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Perforation of a thick plate by rigid projectiles
International Journal of Impact Engineering 28 (2003) 743–759
Perforation of a thick plate by rigid projectiles X.W. Chena, Q.M. Lib,* a
School of Civil and Environmental Engineering, Nanyang Technological University, Nanyang Avenue 639798, Singapore b Department of Mechanical, Aerospace and Manufacturing Engineering, UMIST, P.O. Box 88, Manchester M60 1QD, UK Received 9 January 2002; received in revised form 18 September 2002; accepted 1 November 2002
Abstract Perforation of a thick plate by rigid projectiles with various geometrical characteristics is studied in the present paper. The rigid projectile is subjected to the resistant force from the surrounding medium, which is formulated by the dynamic cavity expansion theory. 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. Simple and explicit formulae are obtained to predict the ballistic limit and residual velocity for the perforation of thick metallic plates, which agree with available experimental results with satisfactory accuracy. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Perforation; Ballistic limit; Thick plate; Non-deformable projectile; Shear plugging
1. Introduction There exist various perforation mechanisms for different target thicknesses, e.g. Backmann and Goldsmith [1] identified eight possible perforation mechanisms for brittle and ductile targets in a range of target thickness. With the increase of plate thickness, the local effect becomes more and more important, accompanied by a reduced effect of structural response on the plate perforation. The perforation of a thick plate is controlled by both the penetration process when the processing zone in front of the projectile nose has not reached the rear free surface and the final perforation process when a failure mechanism is initiated in the target in front of the projectile nose. Failure mechanisms responsible for the final perforation depend on projectile nose shape, impact velocity, target property, target thickness and size. A final failure mechanism, which could be ductile hole *Corresponding author. Fax: +44-161-2003849. E-mail address: [email protected] (Q.M. Li). 0734-743X/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 7 3 4 - 7 4 3 X ( 0 2 ) 0 0 1 5 2 - 5
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Nomenclature A and B dimensionless material constants d diameter of a projectile E Young’s modulus of the target material axial resistant force on the projectile nose Fx h nose height of the projectile H thickness of a circular plate effective thickness of the target at oblique impact Heff ¼ H=cosðb þ dÞ Hn thickness of the plug I impact function, defined by Eq. (7a) M mass of a projectile mass of a plug Mplug ¼ rpd 2 H n =4 N geometry function of the projectile, defined by Eq. (7b) N1 ; N2 and N dimensionless parameters, defined by Eqs. (3a)–(3c) plastic shear force per unit length of a shear plug Q0 ¼ ðH n þ hÞty t time t1 time at the end of perforation V transient velocity of a projectile ballistic limit VBL initial impact velocity of the projectile Vi nominal residual velocity of the projectile and the plug Vr velocity of the projectile when the plug is just formed V ’ W . W ; W; transverse deflection, velocity and acceleration of the projectile and the plug X penetration depth b initial oblique angle H dimensionless thickness of the plate (ratio of plate thickness and projectile diameter) w¼ d g Poisson’s ratio of the target material rpd 2 H Z ¼ ratio of the plug mass to the projectile mass for a thick plate; similarly 4M 2 rpd H pw ¼ Z¼ 4M 4l M dimensionless mass of the projectile l¼ 3 rd r density of the target material yielding stress of the target material sy pffiffiffi ty shear yielding stress (ty ¼ sy = 3 for von-Mises yielding criterion) Johnson’s damage number (FJ ¼ rVi2 =sy ) FJ
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enlargement for sharp nose [2–4], plugging for blunt projectile [5] or fragment for high impact velocity and brittle target, will be responsible for the final perforation of a target plate. Woodward [6] and Woodward and Cimpoeru [7] demonstrated various idealized failure mechanisms for homogeneous plates impacted by various projectiles. Shear plugging and discing failure are usually observed for thick plates, which may be influenced by the tensile opening at the stretched rear surface and the local bending. Multi-stage models have been proposed to study the perforation of relatively thick plates. Awerbuch [8] divided the penetration of a plate target into two stages. In the first stage, only inertia and compressive forces are introduced to decelerate the effective mass of the projectile. The second stage is initiated when a shear plug of the target material is formed, during which the inertia and compressive resistances are replaced by the surrounding shear force. Goldsmith and Finnegan [9] improved Awerbuch’s [8] model by considering the reduction of the shear force in the second stage. Awerbuch’s [8] two-stage model was extended to a three-stage model for the perforation of a plate by a non-deformable projectile [10]. The projectile is subjected to the inertia and compressive resistances of the target material as well as the shear resistance around the plug in the middle stage. This model has been further modified by Ravid and Bodner [11], where a twodimensional model assumes five stages of plate penetration namely, dynamic plastic penetration, bulge formation, bulge advancement, plug formation and projectile exit. The five stages are continuously coupled during penetration and the model predicts not only the exit velocities of the projectile and the plug, but also the bulge and plug shape and the force–time history of the process. Ravid et al. [12] further generalized this analytical model to include various projectile nose shapes, changes in the plastic flow field due to deep penetrations and thermal softening effects, which leads to a variation of failure modes. Liss et al. [13] proposed another five-stage interactive model for the penetration and perforation process, where plastic wave propagation in both the thickness and the radial directions of the plate is considered. A numerical procedure is necessary to solve equations of the motion of the projectile for the five stages. Dikshit and Sundararajan [14] characterized the penetration of an ogival hard projectile into steel plates with lower hardness and various thickness. Dikshit et al. [15] declared that the thickness of target plate strongly affects the effect of plate hardness on ballistic performance. The terms ‘plane strain’ and ‘plane stress’ were introduced for ‘thick’ and ‘thin’ plates to represent ‘constrained’ and ‘unconstrained’ plastic flows, respectively. The transition from plane strain to plane stress conditions occurs when the plastic zone ahead of the penetrating projectile just reaches the bottom surface of the plate. Thus, the penetration of thin plates occurs under plane stress condition while the penetration of thick plates occurs under plane strain condition. Furthermore, several recent analytical models employ point-wise descriptions of the velocity fields. This local model uses the kinematics of an in-viscid fluid in order to generate an approximate velocity field in the target. The origin of this approach dates back to Hill [16], Tate [17] and Ravid and Bodner [11], which has been successfully exploited to study the penetration of both rigid and deformable projectiles (e.g., [12,18,19]). The velocity-field model was further developed by Yarin et al. [20] and Roisman et al. [21,22] to include inertial and elastic–plastic drag effects of a target on the projectile, as well as the finite thickness of the target. It assumes that the target material is incompressible and the target region is divided into an elastic region at a distance from the projectile tip and a rigid-plastic region near the projectile tip. By using the velocity potential f of flow field and method of singularities, the solution procedure presented in
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Yarin et al. [20] leads to a straightforward approximation to accommodate the arbitrary shape of the nose of a projectile. The velocity-field model was further applied by Roisman et al. [21] to describe the phenomenon of ricochet. Yossifon et al. [23] examined two different theoretical approaches that developed from the velocity-field model and showed their good agreements with numerical calculations from AUTODYN-2D in respect to the penetration depth, the ballistic limit and the residual velocity of the projectile. Most other analytical models for the perforation of a metallic target impacted by a rigid projectile, e.g., Taylor [24], Thompson [25], Recht and Ipson [26] and Rosenberg and Forrestal [4], are formulated using energy and momentum balance to predict the ballistic limit or the residual velocity. Many empirical formulae, e.g., De Marre (1886), SRI Formula (1963), BRL formula (1968) (see [27]), Neilson [28] and Wen and Jones [29], etc., are also based on experimental data fitting and basic energy balance in perforating a target plate. Energy and momentum methods make it very easy to treat perforation process. However, it is still difficult to give an accurate calculation for all energy consumption in a penetration and perforation process. The vague definition of projectile nose shape is another challenge problem, which may cause uncertainties in the application of these analytical models and empirical formulae. Meanwhile, most analytical models need more complicated numerical procedures to calculate the ballistic performance. It is a challenge to include most important deformation and failure modes while retaining simplicity and a reasonable degree of accuracy in a plate perforation model. In the present paper, an analytical model is proposed in Section 2 for the penetration and perforation of a thick plate based on dynamic cavity expansion theory and plug formation, which is applied to study the perforation performances of both blunt and sharp nose projectiles in Sections 3 and 4. The obtained formulae embrace early analytical models and give good predictions when compared with experimental results, as shown in Section 5.
