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Shear plugging and perforation of ductile circular plates struck by a blunt projectile
International Journal of Impact Engineering 28 (2003) 513–536
Shear plugging and perforation of ductile circular plates struck by a blunt projectile X.W. Chen, Q.M. Li* Protective Technology Research Center, School of Civil and Environmental Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore Received 12 July 2001; received in revised form 4 April 2002; accepted 18 August 2002
Abstract Shear plugging and perforation of ductile circular plates struck by a blunt rigid projectile is studied in the present paper. Effects of transverse shear, bending and membrane deformations on the perforation process are included in a rigid-plastic analysis while the local indentation/penetration is estimated using the dynamic cavity expansion model. Analytical formulae for the perforation ballistic limit and the residual velocity are obtained for a range of plate thickness, which agree with the available experimental results on the perforation of metallic plates. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Perforation; Ballistic limit; Ductile plate; Blunt projectile; Shear plugging
1. Introduction Both local impact effect and global structural response may be involved in the perforation of a ductile plate subjected to the strike of a non-deformable projectile. In addition to the local indentation or penetration,1 there are basically two types of structural responses in a circular plate, i.e., the bending response and the membrane response. Their contributions to the perforation depend mainly on the thickness of the target and the impact velocity. Generally, local and shear deformations become more important with the increase of plate thickness or impact velocity. Membrane deformation decreases with the increase of plate thickness and bending deformation may reach a maximum at a certain plate thickness [1]. For a thick target, the *Corresponding author. Current address: Department of Mechanical, Aerospace and Manufacturing Engineering, UMIST, P.O. Box 88 Sackville Street, Manchester M60 1QD, UK. Tel. +44-161-200-5740; fax: +44-161-200-3849. E-mail address: [email protected] (Q.M. Li). 1 Indentation is used when the penetration is small. 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 0 7 7 - 5
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Nomenclature a and b material constants in Poncelet’s formula A and B dimensionless material constants cross-sectional area of a projectile A0 d diameter of a projectile D diameter of a circular plate E Young’s modulus of the target material axial resistant force on the projectile nose Fx G mass of a projectile mass of a plug Gpl H thickness of a circular plate thickness of a plug H k empirical parameter in the shear failure criterion of structural members (k=1 in the present paper) Mr ; My radial and circumferential bending moments per unit length of a circular plate plastic bending moment per unit length of a rigid, perfectly plastic circular plate M0 plastic membrane force per unit length of a rigid, perfectly plastic circular plate N0 shear force per unit length of a circular plate Qr Q0 ¼ Hty plastic shear force per unit length of a circular plate Q0 ¼ H ty plastic shear force per unit length of a shear plug t time time at the end of the first phase of motion t1 critical time at the moment of perforation tBL v normal velocity on the surface of a projectile nose V transient velocity of a projectile VBL ballistic limit of a projectile initial impact velocity of a projectile Vi VJump velocity jump of the projectile at the ballistic limit Vpr ; Vplr residual velocities of the projectile and the target plug, respectively nominal residual velocity of the projectile and the plug Vr velocity of the projectile when the plug is just formed V ’ W . transverse deflection, velocity and acceleration fields of a circular plate W, W; ’ . 0 transverse deflection, velocity and acceleration of the projectile and the plug W0 ; W0 ; W ’ . 1 transverse displacement, velocity and acceleration of the plate at r ¼ d=2 W1 ; W1 ; W X penetration depth g Poisson’s ratio of the target material er ; ey radial and circumferential strains at the mid-plane of the cross-section of a circular plate Z ¼ rpd 2 H=4G ratio of the plug mass to the projectile mass for a thin plate Z ¼ rpd 2 H =4G ratio of the plug mass to the projectile mass for a thick plate W dimensionless number in structural model kr ; ky radial and circumferential curvatures in a circular plate l ¼ G=rd 3 dimensionless mass of the projectile
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x location of a bending hinge during the shear sliding phase r density of the target material yielding stress of the target material sy pffiffiffi yielding stress in shear (ty=sy/ 3 for von-Mises yielding criterion) ty Johnson’s damage number (FJ ¼ rVi2 =sy ) FJ w ¼ H=d dimensionless thickness of the plate (ratio of plate thickness and projectile diameter) empirical value in the case of small plate thickness w1
perforation is dominated by the local penetration although the failure mechanism of the final perforation also influences the ballistic limit of a thick target, which depends on the target material, target dimensions, projectile nose and impact velocity. The failure mechanism leading to the final perforation of a target plate could be ductile hole enlargement or petaling for a sharp nose, fragmentation for high impact velocity and brittle target, plugging or adiabatic shearing for a blunt projectile [1–3]. With the increase of the plate thickness, the impact velocity and the bluntness of the projectile, shear plugging becomes a likely failure mode of the final perforation of a plate. Based on the conservation of momentum and energy, Recht and Ipson [4] proposed a shear plugging model to predict the residual velocity according to a given impact velocity while a ballistic limit velocity was obtained from a dimensional analysis, which completely ignored the structural response for relatively thin plates and the local penetration for relatively thick plates. The so-called structural model may be considered as a further development of Recht and Ipson’s model [4], but it is actually an application of rigid-plastic structural dynamics [5] in perforation studies. A structural model, which neglects the local penetration, has given good prediction for thin plate perforation under blunt (flat) projectile impact, as shown in Refs. [6–8]. The shear plugging in a structural model is based on the shear hinge concept in a rigid-plastic model, which may be triggered at the early response stage for a relatively thick structural member [5,9–11]. It is still a challenge to obtain a general structural model of perforation analysis, which incorporate the local failure analysis in the rigid-plastic model because different local response and failure modes, such as dishing, petalling or penetration, may appear for different plate thicknesses and impact velocity [1]. Meanwhile, it is difficult to get simple formulations for a general structural model, which is not in favour of engineering applications. 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. Multi-stage models have been proposed to study the perforation of relatively thick plates when structural response is negligible. Awerbuch [12] 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 compressive resistance is replaced by the surrounding shear force. Goldsmith and Finnegan [13] improved the model of Awerbuch by considering the reduction of the shear force in the second stage. Awerbuch’s two-stage model was extended to a three-stage model for the perforation of a plate by a non-deformable projectile [14]. The projectile is subjected to the inertia and the compressive resistance of the target material as well as the shear resistance around
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the plug in the middle stage. This model has been further modified in Ref. [15], where a twodimensional model assumes five stages of plate penetration, namely dynamic plastic penetration, bulge formation, bulge advancement, plug formation and projectile exit. These 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 shapes as well as the force–time history of the process. This analytical model was further generalized in Ref. [16] to include various projectile nose shapes, changes in the plastic flow field due to deep penetrations and thermal softening effects. Liss et al. [17] 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 motion of the projectile in these five stages. For very thick (essentially semi-infinite) metallic targets impacted by projectiles, a number of penetration models have been proposed [18,19]. Poncelet’s formula is used to calculate the resistant force on a flat projectile nose [20], F ¼ A0 ða þ bV 2 Þ; where A0 is the cross-sectional area of the projectile, V is the impact velocity and a and b are material constants to be determined by experiments (For non-flat nose, a and b depend on the nose shape). Both multi-stage models for perforation and models for deep penetration employed Poncelet’s formula. However, in the study of deep penetration, dynamic cavity expansion model, which was originated from a quasi-static cavity expansion model [21] and subsequently developed and applied in Refs. [22–24], offers a theoretical foundation of Poncelet’s formula. Applications of the dynamic cavity expansion model to predict the penetration depth for various target media, projectile shapes and impact velocities are given in Refs. [25,26]. In the five-stage perforation model in Ref. [17], the shear wave propagation in the target plate outside the plug interface was considered. However, the structural response outside the shear plug is generally neglected in most multi-stage models. Shadbolt et al. [27] hybridized the multi-stage model and the structural model. It was shown that the analytical prediction is improved when Reissner’s plate theory is used as a structural model to replace the simplified plastic membrane and bending model. However, uncertainties of some empirical data and the involvement of numerical procedures increase the difficulty for their practical application. Nevertheless, a similar method but different approach will be employed in the present paper. Experimental studies on the perforation of metallic plates have been summarized in Refs. [1,2]. Experimental results were presented either by the variation of ballistic limit with plate thickness and the hardness of the plate material, or by the variation of the residual velocity of the projectile with impact velocity, which have revealed several interesting phenomena. First, it was shown that the perforation ballistic limit may decrease with the increase of plate thickness in a range of plate thicknesses [28]. Most of the analytical models cannot predict such a local drop of the ballistic limit except the rigid-plastic plate model in Ref. [7], which considered all possible structural responses in a perforation analysis. However, the model in Ref. [7] requires a numerical algorithm to solve non-linear differential equations based on a selected velocity field, which is not favorable to practical applications. Local penetration is not considered in Ref. [7].
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Second, a residual velocity jump at the ballistic limit was noted in Ref. [29] for the perforation of HY-100 steel plates. It shows that the residual velocity may jump to a finite value at the ballistic limit in some cases. Neither Recht and Ipson’s model [4] nor other structural and multi-stage models can predict the velocity jump at the ballistic limit. Forrestal and Hanchak [29] used a rigidplastic beam model to illustrate this phenomenon. Third, a local drop of ballistic limit with the increase of target material hardness was observed in Ref. [30]. This observation is in contradiction with the previous design philosophy, i.e., the higher the hardness, the better the perforation performance. Further experimental evidence for this phenomenon has been shown in Ref. [31]. It is believed that this local drop is due to the initiation of adiabatic shearing regime [1,30,31] although the increased brittleness of the hardened material may also contribute to an easy failure of the target. The transition from shear failure to adiabatic shear failure in a beam was studied in Ref. [32]. However, authors have not noted any analytical model to study this problem for a plate. It is a challenge to develop a simple model, which has a reasonable degree of accuracy and is capable of predicting experimental observations. In the present paper, a hybrid model is proposed to study the perforation of a ductile circular plate due to plug failure when struck by a blunt projectile. The proposed model consists of a plug motion, a rigid-plastic analysis and a local indentation/penetration model, as introduced in Section 2. The simplicity of the rigid-plastic analysis replaces Reissner’s plate theory in the two-stage model in Ref. [27] and makes it possible to get explicit expressions of both ballistic limit and residual velocity in a range of plate thicknesses, as shown in Sections 3 and 4. Although, rigid-plastic model used in other structural models are sometimes more general than the present rigid-plastic model, they generally neglected the local penetration effects and did not give analytical solutions and explicit expressions of the ballistic limit and the residual velocity due to difficulties in solving nonlinear dynamic equations analytically. Experimental results from different sources are compared with the present analytical predictions in Section 5 with satisfactory agreement. Section 6 gives the limitation of the present model and further works in this research subject are suggested.
