is very close to our predicted value. For their triple-plate targets, made of three plates 20 mm thick, we first find that for a single 60 mm target (H/D = 9.7) the effective stress (σr) is 1.53 GPa and the corresponding value for Vbl is 1022 m/s. This value is in excellent agreement with the value of 1025 m/s deduced in Ref. [8]. However, we have to follow the same procedure for this triple-plate target and correct the value of Vbl due to the layering effect. The correction factor for the triple-plate target is (0.974)2, which results in a predicted value of Vbl = 969.5 m/s for the triplelayered targets. Fig. 3 shows the agreement between our model’s predictions and the data for Vr from Ref. [8], for the double and triple-plate targets, with the corrected values for the corresponding Vbl values as discussed above. Clearly, the agreement is excellent for the doubleplate targets enhancing the validity of our approach concerning the layering effect. As for the triple-plate targets, our model accounts very well for the data at the high velocity shots. On the other hand, at impact velocities just over 1000 m/s the model deviates from the data, which resulted in zero residual velocities. In fact, the low velocity experiments suggest that Vbl should be around 1025 m/s, which is practically the same value we calculate for a single plate 60 mm thick (1022 m/s). We assume that these conflicting results reflect the complications arising for laminated targets, and they may be due to differences in the attachment of the plates during their perforation. Namely, the three plates could remain closely attached at the lower impact velocities, while they may have separated somewhat during the high velocity impacts. Obviously, one should use single plates rather than multiple-plate targets, in order to avoid such discrepancies. We assume that experiments with single plates, 60mm thick, will result in a value of Vbl = 1020 m/s and that the residual velocity data would follow the Recht–Ipson model, without the large “jumps” around Vbl, as seen in Fig. 3. On the other hand, when the application demands a multi-plate target, one may expect the data to look as in Fig. 3 for the three-plate targets. We strongly recommend that even with this type of data, one should use the Recht–Ipson model as given by Eq. (2), rather than the empirical relation in Eq. (3), which obscures the physics of the process. Another set of data is from Forrestal et al. [9], where both the cores and the jacketed projectiles are shot at 20 mm thick Al 6082T651 plates, at normal incidence as well as at several obliquities.
The stress–strain curve for this alloy is very similar to that of the 5083-H116 alloy and we chose the same value of Y = 0.4 GPa for the 6082-T651 alloy. Note that our model resulted in a value of Vbl = 517.4 m/s for a 20 mm thick plate with a flow stress of 0.4 GPa, which is very close to the value of Vbl = 514 m/s deduced in Ref. [8] for the normal impacts. Fig. 4a shows the excellent agreement between our predictions for Vr(V0) and the data from Ref. [9], for the steel cores impacting these 20 mm plates at normal incidence. Forrestal et al. [9] note that the trajectories of the projectiles in the inclined plates were very straight, and one may assume that the effective thicknesses of these plates are H/cosβ, where β is the obliquity angle. These effective thicknesses are: Heff = 20.7, 23.1 and 28.3 mm, for a 20 mm thick plate at obliquities of 15°, 30° and 45°, respectively. Using the same procedure for the inclined plates, we find the following values for the effective resisting stress: σr = 1.187, 1.225, and 1.287 GPa, for the three obliquities. These values result in the following values for Vbl from Eq. (4): Vbl = 529, 567 and 644 m/s for β = 15°, 30° and 45°, respectively. Once we have these predicted values for Vbl , we can compare the predicted values for Vr, from Eq (2), with the data for the steel cores at these obliquities, and Fig. 4b shows this comparison.
800 700 600
data 2x20mm model 2x20mm data 3x20mm model 3x20mm
Vr (m/s)
500 400 300 200 100 0 700
800
900 V0 (m/s)
1000
1100
Fig. 3. Comparing our model’s predictions with the data for the double-plate (Vbl = 777 m/s) and for the triple-plate (Vbl = 969.5 m/s) targets.
Fig. 4. Comparing the model’s predictions with data for 20 mm plates of Al6082-T651, (a) normal impacts (Vbl = 417.4 m/s), (b) inclined impacts: β = 15° (Vbl = 529 m/s),β = 30° (Vbl = 587 m/s), β = 45° (Vbl = 644 m/s).
Z. Rosenberg et al. / International Journal of Impact Engineering 97 (2016) 79–86
3.2. Perforations by full (jacketed) projectiles The full 7.62 mm APM2 projectile has a complex structure, which includes a brass jacket around the core and a lead cap at the front. The total weight of the full projectile is about 10.5 g, twice the weight of the steel core, which cannot be ignored when one is considering the interaction of this projectile with metallic plates. On the other hand, these soft parts are striped off quite early in the perforation process, and one usually finds a very deformed brass jacket embedded in the plate near its impact face. Most workers ignore the effect of the jacket on the perforation process and treat the full projectile as if the jacket does not contribute to the process. Still, they do find measurable differences between the values of Vbl for the steel core and the full projectile, as in Forrestal et al. [8] and Holmen et al. [12], for example. Obviously, an analytical model, based on energy considerations, is not easy to construct because of the plastic work involved in the jacket’s deformation, which may vary with the impact velocity and with the strength of the target. Numerical simulations have to consider the attachment of the jacket to the core, which is not easy to simulate. Still, we find several attempts for such simulations from which we can have a rough estimate for the effect of the jacket. For example, Holmen et al. [12] performed simulations for both the steel cores and the full 7.62 mm APM2 projectiles, perforating 20 mm plates of the Al6070-T6 alloy. The simulations were performed for an impact velocity of 903 m/s and the resulting residual velocities were 784 and 754 m/s for the full projectile and the steel core, respectively. Using Eq. (2) we find that the inferred values for the ballistic limit velocities are 448 m/s for the full projectile and 497 m/s for the steel core. The ratio of these values is about 0.9, which means that we can expect a reduction of about 10% in the values of Vbl for the full projectiles, as compared with those for the steel cores. In order to compare this result with experimental data, consider the data in Forrestal et al. [9] for the 20 mm 6082-T651 plates impacted by the full (jacketed) projectile at normal incidence, given in Table 1. A least square fit of these results, using Eq. (2), results in a value of Vbl = 480 m/s, which best-fits the data in this table. This value is smaller by about 7% from the corresponding value of Vbl = 517.4 m/ s, which we derived for the steel cores impacting the same plates. Thus, in the following discussion we shall use a correction factor of 0.93 in order to predict the values of Vbl for the full projectiles, from the calculated values for the steel cores. Note that the cor-
rection factors from the numerical simulations in Ref. [12], and from the data of Ref. [9], are quite close: 0.9 and 0.93, respectively. We chose the experimentally derived factor because it avoids possible uncertainties due to material properties in the simulations. We should also note that the least-square fitting technique, which we used to assess the value of Vbl from Table 1, is preferable to the other technique for Vbl determination, by which one takes the average of the lowest velocity perforation shot and the highest velocity for nonperforation. With the least square fit one uses all the available data rather than only two shots, which may have a large scatter because they are very close to the ballistic limit velocity. The most important question here concerns the reason for the difference between the values of Vbl of the steel core and the full projectile. The simplest explanation is to assume that as the jacket impacts the plate it creates an initial crater before the core reaches the plate and starts to penetrate it. Namely, the jacket reduces the actual thickness of the plate which is being perforated by the core. Through Eq. (4) we can estimate the thickness reduction which will result in a 7% difference between the corresponding Vbl values. As an example, for plate thicknesses of 26 mm and 52 mm we find that one needs thickness reductions of 3 mm and 6 mm, respectively, in order to get the 7% difference in the corresponding Vbl values. In order to check these assessments, we performed two simulations with a full projectile impacting a thick aluminum plate at 500 and 1000 m/s, which are, roughly, the ballistic limits of aluminum plates having thicknesses of 26 and 52 mm, respectively. These simulations showed that the jacket penetrated to depths of 3 and 5 mm before the core started to penetrate the plate, supporting our assumption concerning the effect of the jacket. With the correction factor of 0.93, we can predict the values of Vbl for the full projectiles perforating the 20 mm inclined plates in Ref. [9]. Fig. 5 shows the agreement between the model’s predictions with the data for obliquities of 15° and 30°. We did not bring the comparison for the plates inclined at 45° because the corresponding data is very scattered. We find good agreement between our model and the data for the plates, which are inclined at 15°. A somewhat less agreement is obtained for the low velocity shots at an obliquity of 30°, which imply that the value for Vbl should be higher. On the other hand,
800 data =15 o
700 600
V0(m/s) Vr(m/s)
474 0
500 0
508 105
568 290
573 317
662 464
806 667
917 787
model =30 o
500 400 300 200 100 0 400
Table 1 Residual velocities of full projectiles perforating 20 mm plates of Al 6082-T651.
model =15 o data =30 o
Vr (m/s)
It is clear that the agreement between the model’s predictions and the data is very good for the obliquities of 15° and 30°. A somewhat less agreement is seen for the β = 45° case at impact velocities near the ballistic limit. This discrepancy may be due, at least partly, to the large scatter in the data for Vr at the low velocity range. A very large scatter in the values of Vr is also evident in Ref. [10], for 7.62mm APM2 projectiles impacting 20mm thick 6082-T4 aluminum plates, at an obliquity of 45°. The scatter may be due to a somewhat curved trajectory of the projectile in these targets at high obliquities and for relatively low impact velocities. These curved trajectories, having an S-shape, are due to the asymmetric forces which the front and the back faces of the target exert on the projectile. Such asymmetries are known to take place at high obliquities, especially for impact velocities near the ballistic limits, as shown by Piekutowski et al. [11].