2. Analytical model 2.1. Penetration model A non-deformable projectile with mass M; diameter d and an arbitrary nose shape, as shown in Fig. 1, impacts a thick plate target normally at initial velocity Vi and proceeds to penetrate the target medium at rigid-body velocity V : Based on the dynamic cavity expansion model [30–32], the resulting axial resistant force on the projectile nose can be calculated from the normal compressive stresses, Fx ¼
pd 2 ðAN1 sy þ BN2 rV 2 Þ; 4
ð1Þ
which has the same form as the Poncelet formula. sy and r are yielding stress and density of the target material, respectively. A and B are dimensionless material constants of target materials, in which A is given by 2 E ð2aÞ A¼ 1 þ ln 3 3ð1 gÞsy
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Y →
(_sin , cos )
→
n (cos , sin )
y=y(x)
0 d
X h
Fig. 1. Cross-section of an arbitrary nose.
for an elastic, perfectly plastic material [31], where g is Poisson’s ratio and n Z 1ð3sy =2EÞ 2 2E ðln xÞn dx G ; G¼ A¼ 1þ 3 3sy ð1 xÞ 0
ð2bÞ
for an incompressible elastic, strain-hardening plastic material [32]. B ¼ 1:5 for incompressible materials. However, a numerical procedure is required to calculate parameters A and B in other cases. More details were given in Chen and Li [30]. N1 and N2 in Eq. (1) are two dimensionless parameters relating to nose shape and friction coefficient mm : If an arbitrary nose shape can be represented by the function y ¼ yðxÞ; as shown in Fig. 1, then Z h 8m N1 ¼ 1 þ 2m y dx; ð3aÞ d 0 Z h 8m yy02 N2 ¼ N n þ 2m dx; ð3bÞ 02 d 0 1þy Z 8 h yy03 n N ¼ 2 dx; ð3cÞ d 0 1 þ y02 where h is the height of nose. If the friction is ignored, N1 ¼ 1; N2 ¼ N n : Various formulae of N1 ; N2 and N n for different nose shapes, e.g., cone, ogive, hemispherical nose, and truncated ogive, can be found in Chen and Li [30]. The penetration depth X is determined by dV ð4aÞ ¼ Fx M dt and dX V¼ ð4bÞ dt with the initial condition V ðt ¼ 0Þ ¼ Vi and X ðt ¼ 0Þ ¼ 0: There exists a layer of added mass in front of the projectile nose. Generally, the inertia effect of the added mass should be considered in Eq. (4a), as shown in Yarin et al. [20]. The added mass
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effect is not included in the present study. However, it may become no negligible with the increase of projectile deceleration. 2.2. Motion of the central plug It is assumed that a central plug is formed beneath the projectile at a critical condition when the compressive force on the projectile nose reaches the fully plastic shear force on the plug, as shown in Fig. 2. As soon as the plug is formed, it moves with the projectile under the constant shear resistance, Q0 ; and thus, the acceleration of the projectile and the plug is . ¼ pdQ0 ; ð5aÞ W Mð1 þ Z Þ p ffiffi ffi
where Q0 ¼ H þ h ty ; ty is the yield shear stress of the material (ty ¼ sy = 3 for von-Mises yielding criterion); H n is the residual thickness after penetration by nose tip; Z ¼ ðMplug =MÞ ¼ ðrpd 2 H n =4MÞ and Mplug ¼ ðrpd 2 H n =4Þ is the mass of plug. Eq. (5a) leads to 1 pdQ0 t ’ ð5bÞ V W¼ ð1 þ Z Þ M
and
1 pdQ0 t2 W¼ V t ð5cÞ ð1 þ Z Þ 2M ’ ¼ V =ð1 þ Z Þ at the moment when the plug is formed. V is the velocity of when W ¼ 0 and W the projectile determined by Eqs. (4a) and (4b) at the time when the plug is just formed. 3. Perforation of a thick plate 3.1. Penetration stage If the thickness of the target plate is so large that the projectile stops and embeds in the target medium, the problem can be treated as a pure penetration process. Theoretically, pure penetration
V*
H
h
H*
Plug
Fig. 2. Perforation of thick plates by a non-deformable projectile with an arbitrary nose.
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only corresponds to the case that the plastic zone ahead of the penetrating projectile does not reach the rear surface of the plate, i.e., ‘‘constrained’’ plastic flow. However, it may sometimes be applicable to the case when rear bulging (unconstrained plastic flow) is formed, i.e., the plastic zone ahead of the projectile reaches the rear surface of the plate. The maximum penetration depth is given by X 2 I ¼ N ln 1 þ ð6Þ d p N according to Section 2.1 [30], where I and N are defined as impact function and geometry function, I¼
lFJ AN1
ð7aÞ
and N¼
l BN2
ð7bÞ
with l¼
M rd 3
ð8aÞ
and FJ ¼
rVi2 ðJohnson’s damage numberÞ: sy
ð8bÞ
Perforation will occur if further development of the plastic zone ahead of projectile results in a complete failure of target and the projectile passes through the target. In general, shear plugging will occur when the interactive force between the projectile nose and the plate reaches a critical value of Fx ¼ pdQ0
ð9Þ
which may be satisfied when the bluntness of a projectile is larger enough. For a projectile with sharp nose, plugging stage is neglected and the penetration stage dominates the perforation process to produce a hole enlargement. For thin or medium thick plates, shear plugging may play a more important role during the impact process [33]. The velocity of the projectile is assumed to be V * (V * oVi ) when the plugging condition, Eq. (9), is satisfied, and the corresponding thickness of target in front of projectile nose tip is H ðH oHÞ: Meanwhile, it is assumed that the plugging is initiated from the root of projectile nose, as shown in Fig. 2, which is normally observed for projectiles with blunt nose shape. Thus, we have 4 ðH þhÞ ; ðAN1 þ BN2 FJ * Þ ¼ pffiffiffi d 3 where FJ * ¼ rV 2 =sy : *
ð10Þ
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The penetration depth in the first stage is obtained by integrating Eqs. (4a) and (4b), 3 2 I 1þ 6 ðH H n Þ Hn 2 N 7 6 7 ; ¼w 1 ¼ N ln4 I* 5 d p H 1þ N lFJ * H where w ¼ : It reduces to and I * ¼ d AN1 ðI þ NÞ N: I* ¼ pw Hn 1 exp 2N H
ð11Þ
ð12Þ
Substituting Eq. (12) to Eq. (10), we have the dependence of H n on the initial velocity Vi : 4w H n h AN1 ð1 þ I=NÞ : pffiffiffi þ ¼ ð13Þ pw Hn H 3 H 1 exp 2N H 3.2. Shear plug stage The shear plugging stage follows the penetration stage if there exists a solution of H n =H for Eq. (13), which is usually the case for blunt projectiles, e.g., flat nose and hemispherical nose. The ’ ¼ 0 and W ðtÞ ¼ H in Eqs. (5b) and (5c). Therefore, the ballistic limit is obtained when WðtÞ ballistic limit for a thick plate impacted by a hard projectile is (" ) # 8BN2 wZð1 þ Z * Þ H n h H n AN1 sy pw Hn 2 pffiffiffi 1 1þ þ exp VBL ¼ 1 ð14Þ H H 2N BN2 r H H 3AN1 in which, H =H in Eq. (13) contains VBL : Thus, it is necessary to solve a nonlinear algebra equation for VBL : ’ 1 Þ when W ðt1 Þ ¼ H : The residual velocity for Vi > VBL is obtained as Vr ¼ Wðt vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 u ðV 2 V 2 Þ u i 1 n ; ð15Þ Vr ¼ pw H ð1 þ Z * Þt exp 1 H 2N in which, V1 has the same expression as VBL in Eq. (14). H =H in Eqs. (14) and (15) is calculated from Eq. (13) for a given initial impact velocity Vi : 4. Special cases 4.1. Perforation of a thick plate by a sharp-nosed projectile If the projectile has a larger value of geometry function N; corresponding to a slender projectile body or sharper nose [30], plug may not occur or can be ignored compared to the whole hole
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enlargement. In view of mathematics, it requires that Fx opdQ0 in the penetration process, i.e., H n -0
ð16aÞ
and
pffiffiffi 3AN1 ð1 þ I=NÞ h pw i : ð16bÞ 4 exp 2N Thus the ballistic limit and residual velocity are, respectively, o AN1 sy n pw 2 ¼ VBL 1 ; ð17aÞ exp 2N BN2 r vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 uðV V 2 Þ : ð17bÞ Vr ¼ t i pwBL exp 2N Especially for a sharper projectile perforating a thick metallic plate, the geometry function of projectile N is large enough (I=N-0; referred to [30]), thus simpler formulae can be conducted: h X d
H n -0 and h X d VBL
pffiffiffi 3A ; 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ZAN1 sy ¼ r
and Vr ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 ffi 2 ; Vi VBL
ð18aÞ
ð18bÞ ð19bÞ
ð19cÞ
where Z ¼ rpd 2 H=4M ¼ pw=4l similar to the definition of dimensionless mass Zn : 4.2. Perforation of a thick plate by a flat-nosed projectile For a flat-nosed missile perforating a thick metallic plate (referred to Section 4 of [33]), N1 ¼ N2 ¼ N n ¼ 1; Eq. (13) is simplified to "pffiffiffi # n 3 Hn 1 H 1 ðA þ BFJ Þ : ð20Þ ln ln ¼1 4w 2ZB 2ZB H H pffiffiffi It ispeasily found that a solution of H n =Ho1 exists only if w > 3ðA þ BFJ Þ=4: Alternatively if ffiffiffi wp 3ðA þ BFJ Þ=4; whole shear plugging (i.e., H n ¼ H) without any indention/penetration is applied in Chen and Li [33] to predict the ballistic performance as well as consider the structural response (bending and effect of membrane force). After specifying the projectile and target material, only two dimensionless quantities, i.e., w and FJ are left to determine H n =H: Fig. 3 shows the dependence of H n =H on w with different initial velocities Vi ; in which a hard flat-nosed
X.W. Chen, Q.M. Li / International Journal of Impact Engineering 28 (2003) 743–759 1.0 0.8
H*/H
2.8
vi=0.2 km/s vi=0.3 km/s vi=0.5 km/s vi=0.7 km/s vi=0.9 km/s H*/H at ballistic limit Ballistic limit (km/s)
0.6
2.4 2.0 1.6 1.2
0.4
VBL (km/s)
752
0.8 0.2 0.4 0
0 0
5
10
15
20
25
H/d
Fig. 3. Dependence of H * =H on w with different initial velocities Vi for a flat-nose steel missile perforating the aluminium plate.