2. Analytical model 2.1. 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 equals the fully plastic shear force on the plug, as shown in Fig. 1. 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 . 0 ¼ pdQ0 ; W Gð1 þ Z Þ
ð1aÞ
where Q0 ¼ H ty ; d is the diameter of the projectile, G is the mass of the projectile, ty is the yield shear stress of the material (ty=sy/O3 for von-Mises yielding criterion), H* is the thickness of the
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V*
H
H* Fig. 1. Perforation of a thick plate.
plug, Z ¼ rpd 2 H =4G: Eq. (1a) leads to 1 pdQ0 t ’ V W0 ¼ ; G 1 þ Z 1 pdQ0 t2 V t W0 ¼ ; 2G 1 þ Z
ð1bÞ
ð1cÞ
’ 0 ¼ V =ð1 þ Z Þ are satisfied at the moment when the plug is formed. V is the when W0=0 and W velocity of the projectile when the plug is just formed.2 2.2. Rigid-plastic analysis The plate deformation outside the central plug is controlled by the energy conservation. A general expression of the energy conservation for a rigid-plastic plate considering bending and membrane has been given in Ref. [5], which may be extended when the transverse shear effect is included, i.e.,3 Z x Z x . ’ ’ . ’ G 1 þ Z W0 W0 2prH ðM0 þ N0 W Þk’ y r dr þ pd ðM0 þ N0 W1 Þc WWr dr ¼ 2p d=2
d=2
’ þ pdQ0 W ’0 W ’1 þ 2pxðM0 þ N0 W1 Þc
for the transverse velocity field shown in Fig. 2a, i.e., 8 ’ 0; 0prpd=2; >W < ’ ¼ 2W ’ 1 ðx rÞ=ð2x d Þ; d=2orpx; W > : 0; xorpD=2;
ð2Þ
ð3Þ
’ 1 is the where x represents the location of a bending hinge during the shear sliding phase and W transverse velocity of the plate at r ¼ d=2: H is the thickness of the circular plate; M0 ¼ sy H 2 =4 and N0 ¼ sy H are the fully plastic bending moment and membrane force of a rigid, perfectly plastic circular plate, respectively. 2
In Sections 3 and Appendix A, there is no indentation/penetration, and therefore, H ¼ H; Q0 ¼ Q0 ; Z ¼ Z and V ¼ Vi ; where Vi is the initial impact velocity of the projectile. 3 If penetration occurs before plugging, M0 and Q0 are replaced by M0 ¼ sy H 2 =4 and Q0 at r ¼ d=2; respectively.
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D . W1
r
r
D
. . W W0
. W1
d (b)
(a)
519
. W0
. W
d
D . W0
d (c)
Fig. 2. (a) Transverse velocity profiles for the circular plate (bending hinge locates at r ¼ x), (b) transverse velocity profiles for the circular plate (bending hinge locates at r ¼ D=2), (c) transverse velocity profiles for the circular plate (localized shear only).
The curvatures and the middle surface strain in a circular plate are @2 W 1 @W 1 @W 2 ; er ¼ ; ey ¼ 0; kr ¼ 2 ; ky ¼ @r r @r 2 @r
ð4Þ
2 ’ ’ 1 =frð2x dÞgX0 for d=2orpx; k’ r ¼ d 2 W=dr ’ ¼ 2W ¼ 0; for which lead to k’ y ¼ ðdW=drÞ=r ’ ¼ 2W ’ 1 =ð2x d Þ: d=2orox; k’ r X0 at r ¼ d=2; k’ r p0 at r ¼ x and c
2.3. Penetration model The dynamic cavity expansion analysis yields the following relation between the normal compressive stress sn on the projectile nose and the normal velocity v [24,33–36]: sn ¼ Asy þ Brv2 ;
ð5Þ
where sy and r are the yielding stress and the density of the target material, respectively. A and B are dimensionless material constants. For a flat projectile nose, the normal velocity v at the nose– target interface is the same as the rigid-body velocity V of the projectile and the tangential stress on the projectile nose is zero [25].
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The resultant axial resistant force on the projectile nose can be calculated from the normal compressive stresses pd 2 Asy þ BrV 2 ; ð6Þ Fx ¼ 4 which has the same expression as the Poncelet formula introduced in Section 1. The penetration depth X is determined by dV dX ¼ Fx and V ¼ : ð7a; bÞ G dt dt Parameter A in Eqs. (5) and (6) is given by
2 E 1 þ ln ð8Þ A¼ 3 3ð1 gÞsy for an elastic, perfectly plastic material [33], where g is Poisson’s ratio and n Z 1ð3sy =2E Þ 2 2E ðln xÞn dx A¼ 1þ J ;J ¼ 3 3sy ð1 xÞ 0
ð9Þ
for an incompressible elastic, strain-hardening plastic material [36]. B=1.5 for incompressible materials. However, a numerical procedure is required to calculate parameters A and B in other cases. More discussion is given in [25]. pffiffiffi 3 ðA þ BUJ Þ 3. Perforation of intermediate thick plates, v1 ovp 4 3.1. Introduction Rigid, perfectly plastic analysis has been successfully used to predict the structural response under impact and blast loads. It has been shown that the localized transverse shear may be triggered if the loading intensity is high enough [9,10,37–39]. The formation of the localized transverse shear hinge in a structural member has been studied in Refs. [10,11]. Both experimental and analytical studies on beams and plates show that the localized transverse shear deformation, as long as it is initiated, dominates the early stage of the structural response. The transverse shear response ceases before the bending response and the structural response is gradually dominated by the membrane response when the plate deflects to the same order of the plate thickness [5]. For a relatively thick target, the plate deflection is small during the plugging process. The following analysis focuses on the first phase of the motion of a plate target, where only shear and bending responses are considered. However, a lower bound estimate on the ballistic limit and the residual velocity will be given in Appendix A for a relatively thin plate, where membrane deformation is considered. 3.2. First phase of motion, 0ptpt1 When the membrane effect is neglected, the normality rule of plasticity associated with the cubic yield condition in Fig. 3 requires Mr ¼ M0 at r ¼ d=2; Mr ¼ M0 at r ¼ x; My ¼ M0 and M0 oMr oM0 for d=2orox: In addition, the transverse shear hinge at r ¼ d=2 requires
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M M0
Mr M0
Qr Q0 Fig. 3. Cubic yield condition for the first phase of motion.
Qr ¼ Q0 : According to Eq. (2), the motion of the circular plate is controlled by . 1 ¼ 24½dQ0 ð2x d Þ 8xM0
W rH ð2x þ 3d Þð2x d Þ2
ð10Þ
for the assumed transverse velocity field given by Eq. (3). The location of the bending hinge, x, is stationary during the first response phase, i.e., the shear and bending response phase, when the membrane effect is ignored. The shear resistance per unit length of the plate at position r ¼ x is [5]
1 @ðrMr Þ 2M0 My ¼ ; Q¼ r @r x which, together with the momentum conservation in the impact direction, i.e., Z x ’ 0 þ 2prH ’ dr ; 2pxQt ¼ 4pM0 t ¼ GVi G W Wr 0
requires
Z ’ 4pM0 t ¼ GVi G W0 þ 2prH
x
’ Wr dr :
ð11Þ
0
For the velocity field distribution defined in Eq. (3), Eq. (11) gives . 1 ¼ 6ðdQ0 4M0 Þ ; W rH ð2x d Þðx þ d Þ which leads to ’ 1 ¼ 6ðdQ0 4M0 Þt W rH ð2x d Þðx þ d Þ
ð12Þ
ð13aÞ
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and W1 ¼
3ðdQ0 4M0 Þt2 ; rH ð2x d Þðx þ d Þ
ð13bÞ
’ 1 ¼ 0 at t=0. under the conditions of W1 ¼ 0 and W The position of the bending hinge, x, can be determined from Eqs. (10) and (12), qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 3w þ 1 þ 2 3w 6w2 x ; ð14Þ ¼ pffiffiffi d 2 1 3w pffiffiffi pffiffiffi where w ¼ H=d ¼ 4M0 = 3dQ0 : wo1= 3 is required by Eq. (14). It shows that the position of plastic bending hinge depends on the dimensionless parameter w. The position of the bending hinge determined by Eq. (14) satisfies 12ox=doD=2d if " # 2 D=d 1 1 wopffiffiffi : 2 3 D=d þ 1 þ2 When " # 2 D=d 1 1 1 pffiffiffi pwopffiffiffi; 2 3 D=d þ 1 þ2 3 the plastic pffiffiffi hinge stays at x ¼ D=2 and the velocity field is described by Eq. (3) and Fig. 2b. When wX1= 3; plastic hinge does not exist and the corresponding velocity field is Eq. (3) with x ¼ d=2; as shown by the localized shear in Fig. 2c. However, if the thickness of the plate is small, the membrane effect on the structural response should be included, as presented in Appendix A. According to the discussion in Section 5, results of this section are valid for w > w1 ; where w1 depends on D=d and target material properties. w1 varies between 0.1 and 0.3 for the experiments discussed in Section 5. On the other hand, the total interactive force between the projectile nose and plate target during the penetration/perforation is given by Eq. (6). If FxopdQ0, localized indentation or penetration occurs, which will be studied in Section 4. Otherwise, when shear plugging occurs, FxXpdQ0. The condition for the occurrence of the shear plugging can be expressed in a dimensionless form pffiffi 3 wp 4 ðA þ BFJ Þ where FJ ¼ rV 2 =sy is Johnson’s damage number. Thus, the analytical model in pffiffi 3 this section is valid for w1 pwp 4 ðA þ BFJ Þ: Figs. 2a–c correspond to " # 2 D=d 1 1 w1 owopffiffiffi ; 2 3 D=d þ 1 þ2 " # 2 D=d 1 1 1 pffiffiffi pwopffiffiffi 2 3 D=d þ 1 þ2 3
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and pffiffiffi 3 1 pffiffiffipwp ðA þ BFJ Þ; 4 3 respectively. For the velocity fields in Figs. 2a–c, maximum shear sliding is reached at the end of the first ’ 1 at t ¼ t1 ; where t1 is determined by Eqs. (1b) and (12) as ’0 ¼ W phase of motion when W pffiffiffi 3 rdVi t1 ¼ ð15Þ 4ðZ þ WÞ sy for the velocity fields in Fig. 2a–c where 8 pffiffiffi " # 2 > 3 1 3w ð1 þ ZÞ > D=d 1 > 1 > > > > 2 2x=d 1 x=d þ 1 ; w1 owopffiffi3ffi D=d þ 12 þ2 ; > > > > pffiffiffi > " # < 2 3 1 3w ð1 þ ZÞ D=d 1 1 1 W¼ ; pffiffiffi pwopffiffiffi; 2 > > > D=d 1 D=d þ 2 3 D=d þ 1 þ2 3 > > > p ffiffi ffi > > > 3 1 > > > pffiffiffipwp ðA þ BFJ Þ: : 0; 4 3
ð16Þ
Thus, according to Eqs. (1c) and (13b), the maximum transverse shear displacement at the end of the first phase of motion is pffiffiffi 3FJ d Vi t 1 : ð17Þ ¼ Ws ðt1 Þ ¼ W0 ðt1 Þ W1 ðt1 Þ ¼ 2ð1 þ ZÞ 8ð1 þ ZÞðZ þ WÞ 3.3. Prediction of ballistic limit and residual velocity Elementary failure criterion, Ws ¼ kH; has been used successfully to predict the shear failure in structural members [5,38,39]. Although, other failure criteria, e.g., maximum shear strain failure criterion, are available to predict material failure [7,27], they usually neglect strain rate and temperature effects on the shear failure strain. An elementary failure criterion is capable of including these effects in parameter k, e.g., it has been shown that parameter k contains the temperature effects on the shear failure strain [32]. In the following shear plugging and perforation studies, elementary failure criterion with k ¼ 1 will be used to calculate the ballistic limit VBL and residual velocity Vr after the perforation. Assuming the maximum shear sliding in the target reaches Ws ðt1 Þ ¼ kH at the end of the first phase, i.e., the critical time of the ballistic limit tBL ¼ t1 ; Eqs. (17), (15) and (13b) reduce to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffi 2kwð1 þ ZÞðZ þ WÞ sy pffiffiffi ; ð18Þ VBL ¼ 2 r 3
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tBL
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffi pffiffiffi 3kð1 þ ZÞ rdH ¼ 2ðZ þ WÞ sy
ð19Þ
and W1 ðtBL Þ ¼
WkH : ðZ þ WÞ
ð20Þ
It is easy to show that W1 ðtBL ÞoH=2 is satisfied for all intermediate plates with pffiffi 3 w1 owp 4 ðA þ BFJ Þ; which implies that the initial assumption of ignoring membrane force in the first phase of motion is reasonable. According to Eq. (1b), a velocity jump of the projectile and plug exists at the ballistic limit ’ 0 ðtBL Þ ¼ VJump ¼ W
WVBL > 0; ð1 þ ZÞðZ þ WÞ
ð21Þ
if Wa0 in Eq. (16), which has been observed in experiments [29] and will be discussed in Section 5. If Vi XVBL ; the projectile with the plug would perforate the plate with a residual velocity of ’ 0 ðt1 Þ; where t1 is determined by Ws ðt1 Þ ¼ W0 ðt1 Þ-W1 ðt1 Þ ¼ kH; and W0(t1 ) and Vr ¼ W * * * * * * W1(t1 * ) can be obtained from Eqs. (1c) and (13b), respectively. Thus, pffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ffi 3 rd 2 t1 * ¼ Vi Vi VBL : ð22Þ 4ðZ þ WÞ sy It can be shown that t1 * ptBL ¼ t1 : Submitting Eq. (22) into Eq. (1b), the residual velocity is determined by Vr 2 2W Vi Vr ðZ WÞ Vi 2 Z2 þ ¼ 0; ðZ þ WÞð1 þ ZÞ VBL VBL ðZ þ WÞð1 þ ZÞ2 VBL VBL ðZ þ WÞ2 ð1 þ ZÞ2
ð23Þ
which has the following solution: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 WVi þ Z Vi2 VBL XVJump : Vr ¼ ð24Þ ð1 þ ZÞðZ þ WÞ pffiffi pffiffiffi 3 When 1= 3pwp 4 ðA þ BFJ Þ; pure shear velocity field is assumed and W ¼ 0 according to Eq. (16). Thus, Eqs. (24) and (18) lead to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 ; Vr ¼ Vi2 VBL ð25Þ ð1 þ ZÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffi 2kwZð1 þ ZÞ sy pffiffiffi : ð26Þ VBL ¼ 2 r 3 Eq. (25) is Recht and Ipson’s model [4], which obviously cannot predict the velocity jump at the ballistic limit.
