Oblique perforation of thick metallic plates by rigid projectiles

Transcription

1 Acta Mechanica Sinica (2006) 22: DOI /s RESEARCH PAPER Oblique perforation of thick metallic plates by rigid projectiles Xiaowei Chen Qingming Li Saucheong Fan Received: 17 October 2005 / Revised: 14 March 2006 / Accepted: 3 April 2006 / Published online: 20 June 2006 Springer-Verlag 2006 Abstract Oblique perforation of thick metallic plates by rigid projectiles with various nose shapes is studied in this paper. Two perforation mechanisms, i.e., the hole enlargement for a sharp projectile nose and the plugging formation for a blunt projectile nose, are considered in the proposed analytical model. It is shown that the perforation of a thick plate is dominated by several non-dimensional numbers, i.e., the impact function, the geometry function of projectile, the non-dimensional thickness of target and the impact obliquity. Explicit formulae are obtained to predict the ballistic limit, residual velocity and directional change for the oblique perforation of thick metallic plates. The proposed model is able to predict the critical condition for the occurrence of ricochet. The proposed model is validated by comparing the predictions with other existing models and independent experimental data. The English text was polished by Keren Wang. X. Chen (B) Institute of Structural Mechanics, China Academy of Engineering Physics, P.O. Box , Mianyang , China chenxiaoweintu@yahoo.com Q. Li School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, P.O. Box 88, Pariser Building, Manchester M60 1QD, UK S. Fan Protective Technology Research Center, School of Civil and Environmental Engineering, Nanyang Technological University, Nanyang Avenue, Nanyang, Singapore Keywords Oblique perforation Ballistic limit Metallic plate Rigid projectile Shear plugging 1 Introduction Due to their great importance in military and civil applications, penetration and perforation have been studied intensively. Extensive reviews were conducted by Backman and Goldsmith [1], Brown [2], Anderson and Bodner [3], Corbett et al. [4] and Goldsmith [5], which outlined the advances and progresses in the penetration and perforation mechanics in the 20th century. Especially, Goldsmith [5] gave a detailed review on analytical, numerical and experimental investigations of non-ideal projectile impact on targets, including oblique impact, yaw impact, impact with yaw and obliquity, impact on moving targets, rotating penetrators, impact of tumbling projectiles, soft missile impact, impingement of jets or long rods with relative target motion, and ricochet. In general, the analytical approach is simple, efficient and successful in the studies of the penetration and perforation processes. The perforation of a thick plate by a rigid projectile is usually controlled by both the penetration process and the final failure process. Multistage models have been proposed to study both the normal and oblique perforations of relatively thick plates [6 8]. Zener and Peterson [9] pointed out that the difference between the oblique perforation process and the normal impact is due to the increase in the effective target thickness, H eff = H/cos(β + δ), where δ is the directional change of the penetration path due to the action of the transverse force applied to the projectile. Recht and Ipson [10] and Ipson and Recht [11] further

2 368 X. Chen et al. concluded that the pivoting point of the directional change is close to the impact surface, and thus extended the expression for the normal impact to the oblique impact based on an energy balance. From more recent experimental results, e.g., Fig. 2 of Ref. [12] and Figs of Ref. [13], we find that the direction change indeed occurs at the initial stage of oblique impact of rigid projectiles. It is a challenge to include the dominant deformation and failure modes while retaining simplicity and a reasonable degree of accuracy in a general model of oblique perforation. In this paper, an analytical model based on the dynamic cavity expansion theory and plug formation is proposed in Sect. 2. Its applications to oblique perforations of thick metallic plates by projectiles of given nose-shapes are shown in Sects. 