steel missile of mass 25 g and diameter 7.1 mm perforates a thick 6061-T6 aluminium plate (same as [4] but replacing the conical-nosed missile by flat-nosed missile). Meanwhile, Fig. 3 also n =H at the demonstrates the dependence of ballistic limit VBL on w and the critical curve of HBL n ballistic limit. Only the regime above the critical curve of HBL =H corresponds to the perforation. Indentation/penetration is considered when the dimensionless thickness of plate w > 2:06: Obviously, H n =H drops very quickly and intends to be omitted if the target plate becomes thicker. Assuming the plug is a cylinder with same diameter as the projectile, the residual velocity and ballistic limit of flat-nosed projectiles impacting on thick target plates (corresponding to Eqs. (14) and (15)) are, respectively, (" ) n 2 # Asy 8BwZ Hn H Hn 2 exp 2ZB 1 1 þ pffiffiffi 1 þ Z 1 ; ð21aÞ VBL ¼ Br H H H 3A vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ðVi2 V12 Þ 1 u : u Vr ¼ t Hn Hn exp 2ZB 1 Z 1þ H H
ð21bÞ
Obviously if local shear/plugging dominates in the impact process, i.e., H n -H; Eqs. (21a) and (21b) also lead further to 8sy 2 ¼ pffiffiffi wZð1 þ ZÞ; VBL 3r Vr ¼
1 ð1 þ ZÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Þ: ðVi2 VBL
Eq. (22b) is the same as the formula of Recht and Ipson’s [26] model.
ð22aÞ
ð22bÞ
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5. Experimental analyses Firstly a collection of experimental data from Rosenberg and Forrestal [4], Forrestal and Luk [34], Piekutowski et al. [3] and Roisman et al. [22] are compared with the present analytical predictions on ballistic limit and residual velocity. These experimental results fall in the category of sharp-nosed projectiles and as a result no plugging occurs in perforation. Based on the definition of dimensionless numbers [30], the nose parameters of conical-nose projectile in Rosenberg and Forrestal [4] are c ¼ 1:507; N n ¼ 0:099; N1 ¼ 1:06; N2 ¼ 0:105; and the mass ratio is l ¼ 25:17; and those in Forrestal and Luk [34] are c ¼ 1:785; N n ¼ 0:073; N1 ¼ 1:07; N2 ¼ 0:078; l ¼ 16:72: The nose parameters of ogive–nose projectile in Piekutowski et al. [3] are c ¼ 3 (CRH), N n ¼ 0:106; N1 ¼ 1:09; N2 ¼ 0:11; and the mass ratio is l ¼ 13:96 and those in Roisman et al. [22] are c ¼ 2 (CRH), N n ¼ 0:156; N1 ¼ 1:07; N2 ¼ 0:163; l ¼ 22:50: The friction coefficient uses mm ¼ 0:02 for both ogive and conical nose, as suggested by Forrestal and Luk [31] and Forrestal and Luk [34]. For 6061-T6 aluminium plates used in Rosenberg and Forrestal [4], g ¼ 1=3; Y ¼ 300 MPa; Y =K ¼ 0:0045 (or E ¼ 66:7 GPa) and r ¼ 2710 kg=m3 : For 5083-H131 aluminum plates used in Forrestal and Luk [34], g ¼ 1=3; Y ¼ 276 MPa; E ¼ 70:3 GPa and n ¼ 0:085 and r ¼ 2660 kg=m3 : For 6061-T651 aluminium plates used in Piekutowski et al. [3], g ¼ 1=3; Y ¼ 262 MPa; E ¼ 69 GPa; n ¼ 0:085 and r ¼ 2710 kg=m3 : Roisman et al. [22] also used 6061-T651 aluminium as target material but Y ¼ 345 MPa; which is due to the fact that targets can be made from layered or non-layered material. Simply we use Eq. (2a) to evaluate parameter A and take B ¼ 1:5: Figs. 4–7 indicate the tests by Rosenberg and Forrestal [4], Forrestal and Luk [34], Piekutowski et al. [3] and Roisman et al. [22] and the corresponding theoretical prediction, respectively. The effective thickness of plates at oblique perforation is evaluated based on Heff ¼ H=cosðb þ dÞ; where b is the initial obliquity and d is the angle of the direction change. Roisman et al. [21] confirmed that the angle of directional change d is very small and can be dropped out in the oblique perforation of metal target if initial obliquity bo501: The initial obliquity in
1.6 Analysis Normal missile Arrowhead missile
1.4 1.2
Vr (m/s)
1.0 0.8 0.6 0.4 0.2 0 0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Vi (km/s)
Fig. 4. Prediction of ballistic performance and test data [4].
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Vr (km/s)
Analysis 1.0
H=12.7mm
0.8
H=50.8mm H=76.2mm
0.6 0.4 0.2 0 0
0.2
0.4
0.6
0.8
1.0
1.2
Vi (km/s)
Fig. 5. Prediction of ballistic performance and test data [34].
1.0 Analysis Normal impact
0.8 Vr (km/s)
Oblique impact 0.6 0.4 0.2 0 0
0.2
0.4
0.6
0.8
1.0
Vi (km/s)
Fig. 6. Prediction of ballistic performance and test data [3]. 1.4 Analysis
1.2
Roisman et al.(1999) 1.0 Vr (km/s)
754
0.8 0.6 0.4 0.2 0 0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Vi (km/s)
Fig. 7. Prediction of ballistic performance and test data [22].
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Piekutowski et al. [3] and Roisman et al. [22] is 301 and 451, respectively. Obviously, the theoretical analyses are in excellent agreement with the test data. Authors were not aware of any published experimental results on the perforation of thick plates impacted pffiffiffi by blunt (e.g., flat or hemispherical nose) projectiles, which satisfy the condition of w > 3ðA þ BFJ Þ=4: It is necessary to verify the perforation analysis for thick plates by rigid blunter projectiles when experimental data are available. However, it has been shown that, in a special case when the localized shear plugging dominates the impact process, the present analysis leads to the same results obtained by Recht and Ipson [26] for a flat-nosed projectile impacting a medium plate. Experimental results given in Borvik et al. [5] and Liss and Goldsmith [35] belong to this case, which are compared with the present model in Figs. 8 and 9. Awerbuch and Bodner [36] performed plate perforation experiments using standard lead and armour piercing bullets impacting aluminium and steel plates. Appreciable flattening of the
300 Borvik et al.(1999)
250
Analysis Vr (m/s)
200 150 100 50 0 100
150
200
250
300
350
Vi (m/s)
Fig. 8. Prediction of residual velocity and test data [5].
600 Liss & Goldsmith(1984) 500
Analysis
Vr (m/s)
400 300 200 100 0 0
100
200
300
400
500
600
700
Vi (m/s)
Fig. 9. Prediction of residual velocity and test data with H ¼ 6:4 mm in Ref. [35].