83
500
600
700
800
900
V0 (m/s) Fig. 5. Comparing the model’s predictions with the data for inclined 20 mm plates of 6082-T651 aluminum perforated by full projectiles (Vbl = 492 m/s for β = 15°, and Vbl = 527 m/s for β = 30°).
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the agreement for the high velocity shots, at this obliquity, is very good. We have seen this phenomenon earlier, for steel cores perforating highly inclined plates. Similarly, we assume that the trajectories of the full projectiles in highly inclined plates, at the low velocity shots, may be somewhat curved due to the asymmetric forces which the front and back faces exert on the projectile at large obliquities and low velocities.
we find that the resisting stress is σr = 2Y, from Eqs. (5a) and (5b). Substituting this value for σr together with H = D in Eq. (4), results in the following expression for the ballistic limit velocity (Vbl*) of this plate thickness:
4. Comparison with experimental data of Vbl for various aluminum alloys
from which we find, for example, that: Vbl* = 167.6, 237, 278 and 303 m/s, for the 7.62mm APM2 hard steel cores perforating plates with flow stresses of: Y = 0.2, 0.4, 0.55 and 0.65 GPa, respectively. For larger plate thicknesses ( H ≥ D ) we find, by Eqs. (4) and (5a), for the normalized values of Vbl:
0.5
(6)
⎡H ⎛ ⎛ H⎞⎞ ⎤ = ⎢ ⎜ 1 + 0.4 ln ⎜ ⎟ ⎟ ⎥ ⎝ ⎝ D⎠⎠ ⎦ D ⎣ Vbl* Vbl
0.5
for
H ≥1 D
(7)
and for normalized thicknesses between 1/3 and 1.0 we get, from Eqs. (4) and (5b), the following relation for the normalized ballistic limits:
⎛ H⎞ =⎜ ⎟ Vbl* ⎝ D ⎠ Vbl
0.5
1 H < <1 3 D
for
(8)
Fig. 6 shows the agreement between the data collected in Table A1 for all the single-plate targets, and the normalized curve for Vbl as calculated through Eq. (7). Note that the calculated curve was multiplied by the correction factor of 0.93, in order to obtain the values of Vbl for the full (jacketed) projectiles, as discussed above. One may consider Eqs. (6) and (7) as a universal representation of the normalized Vbl values for different sharp-nosed projectiles perforating ductile metallic plates. Each projectile/plate pair is defined by its corresponding value of Vbl*, through Eq. (6), which is then used to determine the values for Vbl for any plate thickness (for H > D) through Eq. (7). Thus, we expect that Fig. 6 has a “universal” nature and that it should also apply for other ductile metallic plates perforated by 7.62 mm APM2 projectiles. Moreover, we assume that the approach presented here should also account for other armorpiercing sharp-nosed rigid projectiles, although different correction factors may be needed to account for the effect of their jackets, as
5 4.5 4 3.5 *
V bl/Vbl
We are now ready to compare published values of Vbl for various aluminum plates, perforated by the full 7.62 mm APM2 projectiles, with the predicted values as calculated by our model. The analysis follows the same steps outlined above, namely, we first compute the value of σr for the plate/core pair from Eq. (5a), using the dynamic flow stress (Y) of the alloy and the normalized plate thickness (H/D). As discussed above, the value for the flow stress in our model (Y) should be taken from a dynamic test in the Kolsky bar system, at a strain of 0.3 and a strain-rate of about 103 s−1. The next step is to use Eq. (4) in order to find the value of Vbl for the steel core, which is then multiplied by the correction factor of 0.93 in order to obtain the value of Vbl for the full projectile. Table A1 in the Appendix shows the good agreement between the predicted values of Vbl for various aluminum plates, with Vbl data from several sources, most of which are summarized in Refs. [9] and [13]. The table shows that our model captures the main features of the interaction between 7.62 mm APM2 projectiles and aluminum plates, as far as the effects of plate strength and its thickness are concerned. The effect of the projectile’s jacket seems to be represented very well by the correction factor 0f 0.93, which we derived through a single set of experiments. Moreover, the layering issue seems to be represented by the factor of 0.974, as deduced from our numerical simulations. We should also note that the 7075 alloy is considered to be a relatively brittle material, as compared with the other alloys discussed here. Still, we find that the model applies for plates made of this alloy as well. This agreement is probably due to the fact that these plates are much thicker than the diameter of the projectile (20–40 mm vs. 6.2 mm). In contrast, when the projectile has a diameter which is similar to the plate’s thickness, we find that the back surface of the plate shows tensile failure features, besides the ductile hole enlargement perforation process. Such features are clearly seen in the work of Børvik et al. [14] who shot ogive nosed projectiles, with D = 20 mm, at 20 mm thick 7075-T651 aluminum plates. The sectioned plates show a significant back surface failure, which resulted in a low ballistic limit velocity. The tensile failure at the back side of these plates is probably due to significant stress waves which reach the back surface and reflect back as tensile waves. Such waves may lead to the ejection of spalled material, lowering the effective thickness of the plate and resulting in a reduced value of Vbl. In fact, Børvik et al. [14] found that the ballistic limit velocity for a 20 mm thick 7075-T651 plate, perforated by the D = 20 mm ogive-nosed projectile, was lower by 20% than the corresponding value for a 20 mm thick 5083- H116 plate which has a much lower strength. In contrast, when the projectile’s core diameter is significantly smaller than the plate thickness (6.2 mm vs. 20–40 mm as in Table A1), the amplitude of these tensile waves are reduced considerably, and the plates are not expected to fail as easily. In conclusion, the analysis presented here should be applied carefully for brittle materials, such as the 7075 aluminum alloy, especially when relatively thin pates are considered. The resulting values of Vbl can be represented in a normalized form, following the scheme suggested by Rosenberg and Dekel [5]. The normalizing factor is the value of Vbl for a plate of thickness which is equal to the diameter of the projectile (H = D). For this case
⎛ YD ⎞ Vbl* = 2⎜ ⎝ ρ p Leff ⎟⎠
3 2.5 2 1.5 1 0
1
2
3
4
5
6
7
8
9
10
11
12
H/D Fig. 6. Comparing the model for the normalized Vbl values of 7.62mm APM2 projectiles with the data in Table A1.
