Attitude deflection of oblique perforation of concrete targets by a rigid projectile

Attitude deflection of oblique perforation of concrete targets by a rigid projectile

Journal Pre-proof Attitude deflection of oblique perforation of concrete targets by a rigid projectile Zhuo-ping Duan, Shu-rui Li, Zhao-fang Ma, Zhuo-...

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Journal Pre-proof Attitude deflection of oblique perforation of concrete targets by a rigid projectile Zhuo-ping Duan, Shu-rui Li, Zhao-fang Ma, Zhuo-cheng Ou, Feng-lei Huang PII:

S2214-9147(19)30609-9

DOI:

https://doi.org/10.1016/j.dt.2019.09.009

Reference:

DT 537

To appear in:

Defence Technology

Received Date: 12 June 2019 Revised Date:

9 September 2019

Accepted Date: 24 September 2019

Please cite this article as: Duan Z-p, Li S-r, Ma Z-f, Ou Z-c, Huang F-l, Attitude deflection of oblique perforation of concrete targets by a rigid projectile, Defence Technology (2019), doi: https:// doi.org/10.1016/j.dt.2019.09.009. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Production and hosting by Elsevier B.V. on behalf of China Ordnance Society.

Attitude Deflection of Oblique Perforation of Concrete Targets by a Rigid Projectile Zhuo-ping Duana,*, Shu-rui Lia, Zhao-fang Mab,*, Zhuo-cheng Oua, Feng-lei Huanga a

State Key Laboratory of Explosion Science and Technology, Beijing Institute of

Technology, Beijing 100081, PR China b

Beijing institute of Technology, Zhuhai, Zhuhai 519088, PR China

*Corresponding Authors:Zhuo-ping Duan, E-mail: [email protected] Zhao-fang Ma, E-mail: [email protected]

Abstract A perforation model is developed to predict the attitude deflection in the oblique perforation of concrete targets by a rigid projectile, in which the inertial moment of the projectile is introduced, together with taking the attitude deflection during the shear plugging sub-stage into account, and the shape of the plug formed on the rear surface of target is also re-investigated. Moreover, a new classification of concrete targets is proposed based on the target thickness, with which the attitude deflections in different kinds of concrete targets are analyzed. It is found that the numerical results by using the new perforation model are in good agreement with the previous experimental data and simulated results. Furthermore, the variations of the attitude deflection with the initial conditions (the initial attitude angle and the initial impact velocity) are investigated.

Keywords Oblique perforation;Penetration;Concrete target;Attitude deflection;Rigid projectile

1

Attitude deflection of oblique perforation of concrete targets by a

2

rigid projectile

3

Abstract

4

A perforation model is developed to predict the attitude deflection in the oblique

5

perforation of concrete targets by a rigid projectile, in which the inertial moment of

6

the projectile is introduced, together with taking the attitude deflection during the

7

shear plugging sub-stage into account, and the shape of the plug formed on the rear

8

surface of target is also re-investigated. Moreover, a new classification of concrete

9

targets is proposed based on the target thickness, with which the attitude deflections in

10

different kinds of concrete targets are analyzed. It is found that the numerical results

11

by using the new perforation model are in good agreement with the previous

12

experimental data and simulated results. Furthermore, the variations of the attitude

13

deflection with the initial conditions (the initial attitude angle and the initial impact

14

velocity) are investigated.

15

Keywords

16

Oblique perforation;Penetration;Concrete target;Attitude deflection;Rigid

17

projectile

18

1. Introduction

19

Over the past decades, the penetration of concrete targets by a rigid projectile has

20

been concerning continually in the field of impact and penetration, especially the

21

attitude deflection during the oblique perforation process [1-11]. When a rigid

22

projectile penetrates obliquely into a concrete target with finite thickness, the rigid

23

projectile will perforate the target along a deflected path due to the asymmetric

24

resistance, which is different from the normal perforation. Moreover, the well-known

25

spalling phenomenon resulting from the reflected tensile waves will occur on the rear

26

surface of target [12]. 1

27

In order to describe such a complicated oblique perforation process, Chen et al.

28

[3] have developed a three-stage model, in which the perforation process is divided

29

into three sequent sub-stages, namely, the initial cratering, the tunneling and the shear

30

plugging sub-stages. The attitude deflection is characterized quantitatively by both the

31

attitude angle (i.e. the acute angle between the axis of the projectile and the normal

32

direction of the impacting surface of target) and the attitude deflection angle (i.e. the

33

increment of the attitude angle). However, in their model, the attitude deflection was

34

considered only in the initial cratering sub-stage, and hence the attitude angle remains

35

unchanged in both the tunneling and the shear plugging sub-stages, but which is not

36

the case in experimental observations and numerical simulations [6, 13-15].

37

In the oblique perforation experiments of the thick concrete targets, the residual

38

attitude angles (i.e. the attitude angle at the last time of perforation) are generally

39

larger than the initial attitude angles [16, 17]. While in the oblique perforation

40

experiments of the thin concrete targets, the residual attitude angles are found to be

41

less than the initial attitude angles [13, 14]. Moreover, the same phenomenon has also

42

been observed by the numerical simulations of oblique perforation of the thin concrete

43

targets [6, 15], which were investigated by incorporating a combined dynamic

44

constitutive model into the LS-DYNA code, based on the Holmquist-Johnson-Cook

45

(HJC) [18] and the Taylor-Chen-Kuszmaul (TCK) [19] models, with both the

46

compressive and the tensile damage of concrete are described simultaneously.

47

Consequently, it indicates that the directions of the attitude deflection in a thin target

48

are obviously different from that in a thick target, which, however, was rarely

49

mentioned before.

50

The oblique perforation process in a thin concrete target is only composed of two

51

sub-stages, namely, the initial cratering and the shear plugging sub-stages [3]. After it

52

deflects with an increasing attitude angle [4, 20, 21] during the initial cratering

53

sub-stage, the projectile has to deflect oppositely during the subsequent shear

54

plugging sub-stage to make the residual attitude angle less than the initial attitude 2

55

angle. Therefore, the disregard of the attitude deflection in the shear plugging

56

sub-stage makes the model developed by Chen et al. [3] fail to simulate the attitude

57

deflection in oblique perforation of thin concrete targets.