756
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projectile nose was observed, and thus, the plate was actually perforated by deformable projectile. Nevertheless, Awerbuch and Bodner [10,36], and Ravid and Bodner [11] treated the projectiles as a rigid cylinder with the tunnel diameter and calculated the residual velocity based on the multistage models. Yuan et al. [37] compared some data of these experimental results with their plastic wave theory. In the present paper, when the assumption of rigid projectile is applied, Eqs. (22a) and (22b) can be used to calculate the ballistic limit and residual velocity for those experimental results in Awerbuch and Bodner [36] on thick plates, which satisfy either wE1 or w > 1: Table 1 summarizes the experimental and calculated results. It should be noted that for test data on thin plates (e.g., wo1=2) in Awerbuch and Bodner [36], structural responses (e.g., bending effect or membrane deformation) should be taken into account and the relevant issue can be seen in Chen and Li [33].
6. Discussion When calculating the resistant force on the projectile nose, it is assumed that the projectile nose is completely embedded in the target medium, which cannot be satisfied during entry and exit phases. The energy consumption is overestimated during the entry phase and underestimated during the exit phase. However, the reduced energy consumption in the exit phase will compensate the increased energy consumption in the entry phase. This would not cause noticeable difference when the impact and residual velocities are relatively high. Since the initial velocity is always greater than the exit velocity, the model may over-predict the ballistic performance of the target, which only becomes significant when the impact velocity is much higher than the residual velocity and the target thickness is comparable with the nose length of the projectile. There are several empirical formulae and analytical solutions, e.g., De Marre (1886), SRI Formula (1963), BRL formula (1968) (see [27]), Recht and Ipson [26], Neilson [28] and Wen and Jones [29], to predict the ballistic performance of the metallic target. Majorities of the empirical formulae result from curve-fitting practices based on test data and are dimensionally dependent, which provide little physical understanding of the studied problem. Their applications are normally limited by the extent of test parameters and there exists an ambiguity of the projectile nose description. It is necessary to develop a simple but accurate model that can be easily used in the design of a protective structure. The present analysis is actually a two-stage model, i.e., penetration/indention and shear plugging, based on the dynamic cavity expansion and rigid-plastic analysis. The geometric characteristic of the projectile nose shape is considered analytically. Obviously, the formulae obtained in Forrestal et al. [2], Rosenberg and Forrestal [4], Forrestal and Luk [34], Piekutowski et al. [3] and Recht and Ipson [26] are special cases of the present analysis. It implies that the present model unifies these relevant theoretical models on the perforation studies. The present model is applicable to the perforation of thick plates by hard projectiles with arbitrary nose shapes, e.g. conical, ogival, blunt and other more complicated noses. It is also applicable to oblique perforation at small initial obliquity (i.e., bo501). However, the present model still needs to be further developed so that it can be applicable to yaw impact, oblique impact at large obliquity and impact with ricochet.
MS-A-10
AL-6-9.6 AL-6-13 AL-6-19
SA-A-6 6 SA-B-6.35 6.35 SA-D-12 12
S-R
S-R S-R S-R
SAP SAP SAP
9.6 13 19
10
SA-A-6 6 SA-A-8 8 SA-B-6.35 6.35 SA-C-8 8 SA-D-9 9 SA-D-10 10
7.8 7.8 7.8
2.7 2.7 2.7
7.8
7.8 7.8 7.8 7.8 7.8 7.8
980 1200 900
280 280 280
280
980 980 1200 1050 900 900
Plate Plate Yielding thickness density stress of H (mm) r (g/cm3) target sy (MPa)
S-R S-R S-R S-R S-R S-R
Projec- Target tile
10.5 9.8 7.45
8.4 10.1 11
11.9
10.5 10.6 10.4 10 10.5 11.5
835 848 855
845 845 836
855
850 855 854 855 850 845
Average Impact diameter velocity of tunnel Vi (m/s) Davg (mm)
361
453 318 400 351 243 95
Vr (m/s) by Awebuch and Bodner [10]
550–650 499 500–600 492 300–390 418
748 702 712 604 568–585 406
400
500–600 460 350–550 450–470 220–290 155–185
Residual velocity by test Vr (m/s)
660 589 364
Vr (m/s) by Ravid and Bodner [11]
Table 1 Comparison between perforation experiments and theory for steel and aluminium plates [36]
467 476
478 454 463 455
549 565 470
799 653 434
509
548 415 515 431 365 187
392 424 593
151 355 604
341
405 581 475 579 631 779
0.83 0.79 0.75
0.26 0.58 0.63
0.58
0.83 0.86 0.83 0.88 0.86 0.86
0.95 0.91
0.95 0.95 0.94 0.96
1 1 1
1 1 1
1
1 1 1 1 1 1
Vr (m/s) Vr (m/s) Ballistic H n =H H n =H by H n =H by by Yuan by present limit by by test Yuan present present et al. model et al. model [37] model [37] VBL (m/s)
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7. Conclusions This paper presents a two-stage model, i.e., penetration and shear plugging, to study the perforation of thick plates by non-deformable projectiles with different geometrical characteristics. Analytical formulae are given to predict the ballistic performance of perforation of thick plate targets. It unifies several well-known theoretical models on perforation and is available to hard projectiles with different nose shapes.
Acknowledgements The first author would like to acknowledge the Ph.D. scholarship granted by the School of Civil and Environmental Engineering, Nanyang Technological University and the leave of absence offered by the China Academy of Engineering Physics (CAEP).
References [1] Backmann ME, Goldsmith W. The mechanics of penetration of projectiles into targets. Int J Eng Sci 1978;16:1–99. [2] Forrestal MJ, Rosenberg Z, Luk VK, Bless SJ. Perforation of aluminium plates with conical-nosed rods. J Appl Mech Trans ASME 1987;54:230–2. [3] Piekutowski AJ, Forrestal MJ, Poormon KL, Warren TL. Perforation of aluminium plates with ogive-nose steel rods at normal and oblique impacts. Int J Impact Eng 1996;18(7–8):877–87. [4] Rosenberg Z, Forrestal MJ. Perforation of aluminium plates with conical-nosed rods—additional data and discussion. J Appl Mech Trans ASME 1988;55:236–8. [5] Borvik T, Langseth M, Hopperstad OS, Malo KA. Ballistic penetration of steel plates. Int J Impact Eng 1999;22:855–86. [6] Woodward RL. The interrelation of failure modes observed in the penetration of metallic targets. Int J Impact Eng 1984;2:121–9. [7] Woodward RL, Cimpoeru SJ. A study of the perforation of aluminium laminate targets. Int J Impact Eng 1998;21:117–31. [8] Awerbuch J. A mechanics approach to projectile penetration. Israel J Tech 1970;8:375–83. [9] Goldsmith W, Finnegan SA. Penetration and perforation processes in metal targets at and above ballistic velocities. Int J Mech Sci 1971;13:843–66. [10] Awerbuch J, Bodner SR. Analysis of the mechanics of perforation of projectiles in metallic plates. Int J Solids Struct 1974;10:671–84. [11] Ravid M, Bodner SR. Dynamic perforation of viscoplastic plates by rigid projectiles. Int J Eng Sci 1983;21:577–91. [12] Ravid M, Bodner SR, Holcman I. Penetration into thick targets—refinement of a 2D dynamic plasticity approach. Int J Impact Eng 1994;15(4):491–9. [13] Liss J, Goldsmith W, Kelly JM. A phenomenological penetration model of plates. Int J Impact Eng 1983;1(4): 321–41. [14] Dikshit SN, Sundararajan G. The penetration of thick steel plates by ogive shaped projectiles—experiment and analysis. Int J Impact Eng 1992;12(3):373–408. [15] Dikshit SN, Kutumbarao VV, Sundararajan G. The influence of plate hardness on the ballistic penetration of thick steel plates. Int J Impact Eng 1995;16(2):293–320. [16] Hill R. Cavitation and the influence of headshape in attack of thick targets by non-deforming projectiles. J Mech Phys Solids 1980;28:249–63. [17] Tate A. A comment on a paper by Awerbuch and Bodner concerning the mechanics of plate perforation by a projectile. Int J Eng Sci 1979;17:341–4.