Z. Rosenberg et al. / International Journal of Impact Engineering 97 (2016) 79–86
Table 2 Comparing the model’s predictions with data for Vbl (in m/s) of 12.7 mm APM2 projectiles impacting 2139-T8 aluminum plates. H (mm) Vbl (exp) Vbl (calc)
40 668 668
40.9 677 677
52.1 785 787
53.8 796 803
57.2 819 833
64.1 874 896
we show below for the 12.7 mm APM2 projectile. On the other hand, we do not expect this approach to apply for plates, which fail by adiabatic shearing, such as the Ti-6Al-4V alloy, or for projectile which break during impact at very high strength plates. We should note that Eqs. (6) and (7) are different from the corresponding equations derived by Forrestal et al. [9], in which the values for Vbl are proportional to (Hσs)0.5. Their approach is based on the cylindrical cavity expansion analysis to derive the resisting stress (σs), which the plate exerts on the projectile. According to this model the resisting stress has the same value for all plate thicknesses, while in our numerically-based model, the resisting stress (σr) is dependent on H/D, as given by Eqs. (5). The recent model of Masri [15] is similar to ours, in the sense that the resisting stress is dependent on the normalized plate’s thickness. On the other hand, his derivation is based on the spherical cavity expansion analysis, which is similar to the approach in Ref. [9]. In order to check the applicability of our model for other projectiles, consider the data from Cheeseman et al. [16], for 12.7 mm APM2 projectiles perforating 2139-T8 aluminum plates. The structure of these projectiles is similar to the structure of the 7.62 mm projectiles, with a hard steel core encased in a soft metallic jacket. We assume that our approach should apply to these projectiles as well, and the only difference may be due to a somewhat different correction factor for the values of Vbl of the full projectile. In order to determine this factor for the 12.7 mm APM2 projectile we use the data from Ref. [16] for the 39 mm thick plate of the 2139-T8 alloy, which was the thinnest plate they used in their study. The hard steel core of the 12.7 mm projectiles has a diameter of 10.9 mm and a weight of 25.9 g, from which we get ρpLeff = 27.77 g/cm2 for the core. Using this value together with Y = 0.55 GPa for the dynamic flow stress of the 2139-T8 alloy, we find that Vbl = 683 m/s for the hard steel core impacting the 39 mm thick plate. The experimental value of Vbl for the full projectile is 657 m/s, as given in Ref. [16]. Thus, we conclude that the correction factor is 0.962 for this projectile/ target pair. This is a somewhat larger factor than the one we found for the 7.62 mm APM2 projectile (0.93), but it is a reasonable value. Our next step is to use this factor (of 0.962) in order to predict the values of Vbl for the other plates which were tested in Ref. [16]. The procedure we use is the same as before, namely, we first calculate the values of Vbl for the different plates impacted by the hard steel core through our numerically-based model, and multiply these values by 0.962 to get the predicted values for the full 12.7 mm APM2 projectiles. Table 2 shows the results of this analysis together with the experimental results in Ref. [16], and it is clear that the agreement between the two sets is very good. It would be interesting to compare our model’s predictions with other sets of data for this projectile, as well as with data for other AP projectiles, such as the 14.5 mm bullet, which has a tungsten-carbide core. 5. Summary This paper compares predictions from a numerically-based perforation mode and experimental data for the ballistic limit velocities of aluminum plates impacted by 7.62 mm APM2 projectiles. The model applies for sharp-nosed rigid projectiles perforating ductile metallic plates through the hole enlargement process. It is based on the notion of an effective (average) resisting stress, which the plate exerts on a given projectile during the perforation process. This
85
stress depends on the dynamic compressive flow stress of the plate material, and on the normalized thickness of the plate. The value of Vbl, for a given projectile/plate combination, is determined by this effective stress and by the thickness of the plate. The model accounts for perforations by “simple” projectiles such as the hard steel cores of the APM2 projectiles, as we showed here. In order to account for the data of the jacketed 7.62 mm APM2 projectile, we introduced a correction factor (of 0.93), which was derived from a specific set of experiments. The agreement between the model’s predictions and the data of Vbl, for plates of various aluminum alloys, is shown to be very good. In fact, the vast majority of the data fall within 2–3% from our predicted values. This agreement is especially encouraging since we have used data for plates with a large range of dynamic strengths (0.2–0.65 GPa), as well as large range of plate thicknesses (18.6–60 mm). We have also introduced a correction factor of 0.974 in order to account for the possible effect of layering a target by two adjacent plates. This issue needs further study, both empirical and numerical, but it seems that the layering effect is bounded by about 2.6%, as far as the values of Vbl are concerned. As for the residual velocities of the perforating projectiles, we have shown that the physicallybased model of Recht and Ipson [3] accounts for the data analyzed here. This model, which is based on energy conservation considerations, should be preferred over empirical relations which are used to fit a certain set of experiments. We emphasized the importance of having the values of the dynamic compressive flow stresses for the target materials. These values should be obtained by compressive Kolsky bar tests, at strain rates of about 103 s−1 and strains of 0.3, which are the relevant values for the target elements adjacent to the perforating projectile. Moreover, it is very important to characterize the properties of the actual plates that are used for the actual ballistics study. We should note that these values of the compressive strains and strain-rates are the average values experienced by the target elements around the perforating projectile. Finally, we expect that the approach presented here will account for other armor piecing projectiles perforating targets that are made of ductile materials. We do not expect that this approach will account for bluntnosed projectiles or for relatively brittle targets, where the penetration mechanism is mainly through adiabatic shearing, rather than the ductile hole enlargement mechanism. Appendix A In this part of the paper we demonstrate the good agreement between our model’s predictions and the experimental data for Vbl values of 7.62 mm APM2 projectiles impacting aluminum plates, made of various alloys, and having different thicknesses. The agreement is clearly seen in Table A1 below. As mentioned above, published works which deal with projectile perforations do not always supply the dynamic flow stresses for the plates, and we had to look for these values in other sources. As an example, for the ballistics data of the 2139-T8 alloy in Ref. [16], we used the dynamic flow stress (0.57 GPa) from Kolsky bar tests of Vural and Caro [17], at a strain rate of 103 s−1. Tucker et al. [18] show that the dynamic stress–strain curve for the 5083-H131 alloy is very similar to that of the 6061-T651 alloy. Thus, we used the same value of Y = 0.42 GPa for both alloys. For the 2024-T351 plates in Ref. [13], we used a value of Y = 0.55GPa, since published data are found to be in the range of Y = 0.55-0.6GPa. Similarly, published values for the dynamic strength of the 7075-T651 alloy range between 0.65 and 0.7GPa, and we used the value of Y = 0.65GPa in our calculations. For the 5059H131 plates from Ref. [19], we estimated a value of Y = 0.45 GPa. The data for the newly developed 7085-E01 alloy, from Ref. [20], has been analyzed with a value of Y = 0.65 GPa, because the static values for the strength of this alloy, as well as its Brinell hardness, are close to those of the 7075-T651 alloy. For the somewhat weaker
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Table A1 Comparing predicted Vbl values (in m/s) with the data for various aluminum plates. Alloy (flows stress)
H (mm)
Vbl (exp)
Vbl (calc)
Reference
6070-T0 (0.2 GPa) 6070-T4 (0.42 GPa) 6070-T6 (0.48 GPa) 6070-T7 (0.45 GPa) 5083-H116 (0.4 GPa)
20 20 20 20 20 2 × 20 3 × 20 37.8 50.9 54.7 57.2 20 30 40 25.7 26 38.8 51.2 20 2 × 20 26 30 32 38 25.4 26.3 30.4 34.6 37.9 38.9 47.8 51.4 25.4 38.7 51 25.2 32.3 39 40.9 26 30 34 38 18.6 25.4 38 40.5 18.5 25.4 38 40.5
348 506 562 529 492 722 912 712 876 890 927 474 589 698 596 583 754 883 628 909 718 784 818 909 589 610 661 705 763 774 860 913 584 737 874 683 783 860 893 652 715 774 831 598 772 908 938 567 664 842 872
340 493 527 510 481 723 901 734 880 911 946 463 597 710 577 582 746 884 610 919 723 792 824 915 593 607 664 719 761 773 878 917 573 744 882 665 776 871 897 666 716 787 843 586 713 916 953 547 668 858 892
[12]
5083-H131 (0.42 GPa)
5083-H112 (0.37 GPa)
6061-T651 (0.42 GPa)
7075-T651 (0.65 GPa) 7075-T651(0.65 GPa)
5059-H131 (0.45 GPa)
5059-H136 (0.42 GPa)
2139-T8 (0.57 GPa)
2024-T351 (0.55 GPa)
7085-E01 (0.65 GPa)
7085-E02 (0.57 GPa)
num plates, in Ref. [12], we estimated the flow stresses from the stress–strain curves in Ref. [23]. Note that the calculated values for the double-layered (40 mm) and triple-layered (60 mm) include the correction factors for layered targets as discussed above, namely a factor of 0.974 for the double-plate target and 0.95 for the stack of three plates.