58

In addition, there are two more issues need to be justified. On the one hand,

59

Recht and Ipson [20, 22] investigated quantitatively the attitude deflection angle

60

during the initial cratering sub-stage with the initial momentum normal to the

61

penetration path, but their results fail to evaluate quantitatively the effects of the

62

initial impact velocity and the projectile geometry [3]. Later, Chen et al. [3] obtained

63

the attitude deflection angle during the initial cratering sub-stage by analyzing the

64

kinetic energy consumption normal to the penetration path with two dimensionless

65

numbers, namely, the impact function I and the geometry function N [1, 23, 24].

66

However, although their results can describe quantitatively the effects of the initial

67

impact velocity and the initial attitude angle, only the attitude deflection angle at the

68

end time of the initial cratering sub-stage can be estimated, rather than the time

69

history of the attitude deflection angle during all sub-stages of perforation. On the

70

other hand, while the projectile enters the shear plugging sub-stage, a plug will be

71

formed between the projectile nose and the rear surface of the target. Referring to the

72

symmetric cone-shaped plug in normal perforation [25-27], Chen et al. [3] assumed

73

an asymmetric oblique-crossed cone-shaped plug in oblique perforation, with both of

74

their central axes being coincident with the direction of the projectile velocity.

75

However, Jena et al. [28] have pointed out that the shock waves formed at the moment

76

of impact will propagate spherically outward from the point of impact and finally

77

reflect at the nearest surface, so that the damaged area on the rear surface caused by

78

the reflected tensile waves is not in the direction of the projectile velocity in oblique

79

perforation. Moreover, according to the experimental results of oblique perforation of

80

thin concrete targets, the shapes of the damaged areas on the rear surface of targets

81

were symmetric circles when the attack angles (i.e. the acute angle between the axis

82

of the projectile and the direction of the projectile velocity) were negligible [7, 13]. It 3

83

indicates that the asymmetric oblique-crossed cone-shaped plug assumed by Chen et

84

al. [3] is questionable.

85

The purpose of this paper is therefore to develop a perforation model to describe

86

the attitude deflection in oblique perforation of concrete targets, in which the inertia

87

moment of the projectile is introduced, and the attitude deflection during the shear

88

plugging sub-stage is taken into account together with introducing a corresponding

89

deflection mechanism. Moreover, a new classification of the target thickness is

90

presented, and the medium concrete target that has the medium thickness is first

91

proposed. This paper is divided into four sections as follows. After this brief

92

introduction, in Section 2, the perforation model is developed, which the three

93

sub-stages as well as the attitude deflections for different classes of target thickness

94

are analyzed respectively and the determination method of the classes is presented. In

95

Section 3, the numerical results by using the perforation model are presented and

96

compared with a series of the previous experimental data and simulated results,

97

moreover, the variations of the attitude deflection angle versus different initial

98

conditions are investigated and the numerical results of the medium concrete targets

99

are discussed. Finally, some conclusions are drawn out in Section 4.

100

2. Perforation model

101

In the analytical two-dimensional model, a rigid projectile moves in the ballistic

102

plane (i.e. the plane determined by the axis of the projectile and the normal direction

103

of the impacting surface of the target), which is subjected to the oblique target

104

resistance. For the sake of simplicity, the concrete target is assumed to be

105

homogeneous with the homogenized reinforcements and aggregates. As shown in

106

Fig.1, there are usually three kinds of angles in describing such an attitude deflection

107

process, namely, the attitude angle β, the attack angle δ and the oblique angle θ (i.e.

108

the difference between the attitude angle and the attack angle). In this study, we focus

109

mainly on the attitude angle β and define the attitude deflection angle ∆β = β - β0, in

110

which β is the current attitude angle and β0 is the initial attitude angle. In addition, the 4

111

self-rotation of the projectile around its own axis is not taken into account and the

112

attack angle δ is taken to be zero.

113

114 115

Fig.1. The attitude angle β, the attack angle δ and the oblique angle θ.

116

The other geometric parameters of the projectile and the target are shown in

117

Fig.2, where H is the target thickness; d, h, L and lC are the diameter, the nose length,

118

the body length and the distance between the nose tip and the mass centroid of the

119

projectile, respectively; v0 is the initial impact velocity.

120

121 122

Fig.2. Geometric parameters of the rigid projectile and the concrete target.

123

As aforementioned, the total oblique perforation process of the concrete target is

124

divided into three sub-stages, namely, the initial cratering, the tunneling and the shear

125

plugging sub-stages. The projectile does not deflect during the tunneling sub-stage

126

because of the symmetric lateral resistance, but it will deflect in both the initial

5

127

cratering and the shear plugging sub-stages with opposite directions. Therefore, as

128

depicted in

129 130

Fig.3, the attitude deflection angle is positive during the initial cratering sub-stage but negative during the shear plugging sub-stage.

131

132 133

Fig.3 Attitude deflection during three perforation sub-stages.

134

To describe the motion of the projectile during the oblique perforation, in the

135

ballistic plane, three Euler rectangular coordinate systems (x1, y1), (x2, y2) and (x3, y3)

136

are established for the initial cratering, the tunneling and the shear plugging

137

sub-stages, respectively. Each of the three Euler rectangular coordinate systems is

138

determined at the initial time of the corresponding sub-stage, with the origin located at

139

the projectile nose and the xi-axis (i = 1, 2, 3) is along the positive axis of the

140

projectile and (xi, yi) forms a right hand system. Moreover, the motion and the stress

141

analysis of the projectile at the initial time of each perforation sub-stage is shown in

142

Fig.4, where Fni and Fti (i = 1, 2, 3) are the axial resistance and the lateral resistance, 6

143

respectively, and V0i and B0i (i = 1, 2, 3) are the initial projectile velocity and the initial

144

attitude angle of each sub-stage, respectively.