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[18] Tate A. Long rod penetration models—Part I. A flow field model for high speed long rod penetration. Int J Mech Sci 1986a;28:535–48. [19] Tate A. Long rod penetration models—Part II. Extensions to the hydrodynamic theory of penetration. Int J Mech Sci 1986b;28:599–612. [20] Yarin AL, Rubin MB, Roisman IV. Penetration of a rigid projectile into an elastic–plastic target of finite thickness. Int J Impact Eng 1995;16:801–31. [21] Roisman IV, Yarin AL, Rubin MB. Oblique penetration of a rigid projectile into an elastic–plastic target. Int J Impact Eng 1997;19:769–95. [22] Roisman IV, Weber K, Yarin AL, Hohler V, Rubin MB. Oblique penetration of a rigid projectile into a thick elastic–plastic target: theory and experiment. Int J Impact Eng 1999;22:707–26. [23] Yossifon G, Rubin MB, Yarin AL. Penetration of a rigid projectile into a finite thickness elastic–plastic target— comparison between theory and numerical computations. Int J Impact Eng 2001;25:265–90. [24] Taylor GI. The formation and enlargement of a circular hole in a thin plastic sheet. Quart J Mech Appl Math 1948;1:103–24. [25] Thompson WT. An approximate theory of armour penetration. J Appl Phys 1955;26(1):80–2. [26] Recht RF, Ipson TW. Ballistic perforation dynamics. J Appl Mech Trans ASME 1963;30:385–91. [27] Corbett GG, Reid SR, Johnson W. Impact loading of plates and shells by free-flying projectiles: a review. Int J Impact Eng 1996;18:141–230. [28] Neilson AJ. Empirical equations for the perforation of mild steel plates. Int J Impact Eng 1985;3:137–42. [29] Wen HM, Jones N. Semi-empirical equations for the perforation of plates struck by a mass. In: Bulson PS, editor. Structures under shock and impact II. Computational Mechanics Publication, 1992. pp. 369–80. [30] Chen XW, Li QM. Deep penetration of a non-deformable projectile with different geometrical characteristics. Int J Impact Eng 2002;27(6):619–37. [31] Forrestal MJ, Luk VK. Dynamic spherical cavity-expansion in a compressible elastic—plastic solid. Trans ASME J Appl Mech 1988;55:275–9. [32] Luk VK, Forrestal MJ, Amos DE. Dynamics spherical cavity expansion of strain-hardening materials. Trans ASME J Appl Mech 1991;58(1):1–6. [33] Chen XW, Li QM. Shear plugging and perforation of ductile circular plates struck by a blunt projectile. Int J Impact Eng 2002, in press. [34] Forrestal MJ, Luk VK. Perforation of aluminium armor plates with conical-nose projectiles. Mech Mater 1990;10:97–105. [35] Liss J, Goldsmith W. Plate perforation phenomena due to normal impact of blunt cylinders. Int J Impact Eng 1984;2:37–64. [36] Awerbuch J, Bodner SR. Experimental investigation of normal perforation of projectiles in metallic plates. Int J Solids Struct 1974b;10:685–99. [37] Yuan WX, Zhou LT, Ma XQ, Stronge WJ. Plate perforation by deformable projectiles—a plastic wave theory. Int J Impact Eng 1983;1(4):393–412.
Perforation of a thick plate by rigid projectiles X.W. Chena, Q.M. Lib,* a
School of Civil and Environmental Engineering, Nanyang Technological University, Nanyang Avenue 639798, Singapore b Department of Mechanical, Aerospace and Manufacturing Engineering, UMIST, P.O. Box 88, Manchester M60 1QD, UK Received 9 January 2002; received in revised form 18 September 2002; accepted 1 November 2002
Abstract Perforation of a thick plate by rigid projectiles with various geometrical characteristics is studied in the present paper. The rigid projectile is subjected to the resistant force from the surrounding medium, which is formulated by the dynamic cavity expansion theory. 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. Simple and explicit formulae are obtained to predict the ballistic limit and residual velocity for the perforation of thick metallic plates, which agree with available experimental results with satisfactory accuracy. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Perforation; Ballistic limit; Thick plate; Non-deformable projectile; Shear plugging
1. Introduction There exist various perforation mechanisms for different target thicknesses, e.g. Backmann and Goldsmith [1] identified eight possible perforation mechanisms for brittle and ductile targets in a range of target thickness. With the increase of plate thickness, the local effect becomes more and more important, accompanied by a reduced effect of structural response on the plate perforation. The perforation of a thick plate is controlled by both the penetration process when the processing zone in front of the projectile nose has not reached the rear free surface and the final perforation process when a failure mechanism is initiated in the target in front of the projectile nose. Failure mechanisms responsible for the final perforation depend on projectile nose shape, impact velocity, target property, target thickness and size. A final failure mechanism, which could be ductile hole *Corresponding author. Fax: +44-161-2003849. E-mail address: [email protected] (Q.M. Li). 0734-743X/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 7 3 4 - 7 4 3 X ( 0 2 ) 0 0 1 5 2 - 5
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Nomenclature A and B dimensionless material constants d diameter of a projectile E Young’s modulus of the target material axial resistant force on the projectile nose Fx h nose height of the projectile H thickness of a circular plate effective thickness of the target at oblique impact Heff ¼ H=cosðb þ dÞ Hn thickness of the plug I impact function, defined by Eq. (7a) M mass of a projectile mass of a plug Mplug ¼ rpd 2 H n =4 N geometry function of the projectile, defined by Eq. (7b) N1 ; N2 and N dimensionless parameters, defined by Eqs. (3a)–(3c) plastic shear force per unit length of a shear plug Q0 ¼ ðH n þ hÞty t time t1 time at the end of perforation V transient velocity of a projectile ballistic limit VBL initial impact velocity of the projectile Vi nominal residual velocity of the projectile and the plug Vr velocity of the projectile when the plug is just formed V ’ W . W ; W; transverse deflection, velocity and acceleration of the projectile and the plug X penetration depth b initial oblique angle H dimensionless thickness of the plate (ratio of plate thickness and projectile diameter) w¼ d g Poisson’s ratio of the target material rpd 2 H Z ¼ ratio of the plug mass to the projectile mass for a thick plate; similarly 4M 2 rpd H pw ¼ Z¼ 4M 4l M dimensionless mass of the projectile l¼ 3 rd r density of the target material yielding stress of the target material sy pffiffiffi ty shear yielding stress (ty ¼ sy = 3 for von-Mises yielding criterion) Johnson’s damage number (FJ ¼ rVi2 =sy ) FJ
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745
enlargement for sharp nose [2–4], plugging for blunt projectile [5] or fragment for high impact velocity and brittle target, will be responsible for the final perforation of a target plate. Woodward [6] and Woodward and Cimpoeru [7] demonstrated various idealized failure mechanisms for homogeneous plates impacted by various projectiles. Shear plugging and discing failure are usually observed for thick plates, which may be influenced by the tensile opening at the stretched rear surface and the local bending. Multi-stage models have been proposed to study the perforation of relatively thick plates. Awerbuch [8] divided the penetration of a plate target into two stages. In the first stage, only inertia and compressive forces are introduced to decelerate the effective mass of the projectile. The second stage is initiated when a shear plug of the target material is formed, during which the inertia and compressive resistances are replaced by the surrounding shear force. Goldsmith and Finnegan [9] improved Awerbuch’s [8] model by considering the reduction of the shear force in the second stage. Awerbuch’s [8] two-stage model was extended to a three-stage model for the perforation of a plate by a non-deformable projectile [10]. The projectile is subjected to the inertia and compressive resistances of the target material as well as the shear resistance around the plug in the middle stage. This model has been further modified by Ravid and Bodner [11], where a twodimensional model assumes five stages of plate penetration namely, dynamic plastic penetration, bulge formation, bulge advancement, plug formation and projectile exit. The five stages are continuously