[14]
References [8]
[13]
[24]
[25] [19]
[19]
[16]
[13]
[20]
7085-E02 plates from Ref. [20] we chose Y = 0.57GPa, based on their Brinell hardness, as compared with the hardness of the 7085-E01 plates. We should note that the hardness values, as given by Gallardy [20] for each plate thickness, differ by about 7% for both alloys. Thus, we may expect variations of a few percent in Vbl values due to differences in plate strengths. This fact enhances the need to measure the dynamic properties of the actual plate which is used for ballistic testing, as stated above. For the 5083-H112 data in Ref. [13] we estimated a value of Y = 0.37GPa through the static tensile values from Refs. [21] and [22]. For the differently treated 6070 alumi-
[1] Ben-Dor G, Dubinsky A, Elperin T. High-speed penetration dynamics: engineering models and methods. Singapore: World Scientific; 2013. [2] Rosenberg Z, Dekel E. On the deep penetration and plate perforation by rigid projectiles. Int J Solids Struct 2009;46:4169–80. [3] Rosenberg Z, Dekel E. Revisiting the perforation of ductile plates by sharp-nosed rigid projectiles. Int J Solids Struct 2010;47:3022–33. [4] Recht RF, Ipson TW. Ballistic perforation dynamics. J Appl Mech 1963;30:384. [5] Rosenberg Z, Dekel E. Terminal ballistics. Berlin: Springer Science & Business Media; 2012. [6] Lambert JP, Jonas GH. Towards standardization in terminal ballistics testing: velocity representation. BRL-1852. Aberdeen Proving Ground, MD: Army Ballistic Research Lab; 1976. [7] Roisman I, Weber K, Yarin A, Hohler V, Rubin M. Oblique penetration of a rigid projectile into a thick elastic–plastic target: theory and experiment. Int J Impact Eng 1999;22:707–26. [8] Børvik T, Forrestal MJ, Warren TL. Perforation of 5083-H116 aluminum armor plates with ogive-nose rods and 7.62 mm APM2 bullets. Exp Mech 2009;50:969–78. [9] Forrestal MJ, Børvik T, Warren TL, Chen W. Perforation of 6082-T651 aluminum plates with 7.62 mm APM2 bullets at normal and oblique impacts. Exp Mech 2013;54:471–81. [10] Børvik T, Olovsson L, Dey S, Langseth M. Normal and oblique impact of small arms bullets on AA6082-T4 aluminium protective plates. Int J Impact Eng 2011;38:577–89. [11] Piekutowski AJ, Forrestal MJ, Poormon KL, Warren TL. Perforation of aluminum plates with ogive-nose steel rods at normal and oblique impacts. Int J Impact Eng 1996;18:877–87. [12] Holmen JK, Johnsen J, Jupp S, Hopperstad OS, Børvik T. Effects of heat treatment on the ballistic properties of AA6070 aluminium alloy. Int J Impact Eng 2013;57:119–33. [13] Ryan S, Cimpoeru SJ. An evaluation of the Forrestal scaling law for predicting the performance of targets perforated in ductile hole formation. 3rd International Conference on Protective Structures. 2015. [14] Børvik T, Hopperstad OS, Pedersen KO. Quasi-brittle fracture during structural impact of AA7075-T651 aluminium plates. Int J Impact Eng 2010;37:537–51. [15] Masri R. Ballistically equivalent aluminium targets and the effect of hole slenderness ratio on ductile plate perforation. Int J Impact Eng 2015;80:45–55. [16] Cheesman B, Goosh W, Burkins MS. Ballistic evaluation of aluminum 2139-T8. 24th International Symposium Ballistic New Orleans; 2008. p. 651–9. [17] Vural M, Caro J. Experimental analysis and constitutive modeling for the newly developed 2139-T8 alloy. Mater Sci Eng A 2009;520:56–65. [18] Tucker MT, Horstemeyer MF, Whittington WR, Solanki KN, Gullett PM. The effect of varying strain rates and stress states on the plasticity, damage, and fracture of aluminum alloys. Mech Mater 2010;42:895–907. [19] Showalter DD, Placzankis BE, Burkins MS. Ballistic performance testing of aluminum alloy 5059-H131 and 5059-H136 for armor applications. 2008. [20] Gallardy DB. Ballistic evaluation of 7085 aluminum. Aberdeen, MD; 2012. [21] Cui GR, Ma ZY, Li SX. The origin of non-uniform microstructure and its effects on the mechanical properties of a friction stir processed Al–Mg alloy. Acta Mater 2009;57:5718–29. [22] Han M-S. Optimization of corrosion protection potential for stress corrosion cracking and hydrogen embrittlement of 5083-H112 alloy in seawater. Met Mater Int 2008;14:203–11. [23] Johnsen J, Holmen JK, Myhr OR, Hopperstad OS, Børvik T. A nano-scale material model applied in finite element analysis of aluminium plates under impact loading. Comput Mater Sci 2013;79:724–35. [24] Gooch W, Burkins M, Squillacioti R. Ballistic testing of commercial aluminum alloys and alternate processing techniques to increase the availability of aluminum armor. 23rd International Symposium Ballistic Tarragona, Spain. 2007. [25] Forrestal MJ, Børvik T, Warren TL. Perforation of 7075-T651 aluminum armor plates with 7.62 mm APM2 bullets. Exp Mech 2010;50:1245–51.
Contents lists available at ScienceDirect
International Journal of Impact Engineering j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / i j i m p e n g
On the perforation of aluminum plates by 7.62 mm APM2 projectiles Zvi Rosenberg, Roman Kositski *, Erez Dekel RAFAEL, P.O. Box 2250, Haifa, Israel
A R T I C L E
I N F O
Article history: Received 5 March 2016 Received in revised form 17 May 2016 Accepted 1 June 2016 Available online 22 June 2016 Keywords: Plate perforation Armor piercing projectiles Ballistic limit velocity Aluminum plates
A B S T R A C T
The paper is concerned with the perforation process of aluminum plates by 7.62 mm APM2 projectiles. In particular, we compare experimental values of the ballistic limit velocities for various aluminum alloys with predictions from our numerically-based model, which has been published a few years ago. The data include different thicknesses of aluminum plates of several alloys, ranging in their dynamic compressive strengths between 0.2 and 0.65 GPa We introduce an empirical correction factor to account for the effect of the brass jacket around the hard steel cores of these projectiles. We also discuss the issues of layered targets and inclined impacts, which may complicate the analysis. The good agreements between our predictions and the data for the ballistic limit velocities enhance the validity of the model, and suggest that it could also apply for other armor piercing projectiles, as well as for other metallic plates. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction The protection capability of armor plates made of strong aluminum alloys has been the subject of intense research for many years due to their high strength and low density, resulting in lower weights for armored vehicles as compared with traditional steel structures. Many publications deal with the ballistics properties of various metals, including strong aluminum alloys, against armor-piercing (AP) projectiles through empirical data and their numerical simulations. A comprehensive review of these perforation models is given in Ben-Dor et al. [1]. The data include the ballistic limit velocities (Vbl) of various plates perforated by a certain AP projectile, as well as its residual velocity after perforation. In Rosenberg and Dekel [2,3], we analyzed the interaction of rigid projectiles with metallic plates using numerical simulations. We derived a relatively simple model which accounts for the values of Vbl for a large range of plate thicknesses impacted by rigid sharp-nosed projectiles. The basic assumption behind this model is that the perforation process is achieved by the so-called ductile hole enlargement mechanism. This is the process by which sharp-nosed rigid projectiles perforate ductile metallic plates, which do not suffer from failure mechanisms such as spalling, scabbing or discing. In order to account for the residual velocities after plate perforation, we used the model of Recht and Ipson [4], which is based on energy conservation considerations. This physically-based model was found to account for several sets of data, as summarized by Rosenberg and Dekel [5].
* Corresponding author. RAFAEL, P.O. Box 2250, Haifa, Israel. Tel.: 972-73-335-4444; Fax: 73-335-5858. E-mail address: [email protected] (R. Kositski). http://dx.doi.org/10.1016/j.ijimpeng.2016.06.003 0734-743X/© 2016 Elsevier Ltd. All rights reserved.
The purpose of the present paper is to demonstrate the ability of this approach to account for recently published data concerning the impact of 7.62 mm APM2 projectiles at aluminum plates made of different alloys. The data we analyze concern the value of Vbl for a given aluminum plate, and the values of the residual velocities as a function of impact velocities of the 7.62 mm APM2 projectile. We highlight the difference between the values of Vbl for the full (jacketed) projectiles and for their hard steel cores, through an empirically derived factor. We also discuss the issue of laminated targets, which are often used for ballistics studies, as well as the impact of 7.62 mm APM2 projectiles at inclined plates.
2. A short summary of the model 2.1. The Recht–Ipson model for the residual velocity The residual velocity of sharp-nosed rigid projectile perforating a ductile metallic plate can be calculated by the Recht–Ipson model [3], which is based on energy conservation, through:
MV02 MVr2 = +W 2 2
(1)
where M is the mass of the projectile, V0 and Vr are its impact and residual velocities, respectively, and W is the work done on the target in opening a hole for the passage of the projectile. The value of W can be derived by the condition that at the ballistic limit velocity (V0 = Vbl) the residual velocity is zero. Thus, one gets: MVbl2/2 = W, and Eq (1) results in the well-known relation of the Recht–Ipson model:
Vr = (V02 − Vbl2 )
0.5
(2)
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Z. Rosenberg et al. / International Journal of Impact Engineering 97 (2016) 79–86
which should apply for targets perforated by rigid projectiles through the ductile hole enlargement process. In cases where target failure results in back surface debris, the energy conservation equation should include the energy of the debris as well. Such cases are relevant for plates perforated by flat-nosed projectiles, when a cylindrical plug is sheared from the target, or when the target material is relatively brittle and its debris is spalled off from the back surface. As discussed by Rosenberg and Dekel [3], the most important result from Eq. (2) is that the entire physics of the perforation process is manifested by the value of Vbl for the given projectile/plate pair. Once this value is determined, either experimentally or analytically, the residual velocities can be calculated by Eq. (2), as demonstrated in Ref. [5] for various sets of projectile/ plate combinations. In the next section, we summarize the numerically-based model, which was presented in Ref. [2], for the predicted Vbl values, through the relevant properties of the target and the projectile. It is important to note here that several workers use a somewhat different relation than Eq. (2), in order to fit their data for the residual velocities, as follows: 1
Vr = a (V0p − Vblp ) p
(3)
where a and p are empirical constants which are best-fitted to the data. This relation is often related to Recht and Ipson [4] while there is no mention of such a relation in that work. In fact, this is an empirical relation due to Lambert and Jonas [6] who proposed it for the residual velocities of eroding long rod penetrators, after they perforate metallic plates. The application of this empirical relation has been primarily used to better illustrate the relationship between the different data points from experiments or numerical simulations. However, we prefer to use the physically-based relation of Recht and Ipson [4], especially when one deals with pointed projectiles perforating plates of ductile materials. It turns out that most best-fits to the Lambert–Jonas relation for pointed nose projectiles show that the term a in Eq. (3) is close to 1 and p is close to 2, confirming the accuracy of the Recht–Ipson model. As we shall demonstrate here, the use of Eq. (3) for the data of rigid projectiles may obscure important insights, which can be gained by analyzing the data through the physically-based relation in Eq. (2).