145 146

Fig.4 Motion and stress analysis of the projectile at the initial time of each

147 148

perforation sub-stage. The differential equations of motion of the projectile in each sub-stage are

149

JC

150

d 2 xi m 2 = − Fni dt

(1a)

d 2 βi = M C ( Fti ) dt 2

(1b)

151

where i = 1, 2, 3 represent the initial cratering, the tunneling and the shear plugging

152

sub-stages, respectively, m is the mass of the projectile, xi represents the trajectory

153

distance during each sub-stage, MC(Fti) is the torque of Fti to the projectile centroid,

154

and JC is the inertia moment passing through the projectile centroid and normal to the

155

ballistic plane.

156

2.1 Analyses of three sub-stages in oblique perforation

157

2.1.1 Initial cratering sub-stage 7

158

In the initial cratering sub-stage, the axial resistance Fn1 is given by an empirical

159

formula according to the reference [29], and it satisfies the relationship of Eq. (2b)

160

with the lateral resistance Ft1 [3]. According to the stress analysis of the projectile

161

shown in

Fig.4, MC(Ft1), the torque of Ft1 to the projectile centroid, is obtained as

162

Fn1 = c1 x1 ( t )

(2a)

163

Ft1 = Fn1 sin β 0

(2b)

164

M C ( Ft1 ) = Ft1lC

(2c)

165

where c1 is the resistance constant in the initial cratering sub-stage.

166

At the initial time of this sub-stage t = 0, the initial trajectory distance X01 = x1(0)

167

= 0, the initial projectile velocity V01 = v1(0) = v0, the initial angular velocity Ω01 =

168

ω1(0) = 0 and the initial attitude angle B01 = β1(0) =β0. Substituting Eq.(2) into Eq.(1)

169

leads to the initial-value problems for x1(t) and β1(t), which, under the aforementioned

170

initial conditions, gives the solutions as x1 ( t ) =

171

v1 ( t ) =

172

173

174

β1 ( t ) = −

 c  V01 sin  1 t  c1  m  m

(3)

 c  dx1 = V01 cos  1 t  dt  m 

 c  c1lCV01 m cl V t sin  1 t  sin B01 + 1 C 01 c c c  m  JC 1 1 JC 1 m m

(4)

m sin B01 + B01 c1

(5)

At the end time of this stage t = T1, we denote the end trajectory distance, the end

175

projectile velocity and the end attitude angle of the first stage as X1 = x1(T1), V1 = v1(T1)

176

and Β1 = β1(T1), respectively. Moreover, we define the first attitude deflection angle as

177

∆β1 = Β1 − B01. From Eqs.(3)-(5), it reaches

178

X1 =

 c  V01 sin  1 T1  c1  m  m 8

(6)

 c  V1 = V01 cos  1 T1   m 

179

ml V ∆β1 = C 01 JC

180

2   V1   V1   m  sin B01 arccos   − 1−    c1  V01   V01   

(7)

(8)

181

In these equations, the end time of the first stage T1 and the resistance constant c1 are

182

unknown yet and need to be determined in the following.

183

The initial cratering is also divided into the complete and the incomplete cases.

184

For a target thick enough, the projectile will accomplish a complete initial cratering

185

sub-stage, making the trajectory distance reach up to X1 that can be determined by the

186

following empirical formula [1, 24]

X 1 = kd

187

(9)

188

where k = 0.707 + h/d is a dimensionless coefficient. However, in the case of

189

incomplete initial cratering, more details of other sub-stages are necessary to

190

determine X1.

191

2.1.2 Tunneling sub-stage

192

The projectile enters the tunneling sub-stage only after the complete initial

193

cratering. In other words, if the projectile is only able to accomplish an incomplete

194

initial cratering, it will skip over the tunneling sub-stage and enter the shear plugging

195

sub-stage directly. In the tunneling sub-stage, the lateral resistance is symmetric,

196

which leads to Ft2 = 0 and MC(Ft2) = 0. Thus, the projectile moves forward in a

197

straight line under the axial resistance Fn2 that can be estimated by the following

198

empirical formula [2, 29]

199

Fn 2 =

2 πd 2  R + N ∗ ρ ( v2 ( t ) )   4 

(10)

200

where R and ρ are the target strength parameter and density of the concrete target, N*

201

is the geometric factor of the projectile nose. 9

202

At the initial time of the tunneling sub-stage t = T1, which is also the end time of

203

the initial cratering sub-stage, the initial trajectory distance of the second sub-stage

204

X02 = x2(T1) = 0 and the projectile velocity V02 = v2(T1) = V1. Similarly as done for the

205

first sub-stage, at the end time of this sub-stage t = T2, we define the end trajectory

206

distance, the end projectile velocity and the end attitude angle of the second sub-stage

207

as X2 = x2(T2), V2 = v2(T2) and Β2 = β2(T2) =Β1, respectively. Obviously, the second

208

attitude deflection angle ∆β2 = Β2 − Β1 = 0. Substituting Eq.(10) into Eq.(1) for the

209

second sub-stage and then integrating the resulted differential equation, X2 can be

210

derived as X2 =

211

R + N ∗ ρV12 2m ln π d 2 ρ N ∗ R + N ∗ ρV22

(11)

212

The tunnel formed in this stage will impede the lateral motion of the projectile

213

and the attitude deflection in the shear plugging sub-stage in a way, which is called

214

the clamping mechanism of the tunneling sub-stage. If the tunnel is long enough to

215

impede the lateral motion of the projectile completely, the attitude deflection in the

216

shear plugging sub-stage can be negligible, which is called here as the complete

217

clamping mechanism. Contrarily, if the tunnel is not long enough, the attitude

218

deflection of the projectile in the shear plugging sub-stage will depend on the tunnel

219

length, which is called as the incomplete clamping mechanism.

220

2.1.3 Shear plugging sub-stage

221

According to the experimental observation of symmetric damaged areas on the

222

rear surfaces of concrete targets, as shown in Fig.5 [7, 13], a symmetric cone-shaped

223

plug is assumed in this paper, whose central axis is coincident with the normal

224

direction of the target. The cone crater formed in the shear plugging sub-stage is

225

shown as Fig.6(a), which is different from the asymmetric oblique-crossed cone crater

226

proposed by Chen et al. (see Fig.6(b)) [3].

227 10

228

229 230 231

Fig.5 Photographs of damaged areas on the rear surfaces of concrete targets in oblique perforation experiments [7, 13].