3 and 4. The derived explicit formulae cover those for the normal perforation model developed earlier by the authors [14]. Predictions by those formulae are in good agreement with experimental results, as shown in Sect. 5. In addition, it can be used to predict the critical condition for the occurrence of ricochet, as discussed in Sect Analytical model Consider a rigid projectile with mass M,diameter d and an arbitrary nose shape with nose length h impacting a thick metallic plate (thickness H) at initial velocity V 0 with an initial oblique angle β,asshowninfig.1.after the first stage of directional change, the projectile penetrates into the target medium at rigid-body velocity V. In the present model, we follow Refs. [10, 11] to assume that the directional change δ takes place near the front surface when the projectile heading into the target and being subjected to asymmetric resistance. Accordingly, the effective thickness of the target is replaced by H eff = H/cos(β + δ). After the change of projectile direction, there exists only the drag force along the axial direction of the projectile (i.e., the symmetric force) during the penetration process, which can be determined by the dynamic cavity expansion model [15 17]. Subsequently, the motion can be described in two phases, i.e., perforation and shear plugging, as demonstrated in Fig. 1. It is assumed that the plug and projectile travel in the same direction and with the same velocity after perforation. However, for a sharp and slender projectile, the final phase of plugging may not exist. 2.1 Evaluation of initially directional change δ The initial entering process is very complex in the case of oblique impact. As the action line of the asymmetric Fig. 1 The complete perforation process of thick plates by a rigid projectile at initial obliquity angle β resistance does not pass the mass center of the projectile, there exists a time-dependent momentum moment. The projectile enters the target with varying contact area and rotates around its mass center in the same time. The initial trajectory of the nose tip looks like a spiral line before the projectile head submerges into the target. Accompanied with the rotation of the projectile, the asymmetric resistance tends to comply with the central axis of the projectile. After the projectile head enters the target completely, the drag force F becomes axisymmetric and then the trajectory of nose tip becomes a straight line with obliquity (β + δ) in the subsequent stages. However, in the present modelling we approximately decompose the initial spiral line into an arc with angle δ, and a straight line with obliquity (β + δ) in the first stage. In other word, the rotation of the projectile has been taken into account by introducing an arc trajectory and the angle change along the arc is the directional change δ. Meanwhile, the dynamic cavity expansion model is used to analyse the motion of the projectile along the straight line with obliquity (β + δ), including the initial submerging stage when the contact area between projectile nose and target varies. Herein the oblique impact may be regarded as a normal penetration/perforation throughout the target along the direction (β + δ), after considering an initial change of the direction angle. When the motion of the projectile trajectory is assumed in the initial stage, two methods, i.e., momentum