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pffiffiffi 3 4. Perforation of thick plates v > ðA þ BUJ Þ 4 4.1. Penetration stage As discussed in Sections 2.3 and 3.2, local indentation or penetration should be considered if the thickness of the target plate is large enough. The analytical model in this section is applicable for pffiffi 3 w > 4 ðA þ BFJ Þ; which depends on material properties and impact velocity. When the impact velocity is 500 m/s, which is about the upper limit of a sub-ordance impact, the corresponding value of FJ is around 2.5 for metallic targets. For aluminium alloy target 6061-T651, A=4.41 and pffiffi 3 B=1.13 [25]; thus, the value of 4 ðA þ BFJ Þ is about 3.0, which means that the thickness of the 6061-T651 plate should be three times the projectile diameter for the validity of thick plate model. When the global structural response is neglected, the maximum penetration depth(X) can be obtained from Eqs. (6), (7a) and (7b) in Section 2.3 X 2l B ð27Þ ¼ ln 1 þ FJ ; d pB A where l ¼ G=rd 3 [25]. If the thickness of the target plate is much larger than the maximum penetration depth, i.e., 2l B ln 1 þ FJ ; w >> pB A the problem can be treated as a pure penetration process in a semi-infinite target [25]. Otherwise, if the target thickness is in the range of pffiffiffi 3 2l B ðA þ BFJ ÞowB ln 1 þ FJ 4 pB A shear plugging might occur in the second stage of motion following the indentation/penetration process when the interactive force between the projectile nose and the plate in Eq. (6) reaches the maximum shear resistance of the plug Fx ¼ pdQ0 :
ð28Þ
This condition might be met at a certain thickness of plate H* (H oH) when the velocity of the projectile is V* (V oVi ), as shown in Fig. 1. Thus, Eq. (28) gives pd 2 pdH s y ¼ pffiffiffi ; ð29Þ Asy þ BrV 2 * 4 3 and the indentation/penetration depth can be obtained from Eqs. (6), (7a) and (7b): " !# BrV 2 H H 2l BFJ * : ¼ ln 1 þ = 1þ d pB A Asy
ð30Þ
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526
Therefore, the thickness of the final plug H* and the corresponding velocity V* can be obtained from Eqs. (29) and (30) as follows: "pffiffiffi # 3 H 1 H 1 ðA þ BFJ Þ ; ð31aÞ ln ln ¼1 4w 2ZB H 2ZB H V2 *
¼
Vi2
Asy þ Br
H Asy : exp 2ZB 1 H Br
ð31bÞ
Specially, if the indentation/penetration dominates the impact process, i.e., H ooH; Eqs. (31a) and (31b) give "pffiffiffi #, 3 H ðA þ BFJ Þ expð2ZBÞ; ð32aÞ E 4w H Asy Asy 2 2 : ð32bÞ V E Vi þ expð2ZBÞ * Br Br On the other hand, if the shear plugging dominates the impact process, i.e., H*/HB1: "pffiffiffi # 3 H 1 ðA þ BFJ Þ : ln E1 4w H pffiffi 2ZB When w-
3
4
ð33Þ
ðA þ BFJ Þ; Eqs. (33) and (31b) lead to H -H and V * -Vi ; respectively.
4.2. Shear plugging stage The velocity field may be assumed as Fig. 2c for a thick plate and the corresponding relative displacement across the shear hinge is Ws ¼ W0 : Thus, the ballistic limit is obtained when ’ 0 ðt1 Þ ¼ 0 and W0 ðt1 Þ ¼ kH in Eqs. (1b) and (1c), W (" ) 2 # As 8B H H H y 2 exp 2ZB 1 VBL ¼ ð34aÞ 1 þ pffiffiffi kwZ 1 þ Z 1 ; Br H H H 3A in which, H*/H in Eq. (31a) contains VBL. Thus, it is necessary to solve a nonlinear algebra equation for VBL. Following the same procedure as the one used in Section 3.3 for an intermediate ’ 0 ðt1 Þ when W0 ðt1 Þ ¼ kH ; thickness plate, the residual velocity for Vi > VBL is obtained as Vr ¼ W sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2 H H 2 Vi V1 =exp 2ZB 1 1þ Z ; ð34bÞ Vr ¼ H H where V1 has the same expression as VBL in Eq. (34a), but H*/H in Eqs. (34a) and (34b) is determined by Eq. (31a) as a function of the initial impact velocity. Obviously if shear plugging dominates the impact process, i.e., H -H; Eqs. (34a) and (34b) also lead to Recht and Ipson’s [4] results given by Eqs. (25) and (26). If local indentation/penetration dominates the impact process, the residual velocity and the ballistic limit of the projectile become qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 =expð2ZBÞ; Vi VBL ð35aÞ Vr ¼
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527
and 2 VBL ¼
Asy ½expð2ZBÞ 1 ; Br
ð35bÞ
respectively, according to Eqs. (34a) and (34b).
5. Experimental analysis In this section, a collection of experimental data from Refs. [28,29,40–42] are compared with analytical predictions on ballistic limit and residual velocity. These experimental results fall in the plate category between thin plate and intermediate thick plate. Authors are not aware of any published experimental results on the perforation of thick metallic plate impacted by a rigid flat pffiffi 3 nose projectile, which satisfies the condition of w > 4 ðA þ BFJ Þ: Thus, the perforation analysis for thick plate in Section 4 of the present paper should be verified as long as experimental data become available. In practice, residual velocity of the projectile is somewhat different from the residual velocity of the plug. However, the present analytical model assumes that the projectile and plug have the same residual velocity after perforation. In order to compare the analytical results with experimental data, a nominal residual velocity in a test is defined based on the momentum conservation GVpr þ Gpl Vplr ; ð36Þ Vr ¼ G þ Gpl where Vpr and Vplr are the residual velocities of projectile and target plug in a test, respectively. The mass of target plug is Gpl ¼ ZG ¼ prd 2 H=4: 5.1. Prediction of ballistic performance Figs. 4–6 show the ballistic limit and residual velocity of HY-100 steel target reported in Ref. [29] and the corresponding theoretical predictions. There are totally four groups of the tests with a variation of w or H=d from 0.172 (Group 3) to 0.35 (Groups 1, 2 and 4) and two different masses (0.52 kg for Group 4 and 1.56 kg for Groups 1–3), which are listed in Tables 1–4 of Ref. [29]. According to Eq. (14), bending hinge is formed at x=d ¼ 2:31 for the tests of Groups 1, 2 and 4. The analytical model shows that the velocity jumps are VJump=54.5 m/s at ballistic limit of VBL=110.0 m/s for Groups 1 and 2, and VJump=42.9 m/s at ballistic limit of VBL=165.9 m/s for Group 4, respectively. For the tests of Group 3, analytical predictions by medium plate model in Section 3 and the lower bound model in Appendix A agree with experimental results on residual velocity at high velocity perforation. However, the lower bound predictions on the ballistic limit and the jump velocity (i.e., VBL=63.9 m/s and VJump=38.3 m/s) are more close to the test data than those predicted by medium plate model. Fig. 7 shows the test pffiffiffiresults in Ref. [41] and corresponding theoretical predictions. In this case, w ¼ H=d ¼ 0:6 > 1= 3: According to the analytical model in Section 3, velocity jump does not exist at ballistic limit of VBL=174.0 m/s, which agrees with the experimental observations.
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350
Vr (m/s)
300
Analysis in Section 3
250 200 150 100 50 0 0
50
100
150
200 250 Vi (m/s)
300
350
400
Fig. 4. Demonstration of the predicted residual velocity and the test data (Groups I and II are taken from Tables 1 and 2 in Ref. [29]).
400 Group III Analysis in Section 3 Lower bound in Appendix
350
Vr (m/s)
300 250 200 150 100 50 0 0
50
100
150
200
250
300
350
Vi (m/s)
Fig. 5. Demonstration of the predicted residual velocity and the test data (Group III is taken from Table 3 in Ref. [29]).