232

233 234 235

Fig.6 (a) Symmetric cone crater proposed in this study; (b) Asymmetric oblique-crossed cone crater [3].

236

The geometric parameters of the symmetric cone crater are shown in Fig.7,

237

where H* is the plug’s thickness, the cone slope angle α (i.e. the acute angle between

238

the central axis and the conical edge of the plug) is taken to be 66.1°[26], and the

239

shear surface area is As = πl(R + r). Using the relations l2 = (R−r)2 + H*2, r =

240

d/2·sec(β0+∆β1) and R = r + H*tanα, there is

241

  H* AS cos α = π dH *  sec ( β 0 + ∆ β1 ) + tan α  d  

11

(12a)

242 243

Fig.7 Geometric parameters of the symmetric cone crater.

244

In view of von-Mises failure criterion, the plug occurs once its shear stress τf

245

reaches the failure stress of concrete with τ f = f C

246

force of the plug Fτ′ = τ f As ,where As is the shear surface area. As the critical force Fs′

247

is the projection of Fτ′ along the axis of plug and satisfies Fs′ = Fτ′·cosα, the Eq. (12b)

248

can be obtained as Fs′ =

249

3 . At that moment, the shear

1 f C AS cos α 3

(12b)

250

By using the geometric relations as shown in Fig.8 as well as the Newton’s third law,

251

the axial resistance acted on the projectile is Fns = Fs·sec(β0+∆β1) = Fs′·sec(β0+∆β1).

252

Therefore, with Eq.(

253

12b), at the initial time of the shear plugging sub-stage t = T2, the axial resistance Fns

254

can be written as

255

Fns =

  1 H* f C πdH *  sec ( β 0 + ∆ β1 ) + tan α  sec ( β 0 + ∆ β1 ) d 3  

256

12

(13)

257 258

Fig.8 The critical force Fs′of the plug at the initial time of the shear plugging

259

sub-stage.

260

If it enters the shear plugging sub-stage after an incomplete initial cratering, the

261

projectile will deflect again with the opposite direction of the initial cratering

262

sub-stage. Referring to Chen et al. [3], in this sub-stage, with the motion of the

263

projectile, the axial resistance Fn3 can be assumed to decrease linearly from Fns to zero

264

as shown by Eq.(14a), and there exists a relationship between the lateral resistance Ft3

265

and the axial resistance Fn3 as shown by Eq.(14b). It is assumed that the acting point

266

of Ft3 moves gradually to the projectile centroid from the nose tip and the torque arm l

267

= lC – x3(t) decreases linearly from lC. Therefore, the axial resistance Fn3 and the

268

lateral resistance Ft3 are given as

269

Fn3 = Fns − c2 x3 ( t )

(14a)

270

Ft3 = −Fn3 sin ( β0 + ∆β1 )

(14b)

271

M C ( Ft3 ) = Ft3 lC − x3 ( t ) 

(14c)

272

where c2 is the resistance constant in the shear plugging sub-stage.

273

At the initial time of the shear plugging sub-stage t = T2, the initial trajectory

274

distance X03 = x3(T2) = 0, the initial projectile velocity V03 = v3(T2) = V2, the initial 13

275

angular velocity Ω03 = ω3(T2) = 0 and the initial attitude angle B03 = β3(T2) = Β2 = Β1

276

= β0 + ∆β1. Substituting Eq.(14) into Eq.(1) leads to the initial-value problems for x3(t)

277

and β3(t), respectively. Solving the initial-value problems under the initial conditions,

278

the trajectory distance x3(t), the projectile velocity v3(t), the attitude angle β3(t), the

279

angular velocity ω3(t) = dβ3(t)/dt and the acceleration a3(t) = dv3(t)/dt of the projectile

280

during the third sub-stage can be obtained as

281

 m Fns  (t −T2 ) x3 ( t ) = V03 −  e 4 c 2 c 2 2  

282

v3 ( t ) =

 m Fns  (t −T2 ) −  V03 e 4c2 2c2  

sin Β 03 β3 ( t ) = − JC

284

286

 m Fns  −( t −T2 ) − V03 +  e 4 c 2 c 2 2  

c  m Fns  ( t −T2 ) a3 ( t ) = 2  V03 − e m  4c2 2c2 

283

285

c2 m

c2 m

c2 m

c2 m

c2 m

+

 m Fns  −(t −T2 ) + V03 +  e 4 c 2 c 2 2    m Fns  −( t −T2 ) −  V03 + e 4c2 2c2  

Fns (15) c2 c2 m

c2 m

  (16) 

  (17) 

 A1 D1 α1 ( t −T2 ) A2 D2 −α1 ( t −T2 ) t − T2 ) 2 ( − 2 e + D3  2 e 2 α1  α1

(18)

 A D + A2 D2  A1 D1 − A2 D2  − 1 1  + Β 03  ( t − T2 )− α1 α12   

where α12 = c2/m, A1 = (V03/α1 − Fns/c2)/2, A2 = (V03/α1 + Fns/c2)/2, D1 = −c2lC + c2A1 + Fns, D2 = −c2lC − c2A2 + Fns, D3 = −2c2A1A2.

287

The shear plugging sub-stage is finished at the time t = T3 when the plug

288

separates from the projectile, and from then on, there is no longer action between

289

them (the re-contact after separation is not taken into account) and the acceleration

290

of the projectile vanishes. Similarly, we denote the end trajectory distance, the end

291

projectile velocity, the end angular velocity, the end attitude angle and the end

292

attitude deflection angle of the third sub-stage as X3 = x3(T3), V3 = v3(T3), Ω3 =

293

ω3(T3), Β3 = β3(T3) and ∆β3 = Β3 − Β03, respectively. Thus, from Eqs.(15)-(18), we

294

get

295

 m Fns  (T3 −T2 ) X 3 = V03 −  e 4 c 2 c 2 2  

c2 m

 m Fns  −(T3 −T2 ) − V03 +  e 4 c 2 c 2 2  

14

c2 m

+

Fns c2

(19)