coupled during penetration and the model predicts not only the exit velocities of the projectile and the plug, but also the bulge and plug shape and the force–time history of the process. Ravid et al. [12] further generalized this analytical model to include various projectile nose shapes, changes in the plastic flow field due to deep penetrations and thermal softening effects, which leads to a variation of failure modes. Liss et al. [13] proposed another five-stage interactive model for the penetration and perforation process, where plastic wave propagation in both the thickness and the radial directions of the plate is considered. A numerical procedure is necessary to solve equations of the motion of the projectile for the five stages. Dikshit and Sundararajan [14] characterized the penetration of an ogival hard projectile into steel plates with lower hardness and various thickness. Dikshit et al. [15] declared that the thickness of target plate strongly affects the effect of plate hardness on ballistic performance. The terms ‘plane strain’ and ‘plane stress’ were introduced for ‘thick’ and ‘thin’ plates to represent ‘constrained’ and ‘unconstrained’ plastic flows, respectively. The transition from plane strain to plane stress conditions occurs when the plastic zone ahead of the penetrating projectile just reaches the bottom surface of the plate. Thus, the penetration of thin plates occurs under plane stress condition while the penetration of thick plates occurs under plane strain condition. Furthermore, several recent analytical models employ point-wise descriptions of the velocity fields. This local model uses the kinematics of an in-viscid fluid in order to generate an approximate velocity field in the target. The origin of this approach dates back to Hill [16], Tate [17] and Ravid and Bodner [11], which has been successfully exploited to study the penetration of both rigid and deformable projectiles (e.g., [12,18,19]). The velocity-field model was further developed by Yarin et al. [20] and Roisman et al. [21,22] to include inertial and elastic–plastic drag effects of a target on the projectile, as well as the finite thickness of the target. It assumes that the target material is incompressible and the target region is divided into an elastic region at a distance from the projectile tip and a rigid-plastic region near the projectile tip. By using the velocity potential f of flow field and method of singularities, the solution procedure presented in
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746
Yarin et al. [20] leads to a straightforward approximation to accommodate the arbitrary shape of the nose of a projectile. The velocity-field model was further applied by Roisman et al. [21] to describe the phenomenon of ricochet. Yossifon et al. [23] examined two different theoretical approaches that developed from the velocity-field model and showed their good agreements with numerical calculations from AUTODYN-2D in respect to the penetration depth, the ballistic limit and the residual velocity of the projectile. Most other analytical models for the perforation of a metallic target impacted by a rigid projectile, e.g., Taylor [24], Thompson [25], Recht and Ipson [26] and Rosenberg and Forrestal [4], are formulated using energy and momentum balance to predict the ballistic limit or the residual velocity. Many empirical formulae, e.g., De Marre (1886), SRI Formula (1963), BRL formula (1968) (see [27]), Neilson [28] and Wen and Jones [29], etc., are also based on experimental data fitting and basic energy balance in perforating a target plate. Energy and momentum methods make it very easy to treat perforation process. However, it is still difficult to give an accurate calculation for all energy consumption in a penetration and perforation process. The vague definition of projectile nose shape is another challenge problem, which may cause uncertainties in the application of these analytical models and empirical formulae. Meanwhile, most analytical models need more complicated numerical procedures to calculate the ballistic performance. It is a challenge to include most important deformation and failure modes while retaining simplicity and a reasonable degree of accuracy in a plate perforation model. In the present paper, an analytical model is proposed in Section 2 for the penetration and perforation of a thick plate based on dynamic cavity expansion theory and plug formation, which is applied to study the perforation performances of both blunt and sharp nose projectiles in Sections 3 and 4. The obtained formulae embrace early analytical models and give good predictions when compared with experimental results, as shown in Section 5.
2. Analytical model 2.1. Penetration model A non-deformable projectile with mass M; diameter d and an arbitrary nose shape, as shown in Fig. 1, impacts a thick plate target normally at initial velocity Vi and proceeds to penetrate the target medium at rigid-body velocity V : Based on the dynamic cavity expansion model [30–32], the resulting axial resistant force on the projectile nose can be calculated from the normal compressive stresses, Fx ¼
pd 2 ðAN1 sy þ BN2 rV 2 Þ; 4
ð1Þ
which has the same form as the Poncelet formula. sy and r are yielding stress and density of the target material, respectively. A and B are dimensionless material constants of target materials, in which A is given by 2 E ð2aÞ A¼ 1 þ ln 3 3ð1 gÞsy
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747
Y →
(_sin , cos )
→
n (cos , sin )
y=y(x)
0 d
X h
Fig. 1. Cross-section of an arbitrary nose.
for an elastic, perfectly plastic material [31], where g is Poisson’s ratio and n Z 1ð3sy =2EÞ 2 2E ðln xÞn dx G ; G¼ A¼ 1þ 3 3sy ð1 xÞ 0
ð2bÞ
for an incompressible elastic, strain-hardening plastic material [32]. B ¼ 1:5 for incompressible materials. However, a numerical procedure is required to calculate parameters A and B in other cases. More details were given in Chen and Li [30]. N1 and N2 in Eq. (1) are two dimensionless parameters relating to nose shape and friction coefficient mm : If an arbitrary nose shape can be represented by the function y ¼ yðxÞ; as shown in Fig. 1, then Z h 8m N1 ¼ 1 þ 2m y dx; ð3aÞ d 0 Z h 8m yy02 N2 ¼ N n þ 2m dx; ð3bÞ 02 d 0 1þy Z 8 h yy03 n N ¼ 2 dx; ð3cÞ d 0 1 þ y02 where h is the height of nose. If the friction is ignored, N1 ¼ 1; N2 ¼ N n : Various formulae of N1 ; N2 and N n for different nose shapes, e.g., cone, ogive, hemispherical nose, and truncated ogive, can be found in Chen and Li [30]. The penetration depth X is determined by dV ð4aÞ ¼ Fx M dt and dX V¼ ð4bÞ dt with the initial condition V ðt ¼ 0Þ ¼ Vi and X ðt ¼ 0Þ ¼ 0: There exists a layer of added mass in front of the projectile nose. Generally, the inertia effect of the added mass should be considered in Eq. (4a), as shown in Yarin et al. [20]. The added mass
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effect is not included in the present study. However, it may become no negligible with the increase of projectile deceleration. 2.2. Motion of the central plug It is assumed that a central plug is formed beneath the projectile at a critical condition when the compressive force on the projectile nose reaches the fully plastic shear force on the plug, as shown in Fig. 2. As soon as the plug is formed, it moves with the projectile under the constant shear resistance, Q0 ; and thus, the acceleration of the projectile and the plug is . ¼ pdQ0 ; ð5aÞ W Mð1 þ Z Þ p ffiffi ffi
where Q0 ¼ H þ h ty ; ty is the yield shear stress of the material (ty ¼ sy = 3 for von-Mises yielding criterion); H n is the residual thickness after penetration by nose tip; Z ¼ ðMplug =MÞ ¼ ðrpd 2 H n =4MÞ and Mplug ¼ ðrpd 2 H n =4Þ is the mass of plug. Eq. (5a) leads to 1 pdQ0 t ’ ð5bÞ V W¼ ð1 þ Z Þ M
and
1 pdQ0 t2 W¼ V t ð5cÞ ð1 þ Z Þ 2M ’ ¼ V =ð1 þ Z Þ at the moment when the plug is formed. V is the velocity of when W ¼ 0 and W the projectile determined by Eqs. (4a) and (4b) at the time when the plug is just formed. 3. Perforation of a thick plate 3.1. Penetration stage If the thickness of the target plate is so large that the projectile stops and embeds in the target medium, the problem can be treated as a pure penetration process. Theoretically, pure penetration
V*
H
h
H*
Plug
Fig. 2. Perforation of thick plates by a non-deformable projectile with an arbitrary nose.