300
Group IV
Vr (m/s)
250
Analysis in Section 3
200 150 100 50 0 0
50
100
150
200
250
300
350
400
Vs (m/s)
Fig. 6. Demonstration of the predicted residual velocity and the test data (Group IV is taken from Table 4 in Ref. [29]).
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529
300 Borvik et al.[41]
250
Analysis in Section 3 Vr (m/s)
200 150 100 50 0 100
150
200 250 Vi (m/s)
300
350
Fig. 7. Demonstration of the predicted residual velocity and the test data [41].
300 Liss & Goldsmith[40]
250
Analysis in Section 3
Vr (m/s)
200 150 100 50 0 0
50
100
150
200
250
300
Vi (m/s)
Fig. 8. Demonstration of the predicted residual velocity and the test data with H=3.2 mm in Ref. [40].
The phenomena of velocity jump seems also existing in test results of Ref. [40]. Figs. 8 and 9 demonstrate the test results in Ref. [40] and the corresponding theoretical predictions, in which w ¼ H=d ¼ 0:256 for plate thickness of H=3.2 mm and w ¼ H=d ¼ 0:512 for plate thickness of H=6.4 mm, respectively. There is a velocity jump VJump=53.7m/s at ballistic limit of VBL=92.3 m/s for the tests of 3.2 mm plate thickness while the velocity jump is only VJump=0.7 m/s at ballistic limit of VBL=117.2 m/s for the tests of 6.4 mm plate thickness. Residual velocity records are incomplete for other two groups of tests with plate thickness H=9.8 and 12.75 mm, and therefore, it is difficult to compare them with analytical results. The present analytical model predicts quantitatively the occurrence of velocity jump at ballistic limit, which was observed in Ref. [29], where a rigid-plastic beam model was employed to give a qualitative illustration for the observed velocity jump phenomenon. However, it should be noted that, in both Refs. [29,40], there are not sufficient experimental data around the ballistic limit
530
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500
Analysis in Section 3
Vr (m/s)
400 300 200 100 0 0
100
200
300 400 Vi (m/s)
500
600
700
Fig. 9. Demonstration of the predicted residual velocity and the test data with H=6.4 mm in Ref. [40].
15 Borvik et al.[42] 12 Corran et al.[28]
ξ d
9 6 3 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
H/d
Fig. 10. The Position of bending hinges in the plates.
velocity where the velocity jump occurs to offer a solid support for the velocity jump phenomenon. Thus, further experimental works are necessary. Fig. 10 demonstrates the variation of the bending hinge position with the dimensionless parameter w ¼ H=d during the first phase of motion. In addition to parameter w ¼ H=d; the bending hinge position also depends on the diameter pffiffiffi of the circular plate, as discussed in Section 3.2 for an intermediate thick plate of w1 owo1= 3 ¼ 0:58: The upper limit of dimensionless hinge position may be different for different diameters of target plates, which is illustrated in Fig. 10 using experiments in Refs. [28,42]. 5.2. The effect of plate thickness on ballistic performance Figs. 11–14 show the effect of plate thickness on the ballistic performance of various target materials tested in Refs. [28,42]. With the increase of the target plate thickness, both the bendingshear model in Section 3 and the bending-shear-membrane model in Appendix A give good
X.W. Chen, Q.M. Li / International Journal of Impact Engineering 28 (2003) 513–536 400 350
Vbl (m/s)
300 250 200 150
Analysis in Section 3
100
Borvik et al.[42]
50
Lower bound in Appendix
0 0
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8
2
H/d
Fig. 11. Demonstration of the predicted ballistic limit and the test data in Ref. [42].
350 300
Vbl (m/s)
250 200 150 Corran et al.[28]
100
Analysis in Section 3
50
Lower bound in Appendix
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
H/d
Fig. 12. Demonstration of the predicted ballistic limit and the test data for stainless-steel plates in Ref. [28].
300
V bl (m/s)
250 200 150
Corran et al.[28]
100
Analysis in Section 3
50
Lower bound in Appendix
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
H/d
Fig. 13. Demonstration of the predicted ballistic limit and the test data for mild-steel plates in Ref. [28].
531
532
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Vbl (m/s)
250 200 150 100
Corran et al.[28] Analysis in Section 3
50
Lower bound in Appendix 0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
H/d
Fig. 14. Demonstration of the predicted ballistic limit and the test data for aluminum alloy plates in Ref. [28].
predictions. In the case of small plate thickness when wow1 ; the analysis in Appendix A gives a lower bound on ballistic limit. The value of w1 depends on several other parameters, e.g., D=d; which cannot be accurately determined by the present model. Based on the comparison between analytical predictions and experimental results in Figs. 11–14, the value of w1 is estimated to be in the range of 0.1–0.3. According to the bending-shear model in Section 3, there exists a local drop of the ballistic limit when plate thickness increases. This local drop phenomenon is caused by the competition of two opposite effects of plate thickness on the required perforation energy. The model in Ref. [7] is also able to predict the local drop of ballistic limit with plate thickness observed in Ref. [28], which, however, requires a complicated numerical calculation.
6. Remarks Shear plugging failure may be initiated in a broad range of impact problems. Quantitatively, an increase of plate thickness, impact velocity or projectile bluntness in a certain range is in favour of the shear plug formation. Otherwise, other perforation modes, such as petalling, may dominate the ballistic performance of the target or projectile [1,2]. It is difficult to give a simple condition for the occurrence of each failure mode in a perforation problem. Even for a blunt projectile, shear plugging is not the only failure mode to dominate the perforation of a metallic plate. Dishing may be formed for a thin plate subjected to a low velocity impact, which eventually leads to a tensile failure. A structural model including shear, bending, membrane and a proper failure criterion may predict transition between different perforation modes [7], which, however, sometimes requires complicated numerical procedures for predicting ballistic limit and residual velocity. The present model is only valid for the perforation controlled by the shear plugging failure mode with or without the presence of structural bending response and local penetration. The application range is defined by the ratio of the plate thickness to the projectile diameter, the ratio of the plate diameter to the projectile diameter, material parameters and impact velocity, as specified in Sections 3 and 4. In the present paper, blunt projectiles are assumed to be rigid. However, projectile breaking or deforming may occur when a blunt-nosed projectile impacts on thick target. Thus, the proposed
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model is only valid when the projectile has no obvious deformation and damage after the perforation event. It is reasonable to use rigid, perfectly plastic model to study the structural global response. However, the local material response during a penetration/perforation process may involve large plastic deformation, strain rate hardening and temperature or damage softening. A model to include all these factors would be extremely complex and would demand extensive material tests. Although the present analyses do not explicitly consider these factors, all material parameters used for the local deformation and failure in the present model are experimentally based, which have taken their combined effects implicitly, e.g., the parameter k in shear failure criterion may contain the combined effects of temperature and strain and strain rate hardening [32]. The strain and strain rate hardening may be cancelled partially by the thermal and damage softening to reduce their combined effect. However, their respective role may become dominative under certain conditions. For example, the shear plug failure mode may be transformed into an adiabatic shear banding failure with further increase of impact velocity due to enhanced thermal softening. A beam model has been proposed in Ref. [32] to predict the occurrence of adiabatic shear banding. We believe that this failure mode transition is responsible for the observed phenomenon that the ballistic limit may decrease with the increase of target material hardness [30,31]. Nevertheless, it is worthy to further quantify their respective roles in penetration/perforation process.
7. Conclusions A shear plugging model is presented to predict the ballistic limit and residual velocity of a metallic circular plate impacted by a blunt rigid projectile. In addition to the shear around the periphery of a central plug, the proposed model includes plate bending, membrane stretching and local indentation/penetration. The proposed model gives simple formulae and good predictions when compared with various experimental results of perforation tests.
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 China Academy of Engineering Physics (CAEP). Appendix A. Lower Bound Estimate for the Perforation of Ductile Plates (wpw1) For thin plate with wpw1 ; it is assumed that plug failure is still responsible for the final perforation of a ductile circular plate. The transverse velocity field is assumed as ( ’ 0; W 0prpd=2; ’ ¼ ðA:1Þ W ’ 1 ðD 2rÞ=ðD d Þ; d=2orpD=2; W which is the same as Eq. (3) with x ¼ D=2; as shown in Fig. 1b.
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Eq. (2) of energy conservation with membrane effect and the assumed velocity field Eq. (A.1) overestimates the energy consumption in plate response, and thus gives a lower bound estimate for the ballistic limit and residual velocity. Eqs. (2) and (A.1) requires . 1 þ g2 W1 ¼ f ; W ðA:2Þ where
24sy D=d þ 1 24N0 ðD þ d Þ g ¼ ¼ ; rH ðD þ 3d ÞðD d Þ2 rd 2 D=d þ 3 D=d 1 2 2
h i pffiffiffi 24s D=d 1 = 3 wD=d y 24½dQ0 ðD d Þ 4DM0
¼ f ¼ 2 rH ðD þ 3d ÞðD d Þ2 rd D=d þ 3 D=d 1 and
h i pffiffiffi D=d 1 = 3 wD=d f d: ¼ g2 D=d þ 1
ðA:3aÞ
ðA:3bÞ
ðA:3cÞ
’ 1 ¼ 0 are The solutions of Eq. (A.2) under initial conditions W1 ¼ W W1 ¼
f ½1 cosðgtÞ ; g2
ðA:4aÞ
’ 1 ¼ f sinðgtÞ; W g
ðA:4bÞ
. 1 ¼ f cosðgtÞ: W
ðA:4cÞ
’ 0 ðt1 Þ ¼ W ’ 1 ðt1 Þ; and the motion of the The first phase of motion terminates at time t1 when W projectile and plug are controlled by Eqs. (1a)–(1c). A complete analysis of the plate deformation using a similar method was given in Ref. [43]. Following the same procedure in Section 3.3, the ballistic limit can be obtained from the following equations: f 2Zsy f t2BL 2 ½1 cosðgtBL Þ ¼ kH; tBL sinðgtBL Þ þ pffiffiffi g g 3ð1 þ ZÞrd
ðA:5aÞ
f 4Zsy VBL ¼ sinðgtBL Þð1 þ ZÞ þ pffiffiffi tBL ; g 3rd
ðA:5bÞ
f VJump ¼ sinðgtBL Þ: g
ðA:5cÞ
It shows that a velocity jump VJump of the projectile and plug at the ballistic limit also exists when the membrane response in a plate becomes important with the decrease of plate thickness, which has been noted in Ref. [44] for thin plate.
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If Vi XVBL ; the projectile perforates the thin plate and its residual velocity is evaluated by ! 1 4Zsy ðA:6Þ Vr ¼ Vi pffiffiffi t1 * ; ð1 þ ZÞ 3rd where t1* is determined by 2Zsy 1 Vi t1 * pffiffiffi t21 ð1 þ ZÞ 3rd *
!
i fh 1 cos gt ¼ kH 1* g2
where t1 * ptBL :
ðA:7Þ
It should be noted that this lower bound estimate applies only for the plug-induced perforation. With the further decrease of plate thickness, other failure modes, e.g., petaling, may occur, which demand a more general failure criterion and structural response model to study the perforation performance.