V3 =

296

c2 m

 m Fns  (T3 −T2 ) −  V03  e 4 2 c c 2 2  

 m Fns  (T3 −T2 ) − V03 e 4c2 2c2  

297

c2 m

c2 m

 m Fns  −(T3 −T2 ) +  V03 +  e 4 2 c c 2 2  

 m Fns  −(T3 −T2 ) − V03 + e 4c2 2c2  

 mV032 m − FnsV03  4c2 sin Β 03  2 ∆β 3 = − e c2 JC   m 

298

c2 m

c2 m

  (20) 

=0

mV032 Fns2 − + FnslC c2 (T3 −T2 ) 2 2c2 m − c2 m

(21)

(22)

2   mV032 Fns2  ( T3 − T2 ) − − + mlCV03 ( T3 − T2 )   2c2  2   2

 sin Β 03  Fns m − lC   mV03 − Fns  e  J C  2c2 c2  

c2 (T3 −T2 ) m

  F 2  (T − T ) −  mV032 − ns  3 2 + mlCV03  (23) c2  2  

299

Ω3 = −

300

where the resistance constant c2 is unknown and needs to be determined as well.

301

As the analyses presented above are focused on each sub-stage independently, the

302

continuous conditions between the sub-stages are still required. However, referring to

303

Chen et al. [3], the oblique perforations of a thin target and a thick target are

304

composed of different sub-stages. Therefore, for the concrete targets with different

305

thickness, the continuous conditions are also different and their oblique perforation

306

processes should be considered respectively.

307

Chen et al. [3] have divided the target thickness into two classes according to the

308

trajectory distance in the initial cratering sub-stage, namely, the thin and the thick

309

concrete targets. However, It is insufficient while the attitude deflection in the shear

310

plugging sub-stage as well as the clamping mechanism of the tunneling sub-stage are

311

taken into account. Therefore, this paper classifies the target thickness into three

312

classes, among which the medium target between the thin and the thick targets is

313

newly proposed, and the attitude deflection angle ∆βr will be investigated respectively

314

for each class with their individual characteristics and continuous conditions.

315

For a thin concrete target, referring to Chen et al. [3], its oblique perforation

316

consists of only two sub-stages, i.e. an incomplete initial cratering and a shear 15

317

plugging sub-stages. That is, the attitude deflection angle in a thin target is ∆βr = ∆β1 +

318

∆β3 (see Fig.10).

319

While the target thickness reaches up to a certain value, a completed initial

320

cratering sub-stage will be accomplished and a tunneling sub-stage will appear. At this

321

moment, we define that the target transforms into a medium one. Thus, the oblique

322

perforation of a medium concrete target is defined to be composed of three sub-stages,

323

i.e. a complete initial cratering, a tunneling and a shear plugging sub-stages, but

324

among which the tunnel length X2 is limited with just the incomplete clamping

325

mechanism. That is, the attitude deflection angle in a medium target is ∆βr = ∆β1 + ∆β3

326

(X2) (see Fig.11), in which the quantitative description of the incomplete clamping

327

mechanism of the tunneling sub-stage need to be dealt with.

328

When the tunnel length keeps increasing with the target thickness to a certain

329

value, the attitude deflection in the shear plugging sub-stage will be impeded

330

completely. At this moment, it is defined that the target transforms into a thick one.

331

Therefore, the oblique perforation of a thick concrete target is also comprised of three

332

sub-stages, i.e. a complete initial cratering, a tunneling and a shear plugging

333

sub-stages, among which the large enough tunnel length X2 with the complete

334

clamping mechanism leads to ∆β3 = 0. That is, the attitude deflection angle in a thick

335

target is ∆βr = ∆β1 (see

336

Fig.9).

337

2.2 Analyses of concrete targets with different thickness

338

2.2.1 Thick target 16

339

340 341

Fig.9 Attitude deflection in oblique perforation of a thick concrete target.

342

The attitude deflection in oblique perforation of a thick concrete target is

343

depicted as shown in

344

Fig.9. At the end time of the initial cratering sub-stage t = T1, which is also the

345

initial time of the tunneling sub-stage, the axial resistance satisfies Fn1 = Fn2 at this

346

moment, thus we have

347

c1 X1 =

πd 2 R + N * ρV12 ) ( 4

(24)

348

Solving Eqs.(6)-(9) and (24) simultaneously, the end time T1, the end projectile

349

velocity V1 and the end attitude deflection angle ∆β1 of the initial cratering sub-stage

350

as well as the resistance constant c1 can be obtained as

351

V = 2 1

mv02 − ( πkd 3 4 ) R

m + ( πkd 3 4 ) N * ρ

17

(25)

T1 =

352

353

354

ml v ∆β1 = C 0 JC

V  m arccos  1  c1  v0 

2   V1   V1   m  sin β 0 arccos   − 1 −    c1  v0   v0   

  * 2  πd N ρ v0 + R  c1 =   4k  πkd 3 *  N ρ 1+ 4m  

(26)

(27)

(28)

355

From the geometric relation as shown in

356

Fig.9, the end trajectory distance in the tunneling sub-stage X2 and that in the

357

shear plugging sub-stage X3 satisfy, respectively,

358

X 2 = ( H − H * ) sec ( β 0 + ∆β1 ) − X 1

(29)

359

X 3 = H * sec ( β0 + ∆β1 )

(30)

360

Moreover, as the end time of the tunneling sub-stage is just the initial time of the

361

shear plugging sub-stage, the axial resistance satisfies Fn2 = Fn3 at this moment.

362

Substituting Eqs.(10) and (13) into them, one has

363

 πd 2 1 H* ∗ 2 * R + N ρV2 ) = f C πdH sec ( β0 + ∆β1 ) + tan α  sec ( β0 + ∆β1 ) ( 4 d 3  

(31)

364

Therefore, X2 and X3, the end projectile velocities V2 and V3, the end time of the shear

365

plugging sub-stage T3, the resistance constant c2 and the plug thickness H* can be

366

obtained respectively by solving Eqs.(11), (19)-(21) and (29)-(31) simultaneously,

18

367

which the Eq.(11) is approximated simply by the first order Taylor expansion of the

368

logarithm term.

369

2.2.2 Thin target

370

371 372

Fig.10 Attitude deflection in oblique perforation of a thin concrete target.