3 Oblique perforation of thick metallic plates by rigid projectiles 369 method or energy method, may be used in the lateral direction to estimate the directional change. Previous work [10, 11] used the first method. The present model adopts the second approach. It is assumed that the kinetic energy consumption is equal to the mechanical work done by the average lateral force along the arc. Thus the angle of direction change δ can be calculated. It is assumed that the angle of direction change, δ, is small in the initial stage. According to the geometry relation, the two components of the initial velocity perpendicular and parallel to the penetration path are V = V 0 sin δ, V x = V 0 cos δ, (1a) (1b) respectively. Implicitly, the velocity component in the direction defined by δ, i.e. V 0 cos δ, keeps unchanged during the course of direction change. It conflicts with the reality, but the analysis along the direction of obliquity, (β + δ), do not need this assumption. When the head of the projectile enters the target completely, if ignoring any slide in the surface of the target, the distance between the tip of the projectile head and the just submerged cross-section perpendicular to the axisymmetrical line is h + d tan(β + δ)/2 as shown in Fig. 2. However, an apophysis is usually formed close to the impact point in the surface of the target due to the plastic flow of the target material. The influence of the apophysis should be taken into account when we discuss the complete imbedding of the projectile nose into the target. It means, the distance along its central axis by which the projectile imbeds the target, should be greater than h+d tan(β + δ)/2; and to simplify the problem, as shown in Fig. 2, this distance is assumed to be h+d tan(β + δ). It implies that the height of the apophysis is considered to be d tan(β + δ)/2, with respect to the small angle of impact obliquity β. The directional change completes just when the projectile nose imbeds the target. The total turning path during the whole course of direction change is s. It can be approximated as an arc such that s =[h + d tan(β + δ)]δ. (2) In the initial stage, both the direction and magnitude of the drag force change with the varying contact area between the projectile nose and target material. Usually its direction tends to change from asymmetric to axisymmetric. Basically, the instant drag force F in the initial stage consists of the lateral component F and the axial component F x, which are normal to and along with the instant direction of obliquity, (β + δ ), respectively. δ is the instant direction change and its final value is δ. Fig. 2 The demonstration of the distance by which projectile imbeds target along its central axis in the first stage Observations from experiments indicate that the resistant drag increases almost linearly with time until the projectile head submerges into the target. Therein we can regard one half of the initial gross drag force as the average drag force. In this modelling, the average lateral component F avg along the arc s, and the axial component F x along the final direction of obliquity, (β + δ), are used to describe the average drag force F of the first stage. Furthermore, we have to use the geometrical relation (i.e., sin β)to estimate the average lateral drag force, and a similar geometrical relation is also used by Ipson and Recht [11]. Hence, it is reasonable to assume that the average lateral force is, F avg = 1 2 F 0 sin β, (3) where the initial gross drag force is F 0 = 1 4 πd2 (AN 1 σ y + BN 2 ρv 2 0 ) (4) based on the dynamic cavity expansion model [15 17]. In Eq. (4), σ y and ρ are the uni-axial yielding stress and the density of the target material, respectively. A and B are dimensionless material constants of the target plates. N 1 and N 2 are dimensionless parameters related to the nose shape and friction coefficient µ m, as defined in Chen and Li [17]. Thus the kinetic energy consumption is 1 2 MV2 = F avg s. (5) It can be formulated as sin 2 δ = δ sin β π 4 [ h + tan(β + δ) d ]( 1 I + 1 N ), (6)