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749
only corresponds to the case that the plastic zone ahead of the penetrating projectile does not reach the rear surface of the plate, i.e., ‘‘constrained’’ plastic flow. However, it may sometimes be applicable to the case when rear bulging (unconstrained plastic flow) is formed, i.e., the plastic zone ahead of the projectile reaches the rear surface of the plate. The maximum penetration depth is given by X 2 I ¼ N ln 1 þ ð6Þ d p N according to Section 2.1 [30], where I and N are defined as impact function and geometry function, I¼
lFJ AN1
ð7aÞ
and N¼
l BN2
ð7bÞ
with l¼
M rd 3
ð8aÞ
and FJ ¼
rVi2 ðJohnson’s damage numberÞ: sy
ð8bÞ
Perforation will occur if further development of the plastic zone ahead of projectile results in a complete failure of target and the projectile passes through the target. In general, shear plugging will occur when the interactive force between the projectile nose and the plate reaches a critical value of Fx ¼ pdQ0
ð9Þ
which may be satisfied when the bluntness of a projectile is larger enough. For a projectile with sharp nose, plugging stage is neglected and the penetration stage dominates the perforation process to produce a hole enlargement. For thin or medium thick plates, shear plugging may play a more important role during the impact process [33]. The velocity of the projectile is assumed to be V * (V * oVi ) when the plugging condition, Eq. (9), is satisfied, and the corresponding thickness of target in front of projectile nose tip is H ðH oHÞ: Meanwhile, it is assumed that the plugging is initiated from the root of projectile nose, as shown in Fig. 2, which is normally observed for projectiles with blunt nose shape. Thus, we have 4 ðH þhÞ ; ðAN1 þ BN2 FJ * Þ ¼ pffiffiffi d 3 where FJ * ¼ rV 2 =sy : *
ð10Þ
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The penetration depth in the first stage is obtained by integrating Eqs. (4a) and (4b), 3 2 I 1þ 6 ðH H n Þ Hn 2 N 7 6 7 ; ¼w 1 ¼ N ln4 I* 5 d p H 1þ N lFJ * H where w ¼ : It reduces to and I * ¼ d AN1 ðI þ NÞ N: I* ¼ pw Hn 1 exp 2N H
ð11Þ
ð12Þ
Substituting Eq. (12) to Eq. (10), we have the dependence of H n on the initial velocity Vi : 4w H n h AN1 ð1 þ I=NÞ : pffiffiffi þ ¼ ð13Þ pw Hn H 3 H 1 exp 2N H 3.2. Shear plug stage The shear plugging stage follows the penetration stage if there exists a solution of H n =H for Eq. (13), which is usually the case for blunt projectiles, e.g., flat nose and hemispherical nose. The ’ ¼ 0 and W ðtÞ ¼ H in Eqs. (5b) and (5c). Therefore, the ballistic limit is obtained when WðtÞ ballistic limit for a thick plate impacted by a hard projectile is (" ) # 8BN2 wZð1 þ Z * Þ H n h H n AN1 sy pw Hn 2 pffiffiffi 1 1þ þ exp VBL ¼ 1 ð14Þ H H 2N BN2 r H H 3AN1 in which, H =H in Eq. (13) contains VBL : Thus, it is necessary to solve a nonlinear algebra equation for VBL : ’ 1 Þ when W ðt1 Þ ¼ H : The residual velocity for Vi > VBL is obtained as Vr ¼ Wðt vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 u ðV 2 V 2 Þ u i 1 n ; ð15Þ Vr ¼ pw H ð1 þ Z * Þt exp 1 H 2N in which, V1 has the same expression as VBL in Eq. (14). H =H in Eqs. (14) and (15) is calculated from Eq. (13) for a given initial impact velocity Vi : 4. Special cases 4.1. Perforation of a thick plate by a sharp-nosed projectile If the projectile has a larger value of geometry function N; corresponding to a slender projectile body or sharper nose [30], plug may not occur or can be ignored compared to the whole hole
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751
enlargement. In view of mathematics, it requires that Fx opdQ0 in the penetration process, i.e., H n -0
ð16aÞ
and
pffiffiffi 3AN1 ð1 þ I=NÞ h pw i : ð16bÞ 4 exp 2N Thus the ballistic limit and residual velocity are, respectively, o AN1 sy n pw 2 ¼ VBL 1 ; ð17aÞ exp 2N BN2 r vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 uðV V 2 Þ : ð17bÞ Vr ¼ t i pwBL exp 2N Especially for a sharper projectile perforating a thick metallic plate, the geometry function of projectile N is large enough (I=N-0; referred to [30]), thus simpler formulae can be conducted: h X d
H n -0 and h X d VBL
pffiffiffi 3A ; 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ZAN1 sy ¼ r
and Vr ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 ffi 2 ; Vi VBL
ð18aÞ
ð18bÞ ð19bÞ
ð19cÞ
where Z ¼ rpd 2 H=4M ¼ pw=4l similar to the definition of dimensionless mass Zn : 4.2. Perforation of a thick plate by a flat-nosed projectile For a flat-nosed missile perforating a thick metallic plate (referred to Section 4 of [33]), N1 ¼ N2 ¼ N n ¼ 1; Eq. (13) is simplified to "pffiffiffi # n 3 Hn 1 H 1 ðA þ BFJ Þ : ð20Þ ln ln ¼1 4w 2ZB 2ZB H H pffiffiffi It ispeasily found that a solution of H n =Ho1 exists only if w > 3ðA þ BFJ Þ=4: Alternatively if ffiffiffi wp 3ðA þ BFJ Þ=4; whole shear plugging (i.e., H n ¼ H) without any indention/penetration is applied in Chen and Li [33] to predict the ballistic performance as well as consider the structural response (bending and effect of membrane force). After specifying the projectile and target material, only two dimensionless quantities, i.e., w and FJ are left to determine H n =H: Fig. 3 shows the dependence of H n =H on w with different initial velocities Vi ; in which a hard flat-nosed
X.W. Chen, Q.M. Li / International Journal of Impact Engineering 28 (2003) 743–759 1.0 0.8
H*/H
2.8
vi=0.2 km/s vi=0.3 km/s vi=0.5 km/s vi=0.7 km/s vi=0.9 km/s H*/H at ballistic limit Ballistic limit (km/s)
0.6
2.4 2.0 1.6 1.2
0.4
VBL (km/s)
752
0.8 0.2 0.4 0
0 0
5
10
15
20
25
H/d
Fig. 3. Dependence of H * =H on w with different initial velocities Vi for a flat-nose steel missile perforating the aluminium plate.
steel missile of mass 25 g and diameter 7.1 mm perforates a thick 6061-T6 aluminium plate (same as [4] but replacing the conical-nosed missile by flat-nosed missile). Meanwhile, Fig. 3 also n =H at the demonstrates the dependence of ballistic limit VBL on w and the critical curve of HBL n ballistic limit. Only the regime above the critical curve of HBL =H corresponds to the perforation. Indentation/penetration is considered when the dimensionless thickness of plate w > 2:06: Obviously, H n =H drops very quickly and intends to be omitted if the target plate becomes thicker. Assuming the plug is a cylinder with same diameter as the projectile, the residual velocity and ballistic limit of flat-nosed projectiles impacting on thick target plates (corresponding to Eqs. (14) and (15)) are, respectively, (" ) n 2 # Asy 8BwZ Hn H Hn 2 exp 2ZB 1 1 þ pffiffiffi 1 þ Z 1 ; ð21aÞ VBL ¼ Br H H H 3A vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ðVi2 V12 Þ 1 u : u Vr ¼ t Hn Hn exp 2ZB 1 Z 1þ H H
ð21bÞ
Obviously if local shear/plugging dominates in the impact process, i.e., H n -H; Eqs. (21a) and (21b) also lead further to 8sy 2 ¼ pffiffiffi wZð1 þ ZÞ; VBL 3r Vr ¼
1 ð1 þ ZÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Þ: ðVi2 VBL
Eq. (22b) is the same as the formula of Recht and Ipson’s [26] model.
ð22aÞ
ð22bÞ
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753
5. Experimental analyses Firstly a collection of experimental data from Rosenberg and Forrestal [4], Forrestal and Luk [34], Piekutowski et al. [3] and Roisman et al. [22] are compared with the present analytical predictions on ballistic limit and residual velocity. These experimental results fall in the category of sharp-nosed projectiles and as a result no plugging occurs in perforation. Based on the definition of dimensionless numbers [30], the nose parameters of conical-nose projectile in Rosenberg and Forrestal [4] are c ¼ 1:507; N n ¼ 0:099; N1 ¼ 1:06; N2 ¼ 0:105; and the mass ratio is l ¼ 25:17; and those in Forrestal and Luk [34] are c ¼ 1:785; N n ¼ 0:073; N1 ¼ 1:07; N2 ¼ 0:078; l ¼ 16:72: The nose parameters of ogive–nose projectile in Piekutowski et al. [3] are c ¼ 3 (CRH), N n ¼ 0:106; N1 ¼ 1:09; N2 ¼ 0:11; and the mass ratio is l ¼ 13:96 and those in Roisman et al. [22] are c ¼ 2 (CRH), N n ¼ 0:156; N1 ¼ 1:07; N2 ¼ 0:163; l ¼ 22:50: The friction coefficient uses mm ¼ 0:02 for both ogive and conical nose, as suggested by Forrestal and Luk [31] and Forrestal and Luk [34]. For 6061-T6 aluminium plates used in Rosenberg and Forrestal [4], g ¼ 1=3; Y ¼ 300 MPa; Y =K ¼ 0:0045 (or E ¼ 66:7 GPa) and r ¼ 2710 kg=m3 : For 5083-H131 aluminum plates used in Forrestal and Luk [34], g ¼ 1=3; Y ¼ 276 MPa; E ¼ 70:3 GPa and n ¼ 0:085 and r ¼ 2660 kg=m3 : For 6061-T651 aluminium plates used in Piekutowski et al. [3], g ¼ 1=3; Y ¼ 262 MPa; E ¼ 69 GPa; n ¼ 0:085 and r ¼ 2710 kg=m3 : Roisman et al. [22] also used 6061-T651 aluminium as target material but Y ¼ 345 MPa; which is due to the fact that targets can be made from layered or non-layered material. Simply we use Eq. (2a) to evaluate parameter A and take B ¼ 1:5: Figs. 4–7 indicate the tests by Rosenberg and Forrestal [4], Forrestal and Luk [34], Piekutowski et al. [3] and Roisman et al. [22] and the corresponding theoretical prediction, respectively. The effective thickness of plates at oblique perforation is evaluated based on Heff ¼ H=cosðb þ dÞ; where b is the initial obliquity and d is the angle of the direction change. Roisman et al. [21] confirmed that the angle of directional change d is very small and can be dropped out in the oblique perforation of metal target if initial obliquity bo501: The initial obliquity in
1.6 Analysis Normal missile Arrowhead missile
1.4 1.2
Vr (m/s)
1.0 0.8 0.6 0.4 0.2 0 0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Vi (km/s)
Fig. 4. Prediction of ballistic performance and test data [4].