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Shear plugging and perforation of ductile circular plates struck by a blunt projectile X.W. Chen, Q.M. Li* Protective Technology Research Center, School of Civil and Environmental Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore Received 12 July 2001; received in revised form 4 April 2002; accepted 18 August 2002
Abstract Shear plugging and perforation of ductile circular plates struck by a blunt rigid projectile is studied in the present paper. Effects of transverse shear, bending and membrane deformations on the perforation process are included in a rigid-plastic analysis while the local indentation/penetration is estimated using the dynamic cavity expansion model. Analytical formulae for the perforation ballistic limit and the residual velocity are obtained for a range of plate thickness, which agree with the available experimental results on the perforation of metallic plates. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Perforation; Ballistic limit; Ductile plate; Blunt projectile; Shear plugging
1. Introduction Both local impact effect and global structural response may be involved in the perforation of a ductile plate subjected to the strike of a non-deformable projectile. In addition to the local indentation or penetration,1 there are basically two types of structural responses in a circular plate, i.e., the bending response and the membrane response. Their contributions to the perforation depend mainly on the thickness of the target and the impact velocity. Generally, local and shear deformations become more important with the increase of plate thickness or impact velocity. Membrane deformation decreases with the increase of plate thickness and bending deformation may reach a maximum at a certain plate thickness [1]. For a thick target, the *Corresponding author. Current address: Department of Mechanical, Aerospace and Manufacturing Engineering, UMIST, P.O. Box 88 Sackville Street, Manchester M60 1QD, UK. Tel. +44-161-200-5740; fax: +44-161-200-3849. E-mail address: [email protected] (Q.M. Li). 1 Indentation is used when the penetration is small. 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 0 7 7 - 5
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Nomenclature a and b material constants in Poncelet’s formula A and B dimensionless material constants cross-sectional area of a projectile A0 d diameter of a projectile D diameter of a circular plate E Young’s modulus of the target material axial resistant force on the projectile nose Fx G mass of a projectile mass of a plug Gpl H thickness of a circular plate thickness of a plug H k empirical parameter in the shear failure criterion of structural members (k=1 in the present paper) Mr ; My radial and circumferential bending moments per unit length of a circular plate plastic bending moment per unit length of a rigid, perfectly plastic circular plate M0 plastic membrane force per unit length of a rigid, perfectly plastic circular plate N0 shear force per unit length of a circular plate Qr Q0 ¼ Hty plastic shear force per unit length of a circular plate Q0 ¼ H ty plastic shear force per unit length of a shear plug t time time at the end of the first phase of motion t1 critical time at the moment of perforation tBL v normal velocity on the surface of a projectile nose V transient velocity of a projectile VBL ballistic limit of a projectile initial impact velocity of a projectile Vi VJump velocity jump of the projectile at the ballistic limit Vpr ; Vplr residual velocities of the projectile and the target plug, respectively nominal residual velocity of the projectile and the plug Vr velocity of the projectile when the plug is just formed V ’ W . transverse deflection, velocity and acceleration fields of a circular plate W, W; ’ . 0 transverse deflection, velocity and acceleration of the projectile and the plug W0 ; W0 ; W ’ . 1 transverse displacement, velocity and acceleration of the plate at r ¼ d=2 W1 ; W1 ; W X penetration depth g Poisson’s ratio of the target material er ; ey radial and circumferential strains at the mid-plane of the cross-section of a circular plate Z ¼ rpd 2 H=4G ratio of the plug mass to the projectile mass for a thin plate Z ¼ rpd 2 H =4G ratio of the plug mass to the projectile mass for a thick plate W dimensionless number in structural model kr ; ky radial and circumferential curvatures in a circular plate l ¼ G=rd 3 dimensionless mass of the projectile
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x location of a bending hinge during the shear sliding phase r density of the target material yielding stress of the target material sy pffiffiffi yielding stress in shear (ty=sy/ 3 for von-Mises yielding criterion) ty Johnson’s damage number (FJ ¼ rVi2 =sy ) FJ w ¼ H=d dimensionless thickness of the plate (ratio of plate thickness and projectile diameter) empirical value in the case of small plate thickness w1
perforation is dominated by the local penetration although the failure mechanism of the final perforation also influences the ballistic limit of a thick target, which depends on the target material, target dimensions, projectile nose and impact velocity. The failure mechanism leading to the final perforation of a target plate could be ductile hole enlargement or petaling for a sharp nose, fragmentation for high impact velocity and brittle target, plugging or adiabatic shearing for a blunt projectile [1–3]. With the increase of the plate thickness, the impact velocity and the bluntness of the projectile, shear plugging becomes a likely failure mode of the final perforation of a plate. Based on the conservation of momentum and energy, Recht and Ipson [4] proposed a shear plugging model to predict the residual velocity according to a given impact velocity while a ballistic limit velocity was obtained from a dimensional analysis, which completely ignored the structural response for relatively thin plates and the local penetration for relatively thick plates. The so-called structural model may be considered as a further development of Recht and Ipson’s model [4], but it is actually an application of rigid-plastic structural dynamics [5] in perforation studies. A structural model, which neglects the local penetration, has given good prediction for thin plate perforation under blunt (flat) projectile impact, as shown in Refs. [6–8]. The shear plugging in a structural model is based on the shear hinge concept in a rigid-plastic model, which may be triggered at the early response stage for a relatively thick structural member [5,9–11]. It is still a challenge to obtain a general structural model of perforation analysis, which incorporate the local failure analysis in the rigid-plastic model because different local response and failure modes, such as dishing, petalling or penetration, may appear for different plate thicknesses and impact velocity [1]. Meanwhile, it is difficult to get simple formulations for a general structural model, which is not in favour of engineering applications. 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. Multi-stage models have been proposed to study the perforation of relatively thick plates when structural response is negligible. Awerbuch [12] 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 compressive resistance is replaced by the surrounding shear force. Goldsmith and Finnegan [13] improved the model of Awerbuch by considering the reduction of the shear force in the second stage. Awerbuch’s two-stage model was extended to a three-stage model for the perforation of a plate by a non-deformable projectile [14]. The projectile is subjected to the inertia and the compressive resistance of the target material as well as the shear resistance around
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the plug in the middle stage. This model has been further modified in Ref. [15], where a twodimensional model assumes five stages of plate penetration, namely dynamic plastic penetration, bulge formation, bulge advancement, plug formation and projectile exit. These 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 shapes as well as the force–time history of the process. This analytical model was further generalized in Ref. [16] to include various projectile nose shapes, changes in the plastic flow field due to deep penetrations and thermal softening effects. Liss et al. [17] 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 motion of the projectile in these five stages. For very thick (essentially semi-infinite) metallic targets impacted by projectiles, a number of penetration models have been proposed [18,19]. Poncelet’s formula is used to calculate the resistant force on a flat projectile nose [20], F ¼ A0 ða þ bV 2 Þ; where A0 is the cross-sectional area of the projectile, V is the impact velocity and a and b are material constants to be determined by experiments (For non-flat nose, a and b depend on the nose shape). Both multi-stage models for perforation and models for deep penetration employed Poncelet’s formula. However, in the study of deep penetration, dynamic cavity expansion model, which was originated from a quasi-static cavity expansion model [21] and subsequently developed and applied in Refs. [22–24], offers a theoretical foundation of Poncelet’s formula. Applications of the dynamic cavity expansion model to predict the penetration depth for various target media, projectile shapes and impact velocities are given in Refs. [25,26]. In the five-stage perforation model in Ref. [17], the shear wave propagation in the target plate outside the plug interface was considered. However, the structural response outside the shear plug is generally neglected in most multi-stage models. Shadbolt et al. [27] hybridized the multi-stage model and the structural model. It was shown that the analytical prediction is improved when Reissner’s plate theory is used as a structural model to replace the simplified plastic membrane and bending model. However, uncertainties of some empirical data and the involvement of numerical procedures increase the difficulty for their practical application. Nevertheless, a similar method but different approach will be employed in the present paper. Experimental studies on the perforation of metallic plates have been summarized in Refs. [1,2]. Experimental results were presented either by the variation of ballistic limit with plate thickness and the hardness of the plate material, or by the variation of the residual velocity of the projectile with impact velocity, which have revealed several interesting phenomena. First, it was shown that the perforation ballistic limit may decrease with the increase of plate thickness in a range of plate thicknesses [28]. Most of the analytical models cannot predict such a local drop of the ballistic limit except the rigid-plastic plate model in Ref. [7], which considered all possible structural responses in a perforation analysis. However, the model in Ref. [7] requires a numerical algorithm to solve non-linear differential equations based on a selected velocity field, which is not favorable to practical applications. Local penetration is not considered in Ref. [7].
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Second, a residual velocity jump at the ballistic limit was noted in Ref. [29] for the perforation of HY-100 steel plates. It shows that the residual velocity may jump to a finite value at the ballistic limit in some cases. Neither Recht and Ipson’s model [4] nor other structural and multi-stage models can predict the velocity jump at the ballistic limit. Forrestal and Hanchak [29] used a rigidplastic beam model to illustrate this phenomenon. Third, a local drop of ballistic limit with the increase of target material hardness was observed in Ref. [30]. This observation is in contradiction with the previous design philosophy, i.e., the higher the hardness, the better the perforation performance. Further experimental evidence for this phenomenon has been shown in Ref. [31]. It is believed that this local drop is due to the initiation of adiabatic shearing regime [1,30,31] although the increased brittleness of the hardened material may also contribute to an easy failure of the target. The transition from shear failure to adiabatic shear failure in a beam was studied in Ref. [32]. However, authors have not noted any analytical model to study this problem for a plate. It is a challenge to develop a simple model, which has a reasonable degree of accuracy and is capable of predicting experimental observations. In the present paper, a hybrid model is proposed to study the perforation of a ductile circular plate due to plug failure when struck by a blunt projectile. The proposed model consists of a plug motion, a rigid-plastic analysis and a local indentation/penetration model, as introduced in Section 2. The simplicity of the rigid-plastic analysis replaces Reissner’s plate theory in the two-stage model in Ref. [27] and makes it possible to get explicit expressions of both ballistic limit and residual velocity in a range of plate thicknesses, as shown in Sections 3 and 4. Although, rigid-plastic model used in other structural models are sometimes more general than the present rigid-plastic model, they generally neglected the local penetration effects and did not give analytical solutions and explicit expressions of the ballistic limit and the residual velocity due to difficulties in solving nonlinear dynamic equations analytically. Experimental results from different sources are compared with the present analytical predictions in Section 5 with satisfactory agreement. Section 6 gives the limitation of the present model and further works in this research subject are suggested.