4 370 X. Chen et al. where I and N are defined as impact function and geometry function in Chen and Li [14,17], i.e., I = λ J AN 1, N = λ, BN 2 with λ = M ρd 3, (7a) (7b) (8a) J = ρv2 0, (8b) σ y in which J is Johnson s damage number. Furthermore, for the perforation under relatively high impact velocity, δ = o(β), we have, δ = π ( 1 4 I + 1 )[ ] h N d + tan β sin β. (9) For other special cases, e.g. N I or blunt head (h = 0), much simpler expressions for δ can be formulated directly from Eq. (6). Obviously, Eq. (6) and Eq. (9) give a quantitative description of the effects of the projectile geometry, target material, impact velocity and initial obliquity, etc., which are quite similar to the case of oblique perforation of concrete panels [18]. 2.2 Penetration process Accompanied with the initial direction change, the rigid projectile penetrates into the target plate along X direction at an obliquity of (β + δ) and the target resistance on the projectile is determined by the dynamic cavity expansion model. The resultant axial resistant force and the momentum equations are the same as Eq. (1) and (4a, b) in Ref. [14], except that the initial conditions are correspondingly changed to V(t = 0) = V 0 cos δ, X(t = 0) = 0, (10a) (10b) where t = 0 means the initial time when the projectile touches the surface. 2.3 Motion of the central plug It is assumed that the plug travels in the same direction and with the same velocity as the striker after perforation. In addition, the criterion for a plug to be formed ahead of the projectile is that when the total resisting force on the projectile nose is equal to the total shear force acted along the separating surfaces of the plug, as shown in Fig. 1. As soon as the plug is formed, it separates from the parent target and moves with the projectile under the constant shear resistance, Q 0. Similarly, the dynamic equations of the central plug are almost the same as that in the configuration of normal perforation. Hence, the acceleration, velocity and displacement of the projectile and the plug, i.e., V Q, V Q and X Q, are formulated by equations (5a) (5b) and (5c) in Ref. [14], except that the constant shear resistance is replaced by Q 0 =[H sec(β + δ) + h]τ y, (11) where von-mises yielding criterion is used, τ y = σ y / 3. H is the residual thickness at the moment when the plug is formed (as shown in Fig. 1); η = M plug /M and M plug is the mass of the plug. 3 Oblique perforation of a thick plate 3.1 Oblique penetration stage When the target plate is so thick that the projectile stops and is embedded within the target medium, the problem can be treated as a pure oblique penetration process. The maximum oblique penetration depth is given by ( 1 + I cos2 δ N ), (12) X d = 2 π N ln according to Sect. 2.2, which is comparable to the results obtained by Chen and Li [17] for normal impact. Oblique perforation will occur if, subsequently, a plastic zone is developed ahead of the projectile and a complete failure of the target follows. Without losing any generality, we assume that the whole process of oblique perforation consists of the initial directional change and the subsequent tunnelling as well as the final plugging. Similar to Ref. [14], we may conclude that the shear plugging occurs at the critical condition, (AN 1 + BN 2 J ) = 4 [H sec(β + δ) + h], (13) 3 d where J = ρv 2/σ y and V is the velocity of the projectile at the time when the plug is just formed. The oblique penetration depth during the hole enlargement stage can be obtained as ) (1 H (H H ) sec(β + δ) = χ sec(β + δ) d H [( = (I cos 2 π N ln δ/n) ) ], (14) (1 + (I /N))

5 Oblique perforation of thick metallic plates by rigid projectiles 371 where χ = H/d and I = λ J /AN 1. It can then be shown that (I cos 2 δ + N) I = exp [ πχ sec(β + δ)/2n (1 (H ] N. (15) /H)) Substituting Eq. (15) into Eq. (13) yields the following expression which relates H to the initial velocity V 0 4χ sec(β + δ) 3 [ H H + ] h H sec(β + δ) AN 1 (1 + I cos 2 δ/n) = exp [ πχ sec(β + δ)/2n (1 (H ]. (16) /H)) 3.2 Shear plugging stage The shear plugging stage follows the penetration stage if there exists a real solution of H /H for Eq. (14) or Eq. (16). Usually, it occurs in the cases of blunt projectiles, e.g., flat-nose, hemispherical nose, etc. The ballistic limit is reached when V Q (t) = 0 and X Q (t) = H sec(β + δ), which are defined in Sect Hence, the ballistic limit for a thick plate impacted by a hard projectile is VBL 2 = AN 1σ y sec 2 δ BN 2 ρ {[ 1 + 8BN 2χηsec 2 (β + δ)(1 + η ) 3AN1 ( H ) H + h H ] H sec(β + δ) H [ )] } πχ sec(β + δ) exp (1 H 1, (17) 2N H where η = ρπd 2 H/4M = πχ/4λ. Since V BL is already included in the impact function I of Eq. (16), it is necessary to solve the nonlinear algebraic equations, i.e., Eqs. (6), (16), (17), for V BL, H /H and δ. Provided that V 0 > V BL, the residual velocity V r can be obtained by setting V r = V Q (t 1 ) such that X Q (t 1 ) = H sec(β + δ), i.e., 1 V r = (1 + η ) (V0 2 V2 1 ) cos2 δ [ exp (πχ sec(β + δ)/2n) (1 H H )], (18) in which V 1 has the same expression as V BL in Eq. (17). The value of H /H in Eqs. (17) and (18) can be obtained from Eq. (16) for a given initial impact velocity V 0. 