2.2. The numerically-based model for Vbl The perforation process of metallic plates by rigid projectiles is dominated by the free surfaces of these plates, resulting in a timevarying resisting stress, which acts on the projectile during perforation. As discussed in Ref. [3], a simple penetration model that is based on this time-varying resisting stress, is difficult to construct. Instead, we defined an effective (constant) resisting stress (σr), which the finite plate exerts on the projectile. The basic idea behind this concept is that the work done by this effective stress, throughout the perforation process, is equal the work done by the actual time-varying resisting stress. Thus, one can write for this work: W = πr2σrH, where r is the radius of the projectile, and H is the thickness of the plate. Using this relation together with the one derived from the Recht–Ipson model: W = MVbl2/2, we get:
⎛ 2σ H ⎞ Vbl = ⎜ r ⎟ ⎝ ρ p Leff ⎠
0.5
(4)
where ρp and Leff are the projectile’s density and its effective length, respectively, and their product is determined through: M = πr2ρpLeff. Through a series of numerical simulations, Rosenberg and Dekel [3] derived the dependence of the effective resisting stress (σr) on
the plate’s properties and on the projectile’s diameter (D). They found that σr is dependent on the target’s dynamic compressive strength (Y) and on its normalized thickness (H/D), through the following relations:
σr ⎛ H⎞ = 2 + 0.8ln ⎜ ⎟ ⎝ D⎠ Y σr =2 Y
1 H < <1 3 D
H σr 2 = +4 Y 3 D
H 1 ≤ D 3
H ≥1 D
(5a)
(5b)
(5c)
These relations are due to the different states of stress which the plate experiences during its perforation, as discussed in Ref. [5]. In particular, Eq. (5b) applies for plate thicknesses where the state in the plate is that of plane stress, while Eq. (5a) describes the transition to the more complex three-dimensional stress state. In the present paper, we shall deal with projectile/plate combinations for which H/D > 1.0, and we use Eq. (5a) throughout the paper. Rosenberg and Dekel [5] pointed out that the value for Y should be the dynamic compressive flow stress of the target material, at a strain of about 0.3 and strain rates of the order of 103 s−1. These are the typical values of the rates and strains in the target elements adjacent to the perforating projectile, which experience large compression during the perforation process. Dynamic compressive flow stresses are easily obtained by Kolsky bar tests, which result in compressive stress–strain curves for large strains at the relevant rates. Moreover, these tests should be performed on specimens taken from the plates that are used for the ballistics experiments, in order to avoid possible variations in the properties of plates taken from different lots. In order to understand why the relevant dynamic flow stress (Y) has to be determined at a strain of about 0.3, consider the numerical simulations of Rosenberg and Dekel [3] for rigid projectiles perforating plates of stainless steel 304L. This material has a very large hardening coefficient, which increases its flow stress from a value of 0.34 GPa at yield, to more than 2 GPa at its ultimate value. These simulations showed that the SS304L plates behave in the same way as plates having a constant (von-Mises) strength of Y = 0.82 GPa. Using the constitutive relation for SS304L, with its strong hardening property, one finds that this intermediate value of flow stress corresponds to a strain of 0.3. This result indicates that the average strain of the target elements around the projectile is about 0.3, which is the reason for choosing the value for Y at this strain from dynamic compressive stress–strain curve. In the same way, one may consider the rate of 10−3 s−1 as representing an average strain rate for these elements. Note that the value of 0.3 for the average strain does not mean that the material fails at this strain. In fact, the proposed model applies for ductile materials for which the strain to failure is usually much larger. In order to demonstrate the validity of our approach, consider the data of Roisman et al. [7], who shot tungsten-alloy ogivenosed projectiles, with M = 88 g and D = 11.3 mm, at stacks of three 6061-T651 aluminum plates with a total thickness of 120 mm. The experiments included normal impacts at stacks of three 40 mm thick plates, as well as inclined impacts at an obliquity of 30° with two plates of 40 mm and one of 25 mm. From Eq. (5a), we find that that σr = 3.89Y for this case and with Y = 0.42 GPa, as measured for the 6061-T651 alloy in our Kolsly bar system, we get from Eq. (4): Vbl = 668 m/s for this case. With this value of Vbl we can calculate the Vr(V0) curve from Eq. (2). Fig. 1 shows the excellent agreement between the model’s predictions and the data from Ref. [7], as far as Vbl and Vr(V0) are concerned.
Z. Rosenberg et al. / International Journal of Impact Engineering 97 (2016) 79–86
800
1000 900
81
data model
700
data model
800
600 700
500 Vr (m/s)
Vr (m/s)
600 500 400
400 300
300
200
200
100
100 0 600
700
800
900 V0 (m/s)
1000
1100
1200
Fig. 1. The agreement between our model (Vbl = 668 m/s) and the data from Ref. [7].
0 500
600
700 V0 (m/s)
800
900
Fig. 2. Comparing the model predictions (Vbl = 517.4 m/s) with the data for 20 mm plates of 5083-H116 aluminum perforated by the steel cores.
3. Comparing the model’s predictions with experimental data In this section we compare the model’s predictions with several sets of data, which were published recently, for the perforation of aluminum plates by 7.62 mm APM2 projectiles. We start with the data for the hard steel cores impacting aluminum plates of different alloys, since this is the case where the projectile is a truly rigid one with well-defined dimensions. The case of a full projectile, which includes the soft jacket surrounding the hard steel core, is discussed later where we account for the effect of the jacket on the value of Vbl. We shall also address the issue of target lamination since, often, the data are given for targets that consist of two or three plates that are held together. 3.1. Perforations by the hard steel cores Data for the perforation of Al 5083-H116 plates by both the hard steel cores and the full (jacketed) 7.62 mm APM2 projectile is presented in Ref. [8]. The diameter of the core is 6.17 mm and its weight is 5.25 g from which we get: ρpLeff = 17.57 g/cm2, which should be used in Eq. (4) for the derivation of the corresponding Vbl values. For consistency of units in Eq. (4), one should use GPa for the strength term (σr) and H should be given in centimeters. The values for Vbl are then obtained in units of km/s. The compressive stress–strain curve for this alloy, as given in Ref. [8], shows that at a strain of about 0.3, the flow stress is about Y = 0.4 GPa. Using this value in Eq. (5a), together with H/D = 3.24 for the 20 mm plates, results in σr = 1.176 GPa for this plate/projectile combination. Inserting this value in Eq. (4), together with the value of ρpLeff, results in a predicted value of Vbl = 517.4 m/s for this case, which is in excellent agreement with Vbl = 513 m/s, as deduced in Ref. [8] from the residual velocity measurements. Fig. 2 shows that Eq. (2) for Vr, together with our predicted value for Vbl, accounts for the residual velocity data in a very satisfactory way. This agreement enhances the claim of Rosenberg and Dekel [5], that the whole Vr(V0) curve, for a given projectile/plate configuration, can be calculated through Eqs. (4) and (5) and the Recht–Ipson relation for Vr, as given by Eq. (2). Our next step is to consider the data in Ref. [8] for targets that were made of two adjacent 20 mm plates. Obviously, if the plates are strongly fastened to each other, they can be treated as a single plate which is 40 mm thick. However, as we do not know how well
these plates were held together, before and during the perforation process, we assume that a small interval may have existed between the two plates. In order to assess the effect of such intervals on the ballistic limit velocity, we performed two sets of simulations, one with a single aluminum plate, 40 mm thick and another with two 20 mm plates, separated by an interval of 0.5 mm. The targets were modeled as elastic perfectly plastic, with yield strength of 0.4 GPa. The ogive nosed steel projectile in these simulations had a diameter of 6.2 mm and a mass of 5.3 g, which are very close to those for the hard steel core of a 7.62 mm APM2 projectile. We should note that these simulations were performed in order to check the influence of layering, and the hard steel core steel was chosen in order to avoid the complications arising from the jacket of the full AP projectile. The impact velocity in these simulations was 1000 m/s and the resulting residual velocities were 603 m/s and 629 m/s, for the single and the double-plate targets, respectively. Using the Recht– Ipson model, as depicted by Eq. (2), we get a difference of about 2.6% for the inferred ballistic limits of the single and double-plate configurations. We shall take this difference into account for the doubleplate targets in Ref. [8]. Namely, we shall multiply the calculated values of Vbl for the single plate by a factor of 0.974, in order to predict the value for the double-plate configuration. For the tripleplate targets, we shall use a correction factor of (0.974)2 = 0.95. Note, that these corrections can be smaller if the interval between the plates is smaller, and we may consider the 2.6% difference as an upper bound for such a correction. Another set of simulations for the two arrangements discussed here, was performed at an impact velocity of 800 m/s, which is close to the ballistic limit velocity of this configuration. The inferred values of Vbl for the single and doubleplate targets differed by only 1.8%, which shows that an interval of 0.5 mm between the plates may have a smaller effect at lower impact velocities. The first step in our calculation is to find the value of Vbl for a single plate 40 mm thick, which corresponds to H/D = 6.48. Inserting this value and Y = 0.4 GPa in Eq. (5a) results in: σr = 1.4 GPa for this plate/core combination. With this value for σr, Eq. (4) results in a predicted value of Vbl = 797.8 m/s for the 40 mm plate. Multiplying this value by the correction factor of 0.974 from our simulations, we get a predicted value of Vbl = 777 m/s for the doubleplate target. Note that Børvik et al. [8] deduced a value of Vbl = 767 m/s