373

The attitude deflection in oblique perforation of a thin concrete target is shown in

374

Fig.10. Similarity, at the end time of the initial cratering sub-stage, which is also the

375

initial time of the shear plugging sub-stage, the axial resistance satisfies Fn1 = Fn3 at

376

this moment. Thus from Eqs.(2a) and (13), there is

377

c1 X1 =

  1 H* f C πdH * sec ( β0 + ∆β1 ) + tan α  sec ( β0 + ∆β1 ) d 3  

(32)

378

Moreover, as shown in Fig.10, the end trajectory distance of the initial cratering

379

sub-stage X1 and that of the shear plugging sub-stage X3 can be written as

380

X1 = ( H − H* ) sec( β0 +∆β1 )

(33)

381

X 3 = H * sec ( β0 + ∆β1 + ∆β3 )

(34)

382

As the resistance constant c1 given by Eq.(28) is only determined by the

383

geometric and the material parameters of the projectile and the target, it is

384

independent of the trajectory distance during the initial cratering sub-stage and the

385

target thickness. Therefore, the Eq.(28) is also adapted for the incomplete initial

386

cratering in the oblique perforation of the thin concrete target [3]. Solving Eqs.(6)-(8), 19

387

(19)-(23) and (32)-(34) simultaneously, the end time of the initial cratering sub-stage

388

T1 and the shear plugging sub-stage T3, the end trajectory distances X1 and X3, the end

389

projectile velocities V1 and V3, the end attitude deflection angles ∆β1 and ∆β3, the end

390

angular velocity Ω3, the resistance constant c2 and the plug thickness H* can be

391

obtained.

392

2.2.3 Medium target

393

394 395

Fig.11 Attitude deflection in oblique perforation of a medium concrete target.

396

The attitude deflection in oblique perforation of a medium concrete target is

397

shown in Fig.11. The analysis of the complete initial cratering and the tunneling

398

sub-stages for the medium target is the same as that for the thick target, by which the

399

end projectile velocity V2 and the end trajectory distance X2 at the end time of the

400

tunneling sub-stage can be obtained. In order to determine the attitude deflection

401

angle and the angular velocity in the shear plugging sub-stage affected by the

402

tunneling sub-stage, an oblique perforation process of a fictitious thin concrete target

403

is assumed to consist of a complete initial cratering and a shear plugging sub-stages,

404

in which we denote the initial impact velocity as v0t and the projectile velocity at the

405

end time of the complete initial cratering sub-stage as V1t = V2. Thus, using Eq.(25),

406

v0t can be obtained as

407

v0 t = 1 + ( πkd 3 4 m ) N * ρ  V22 + ( πkd 3 4m ) R 20

(35)

408

Moreover, to describe the effects of the tunnel length X2 on the attitude deflection

409

angle ∆β3, a coefficient δx is introduced as

410

δx =

v0 t  X2  1 −  v0  XL 

(36)

411

where XL is the minimum tunnel length to make ∆β3 = 0. Empirically, XL = 2d for the

412

projectiles with the ratio of the length to the diameter larger than 4. Using Eqs.(22)

413

and (23), the attitude deflection angle ∆β3 and the end angular velocity Ω3 in the

414

oblique perforation of the medium concrete target can be obtained as follows  mV22 m − FnsV2  2 4 c2 sin Β 2  ∆β3 = −δ x e c2 JC   m 

415

c2 (T3 −T2 ) m

mV22 Fns2 − + FnslC 2 2c2 − c2 m

(37)

2   mV22 Fns2  (T3 − T2 ) − − + mlCV2 (T3 − T2 )   2c2  2  2 

416

417

Ω 3 = −δ x

sin Β 2 JC

 F  m  ns − lC   mV2 − Fns e c2   2c2 

c2 (T3 −T2 ) m

  F 2  (T − T ) −  mV22 − ns  3 2 + mlCV2  (38) c2  2  

2.3 Determination of classes for concrete targets

418

As aforementioned, the tunneling sub-stage only follows a complete initial

419

cratering sub-stage. Thus the existence of the tunneling sub-stage can be used to

420

distinguish a thin concrete target from a medium or a thick concrete target. A critical

421

target thickness HL is then introduced as the minimum thickness of the target that

422

consists of a complete initial cratering and a shear plugging sub-stages but without a

423

tunneling sub-stage, as shown in Fig.12. Therefore, a concrete target with the

424

thickness H less than HL is defined as a thin concrete target, and, vice versa, either a

425

thick or a medium target.

426

21

427 428

Fig.12 Attitude deflection in the oblique perforation of a concrete target with the

429

critical thickness HL.

430

It can be seen from Fig.12 that the critical target thickness HL can be written as

H L = kd cos ( β0 + ∆β1L ) + H L*

431

(39)

432

At the end time of the complete initial cratering sub-stage, the end projectile velocity

433

V1L, the end attitude deflection angle ∆β1L and the resistance constant c1 can be

434

obtained from Eqs.(25), (27) and (28). Moreover, as the end time of the complete

435

initial cratering sub-stage is also the initial time of the shear plugging sub-stage, so

436

that Fn1 = Fn3 at this moment. By virtue of Eqs.(2a) and (13), ∆β1L and H*L can be

437

solved, and then the critical target thickness HL can be derived from Eq. (39).

438

As XL is the minimum tunnel length to impede completely the attitude deflection

439

in the shear plugging sub-stage (to make ∆β3 = 0), it can be used to distinguish a

440

medium concrete target from a thick target. Therefore, a concrete target with the

441

tunnel length X2 less than XL (0 < X2 < XL) belongs to a medium concrete target, while

442

a target with X2 ≥ XL belongs to a thick concrete target.

443

3. Numerical results and discussions

444

The perforation model is used and the numerical results are compared with the

445

experimental data of the oblique perforation of both thick and thin concrete targets, as

446

well as the simulated results of the oblique perforation of thin concrete targets [6, 14,

447

16]. 22

448

In the oblique perforation experiments of thick concrete targets reported by Fan

449

et al. [16], for the tested ogive-nose projectiles, the mass m = 3.64 kg, the diameter d

450

= 50 mm, the caliber-radius-head CRH = 3, and the inertia moment JC = 0.018 kg·m2.