4 Special cases 4.1 Normal perforation of thick plate by rigid projectile In the case of normal perforation of thick plate by rigid projectile, i.e., β = δ = 0, simple formulae can be easily deduced and are the same as those obtained previously by Chen and Li [14]. For example, Eqs. (13), (14) and (15) in Ref. [14] correspond to Eqs. (16) (17) and (18) in the present paper when β = δ = Oblique perforation of thick plate by sharp-nosed projectile In the case of the projectile being slender or sharpnosed [19], which means a large value of the geometry function N, plugging may not occur or can be ignored when compared to the dominant hole-enlargement process. In mathematical terms, it requires that F x < Q 0 during the penetration process, i.e., H 0, (19a) h d 3AN1 (1 + I cos 2 δ/n) 4 exp [πχ sec(β + δ)/2n], (19b) according to Eq. (16). Hence the square of the ballistic limit becomes VBL 2 = AN 1σ y sec 2 δ { [ πχ sec(β + δ) exp BN 2 ρ 2N and the residual velocity is (V0 2 V r = V2 1 ) cos2 δ exp [πχ sec(β + δ)/2n]. ] } 1, (20a) (20b) When the geometry function of projectile N is large enough (I/N 0), simple expressions can be deduced from Eqs. (16)(17) and (18), H 0, h d (21a) 3A 4. (21b) V BL = sec δ V r = 2ηAN 1 σ y sec(β + δ), (22a) ρ (V 2 0 V2 1 ) cos2 δ. (22b) 5 Experimental analyses In this section, the experimental data from Refs. [12, 13, 20, 21] are compared with the present analytical

6 372 X. Chen et al. predictions on ballistic limit and residual velocity. Those experimental results fall in the category of sharp-nosed projectiles and thus no plugging occurs during the perforation process. Piekutowski et al. [12] used 6061-T651 aluminium plates. The material properties are: γ = 1/3, σ y = 262 MPa, E = 69 GPa, ρ = 2, 710 kg/m 3 and n = (the hardening parameter). Roisman et al. [13] also used 6061-T651 aluminium as the target material but the average σ y is 295 MPa, which are due to the fact that targets can be made from layered or non-layered material. Gupta and Madhu [20,21] used mild steel MS1, MS2, MS3 and RHA steel. In the following analyses, Young s modulus E = 200 GPa is adopted for the steel but the yield strengths are σ y = 280, 350, 327 and 650 MPa, respectively. Gupta and Madhu [21] also used aluminium targets AL1, AL2, AL3. In the following analyses, we take Young s modulus E = 70 GPa but the yield strength are σ y = 260, 270 and 100 MPa, respectively. Based on the definition of non-dimensional numbers [17], the parameters associated with ogive-nose projectile in Ref. [12] are ψ = 3(CRH), N = 0.106, N 1 = 1.09, N 2 = 0.11, λ = and those in Ref. [13] are ψ = 2(CRH), N = 0.156, N 1 = 1.07, N 2 = 0.163, λ = We take the nose factors of the ogive-nose projectile in Refs. [20,21] as ψ = 3(CRH), N = 0.106, N 1 = 1.09, N 2 = 0.11; while λ =2.80, 2.80, 2.80, 2.68, 8.42, 8.05 and 9.40 for different target materials MS1, MS2, MS3, RHA, AL1, AL2 and AL3, respectively. The friction coefficient is µ m = 0.02 for all ogive noses, as suggested by Forrestal and Luk of [18]. Assuming that the materials are incompressible elastic, perfectly plastic, we can calculate the non-dimensional parameter A and take B = 1.5 [17]. Figures 3 and 4 plot the residual velocity versus impact velocity for the tests of Refs. [12,13] and the corresponding predictions by the present model. Figure 5 demonstrates the predicted correlation of directional change with initial impact velocity at angles of obliquity of 30 and 45, and the calculated results by Roisman et al. [13], respectively. Obviously, the theoretical analyses are in good agreement with the test data. For the