2. Analytical model 2.1. 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 equals the fully plastic shear force on the plug, as shown in Fig. 1. 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 . 0 ¼ pdQ0 ; W Gð1 þ Z Þ
ð1aÞ
where Q0 ¼ H ty ; d is the diameter of the projectile, G is the mass of the projectile, ty is the yield shear stress of the material (ty=sy/O3 for von-Mises yielding criterion), H* is the thickness of the
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V*
H
H* Fig. 1. Perforation of a thick plate.
plug, Z ¼ rpd 2 H =4G: Eq. (1a) leads to 1 pdQ0 t ’ V W0 ¼ ; G 1 þ Z 1 pdQ0 t2 V t W0 ¼ ; 2G 1 þ Z
ð1bÞ
ð1cÞ
’ 0 ¼ V =ð1 þ Z Þ are satisfied at the moment when the plug is formed. V is the when W0=0 and W velocity of the projectile when the plug is just formed.2 2.2. Rigid-plastic analysis The plate deformation outside the central plug is controlled by the energy conservation. A general expression of the energy conservation for a rigid-plastic plate considering bending and membrane has been given in Ref. [5], which may be extended when the transverse shear effect is included, i.e.,3 Z x Z x . ’ ’ . ’ G 1 þ Z W0 W0 2prH ðM0 þ N0 W Þk’ y r dr þ pd ðM0 þ N0 W1 Þc WWr dr ¼ 2p d=2
d=2
’ þ pdQ0 W ’0 W ’1 þ 2pxðM0 þ N0 W1 Þc
for the transverse velocity field shown in Fig. 2a, i.e., 8 ’ 0; 0prpd=2; >W < ’ ¼ 2W ’ 1 ðx rÞ=ð2x d Þ; d=2orpx; W > : 0; xorpD=2;
ð2Þ
ð3Þ
’ 1 is the where x represents the location of a bending hinge during the shear sliding phase and W transverse velocity of the plate at r ¼ d=2: H is the thickness of the circular plate; M0 ¼ sy H 2 =4 and N0 ¼ sy H are the fully plastic bending moment and membrane force of a rigid, perfectly plastic circular plate, respectively. 2
In Sections 3 and Appendix A, there is no indentation/penetration, and therefore, H ¼ H; Q0 ¼ Q0 ; Z ¼ Z and V ¼ Vi ; where Vi is the initial impact velocity of the projectile. 3 If penetration occurs before plugging, M0 and Q0 are replaced by M0 ¼ sy H 2 =4 and Q0 at r ¼ d=2; respectively.
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D . W1
r
r
D
. . W W0
. W1
d (b)
(a)
519
. W0
. W
d
D . W0
d (c)
Fig. 2. (a) Transverse velocity profiles for the circular plate (bending hinge locates at r ¼ x), (b) transverse velocity profiles for the circular plate (bending hinge locates at r ¼ D=2), (c) transverse velocity profiles for the circular plate (localized shear only).
The curvatures and the middle surface strain in a circular plate are @2 W 1 @W 1 @W 2 ; er ¼ ; ey ¼ 0; kr ¼ 2 ; ky ¼ @r r @r 2 @r
ð4Þ
2 ’ ’ 1 =frð2x dÞgX0 for d=2orpx; k’ r ¼ d 2 W=dr ’ ¼ 2W ¼ 0; for which lead to k’ y ¼ ðdW=drÞ=r ’ ¼ 2W ’ 1 =ð2x d Þ: d=2orox; k’ r X0 at r ¼ d=2; k’ r p0 at r ¼ x and c
2.3. Penetration model The dynamic cavity expansion analysis yields the following relation between the normal compressive stress sn on the projectile nose and the normal velocity v [24,33–36]: sn ¼ Asy þ Brv2 ;
ð5Þ
where sy and r are the yielding stress and the density of the target material, respectively. A and B are dimensionless material constants. For a flat projectile nose, the normal velocity v at the nose– target interface is the same as the rigid-body velocity V of the projectile and the tangential stress on the projectile nose is zero [25].
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The resultant axial resistant force on the projectile nose can be calculated from the normal compressive stresses pd 2 Asy þ BrV 2 ; ð6Þ Fx ¼ 4 which has the same expression as the Poncelet formula introduced in Section 1. The penetration depth X is determined by dV dX ¼ Fx and V ¼ : ð7a; bÞ G dt dt Parameter A in Eqs. (5) and (6) is given by
2 E 1 þ ln ð8Þ A¼ 3 3ð1 gÞsy for an elastic, perfectly plastic material [33], where g is Poisson’s ratio and n Z 1ð3sy =2E Þ 2 2E ðln xÞn dx A¼ 1þ J ;J ¼ 3 3sy ð1 xÞ 0
ð9Þ
for an incompressible elastic, strain-hardening plastic material [36]. B=1.5 for incompressible materials. However, a numerical procedure is required to calculate parameters A and B in other cases. More discussion is given in [25]. pffiffiffi 3 ðA þ BUJ Þ 3. Perforation of intermediate thick plates, v1 ovp 4 3.1. Introduction Rigid, perfectly plastic analysis has been successfully used to predict the structural response under impact and blast loads. It has been shown that the localized transverse shear may be triggered if the loading intensity is high enough [9,10,37–39]. The formation of the localized transverse shear hinge in a structural member has been studied in Refs. [10,11]. Both experimental and analytical studies on beams and plates show that the localized transverse shear deformation, as long as it is initiated, dominates the early stage of the structural response. The transverse shear response ceases before the bending response and the structural response is gradually dominated by the membrane response when the plate deflects to the same order of the plate thickness [5]. For a relatively thick target, the plate deflection is small during the plugging process. The following analysis focuses on the first phase of the motion of a plate target, where only shear and bending responses are considered. However, a lower bound estimate on the ballistic limit and the residual velocity will be given in Appendix A for a relatively thin plate, where membrane deformation is considered. 3.2. First phase of motion, 0ptpt1 When the membrane effect is neglected, the normality rule of plasticity associated with the cubic yield condition in Fig. 3 requires Mr ¼ M0 at r ¼ d=2; Mr ¼ M0 at r ¼ x; My ¼ M0 and M0 oMr oM0 for d=2orox: In addition, the transverse shear hinge at r ¼ d=2 requires
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M M0
Mr M0
Qr Q0 Fig. 3. Cubic yield condition for the first phase of motion.
Qr ¼ Q0 : According to Eq. (2), the motion of the circular plate is controlled by . 1 ¼ 24½dQ0 ð2x d Þ 8xM0
W rH ð2x þ 3d Þð2x d Þ2
ð10Þ
for the assumed transverse velocity field given by Eq. (3). The location of the bending hinge, x, is stationary during the first response phase, i.e., the shear and bending response phase, when the membrane effect is ignored. The shear resistance per unit length of the plate at position r ¼ x is [5]
1 @ðrMr Þ 2M0 My ¼ ; Q¼ r @r x which, together with the momentum conservation in the impact direction, i.e., Z x ’ 0 þ 2prH ’ dr ; 2pxQt ¼ 4pM0 t ¼ GVi G W Wr 0
requires
Z ’ 4pM0 t ¼ GVi G W0 þ 2prH
x
’ Wr dr :
ð11Þ
0
For the velocity field distribution defined in Eq. (3), Eq. (11) gives . 1 ¼ 6ðdQ0 4M0 Þ ; W rH ð2x d Þðx þ d Þ which leads to ’ 1 ¼ 6ðdQ0 4M0 Þt W rH ð2x d Þðx þ d Þ
ð12Þ
ð13aÞ
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and W1 ¼
3ðdQ0 4M0 Þt2 ; rH ð2x d Þðx þ d Þ
ð13bÞ
’ 1 ¼ 0 at t=0. under the conditions of W1 ¼ 0 and W The position of the bending hinge, x, can be determined from Eqs. (10) and (12), qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 3w þ 1 þ 2 3w 6w2 x ; ð14Þ ¼ pffiffiffi d 2 1 3w pffiffiffi pffiffiffi where w ¼ H=d ¼ 4M0 = 3dQ0 : wo1= 3 is required by Eq. (14). It shows that the position of plastic bending hinge depends on the dimensionless parameter w. The position of the bending hinge determined by Eq. (14) satisfies 12ox=doD=2d if " # 2 D=d 1 1 wopffiffiffi : 2 3 D=d þ 1 þ2 When " # 2 D=d 1 1 1 pffiffiffi pwopffiffiffi; 2 3 D=d þ 1 þ2 3 the plastic pffiffiffi hinge stays at x ¼ D=2 and the velocity field is described by Eq. (3) and Fig. 2b. When wX1= 3; plastic hinge does not exist and the corresponding velocity field is Eq. (3) with x ¼ d=2; as shown by the localized shear in Fig. 2c. However, if the thickness of the plate is small, the membrane effect on the structural response should be included, as presented in Appendix A. According to the discussion in Section 5, results of this section are valid for w > w1 ; where w1 depends on D=d and target material properties. w1 varies between 0.1 and 0.3 for the experiments discussed in Section 5. On the other hand, the total interactive force between the projectile nose and plate target during the penetration/perforation is given by Eq. (6). If FxopdQ0, localized indentation or penetration occurs, which will be studied in Section 4. Otherwise, when shear plugging occurs, FxXpdQ0. The condition for the occurrence of the shear plugging can be expressed in a dimensionless form pffiffi 3 wp 4 ðA þ BFJ Þ where FJ ¼ rV 2 =sy is Johnson’s damage number. Thus, the analytical model in pffiffi 3 this section is valid for w1 pwp 4 ðA þ BFJ Þ: Figs. 2a–c correspond to " # 2 D=d 1 1 w1 owopffiffiffi ; 2 3 D=d þ 1 þ2 " # 2 D=d 1 1 1 pffiffiffi pwopffiffiffi 2 3 D=d þ 1 þ2 3
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and pffiffiffi 3 1 pffiffiffipwp ðA þ BFJ Þ; 4 3 respectively. For the velocity fields in Figs. 2a–c, maximum shear sliding is reached at the end of the first ’ 1 at t ¼ t1 ; where t1 is determined by Eqs. (1b) and (12) as ’0 ¼ W phase of motion when W pffiffiffi 3 rdVi t1 ¼ ð15Þ 4ðZ þ WÞ sy for the velocity fields in Fig. 2a–c where 8 pffiffiffi " # 2 > 3 1 3w ð1 þ ZÞ > D=d 1 > 1 > > > > 2 2x=d 1 x=d þ 1 ; w1 owopffiffi3ffi D=d þ 12 þ2 ; > > > > pffiffiffi > " # < 2 3 1 3w ð1 þ ZÞ D=d 1 1 1 W¼ ; pffiffiffi pwopffiffiffi; 2 > > > D=d 1 D=d þ 2 3 D=d þ 1 þ2 3 > > > p ffiffi ffi > > > 3 1 > > > pffiffiffipwp ðA þ BFJ Þ: : 0; 4 3
ð16Þ
Thus, according to Eqs. (1c) and (13b), the maximum transverse shear displacement at the end of the first phase of motion is pffiffiffi 3FJ d Vi t 1 : ð17Þ ¼ Ws ðt1 Þ ¼ W0 ðt1 Þ W1 ðt1 Þ ¼ 2ð1 þ ZÞ 8ð1 þ ZÞðZ þ WÞ 3.3. Prediction of ballistic limit and residual velocity Elementary failure criterion, Ws ¼ kH; has been used successfully to predict the shear failure in structural members [5,38,39]. Although, other failure criteria, e.g., maximum shear strain failure criterion, are available to predict material failure [7,27], they usually neglect strain rate and temperature effects on the shear failure strain. An elementary failure criterion is capable of including these effects in parameter k, e.g., it has been shown that parameter k contains the temperature effects on the shear failure strain [32]. In the following shear plugging and perforation studies, elementary failure criterion with k ¼ 1 will be used to calculate the ballistic limit VBL and residual velocity Vr after the perforation. Assuming the maximum shear sliding in the target reaches Ws ðt1 Þ ¼ kH at the end of the first phase, i.e., the critical time of the ballistic limit tBL ¼ t1 ; Eqs. (17), (15) and (13b) reduce to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffi 2kwð1 þ ZÞðZ þ WÞ sy pffiffiffi ; ð18Þ VBL ¼ 2 r 3
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tBL
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffi pffiffiffi 3kð1 þ ZÞ rdH ¼ 2ðZ þ WÞ sy
ð19Þ
and W1 ðtBL Þ ¼
WkH : ðZ þ WÞ
ð20Þ
It is easy to show that W1 ðtBL ÞoH=2 is satisfied for all intermediate plates with pffiffi 3 w1 owp 4 ðA þ BFJ Þ; which implies that the initial assumption of ignoring membrane force in the first phase of motion is reasonable. According to Eq. (1b), a velocity jump of the projectile and plug exists at the ballistic limit ’ 0 ðtBL Þ ¼ VJump ¼ W
WVBL > 0; ð1 þ ZÞðZ þ WÞ
ð21Þ
if Wa0 in Eq. (16), which has been observed in experiments [29] and will be discussed in Section 5. If Vi XVBL ; the projectile with the plug would perforate the plate with a residual velocity of ’ 0 ðt1 Þ; where t1 is determined by Ws ðt1 Þ ¼ W0 ðt1 Þ-W1 ðt1 Þ ¼ kH; and W0(t1 ) and Vr ¼ W * * * * * * W1(t1 * ) can be obtained from Eqs. (1c) and (13b), respectively. Thus, pffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ffi 3 rd 2 t1 * ¼ Vi Vi VBL : ð22Þ 4ðZ þ WÞ sy It can be shown that t1 * ptBL ¼ t1 : Submitting Eq. (22) into Eq. (1b), the residual velocity is determined by Vr 2 2W Vi Vr ðZ WÞ Vi 2 Z2 þ ¼ 0; ðZ þ WÞð1 þ ZÞ VBL VBL ðZ þ WÞð1 þ ZÞ2 VBL VBL ðZ þ WÞ2 ð1 þ ZÞ2
ð23Þ
which has the following solution: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 WVi þ Z Vi2 VBL XVJump : Vr ¼ ð24Þ ð1 þ ZÞðZ þ WÞ pffiffi pffiffiffi 3 When 1= 3pwp 4 ðA þ BFJ Þ; pure shear velocity field is assumed and W ¼ 0 according to Eq. (16). Thus, Eqs. (24) and (18) lead to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 ; Vr ¼ Vi2 VBL ð25Þ ð1 þ ZÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffi 2kwZð1 þ ZÞ sy pffiffiffi : ð26Þ VBL ¼ 2 r 3 Eq. (25) is Recht and Ipson’s model [4], which obviously cannot predict the velocity jump at the ballistic limit.