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from their data, which is very close to our predicted value. For their triple-plate targets, made of three plates 20 mm thick, we first find that for a single 60 mm target (H/D = 9.7) the effective stress (σr) is 1.53 GPa and the corresponding value for Vbl is 1022 m/s. This value is in excellent agreement with the value of 1025 m/s deduced in Ref. [8]. However, we have to follow the same procedure for this triple-plate target and correct the value of Vbl due to the layering effect. The correction factor for the triple-plate target is (0.974)2, which results in a predicted value of Vbl = 969.5 m/s for the triplelayered targets. Fig. 3 shows the agreement between our model’s predictions and the data for Vr from Ref. [8], for the double and triple-plate targets, with the corrected values for the corresponding Vbl values as discussed above. Clearly, the agreement is excellent for the doubleplate targets enhancing the validity of our approach concerning the layering effect. As for the triple-plate targets, our model accounts very well for the data at the high velocity shots. On the other hand, at impact velocities just over 1000 m/s the model deviates from the data, which resulted in zero residual velocities. In fact, the low velocity experiments suggest that Vbl should be around 1025 m/s, which is practically the same value we calculate for a single plate 60 mm thick (1022 m/s). We assume that these conflicting results reflect the complications arising for laminated targets, and they may be due to differences in the attachment of the plates during their perforation. Namely, the three plates could remain closely attached at the lower impact velocities, while they may have separated somewhat during the high velocity impacts. Obviously, one should use single plates rather than multiple-plate targets, in order to avoid such discrepancies. We assume that experiments with single plates, 60mm thick, will result in a value of Vbl = 1020 m/s and that the residual velocity data would follow the Recht–Ipson model, without the large “jumps” around Vbl, as seen in Fig. 3. On the other hand, when the application demands a multi-plate target, one may expect the data to look as in Fig. 3 for the three-plate targets. We strongly recommend that even with this type of data, one should use the Recht–Ipson model as given by Eq. (2), rather than the empirical relation in Eq. (3), which obscures the physics of the process. Another set of data is from Forrestal et al. [9], where both the cores and the jacketed projectiles are shot at 20 mm thick Al 6082T651 plates, at normal incidence as well as at several obliquities.
The stress–strain curve for this alloy is very similar to that of the 5083-H116 alloy and we chose the same value of Y = 0.4 GPa for the 6082-T651 alloy. Note that our model resulted in a value of Vbl = 517.4 m/s for a 20 mm thick plate with a flow stress of 0.4 GPa, which is very close to the value of Vbl = 514 m/s deduced in Ref. [8] for the normal impacts. Fig. 4a shows the excellent agreement between our predictions for Vr(V0) and the data from Ref. [9], for the steel cores impacting these 20 mm plates at normal incidence. Forrestal et al. [9] note that the trajectories of the projectiles in the inclined plates were very straight, and one may assume that the effective thicknesses of these plates are H/cosβ, where β is the obliquity angle. These effective thicknesses are: Heff = 20.7, 23.1 and 28.3 mm, for a 20 mm thick plate at obliquities of 15°, 30° and 45°, respectively. Using the same procedure for the inclined plates, we find the following values for the effective resisting stress: σr = 1.187, 1.225, and 1.287 GPa, for the three obliquities. These values result in the following values for Vbl from Eq. (4): Vbl = 529, 567 and 644 m/s for β = 15°, 30° and 45°, respectively. Once we have these predicted values for Vbl , we can compare the predicted values for Vr, from Eq (2), with the data for the steel cores at these obliquities, and Fig. 4b shows this comparison.
800 700 600
data 2x20mm model 2x20mm data 3x20mm model 3x20mm
Vr (m/s)
500 400 300 200 100 0 700
800
900 V0 (m/s)
1000
1100
Fig. 3. Comparing our model’s predictions with the data for the double-plate (Vbl = 777 m/s) and for the triple-plate (Vbl = 969.5 m/s) targets.
Fig. 4. Comparing the model’s predictions with data for 20 mm plates of Al6082-T651, (a) normal impacts (Vbl = 417.4 m/s), (b) inclined impacts: β = 15° (Vbl = 529 m/s),β = 30° (Vbl = 587 m/s), β = 45° (Vbl = 644 m/s).
Z. Rosenberg et al. / International Journal of Impact Engineering 97 (2016) 79–86
3.2. Perforations by full (jacketed) projectiles The full 7.62 mm APM2 projectile has a complex structure, which includes a brass jacket around the core and a lead cap at the front. The total weight of the full projectile is about 10.5 g, twice the weight of the steel core, which cannot be ignored when one is considering the interaction of this projectile with metallic plates. On the other hand, these soft parts are striped off quite early in the perforation process, and one usually finds a very deformed brass jacket embedded in the plate near its impact face. Most workers ignore the effect of the jacket on the perforation process and treat the full projectile as if the jacket does not contribute to the process. Still, they do find measurable differences between the values of Vbl for the steel core and the full projectile, as in Forrestal et al. [8] and Holmen et al. [12], for example. Obviously, an analytical model, based on energy considerations, is not easy to construct because of the plastic work involved in the jacket’s deformation, which may vary with the impact velocity and with the strength of the target. Numerical simulations have to consider the attachment of the jacket to the core, which is not easy to simulate. Still, we find several attempts for such simulations from which we can have a rough estimate for the effect of the jacket. For example, Holmen et al. [12] performed simulations for both the steel cores and the full 7.62 mm APM2 projectiles, perforating 20 mm plates of the Al6070-T6 alloy. The simulations were performed for an impact velocity of 903 m/s and the resulting residual velocities were 784 and 754 m/s for the full projectile and the steel core, respectively. Using Eq. (2) we find that the inferred values for the ballistic limit velocities are 448 m/s for the full projectile and 497 m/s for the steel core. The ratio of these values is about 0.9, which means that we can expect a reduction of about 10% in the values of Vbl for the full projectiles, as compared with those for the steel cores. In order to compare this result with experimental data, consider the data in Forrestal et al. [9] for the 20 mm 6082-T651 plates impacted by the full (jacketed) projectile at normal incidence, given in Table 1. A least square fit of these results, using Eq. (2), results in a value of Vbl = 480 m/s, which best-fits the data in this table. This value is smaller by about 7% from the corresponding value of Vbl = 517.4 m/ s, which we derived for the steel cores impacting the same plates. Thus, in the following discussion we shall use a correction factor of 0.93 in order to predict the values of Vbl for the full projectiles, from the calculated values for the steel cores. Note that the cor-
rection factors from the numerical simulations in Ref. [12], and from the data of Ref. [9], are quite close: 0.9 and 0.93, respectively. We chose the experimentally derived factor because it avoids possible uncertainties due to material properties in the simulations. We should also note that the least-square fitting technique, which we used to assess the value of Vbl from Table 1, is preferable to the other technique for Vbl determination, by which one takes the average of the lowest velocity perforation shot and the highest velocity for nonperforation. With the least square fit one uses all the available data rather than only two shots, which may have a large scatter because they are very close to the ballistic limit velocity. The most important question here concerns the reason for the difference between the values of Vbl of the steel core and the full projectile. The simplest explanation is to assume that as the jacket impacts the plate it creates an initial crater before the core reaches the plate and starts to penetrate it. Namely, the jacket reduces the actual thickness of the plate which is being perforated by the core. Through Eq. (4) we can estimate the thickness reduction which will result in a 7% difference between the corresponding Vbl values. As an example, for plate thicknesses of 26 mm and 52 mm we find that one needs thickness reductions of 3 mm and 6 mm, respectively, in order to get the 7% difference in the corresponding Vbl values. In order to check these assessments, we performed two simulations with a full projectile impacting a thick aluminum plate at 500 and 1000 m/s, which are, roughly, the ballistic limits of aluminum plates having thicknesses of 26 and 52 mm, respectively. These simulations showed that the jacket penetrated to depths of 3 and 5 mm before the core started to penetrate the plate, supporting our assumption concerning the effect of the jacket. With the correction factor of 0.93, we can predict the values of Vbl for the full projectiles perforating the 20 mm inclined plates in Ref. [9]. Fig. 5 shows the agreement between the model’s predictions with the data for obliquities of 15° and 30°. We did not bring the comparison for the plates inclined at 45° because the corresponding data is very scattered. We find good agreement between our model and the data for the plates, which are inclined at 15°. A somewhat less agreement is obtained for the low velocity shots at an obliquity of 30°, which imply that the value for Vbl should be higher. On the other hand,
800 data =15 o
700 600
V0(m/s) Vr(m/s)
474 0
500 0
508 105
568 290
573 317
662 464
806 667
917 787
model =30 o
500 400 300 200 100 0 400
Table 1 Residual velocities of full projectiles perforating 20 mm plates of Al 6082-T651.