case of 80 mm thick 6061-T651 aluminum plates used in Refs. [13], the correlations of directional change and ballistic limit with the initial angle of obliquity are shown in Fig. 6. Note that the predictions are for ideal cases. In fact, the assumption of rigid projectile would most likely be violated when the impact velocity exceeds 1,000 m/s. Figures 7 and 8 show the non-dimensional velocity drop (V 0 V r )/V 0 versus the impact obliquity, respectively, for mild steel MS3 and aluminium AL3 of different Fig. 3 Prediction of ballistic performance and test data (Referred to Piekutowski et al. 1996) Fig. 4 Prediction of ballistic performance and test data (Referred to Roisman et al. 1999) Fig. 5 The variation of direction change angle with initial obliquity and impact velocity plate thicknesses. The corresponding impact velocities are the same as those in Figs. 12 and 15 of Ref. [21]. The deviations could be partially due to the uncertainty of material parameters and projectile geometry used in the present calculation. The exit angles are plotted against the impact angle for plates MS3 and AL3 of various thicknesses in Figs. 9 and 10, respectively. Obviously,

7 Oblique perforation of thick metallic plates by rigid projectiles 373 Fig. 6 Prediction of ballistic limit and direction change of 80 mm thick Al-6061-T651 plate under oblique impact Fig. 8 Non-dimensional velocity drop versus angle of impact for different plate thickness of aluminum (AL3). β p points for (1) H = 40 mm, (2) H = 30 mm, (3) H = 20 mm and (4) H = 10 mm Fig. 7 Non-dimensional velocity drop versus angle of impact for different plate thickness of mild steel (MS3). β p points for (1) H = 25 mm, (2) H = 20 mm, (3) H = 16 mm, (4) H = 12 mm and (5) H = 10 mm Fig. 9 Exit angle versus impact angle of projectile for different plate thicknesses of mild steel (MS3) the directional changes for the mild steel plate MS3 appears greater than those for the aluminium plate AL3 due to its higher yielding stress and Young s modulus. The directional change is found to be insensitive to the plate thickness for MS3 but it is sensitive for AL3. We noted that most of the exit angles in the perforation tests of AL3 plates are less than the impact angles [21]. 6 Discussions The present model assumes a circular arc for the initial lateral turning path, and it is valid for small directional change δ, which corresponds to small angle of impact obliquity β. However, the analyses in Sect. 5 show that the present formulation remains applicable even if π/4 <β<π/3, as long as a non-trivial solution of δ(δ = 0) exists in Eq. (6). For β beyond that range, different phenomena will come into play. Ricochet of Fig. 10 Exit angle versus impact angle of projectile for different plate thicknesses of aluminum (AL3) projectile may occur rather than perforation. Although the present model cannot describe the phenomenon of ricochet, it is still capable of predicting the critical angle for the occurrence of ricochet. From Eq. (6), we note that with increasing impact obliquity β, the angle of directional change δ in Eq. (6) may become very large.

8 374 X. Chen et al. When β approaches a certain critical value β c,eq.(6) has no reasonable solution for δ, which implies that other phenomena may occur when β>β c. There exist upper bounds for β and δ (i.e., β c and δ c ). For example, in the experimental analyses of Sect. 5, corresponding to Ref. [21], the upper bounds are β c = 46 and δ c = 26 for mild steel MS3 (see Fig. 9), and β c = 64 and δ c = 11 for aluminium AL3 (see Fig. 10). On the other hand, there exists a maximum obliquity β p for a target plate of a given thickness corresponding to the existence of perforation ballistic limit, at which the non-dimensional velocity drop is 1.0. The maximum obliquity β p varies with the thickness of the target. Obviously, β p for thick plates will be smaller than that for thin plates, as shown in Figs. 7 and 8. Furthermore, we notice that the critical value of β p approaches the critical limit β c when the thickness of target plates is reduced. Meanwhile, both Figs. 7 and 8 show that for a target plate of a given thickness, there is a corresponding narrow range of β (near β p ) where the non-dimensional velocity drop (V 0 V r )/V 0 is very sensitive to β. Hence, we may consider that the ricochet occurs if the impact obliquity β exceeds the upper limit β c, i.e., β> β c, and X(δ c )/d H/d cos(β + δ c ), where X(δ c )/d is calculated according to Eq.