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pffiffiffi 3 4. Perforation of thick plates v > ðA þ BUJ Þ 4 4.1. Penetration stage As discussed in Sections 2.3 and 3.2, local indentation or penetration should be considered if the thickness of the target plate is large enough. The analytical model in this section is applicable for pffiffi 3 w > 4 ðA þ BFJ Þ; which depends on material properties and impact velocity. When the impact velocity is 500 m/s, which is about the upper limit of a sub-ordance impact, the corresponding value of FJ is around 2.5 for metallic targets. For aluminium alloy target 6061-T651, A=4.41 and pffiffi 3 B=1.13 [25]; thus, the value of 4 ðA þ BFJ Þ is about 3.0, which means that the thickness of the 6061-T651 plate should be three times the projectile diameter for the validity of thick plate model. When the global structural response is neglected, the maximum penetration depth(X) can be obtained from Eqs. (6), (7a) and (7b) in Section 2.3 X 2l B ð27Þ ¼ ln 1 þ FJ ; d pB A where l ¼ G=rd 3 [25]. If the thickness of the target plate is much larger than the maximum penetration depth, i.e., 2l B ln 1 þ FJ ; w >> pB A the problem can be treated as a pure penetration process in a semi-infinite target [25]. Otherwise, if the target thickness is in the range of pffiffiffi 3 2l B ðA þ BFJ ÞowB ln 1 þ FJ 4 pB A shear plugging might occur in the second stage of motion following the indentation/penetration process when the interactive force between the projectile nose and the plate in Eq. (6) reaches the maximum shear resistance of the plug Fx ¼ pdQ0 :
ð28Þ
This condition might be met at a certain thickness of plate H* (H oH) when the velocity of the projectile is V* (V oVi ), as shown in Fig. 1. Thus, Eq. (28) gives pd 2 pdH s y ¼ pffiffiffi ; ð29Þ Asy þ BrV 2 * 4 3 and the indentation/penetration depth can be obtained from Eqs. (6), (7a) and (7b): " !# BrV 2 H H 2l BFJ * : ¼ ln 1 þ = 1þ d pB A Asy
ð30Þ
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Therefore, the thickness of the final plug H* and the corresponding velocity V* can be obtained from Eqs. (29) and (30) as follows: "pffiffiffi # 3 H 1 H 1 ðA þ BFJ Þ ; ð31aÞ ln ln ¼1 4w 2ZB H 2ZB H V2 *
¼
Vi2
Asy þ Br
H Asy : exp 2ZB 1 H Br
ð31bÞ
Specially, if the indentation/penetration dominates the impact process, i.e., H ooH; Eqs. (31a) and (31b) give "pffiffiffi #, 3 H ðA þ BFJ Þ expð2ZBÞ; ð32aÞ E 4w H Asy Asy 2 2 : ð32bÞ V E Vi þ expð2ZBÞ * Br Br On the other hand, if the shear plugging dominates the impact process, i.e., H*/HB1: "pffiffiffi # 3 H 1 ðA þ BFJ Þ : ln E1 4w H pffiffi 2ZB When w-
3
4
ð33Þ
ðA þ BFJ Þ; Eqs. (33) and (31b) lead to H -H and V * -Vi ; respectively.
4.2. Shear plugging stage The velocity field may be assumed as Fig. 2c for a thick plate and the corresponding relative displacement across the shear hinge is Ws ¼ W0 : Thus, the ballistic limit is obtained when ’ 0 ðt1 Þ ¼ 0 and W0 ðt1 Þ ¼ kH in Eqs. (1b) and (1c), W (" ) 2 # As 8B H H H y 2 exp 2ZB 1 VBL ¼ ð34aÞ 1 þ pffiffiffi kwZ 1 þ Z 1 ; Br H H H 3A in which, H*/H in Eq. (31a) contains VBL. Thus, it is necessary to solve a nonlinear algebra equation for VBL. Following the same procedure as the one used in Section 3.3 for an intermediate ’ 0 ðt1 Þ when W0 ðt1 Þ ¼ kH ; thickness plate, the residual velocity for Vi > VBL is obtained as Vr ¼ W sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2 H H 2 Vi V1 =exp 2ZB 1 1þ Z ; ð34bÞ Vr ¼ H H where V1 has the same expression as VBL in Eq. (34a), but H*/H in Eqs. (34a) and (34b) is determined by Eq. (31a) as a function of the initial impact velocity. Obviously if shear plugging dominates the impact process, i.e., H -H; Eqs. (34a) and (34b) also lead to Recht and Ipson’s [4] results given by Eqs. (25) and (26). If local indentation/penetration dominates the impact process, the residual velocity and the ballistic limit of the projectile become qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 =expð2ZBÞ; Vi VBL ð35aÞ Vr ¼
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527
and 2 VBL ¼
Asy ½expð2ZBÞ 1 ; Br
ð35bÞ
respectively, according to Eqs. (34a) and (34b).
5. Experimental analysis In this section, a collection of experimental data from Refs. [28,29,40–42] are compared with analytical predictions on ballistic limit and residual velocity. These experimental results fall in the plate category between thin plate and intermediate thick plate. Authors are not aware of any published experimental results on the perforation of thick metallic plate impacted by a rigid flat pffiffi 3 nose projectile, which satisfies the condition of w > 4 ðA þ BFJ Þ: Thus, the perforation analysis for thick plate in Section 4 of the present paper should be verified as long as experimental data become available. In practice, residual velocity of the projectile is somewhat different from the residual velocity of the plug. However, the present analytical model assumes that the projectile and plug have the same residual velocity after perforation. In order to compare the analytical results with experimental data, a nominal residual velocity in a test is defined based on the momentum conservation GVpr þ Gpl Vplr ; ð36Þ Vr ¼ G þ Gpl where Vpr and Vplr are the residual velocities of projectile and target plug in a test, respectively. The mass of target plug is Gpl ¼ ZG ¼ prd 2 H=4: 5.1. Prediction of ballistic performance Figs. 4–6 show the ballistic limit and residual velocity of HY-100 steel target reported in Ref. [29] and the corresponding theoretical predictions. There are totally four groups of the tests with a variation of w or H=d from 0.172 (Group 3) to 0.35 (Groups 1, 2 and 4) and two different masses (0.52 kg for Group 4 and 1.56 kg for Groups 1–3), which are listed in Tables 1–4 of Ref. [29]. According to Eq. (14), bending hinge is formed at x=d ¼ 2:31 for the tests of Groups 1, 2 and 4. The analytical model shows that the velocity jumps are VJump=54.5 m/s at ballistic limit of VBL=110.0 m/s for Groups 1 and 2, and VJump=42.9 m/s at ballistic limit of VBL=165.9 m/s for Group 4, respectively. For the tests of Group 3, analytical predictions by medium plate model in Section 3 and the lower bound model in Appendix A agree with experimental results on residual velocity at high velocity perforation. However, the lower bound predictions on the ballistic limit and the jump velocity (i.e., VBL=63.9 m/s and VJump=38.3 m/s) are more close to the test data than those predicted by medium plate model. Fig. 7 shows the test pffiffiffiresults in Ref. [41] and corresponding theoretical predictions. In this case, w ¼ H=d ¼ 0:6 > 1= 3: According to the analytical model in Section 3, velocity jump does not exist at ballistic limit of VBL=174.0 m/s, which agrees with the experimental observations.
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350
Vr (m/s)
300
Analysis in Section 3
250 200 150 100 50 0 0
50
100
150
200 250 Vi (m/s)
300
350
400
Fig. 4. Demonstration of the predicted residual velocity and the test data (Groups I and II are taken from Tables 1 and 2 in Ref. [29]).
400 Group III Analysis in Section 3 Lower bound in Appendix
350
Vr (m/s)
300 250 200 150 100 50 0 0
50
100
150
200
250
300
350
Vi (m/s)
Fig. 5. Demonstration of the predicted residual velocity and the test data (Group III is taken from Table 3 in Ref. [29]).
300
Group IV
Vr (m/s)
250
Analysis in Section 3
200 150 100 50 0 0
50
100
150
200
250
300
350
400
Vs (m/s)
Fig. 6. Demonstration of the predicted residual velocity and the test data (Group IV is taken from Table 4 in Ref. [29]).
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529
300 Borvik et al.[41]
250
Analysis in Section 3 Vr (m/s)
200 150 100 50 0 100
150
200 250 Vi (m/s)
300
350
Fig. 7. Demonstration of the predicted residual velocity and the test data [41].
300 Liss & Goldsmith[40]
250
Analysis in Section 3
Vr (m/s)
200 150 100 50 0 0
50
100
150
200
250
300
Vi (m/s)
Fig. 8. Demonstration of the predicted residual velocity and the test data with H=3.2 mm in Ref. [40].
The phenomena of velocity jump seems also existing in test results of Ref. [40]. Figs. 8 and 9 demonstrate the test results in Ref. [40] and the corresponding theoretical predictions, in which w ¼ H=d ¼ 0:256 for plate thickness of H=3.2 mm and w ¼ H=d ¼ 0:512 for plate thickness of H=6.4 mm, respectively. There is a velocity jump VJump=53.7m/s at ballistic limit of VBL=92.3 m/s for the tests of 3.2 mm plate thickness while the velocity jump is only VJump=0.7 m/s at ballistic limit of VBL=117.2 m/s for the tests of 6.4 mm plate thickness. Residual velocity records are incomplete for other two groups of tests with plate thickness H=9.8 and 12.75 mm, and therefore, it is difficult to compare them with analytical results. The present analytical model predicts quantitatively the occurrence of velocity jump at ballistic limit, which was observed in Ref. [29], where a rigid-plastic beam model was employed to give a qualitative illustration for the observed velocity jump phenomenon. However, it should be noted that, in both Refs. [29,40], there are not sufficient experimental data around the ballistic limit
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500
Analysis in Section 3
Vr (m/s)
400 300 200 100 0 0
100
200
300 400 Vi (m/s)
500
600
700
Fig. 9. Demonstration of the predicted residual velocity and the test data with H=6.4 mm in Ref. [40].
15 Borvik et al.[42] 12 Corran et al.[28]
ξ d
9 6 3 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
H/d
Fig. 10. The Position of bending hinges in the plates.
velocity where the velocity jump occurs to offer a solid support for the velocity jump phenomenon. Thus, further experimental works are necessary. Fig. 10 demonstrates the variation of the bending hinge position with the dimensionless parameter w ¼ H=d during the first phase of motion. In addition to parameter w ¼ H=d; the bending hinge position also depends on the diameter pffiffiffi of the circular plate, as discussed in Section 3.2 for an intermediate thick plate of w1 owo1= 3 ¼ 0:58: The upper limit of dimensionless hinge position may be different for different diameters of target plates, which is illustrated in Fig. 10 using experiments in Refs. [28,42]. 5.2. The effect of plate thickness on ballistic performance Figs. 11–14 show the effect of plate thickness on the ballistic performance of various target materials tested in Refs. [28,42]. With the increase of the target plate thickness, both the bendingshear model in Section 3 and the bending-shear-membrane model in Appendix A give good
X.W. Chen, Q.M. Li / International Journal of Impact Engineering 28 (2003) 513–536 400 350
Vbl (m/s)
300 250 200 150
Analysis in Section 3
100
Borvik et al.[42]
50
Lower bound in Appendix
0 0
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8
2
H/d
Fig. 11. Demonstration of the predicted ballistic limit and the test data in Ref. [42].
350 300
Vbl (m/s)
250 200 150 Corran et al.[28]
100
Analysis in Section 3
50
Lower bound in Appendix
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
H/d
Fig. 12. Demonstration of the predicted ballistic limit and the test data for stainless-steel plates in Ref. [28].
300
V bl (m/s)
250 200 150
Corran et al.[28]
100
Analysis in Section 3
50
Lower bound in Appendix
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
H/d
Fig. 13. Demonstration of the predicted ballistic limit and the test data for mild-steel plates in Ref. [28].
531
532
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Vbl (m/s)
250 200 150 100
Corran et al.[28] Analysis in Section 3
50
Lower bound in Appendix 0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
H/d
Fig. 14. Demonstration of the predicted ballistic limit and the test data for aluminum alloy plates in Ref. [28].
predictions. In the case of small plate thickness when wow1 ; the analysis in Appendix A gives a lower bound on ballistic limit. The value of w1 depends on several other parameters, e.g., D=d; which cannot be accurately determined by the present model. Based on the comparison between analytical predictions and experimental results in Figs. 11–14, the value of w1 is estimated to be in the range of 0.1–0.3. According to the bending-shear model in Section 3, there exists a local drop of the ballistic limit when plate thickness increases. This local drop phenomenon is caused by the competition of two opposite effects of plate thickness on the required perforation energy. The model in Ref. [7] is also able to predict the local drop of ballistic limit with plate thickness observed in Ref. [28], which, however, requires a complicated numerical calculation.
6. Remarks Shear plugging failure may be initiated in a broad range of impact problems. Quantitatively, an increase of plate thickness, impact velocity or projectile bluntness in a certain range is in favour of the shear plug formation. Otherwise, other perforation modes, such as petalling, may dominate the ballistic performance of the target or projectile [1,2]. It is difficult to give a simple condition for the occurrence of each failure mode in a perforation problem. Even for a blunt projectile, shear plugging is not the only failure mode to dominate the perforation of a metallic plate. Dishing may be formed for a thin plate subjected to a low velocity impact, which eventually leads to a tensile failure. A structural model including shear, bending, membrane and a proper failure criterion may predict transition between different perforation modes [7], which, however, sometimes requires complicated numerical procedures for predicting ballistic limit and residual velocity. The present model is only valid for the perforation controlled by the shear plugging failure mode with or without the presence of structural bending response and local penetration. The application range is defined by the ratio of the plate thickness to the projectile diameter, the ratio of the plate diameter to the projectile diameter, material parameters and impact velocity, as specified in Sections 3 and 4. In the present paper, blunt projectiles are assumed to be rigid. However, projectile breaking or deforming may occur when a blunt-nosed projectile impacts on thick target. Thus, the proposed
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model is only valid when the projectile has no obvious deformation and damage after the perforation event. It is reasonable to use rigid, perfectly plastic model to study the structural global response. However, the local material response during a penetration/perforation process may involve large plastic deformation, strain rate hardening and temperature or damage softening. A model to include all these factors would be extremely complex and would demand extensive material tests. Although the present analyses do not explicitly consider these factors, all material parameters used for the local deformation and failure in the present model are experimentally based, which have taken their combined effects implicitly, e.g., the parameter k in shear failure criterion may contain the combined effects of temperature and strain and strain rate hardening [32]. The strain and strain rate hardening may be cancelled partially by the thermal and damage softening to reduce their combined effect. However, their respective role may become dominative under certain conditions. For example, the shear plug failure mode may be transformed into an adiabatic shear banding failure with further increase of impact velocity due to enhanced thermal softening. A beam model has been proposed in Ref. [32] to predict the occurrence of adiabatic shear banding. We believe that this failure mode transition is responsible for the observed phenomenon that the ballistic limit may decrease with the increase of target material hardness [30,31]. Nevertheless, it is worthy to further quantify their respective roles in penetration/perforation process.
7. Conclusions A shear plugging model is presented to predict the ballistic limit and residual velocity of a metallic circular plate impacted by a blunt rigid projectile. In addition to the shear around the periphery of a central plug, the proposed model includes plate bending, membrane stretching and local indentation/penetration. The proposed model gives simple formulae and good predictions when compared with various experimental results of perforation tests.
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 China Academy of Engineering Physics (CAEP). Appendix A. Lower Bound Estimate for the Perforation of Ductile Plates (wpw1) For thin plate with wpw1 ; it is assumed that plug failure is still responsible for the final perforation of a ductile circular plate. The transverse velocity field is assumed as ( ’ 0; W 0prpd=2; ’ ¼ ðA:1Þ W ’ 1 ðD 2rÞ=ðD d Þ; d=2orpD=2; W which is the same as Eq. (3) with x ¼ D=2; as shown in Fig. 1b.
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Eq. (2) of energy conservation with membrane effect and the assumed velocity field Eq. (A.1) overestimates the energy consumption in plate response, and thus gives a lower bound estimate for the ballistic limit and residual velocity. Eqs. (2) and (A.1) requires . 1 þ g2 W1 ¼ f ; W ðA:2Þ where
24sy D=d þ 1 24N0 ðD þ d Þ g ¼ ¼ ; rH ðD þ 3d ÞðD d Þ2 rd 2 D=d þ 3 D=d 1 2 2
h i pffiffiffi 24s D=d 1 = 3 wD=d y 24½dQ0 ðD d Þ 4DM0
¼ f ¼ 2 rH ðD þ 3d ÞðD d Þ2 rd D=d þ 3 D=d 1 and
h i pffiffiffi D=d 1 = 3 wD=d f d: ¼ g2 D=d þ 1
ðA:3aÞ
ðA:3bÞ
ðA:3cÞ
’ 1 ¼ 0 are The solutions of Eq. (A.2) under initial conditions W1 ¼ W W1 ¼
f ½1 cosðgtÞ ; g2
ðA:4aÞ
’ 1 ¼ f sinðgtÞ; W g
ðA:4bÞ
. 1 ¼ f cosðgtÞ: W
ðA:4cÞ
’ 0 ðt1 Þ ¼ W ’ 1 ðt1 Þ; and the motion of the The first phase of motion terminates at time t1 when W projectile and plug are controlled by Eqs. (1a)–(1c). A complete analysis of the plate deformation using a similar method was given in Ref. [43]. Following the same procedure in Section 3.3, the ballistic limit can be obtained from the following equations: f 2Zsy f t2BL 2 ½1 cosðgtBL Þ ¼ kH; tBL sinðgtBL Þ þ pffiffiffi g g 3ð1 þ ZÞrd
ðA:5aÞ
f 4Zsy VBL ¼ sinðgtBL Þð1 þ ZÞ þ pffiffiffi tBL ; g 3rd
ðA:5bÞ
f VJump ¼ sinðgtBL Þ: g
ðA:5cÞ
It shows that a velocity jump VJump of the projectile and plug at the ballistic limit also exists when the membrane response in a plate becomes important with the decrease of plate thickness, which has been noted in Ref. [44] for thin plate.
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If Vi XVBL ; the projectile perforates the thin plate and its residual velocity is evaluated by ! 1 4Zsy ðA:6Þ Vr ¼ Vi pffiffiffi t1 * ; ð1 þ ZÞ 3rd where t1* is determined by 2Zsy 1 Vi t1 * pffiffiffi t21 ð1 þ ZÞ 3rd *
!
i fh 1 cos gt ¼ kH 1* g2
where t1 * ptBL :
ðA:7Þ
It should be noted that this lower bound estimate applies only for the plug-induced perforation. With the further decrease of plate thickness, other failure modes, e.g., petaling, may occur, which demand a more general failure criterion and structural response model to study the perforation performance.
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