X.W. Chen, Q.M. Li / International Journal of Impact Engineering 28 (2003) 743–759 1.2
Vr (km/s)
Analysis 1.0
H=12.7mm
0.8
H=50.8mm H=76.2mm
0.6 0.4 0.2 0 0
0.2
0.4
0.6
0.8
1.0
1.2
Vi (km/s)
Fig. 5. Prediction of ballistic performance and test data [34].
1.0 Analysis Normal impact
0.8 Vr (km/s)
Oblique impact 0.6 0.4 0.2 0 0
0.2
0.4
0.6
0.8
1.0
Vi (km/s)
Fig. 6. Prediction of ballistic performance and test data [3]. 1.4 Analysis
1.2
Roisman et al.(1999) 1.0 Vr (km/s)
754
0.8 0.6 0.4 0.2 0 0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Vi (km/s)
Fig. 7. Prediction of ballistic performance and test data [22].
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755
Piekutowski et al. [3] and Roisman et al. [22] is 301 and 451, respectively. Obviously, the theoretical analyses are in excellent agreement with the test data. Authors were not aware of any published experimental results on the perforation of thick plates impacted pffiffiffi by blunt (e.g., flat or hemispherical nose) projectiles, which satisfy the condition of w > 3ðA þ BFJ Þ=4: It is necessary to verify the perforation analysis for thick plates by rigid blunter projectiles when experimental data are available. However, it has been shown that, in a special case when the localized shear plugging dominates the impact process, the present analysis leads to the same results obtained by Recht and Ipson [26] for a flat-nosed projectile impacting a medium plate. Experimental results given in Borvik et al. [5] and Liss and Goldsmith [35] belong to this case, which are compared with the present model in Figs. 8 and 9. Awerbuch and Bodner [36] performed plate perforation experiments using standard lead and armour piercing bullets impacting aluminium and steel plates. Appreciable flattening of the
300 Borvik et al.(1999)
250
Analysis Vr (m/s)
200 150 100 50 0 100
150
200
250
300
350
Vi (m/s)
Fig. 8. Prediction of residual velocity and test data [5].
600 Liss & Goldsmith(1984) 500
Analysis
Vr (m/s)
400 300 200 100 0 0
100
200
300
400
500
600
700
Vi (m/s)
Fig. 9. Prediction of residual velocity and test data with H ¼ 6:4 mm in Ref. [35].
756
X.W. Chen, Q.M. Li / International Journal of Impact Engineering 28 (2003) 743–759
projectile nose was observed, and thus, the plate was actually perforated by deformable projectile. Nevertheless, Awerbuch and Bodner [10,36], and Ravid and Bodner [11] treated the projectiles as a rigid cylinder with the tunnel diameter and calculated the residual velocity based on the multistage models. Yuan et al. [37] compared some data of these experimental results with their plastic wave theory. In the present paper, when the assumption of rigid projectile is applied, Eqs. (22a) and (22b) can be used to calculate the ballistic limit and residual velocity for those experimental results in Awerbuch and Bodner [36] on thick plates, which satisfy either wE1 or w > 1: Table 1 summarizes the experimental and calculated results. It should be noted that for test data on thin plates (e.g., wo1=2) in Awerbuch and Bodner [36], structural responses (e.g., bending effect or membrane deformation) should be taken into account and the relevant issue can be seen in Chen and Li [33].
6. Discussion When calculating the resistant force on the projectile nose, it is assumed that the projectile nose is completely embedded in the target medium, which cannot be satisfied during entry and exit phases. The energy consumption is overestimated during the entry phase and underestimated during the exit phase. However, the reduced energy consumption in the exit phase will compensate the increased energy consumption in the entry phase. This would not cause noticeable difference when the impact and residual velocities are relatively high. Since the initial velocity is always greater than the exit velocity, the model may over-predict the ballistic performance of the target, which only becomes significant when the impact velocity is much higher than the residual velocity and the target thickness is comparable with the nose length of the projectile. There are several empirical formulae and analytical solutions, e.g., De Marre (1886), SRI Formula (1963), BRL formula (1968) (see [27]), Recht and Ipson [26], Neilson [28] and Wen and Jones [29], to predict the ballistic performance of the metallic target. Majorities of the empirical formulae result from curve-fitting practices based on test data and are dimensionally dependent, which provide little physical understanding of the studied problem. Their applications are normally limited by the extent of test parameters and there exists an ambiguity of the projectile nose description. It is necessary to develop a simple but accurate model that can be easily used in the design of a protective structure. The present analysis is actually a two-stage model, i.e., penetration/indention and shear plugging, based on the dynamic cavity expansion and rigid-plastic analysis. The geometric characteristic of the projectile nose shape is considered analytically. Obviously, the formulae obtained in Forrestal et al. [2], Rosenberg and Forrestal [4], Forrestal and Luk [34], Piekutowski et al. [3] and Recht and Ipson [26] are special cases of the present analysis. It implies that the present model unifies these relevant theoretical models on the perforation studies. The present model is applicable to the perforation of thick plates by hard projectiles with arbitrary nose shapes, e.g. conical, ogival, blunt and other more complicated noses. It is also applicable to oblique perforation at small initial obliquity (i.e., bo501). However, the present model still needs to be further developed so that it can be applicable to yaw impact, oblique impact at large obliquity and impact with ricochet.
MS-A-10
AL-6-9.6 AL-6-13 AL-6-19
SA-A-6 6 SA-B-6.35 6.35 SA-D-12 12
S-R
S-R S-R S-R
SAP SAP SAP
9.6 13 19
10
SA-A-6 6 SA-A-8 8 SA-B-6.35 6.35 SA-C-8 8 SA-D-9 9 SA-D-10 10
7.8 7.8 7.8
2.7 2.7 2.7
7.8
7.8 7.8 7.8 7.8 7.8 7.8
980 1200 900
280 280 280
280
980 980 1200 1050 900 900
Plate Plate Yielding thickness density stress of H (mm) r (g/cm3) target sy (MPa)
S-R S-R S-R S-R S-R S-R
Projec- Target tile
10.5 9.8 7.45
8.4 10.1 11
11.9
10.5 10.6 10.4 10 10.5 11.5
835 848 855
845 845 836
855
850 855 854 855 850 845
Average Impact diameter velocity of tunnel Vi (m/s) Davg (mm)
361
453 318 400 351 243 95
Vr (m/s) by Awebuch and Bodner [10]
550–650 499 500–600 492 300–390 418
748 702 712 604 568–585 406
400
500–600 460 350–550 450–470 220–290 155–185
Residual velocity by test Vr (m/s)
660 589 364
Vr (m/s) by Ravid and Bodner [11]
Table 1 Comparison between perforation experiments and theory for steel and aluminium plates [36]
467 476
478 454 463 455
549 565 470
799 653 434
509
548 415 515 431 365 187
392 424 593
151 355 604
341
405 581 475 579 631 779
0.83 0.79 0.75
0.26 0.58 0.63
0.58
0.83 0.86 0.83 0.88 0.86 0.86
0.95 0.91
0.95 0.95 0.94 0.96
1 1 1
1 1 1
1
1 1 1 1 1 1
Vr (m/s) Vr (m/s) Ballistic H n =H H n =H by H n =H by by Yuan by present limit by by test Yuan present present et al. model et al. model [37] model [37] VBL (m/s)
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7. Conclusions This paper presents a two-stage model, i.e., penetration and shear plugging, to study the perforation of thick plates by non-deformable projectiles with different geometrical characteristics. Analytical formulae are given to predict the ballistic performance of perforation of thick plate targets. It unifies several well-known theoretical models on perforation and is available to hard projectiles with different nose shapes.
Acknowledgements The first author would like to acknowledge the Ph.D. scholarship granted by the School of Civil and Environmental Engineering, Nanyang Technological University and the leave of absence offered by the China Academy of Engineering Physics (CAEP).
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