(13) with δ = δ c. Otherwise, the projectile can still perforate the target even if β>β c but X(δ c )/d > H/d cos(β + δ c ). For thick plates, if the impact obliquity β lies between the maximum obliquity β p and the upper limit β c, i.e., β p <β<β c, the projectile may perforate or be embedded in the target, depending on its impact velocity V 0, i.e., corresponding to X(δ)/d > H/d cos(β + δ) or X(δ)/d H/d cos(β + δ), where the solution of Eq. (6) for the angle of direction change δ exists. Accordingly, the critical ricochet angles for MS3 and AL3 in Figs. 9 and 10 are 46 and 64, respectively. It is comparable to the experimental values for the critical ricochet angles of MS3 in Refs. [20,21], which are 62,59,51,51 and 50 for MS3 plates of thickness 10, 12, 16, 20 and 25 mm, respectively. Within the test range of impact obliquity, i.e., β 60, Gupta and Madhu [21] reported no ricochet occurring for all AL3 specimens. Table 1 lists the test data registered by Roisman et al. [22] and their predictions for the residual velocity, and compares them with the predictions of the present model. Most of those tests are oblique perforations at large obliquity, i.e., β>50, close to the critical angle of ricochet. Three kinds of target materials, i.e., Al 2024-T3, Cu and Mild steel are used in the test. The projectiles are either cone-nosed or ovoid-nosed shape. Similarly, for the corresponding friction coefficient we use µ m = 0.10 for the cone-nosed projectiles, as suggested by Forrestal and Luk [15] and discussed in Ref. [17]. To simplify the calculations of the problem, we use the ogive-nose (CRH = 2) to replace the ovoid-nose, which is in contrast to Roisman et al. [13]. For the three different types of targets, the present model predicts their limit pairs to be β c = 56,48,46 and δ c = 18,21,18, respectively. The discrepancies between the predicted and experimental critical angles in Table 1 may be caused by our approximation of ovoid nose being replaced by ogiveshape. They are found to be rather sensitive to the geometry function N of the projectiles because their corresponding N values are small due to small projectile mass, i.e., N = 21.0, 8.0 and 7.0, respectively. Meanwhile, the occurrence of ricochet is very sensitive to the initial obliquity near the critical angle. In fact, the predictions made by Roisman et al. [22] do not match their experimental results too. They attributed the discrepancies between the predicted and experimental residual velocities to such phenomena as: projectile failure or breakup during penetration; inaccuracy of the reported obliquity angles; and the necessity to estimate the target yield stress due to lack of data. It is worth mentioning that plugging does not appear in the comparison with the experimental data for sharp nose projectiles, although the present model includes two stages in general, i.e., penetration and shear plugging. Authors are not aware of any published experimental results on the oblique perforation of thick plates impacted by blunt (e.g., flat or hemispherical nose) projectiles. It is necessary to verify the oblique perforation analysis for thick plates by rigid blunter projectiles when experimental data become available. However, it had been shown in Ref. [14] that, in a special case when the localized shear plugging dominates the normal impact process (i.e., H H), the present analysis leads to the same results obtained by Recht and Ipson [10] for the case of flat-nosed projectile impacting a medium plate. A lot of published experimental results have been compared with the present model in Ref. [14]. The present paper extends the model for normal penetration and perforation of ductile metallic plates to the scenario