model =15 o data =30 o
Vr (m/s)
It is clear that the agreement between the model’s predictions and the data is very good for the obliquities of 15° and 30°. A somewhat less agreement is seen for the β = 45° case at impact velocities near the ballistic limit. This discrepancy may be due, at least partly, to the large scatter in the data for Vr at the low velocity range. A very large scatter in the values of Vr is also evident in Ref. [10], for 7.62mm APM2 projectiles impacting 20mm thick 6082-T4 aluminum plates, at an obliquity of 45°. The scatter may be due to a somewhat curved trajectory of the projectile in these targets at high obliquities and for relatively low impact velocities. These curved trajectories, having an S-shape, are due to the asymmetric forces which the front and the back faces of the target exert on the projectile. Such asymmetries are known to take place at high obliquities, especially for impact velocities near the ballistic limits, as shown by Piekutowski et al. [11].
83
500
600
700
800
900
V0 (m/s) Fig. 5. Comparing the model’s predictions with the data for inclined 20 mm plates of 6082-T651 aluminum perforated by full projectiles (Vbl = 492 m/s for β = 15°, and Vbl = 527 m/s for β = 30°).
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the agreement for the high velocity shots, at this obliquity, is very good. We have seen this phenomenon earlier, for steel cores perforating highly inclined plates. Similarly, we assume that the trajectories of the full projectiles in highly inclined plates, at the low velocity shots, may be somewhat curved due to the asymmetric forces which the front and back faces exert on the projectile at large obliquities and low velocities.
we find that the resisting stress is σr = 2Y, from Eqs. (5a) and (5b). Substituting this value for σr together with H = D in Eq. (4), results in the following expression for the ballistic limit velocity (Vbl*) of this plate thickness:
4. Comparison with experimental data of Vbl for various aluminum alloys
from which we find, for example, that: Vbl* = 167.6, 237, 278 and 303 m/s, for the 7.62mm APM2 hard steel cores perforating plates with flow stresses of: Y = 0.2, 0.4, 0.55 and 0.65 GPa, respectively. For larger plate thicknesses ( H ≥ D ) we find, by Eqs. (4) and (5a), for the normalized values of Vbl:
0.5
(6)
⎡H ⎛ ⎛ H⎞⎞ ⎤ = ⎢ ⎜ 1 + 0.4 ln ⎜ ⎟ ⎟ ⎥ ⎝ ⎝ D⎠⎠ ⎦ D ⎣ Vbl* Vbl
0.5
for
H ≥1 D
(7)
and for normalized thicknesses between 1/3 and 1.0 we get, from Eqs. (4) and (5b), the following relation for the normalized ballistic limits:
⎛ H⎞ =⎜ ⎟ Vbl* ⎝ D ⎠ Vbl
0.5
1 H < <1 3 D
for
(8)
Fig. 6 shows the agreement between the data collected in Table A1 for all the single-plate targets, and the normalized curve for Vbl as calculated through Eq. (7). Note that the calculated curve was multiplied by the correction factor of 0.93, in order to obtain the values of Vbl for the full (jacketed) projectiles, as discussed above. One may consider Eqs. (6) and (7) as a universal representation of the normalized Vbl values for different sharp-nosed projectiles perforating ductile metallic plates. Each projectile/plate pair is defined by its corresponding value of Vbl*, through Eq. (6), which is then used to determine the values for Vbl for any plate thickness (for H > D) through Eq. (7). Thus, we expect that Fig. 6 has a “universal” nature and that it should also apply for other ductile metallic plates perforated by 7.62 mm APM2 projectiles. Moreover, we assume that the approach presented here should also account for other armorpiercing sharp-nosed rigid projectiles, although different correction factors may be needed to account for the effect of their jackets, as
5 4.5 4 3.5 *
V bl/Vbl
We are now ready to compare published values of Vbl for various aluminum plates, perforated by the full 7.62 mm APM2 projectiles, with the predicted values as calculated by our model. The analysis follows the same steps outlined above, namely, we first compute the value of σr for the plate/core pair from Eq. (5a), using the dynamic flow stress (Y) of the alloy and the normalized plate thickness (H/D). As discussed above, the value for the flow stress in our model (Y) should be taken from a dynamic test in the Kolsky bar system, at a strain of 0.3 and a strain-rate of about 103 s−1. The next step is to use Eq. (4) in order to find the value of Vbl for the steel core, which is then multiplied by the correction factor of 0.93 in order to obtain the value of Vbl for the full projectile. Table A1 in the Appendix shows the good agreement between the predicted values of Vbl for various aluminum plates, with Vbl data from several sources, most of which are summarized in Refs. [9] and [13]. The table shows that our model captures the main features of the interaction between 7.62 mm APM2 projectiles and aluminum plates, as far as the effects of plate strength and its thickness are concerned. The effect of the projectile’s jacket seems to be represented very well by the correction factor 0f 0.93, which we derived through a single set of experiments. Moreover, the layering issue seems to be represented by the factor of 0.974, as deduced from our numerical simulations. We should also note that the 7075 alloy is considered to be a relatively brittle material, as compared with the other alloys discussed here. Still, we find that the model applies for plates made of this alloy as well. This agreement is probably due to the fact that these plates are much thicker than the diameter of the projectile (20–40 mm vs. 6.2 mm). In contrast, when the projectile has a diameter which is similar to the plate’s thickness, we find that the back surface of the plate shows tensile failure features, besides the ductile hole enlargement perforation process. Such features are clearly seen in the work of Børvik et al. [14] who shot ogive nosed projectiles, with D = 20 mm, at 20 mm thick 7075-T651 aluminum plates. The sectioned plates show a significant back surface failure, which resulted in a low ballistic limit velocity. The tensile failure at the back side of these plates is probably due to significant stress waves which reach the back surface and reflect back as tensile waves. Such waves may lead to the ejection of spalled material, lowering the effective thickness of the plate and resulting in a reduced value of Vbl. In fact, Børvik et al. [14] found that the ballistic limit velocity for a 20 mm thick 7075-T651 plate, perforated by the D = 20 mm ogive-nosed projectile, was lower by 20% than the corresponding value for a 20 mm thick 5083- H116 plate which has a much lower strength. In contrast, when the projectile’s core diameter is significantly smaller than the plate thickness (6.2 mm vs. 20–40 mm as in Table A1), the amplitude of these tensile waves are reduced considerably, and the plates are not expected to fail as easily. In conclusion, the analysis presented here should be applied carefully for brittle materials, such as the 7075 aluminum alloy, especially when relatively thin pates are considered. The resulting values of Vbl can be represented in a normalized form, following the scheme suggested by Rosenberg and Dekel [5]. The normalizing factor is the value of Vbl for a plate of thickness which is equal to the diameter of the projectile (H = D). For this case
⎛ YD ⎞ Vbl* = 2⎜ ⎝ ρ p Leff ⎟⎠
3 2.5 2 1.5 1 0
1
2
3
4
5
6
7
8
9
10
11
12
H/D Fig. 6. Comparing the model for the normalized Vbl values of 7.62mm APM2 projectiles with the data in Table A1.