451

For the tested thick targets, the compressive strength fc = 42 MPa, the density ρ =

452

2280 kg/m3, and the target thickness H = 250 mm, 300 mm, 400 mm, 550 mm,

453

respectively. With the same initial attitude angle β0 = 30°, the experimental data and

454

the numerical results of the residual velocity (i.e. the projectile velocity at the last

455

time of perforation) and the attitude deflection angle are shown in Fig.13.

456

457 458

Fig.13 The experimental data and the numerical results of the oblique perforation

459

experiments of thick concrete targets [16]: (a) residual velocity; (b) attitude deflection

460

angle.

461

In the numerical simulations of the oblique perforation of thin concrete targets

462

that investigated by LS-DYNA software and reported by Liu et al. [6], the mass of the

463

numerical ogive-nose projectile is m = 100 kg, the diameter d = 189 mm, the length L

464

= 1050 mm, CRH = 3.3, and the inertia moment JC = 9.15 kg·m2. For the numerical 23

465

thin targets, the compressive strength fc = 30 MPa, the density ρ = 2500 kg/m3, and the

466

target thickness H = 180 mm. The initial attitude angle β0 varies from 15° to 45° and

467

the initial impact velocity v0 varies from 200 m/s to 500 m/s. The simulated results

468

and the numerical results of the attitude deflection angle are shown in Fig.14, which

469

the negative attitude deflection angles indicate the projectiles deflect with decreasing

470

attitude angle.

471

472 473 474

Fig.14 The simulated results and the numerical results of the oblique perforation simulations of thin concrete targets [6].

475

In the two oblique perforation experiments of multilayered thin concrete targets

476

performed by Ma et al. [14], for the tested truncated-ogive-nose projectiles, the mass

477

m = 290 kg, the diameter d = 250 mm, CRH = 1.56, and the inertia moment JC =

478

34.53 kg·m2. For the tested thin targets, the compressive strength fc = 46 MPa, the

479

density ρ = 2500 kg/m3, the target thicknesses of the first layers are both 300mm

480

while others are 180 mm, and the target spacing is 3.5 m. The initial impact velocities

481

and the initial attitude angles of the two tests (numbered No.290-1 and No.290-2) are

482

v0 = 833 m/s, β0 = 15° and v0 = 688 m/s, β0 = 14.7°, respectively. There were no attack

483

angles when the projectiles impacted on the first layers of targets.

484

The ballistic deflections of the two tests recorded by a High Speed Photograph

485

System are shown in Fig.15 [14], in which the blue lines are the projectile axes when

486

they impacted on the first layers of targets, the purple and the red lines are the

487

connection of the impacting points on each layer of targets during the penetration 24

488

process. These observations indicate that the projectiles deflected with decreasing

489

attitude angle during the penetration process. As the perforation processes of the first

490

two or three layers of targets are less affected by the attack angles, only them are

491

calculated. Both the experimental data and the numerical results of the residual

492

velocity Vr and the attitude deflection angle ∆βr are listed in Table 1 and Table 2,

493

respectively, in which the errors are defined as ERr = |Vr,exp − Vr,num|/|Vr,exp|×100% and

494

ERa = |∆βr,exp − ∆βr,num|, respectively.

495

496 497

Fig.15 The ballistic deflections recorded in the oblique perforation experiments

498

of multilayered thin concrete targets [14].

499

Table 1. The experimental data and the numerical results of the residual velocity Vr in the oblique perforation

500

experiments of multilayered thin concrete targets.

No. 290-1

Target number

H /mm

v0 /m·s-1

Vr,exp [14]/m·s-1 Vr,num/m·s-1

1

300

833

820

827

0.9

2

180

820

800

818

2.3

3

180

800

771

798

3.5

1

300

688

684

681

0.4

2

180

684

669

679

1.5

ERr/%

No. 290-2

501

Table 2. The experimental data and the numerical results of the attitude deflection angle ∆βr in the oblique

502

perforation experiments of multilayered thin concrete targets.

25

No. 290-1

Target number

H/mm

β0/(°)

∆βr,exp [14]/(°)

∆βr,num /(°)

ERa /(°)

1

300

15.0

-0.9

-1

0.1

2

180

14.1

-0.1

-0.3

0.2

3

180

14.0

-0.2

-0.3

0.1

1

300

14.7

-1.6

-1.4

0.2

2

180

13.1

-0.4

-0.42

0.02

No. 290-2

503

As shown in Fig.13, the prediction of the perforation model is in good agreement

504

with the experimental data for the thick concrete targets, which indicates that the

505

inertia moment JC is effective in describing the attitude deflection during the initial

506

cratering sub-stage. Moreover, it can also be found from Fig.14, Table 1 and Table 2

507

that considering the attitude deflection during the shear plugging sub-stage and

508

introducing the corresponding deflection mechanism make it possible to describe the

509

phenomena in the oblique perforation of thin concrete targets, which the residual

510

attitude angle is less than the initial attitude angle and the penetration path of the

511

projectile tends to be the shortest path, i.e. along the normal direction of the target.

512

With the geometric and the material parameters of the numerical projectile (m =

513

100 kg) and the concrete target given by Liu et al. [6], the variations of the attitude

514

deflection angle ∆βr versus the dimensionless target thickness χ (= H/d), the initial

515

attitude angle β0 and the initial impact velocity v0 are investigated respectively. The

516

numerical results are shown in Fig.16-18, respectively. The signs of ∆βr represent the

517

deflection directions, i.e. ∆βr > 0 indicates that the projectile deflects with increasing

518

attitude angle while ∆βr < 0 indicates the decreasing attitude angle. Moreover, while

519

the signs of ∆β1 (> 0) and ∆β3 (< 0) both represent the deflection directions, the

520

increase of ∆β1 and the decrease of ∆β3 indicate a more violent attitude deflection in

521

the initial cratering sub-stage and the shear plugging sub-stage, respectively.