of oblique impact after Chen and Li [14,17]. It clearly shows that the perforation process is dominated by several non-dimensional numbers, i.e., the impact function I, the geometry function N, the nondimensional thickness of target χ and the impact obliquity β. The present model is basically applicable to both normal and oblique perforation of thick plates by rigid projectiles of any common nose shapes provided that the dynamic cavity theory is valid. However, due to those assumptions made, it is only valid when the projectile deformation is not significant, which can normally be satisfied in the sub-ordnance velocity range,

9 Oblique perforation of thick metallic plates by rigid projectiles 375 Table 1 Comparison of theoretical prediction with experimental values of residual velocity for oblique perforation at large obliquity Configuration Residual velocity Delta angle/( ) Nose Target Impact velocity Impact Thickness of V r /(m s 1 ) pre- V r /(m s 1 ) V r /(m s 1 ) preshape material V 0 /(m s 1 ) obliquity β/( )platesh/mm dicted in ref. [22] tested in ref. [22] sent model present model Cone Al 2024-T Cone Cu Ricochet Ricochet (Ricochet occurs Ricochet Ricochet at β = 66 ) Ricochet Ricochet (Ricochet occurs Ricochet Ricochet at β = 64.5 ) Ovoid Mild steel Ricochet Ricochet (Ricochet occurs 0 Ricochet at β = 65 ) Ricochet Note: V r = 0 indicates that projectile is embedded in the target. e.g., V 0 < 1, 000 m/s of semi-armour piercing by KE penetrator. Beyond this impact velocity, the projectile deformation and erosion may become significant and thus the assumption of rigid projectile will be invalid. In addition, the present model yields better predictions of ballistic performance of rigid projectile for higher values of the geometry function N, e.g., when N > Conclusions This paper presents a two-stage model, i.e., penetration and shear plugging, to describe the oblique perforation of thick metallic plates by rigid projectiles with different nose shapes. Explicit formulae are deduced to predict the ballistic performance of oblique perforation. Critical condition for occurrence of ricochet is discussed. The formulae can be easily applied to various nose shapes and can be extended to other target materials (e.g., concrete, soil and ceramic targets etc.) as long as the dynamic cavity theory is valid. References 1. Backman, M.E., Goldsmith, W.: The mechanics of penetration of projectiles into targets. Int. J. Eng. Sci. 16, 1 99 (1978) 2. Brown, S.J.: Energy release protection for pressurized systems. Part II: review of studies into impact terminal ballistics. Appl. Mech. Rev. 39, (1986) 3. Anderson, C.E. Jr, Bodner, S.R.: Ballistic impact: the status of analytical and numerical modeling. Int. J. Impact Eng. 7, 9 35 (1988) 4. Corbett, G.G., Reid, S.R., Johnson, W.: Impact loading of plates and shells by free-flying projectiles: a review. Int. J. Impact Eng. 18, (1996) 5. Goldsmith, W.: Review: non-ideal projectile impact on targets. Int. J. Impact Eng. 22, (1999) 6. Awerbuch, J.: A mechanics approach to projectile penetration. Israel J. Tech. 8, (1970) 7. Awerbuch, J., Bodner S.R.: Analysis of the mechanics of perforation of projectiles in metallic plates. Int. J. Solid Struct. 10, (1974) 8. Awerbuch, J., Bodner S.R.: An investigation of oblique perforation of metallic plates. Exp. Mech. 17, (1977) 9. Zener, C., Peterson, R.E.: Mechanics of armor penetration. Watertown arsenal report no.710/492, Recht, R.F., Ipson, T.W.: Ballistic perforation dynamics. J. Appl. Mech. Trans. ASME 30, (1963) 11. Ipson, T.W., Recht, R.F.: Ballistic penetration resistance and its measurement. Exp. Mech. 15, (1975) 12. Piekutowski, A.J., Forrestal, M.J., Poormon, K.L., Warren, T.L.: Perforation of aluminium plates with ogive-nose steel rods at normal and oblique impacts. Int. J. Impact Eng. 18(7 8) (1996) 13. Roisman, I.V., Weber, K., Yarin, A.L., Hohler, V., Rubin, M.B.: Oblique penetration of a rigid projectile into a thick elastic-plastic target: theory and experiment. Int. J. Impact Eng. 22, (1999)

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