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Table 2 Comparing the model’s predictions with data for Vbl (in m/s) of 12.7 mm APM2 projectiles impacting 2139-T8 aluminum plates. H (mm) Vbl (exp) Vbl (calc)
40 668 668
40.9 677 677
52.1 785 787
53.8 796 803
57.2 819 833
64.1 874 896
we show below for the 12.7 mm APM2 projectile. On the other hand, we do not expect this approach to apply for plates, which fail by adiabatic shearing, such as the Ti-6Al-4V alloy, or for projectile which break during impact at very high strength plates. We should note that Eqs. (6) and (7) are different from the corresponding equations derived by Forrestal et al. [9], in which the values for Vbl are proportional to (Hσs)0.5. Their approach is based on the cylindrical cavity expansion analysis to derive the resisting stress (σs), which the plate exerts on the projectile. According to this model the resisting stress has the same value for all plate thicknesses, while in our numerically-based model, the resisting stress (σr) is dependent on H/D, as given by Eqs. (5). The recent model of Masri [15] is similar to ours, in the sense that the resisting stress is dependent on the normalized plate’s thickness. On the other hand, his derivation is based on the spherical cavity expansion analysis, which is similar to the approach in Ref. [9]. In order to check the applicability of our model for other projectiles, consider the data from Cheeseman et al. [16], for 12.7 mm APM2 projectiles perforating 2139-T8 aluminum plates. The structure of these projectiles is similar to the structure of the 7.62 mm projectiles, with a hard steel core encased in a soft metallic jacket. We assume that our approach should apply to these projectiles as well, and the only difference may be due to a somewhat different correction factor for the values of Vbl of the full projectile. In order to determine this factor for the 12.7 mm APM2 projectile we use the data from Ref. [16] for the 39 mm thick plate of the 2139-T8 alloy, which was the thinnest plate they used in their study. The hard steel core of the 12.7 mm projectiles has a diameter of 10.9 mm and a weight of 25.9 g, from which we get ρpLeff = 27.77 g/cm2 for the core. Using this value together with Y = 0.55 GPa for the dynamic flow stress of the 2139-T8 alloy, we find that Vbl = 683 m/s for the hard steel core impacting the 39 mm thick plate. The experimental value of Vbl for the full projectile is 657 m/s, as given in Ref. [16]. Thus, we conclude that the correction factor is 0.962 for this projectile/ target pair. This is a somewhat larger factor than the one we found for the 7.62 mm APM2 projectile (0.93), but it is a reasonable value. Our next step is to use this factor (of 0.962) in order to predict the values of Vbl for the other plates which were tested in Ref. [16]. The procedure we use is the same as before, namely, we first calculate the values of Vbl for the different plates impacted by the hard steel core through our numerically-based model, and multiply these values by 0.962 to get the predicted values for the full 12.7 mm APM2 projectiles. Table 2 shows the results of this analysis together with the experimental results in Ref. [16], and it is clear that the agreement between the two sets is very good. It would be interesting to compare our model’s predictions with other sets of data for this projectile, as well as with data for other AP projectiles, such as the 14.5 mm bullet, which has a tungsten-carbide core. 5. Summary This paper compares predictions from a numerically-based perforation mode and experimental data for the ballistic limit velocities of aluminum plates impacted by 7.62 mm APM2 projectiles. The model applies for sharp-nosed rigid projectiles perforating ductile metallic plates through the hole enlargement process. It is based on the notion of an effective (average) resisting stress, which the plate exerts on a given projectile during the perforation process. This
85
stress depends on the dynamic compressive flow stress of the plate material, and on the normalized thickness of the plate. The value of Vbl, for a given projectile/plate combination, is determined by this effective stress and by the thickness of the plate. The model accounts for perforations by “simple” projectiles such as the hard steel cores of the APM2 projectiles, as we showed here. In order to account for the data of the jacketed 7.62 mm APM2 projectile, we introduced a correction factor (of 0.93), which was derived from a specific set of experiments. The agreement between the model’s predictions and the data of Vbl, for plates of various aluminum alloys, is shown to be very good. In fact, the vast majority of the data fall within 2–3% from our predicted values. This agreement is especially encouraging since we have used data for plates with a large range of dynamic strengths (0.2–0.65 GPa), as well as large range of plate thicknesses (18.6–60 mm). We have also introduced a correction factor of 0.974 in order to account for the possible effect of layering a target by two adjacent plates. This issue needs further study, both empirical and numerical, but it seems that the layering effect is bounded by about 2.6%, as far as the values of Vbl are concerned. As for the residual velocities of the perforating projectiles, we have shown that the physicallybased model of Recht and Ipson [3] accounts for the data analyzed here. This model, which is based on energy conservation considerations, should be preferred over empirical relations which are used to fit a certain set of experiments. We emphasized the importance of having the values of the dynamic compressive flow stresses for the target materials. These values should be obtained by compressive Kolsky bar tests, at strain rates of about 103 s−1 and strains of 0.3, which are the relevant values for the target elements adjacent to the perforating projectile. Moreover, it is very important to characterize the properties of the actual plates that are used for the actual ballistics study. We should note that these values of the compressive strains and strain-rates are the average values experienced by the target elements around the perforating projectile. Finally, we expect that the approach presented here will account for other armor piecing projectiles perforating targets that are made of ductile materials. We do not expect that this approach will account for bluntnosed projectiles or for relatively brittle targets, where the penetration mechanism is mainly through adiabatic shearing, rather than the ductile hole enlargement mechanism. Appendix A In this part of the paper we demonstrate the good agreement between our model’s predictions and the experimental data for Vbl values of 7.62 mm APM2 projectiles impacting aluminum plates, made of various alloys, and having different thicknesses. The agreement is clearly seen in Table A1 below. As mentioned above, published works which deal with projectile perforations do not always supply the dynamic flow stresses for the plates, and we had to look for these values in other sources. As an example, for the ballistics data of the 2139-T8 alloy in Ref. [16], we used the dynamic flow stress (0.57 GPa) from Kolsky bar tests of Vural and Caro [17], at a strain rate of 103 s−1. Tucker et al. [18] show that the dynamic stress–strain curve for the 5083-H131 alloy is very similar to that of the 6061-T651 alloy. Thus, we used the same value of Y = 0.42 GPa for both alloys. For the 2024-T351 plates in Ref. [13], we used a value of Y = 0.55GPa, since published data are found to be in the range of Y = 0.55-0.6GPa. Similarly, published values for the dynamic strength of the 7075-T651 alloy range between 0.65 and 0.7GPa, and we used the value of Y = 0.65GPa in our calculations. For the 5059H131 plates from Ref. [19], we estimated a value of Y = 0.45 GPa. The data for the newly developed 7085-E01 alloy, from Ref. [20], has been analyzed with a value of Y = 0.65 GPa, because the static values for the strength of this alloy, as well as its Brinell hardness, are close to those of the 7075-T651 alloy. For the somewhat weaker
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Table A1 Comparing predicted Vbl values (in m/s) with the data for various aluminum plates. Alloy (flows stress)
H (mm)
Vbl (exp)
Vbl (calc)
Reference
6070-T0 (0.2 GPa) 6070-T4 (0.42 GPa) 6070-T6 (0.48 GPa) 6070-T7 (0.45 GPa) 5083-H116 (0.4 GPa)
20 20 20 20 20 2 × 20 3 × 20 37.8 50.9 54.7 57.2 20 30 40 25.7 26 38.8 51.2 20 2 × 20 26 30 32 38 25.4 26.3 30.4 34.6 37.9 38.9 47.8 51.4 25.4 38.7 51 25.2 32.3 39 40.9 26 30 34 38 18.6 25.4 38 40.5 18.5 25.4 38 40.5
348 506 562 529 492 722 912 712 876 890 927 474 589 698 596 583 754 883 628 909 718 784 818 909 589 610 661 705 763 774 860 913 584 737 874 683 783 860 893 652 715 774 831 598 772 908 938 567 664 842 872
340 493 527 510 481 723 901 734 880 911 946 463 597 710 577 582 746 884 610 919 723 792 824 915 593 607 664 719 761 773 878 917 573 744 882 665 776 871 897 666 716 787 843 586 713 916 953 547 668 858 892
[12]
5083-H131 (0.42 GPa)
5083-H112 (0.37 GPa)
6061-T651 (0.42 GPa)
7075-T651 (0.65 GPa) 7075-T651(0.65 GPa)
5059-H131 (0.45 GPa)
5059-H136 (0.42 GPa)
2139-T8 (0.57 GPa)
2024-T351 (0.55 GPa)
7085-E01 (0.65 GPa)
7085-E02 (0.57 GPa)
num plates, in Ref. [12], we estimated the flow stresses from the stress–strain curves in Ref. [23]. Note that the calculated values for the double-layered (40 mm) and triple-layered (60 mm) include the correction factors for layered targets as discussed above, namely a factor of 0.974 for the double-plate target and 0.95 for the stack of three plates.
[14]
References [8]
[13]
[24]
[25] [19]
[19]
[16]
[13]
[20]
7085-E02 plates from Ref. [20] we chose Y = 0.57GPa, based on their Brinell hardness, as compared with the hardness of the 7085-E01 plates. We should note that the hardness values, as given by Gallardy [20] for each plate thickness, differ by about 7% for both alloys. Thus, we may expect variations of a few percent in Vbl values due to differences in plate strengths. This fact enhances the need to measure the dynamic properties of the actual plate which is used for ballistic testing, as stated above. For the 5083-H112 data in Ref. [13] we estimated a value of Y = 0.37GPa through the static tensile values from Refs. [21] and [22]. For the differently treated 6070 alumi-
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