522

The variation of ∆βr versus χ for four initial attitude angles β0 = 10°, 15°, 20°, 25°

523

under the same initial impact velocity v0 = 500 m/s are shown in Fig.16. It can be seen 26

524

clearly that there exist two critical target thicknesses χL and χLT on each curve, cross

525

which the class of the target will transform and the attitude deflection angle will vary

526

versus χ in a different way. Both of them decrease with the increase of β0. Typically,

527

the two critical target thicknesses χL = 3.84, χLT = 5.74 for β0 = 15°. Moreover, it can

528

be found that, with the increase of χ, ∆βr decreases in the thin target, but increases in

529

the medium target and then keeps constant in the thick target.

530

In the medium target, the attitude deflection angle ∆β1 of the complete initial

531

cratering sub-stage remains unchanged with the variation of χ. However, the increase

532

of the tunnel length X2, resulting from the increase of χ, and the enhanced clamping

533

mechanism will alleviate the attitude deflection in the shear plugging sub-stage, so

534

that ∆β3 increases with χ. Therefore, the attitude deflection angle ∆βr increases with χ

535

in the medium target.

536

537 538 539

Fig.16 Variation of the attitude deflection angle ∆βr versus the dimensionless target thickness χ in the oblique perforation of concrete targets.

540

The variation of ∆βr versus β0 for seven target thicknesses χ = 1.2, 1.6 (thin

541

targets), 4.3, 4.5, 4.8 (medium targets), 7.0 and 8.0 (thick targets) under the same

542

initial impact velocity v0 = 500 m/s are shown in Fig.17. It is found that with the

543

increase of β0, ∆βr decreases in the thin target and increases in the thick target, which

544

both indicate a more violent attitude deflection. But in the medium target, ∆βr

545

decreases first and then increases. Moreover, as shown in Fig.17, there exists a critical 27

546

initial attitude angle β0L on each curve for the medium targets, and, for β0 > β0L, the

547

medium target will transform into a thick target. Typically, the critical initial attitude

548

angle β0L = 47.5° for χ = 4.5.

549

In the medium target, the increased β0 results in the increase of both ∆β1 and X2 as

550

well as the decrease of ∆β3, however, the increased X2 leads contrarily to the increase

551

of ∆β3. When β0 is small and the clamping mechanism under a small X2 is not

552

significant, the variation of ∆β3 is dominated by β0. But with the increase of β0, the

553

increased X2 will gradually dominate the variation of ∆β3. Therefore, ∆β3 as well as

554

∆βr first decreases and then increases in the medium target. When β0 increases to β0L,

555

X2 becomes large enough such that ∆β3 increases to zero, and then the medium target

556

transforms into the thick one.

557

558 559 560

Fig.17 Variation of the attitude deflection angle ∆βr versus the initial attitude angle β0 in the oblique perforation of concrete targets.

561

The variation of ∆βr versus v0 for six target thicknesses χ = 2.2, 2.8 (thin targets),

562

5.1, 5.4 (medium targets), 7.0 and 8.0 (thick targets) under the same initial attitude

563

angle β0 = 15° are shown in Fig.18. It is found that with the increase of v0, ∆βr 28

564

increases in the thin target but decreases in the thick target, which both indicate that

565

the attitude deflection is alleviated. However, in the medium targets, the variation of

566

∆βr depends on the target thickness. For example, for a thinner medium target (χ =

567

5.1), ∆βr increases monotonously, while, for a thicker medium target (χ = 5.4), ∆βr

568

increases first and then decreases.

569

In the medium target, both ∆β1 and X2 will decrease with v0. Similarly, while the

570

increased v0 leads to the increase of ∆β3, the decreased X2 leads to the decrease of ∆β3

571

contrarily. For a thinner medium target (χ = 5.1), the negligible clamping mechanism

572

under a small X2 dominates the variation of ∆β3 with v0, so that the attitude deflection

573

angle ∆βr increases monotonously. However, for a thicker medium target (χ = 5.4), the

574

remarkable clamping mechanism under a larger X2 leads to that the variation of ∆β3 is

575

gradually changed into a X2-dominated one, so that ∆β3 as well as the attitude

576

deflection angle ∆βr first increase and then decrease with v0.

577 578 579

Fig.18 Variation of the attitude deflection angle ∆βr versus the initial impact velocity v0 in the oblique perforation of concrete targets.

580

In a word, the attitude deflection in a medium concrete target is under the

581

deflection mechanisms of both the initial cratering and the shear plugging sub-stages

582

as well as the incomplete clamping mechanism of the tunneling sub-stage. The

583

quantitative description of the incomplete clamping mechanism of the tunneling

584

sub-stage can be realized by the introduced coefficient δx, and the variations of the

29

585

attitude deflection angle ∆βr versus different initial conditions can be predicted for a

586

medium target.

587

4. Conclusion

588

A perforation model has been developed to predict the attitude deflection in the

589

oblique perforation of concrete targets by a rigid projectile, with broader applications,

590

and the numerical results are in good agreement with a series of experimental data and

591

simulated results. Some conclusions can be drawn out as follows.

592

(1) The inertial moment of a projectile is an important parameter in predicting the

593

time history of the attitude deflection angle during the initial cratering sub-stage,

594

by which, the perforation model can predict accurately the attitude deflection in

595

the oblique perforation of thick concrete targets.

596

(2) The shape revision of the plug formed on the rear surface of the target is

597

reasonable. Moreover, the introduced deflection mechanism in the shear

598

plugging sub-stage is of great significance in describing the attitude deflection in

599

the oblique perforation of thin concrete targets, by which, the phenomenon that

600

the residual attitude angle is less than the initial attitude angle can be predicted

601

well.

602

(3) The incomplete clamping mechanism during the tunneling sub-stage can be

603

described quantitatively by an introduced coefficient δx, by which the attitude

604

deflection in the oblique perforation of medium concrete targets can be predicted

605

well.

606

(4) The classification for the concrete targets in the oblique perforation is necessary

607

to predict well the attitude deflection angle, which can be realized by defining

608

the critical target thickness HL and the critical tunnel length XL. Thus, the targets

609

with the thickness less than HL belong to the thin targets, that with the thickness

610

larger than HL but the tunnel length larger than XL belong to the thick targets, and

611

the rest belong to the medium targets. 30

612 613

Acknowledgments This work was supported by the National Natural Science Foundation of China

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[grant numbers 11521062].

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Declarations of interest

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No conflict of interest exits in the submission of this manuscript.

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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: