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A new motion model of rifle bullet penetration into ballistic gelatin
Accepted Manuscript Title: A new motion model of rifle bullet penetration into ballistic gelatin Author: Liu Susu, Xu Cheng, Wen Yaoke, Zhang Xiaoyun PII: DOI: Reference:
S0734-743X(16)30031-8 http://dx.doi.org/doi: 10.1016/j.ijimpeng.2016.02.003 IE 2647
To appear in:
International Journal of Impact Engineering
Received date: Revised date: Accepted date:
25-9-2015 29-1-2016 8-2-2016
Please cite this article as: Liu Susu, Xu Cheng, Wen Yaoke, Zhang Xiaoyun, A new motion model of rifle bullet penetration into ballistic gelatin, International Journal of Impact Engineering (2016), http://dx.doi.org/doi: 10.1016/j.ijimpeng.2016.02.003. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.
1
A new motion model of rifle bullet penetration into ballistic gelatin
2
LIU Susu，XU Cheng, WEN Yaoke, ZHANG Xiaoyun
3
(School of Mechanical Engineering, Nanjing University of Science & Technology, Nanjing
4
210094,China )
5 6 7 8 9 10
Corresponding author: Xu Cheng Tel: +8613016962934
E-mail: [email protected]
Liu Susu Tel: +8613951760693
E-mail: [email protected]
Highlights
11
New frameworks for drag and lift coefficients are proposed.
12
A penetration model for rifle bullets penetrating ballistic gelatin is established.
13 14 15
The motion model is well consistent with the penetration tests and simulated results.
16
Abstract: An accurate description of the motion of bullets in ballistic gelatin
17
penetration can only be given if a corresponding mathematical model is derived. In
18
this paper, change of the effective wetted area of the bullet is studied well with the
19
increasing of angle of yaw in the penetration process. By introducing an area detached
20
ratio and the influence of slenderness, a novel framework is proposed for drag and lift 1
Page 1 of 33
21
coefficients. Further, a new motion model of rifle bullet is established based on the
22
new frameworks and validated by comparison with the results from experiment data
23
and FEA. The comparative analysis shows that results of the new motion model have
24
a better fit with experiment data than that of the traditional models in previous
25
literatures and the proposed framework for drag and lift coefficients is better than the
26
traditional ones in literatures by comparison with the numerical results. In addition,
27
the calculation of the new motion model are in great accordance with FEA in terms of
28
penetration depth, deflection path, yaw angle, velocity, lift force and drag force at
29
different initial conditions. Benefitting from the motion model based on the new
30
frameworks for drag and lift coefficients, the behavior of rifle bullet in gelatin
31
penetration can be characterized accurately, the prediction of the distribution of
32
energy deposited along the penetration trajectory and the potential for incapacitation
33
of rifle bullets may become possible.
34
Keyword: penetration, drag and lift coefficient, ballistic gelatin, motion model Notations , ,
positive constant
diameter of the bullet
angle of yaw
drag force
maximum angle of yaw in the reference diameter narrow channel angle of yaw in damped oscillation
the length to a diameter ratio
2
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phase ratio of length of bullet to yawing angular velocity length of ogive head yawing angular acceleration
drag force
angle of incidence
H
length of ogive head of bullet
media density
symbol of function
area detachment ratio
transverse moment of inertia
cross-sectional area of rifle bullet
length of rifle bullet
effective wetted area at
lift force tangent line at the separation
the effective wetted area of bullet
line-0,line-1 point
the maximum effective wetted area overturning moment. of bullet in narrow channel the maximum effective wetted area damping moment of bullet in second phase constant used in the Lift coefficient
increment of wetted area
lift coefficient
instantaneous penetration
drag coefficient
reference area
a constant, expressing the t
time after impact
influences specific to head-shape of
3
Page 3 of 33
the bullet an drag coefficient at
critical velocity
damping coefficient
instantaneous velocity displacement of bullet in X
symbol of function and Y coordinate velocity of bullet in X and Y overturning moment coefficient direction Acceleration of bullet in X and a detached diameter Y direction
35
1 Introduction
36
By now, ballistic gelatin (hereafter referred to as gelatin) has become popular as
37
a tissue simulant in wound ballistics [1-4] after a large number of experimental tests
38
were carried out in different countries. That is because the gelatin is homogeneous
39
and presents the same physical characteristics block after block. Besides, it is
40
transparent so that any changes inside can be recorded by high-speed movies. In
41
addition, its retarding properties are similar to those of skeletal muscle, especially 10%
42
gelatin [5-11]. Therefore, it is significant to characterize the behavior of a bullet
43
during gelatin penetration and understand the mechanisms acting between bullets and
44
bionic material.
45
In the study of ballistic penetration problems, one of major concerns is the
46
penetration resistance and its mathematical model. Allen[12] proposed a general 4
Page 4 of 33
47
model, composing of an inertia component, a viscous component and the natural
48
strength component of the target material, as follows: （1）
49 50
where
,
and
are positive constants. In some practical analysis, only the main
51
components are selected in mathematical models of penetration to simplify the
52
penetration problem. And the selection mainly depends on a critical velocity
53
an abrupt transition is believed to occur [12]. At a very low velocity
54
of material strength is greater than others and the resistance of the bullet is assumed to be
55
a constant [13]. While the major concern is the inertial component at a very high velocity
56
[14,15]. Some empirical models also consider the influence of more than one
57
component [16,17,18]. As to the gelatin penetration, Sturdivan[19] presents a general
58
mathematical model by considering viscous and inertia components jointly. Further,
59
Segletes[20] introduced a rate-based strength to bridge the gap between pure viscous
60
and pure strength-based velocity retardation models. And the mathematical model
61
used by Peters[16,17,18] is a special case of Segletes’.
, at which
, the influence
62
The yawing motion of projectiles in dense media is strikingly different from
63
free-flight yawing motion in air and has been investigated previously from the
64
perspective of both theoretical and experimental results [21-26]. Roecker [21]
65
proposed one important result, which formed the current theoretical foundation for
66
characterizing yawing behavior of projectile in dense media. Flis[22] obtained an
67
analytical solution to the nonlinear form of the governing equation. It can provide a 5
Page 5 of 33
68
relatively simple analytical form for Roecker’s numerical results and extend
69
Roecker’s analytical solution of the linear governing equation. Furthermore,
70
Weinacht[23] presented a complete set of analytical solutions for the linear and
71
nonlinear yaw growth of a projectile impacting and traversing dense media.
72
Mo [27] presented a surface pressure model to predict the translational and
73
yawing motion of rifle bullets in gelatin penetration. The resultant force and moment
74
on a projectile were achieved through numerical integration of surface pressure, and
75
the trace of a projectile was calculated by solving spatial motion equations deduced
76
from the mass center motion equations and Euler equations. K. LIU [28] established a
77
two-dimensional motion model and investigated the model parameters of rifle bullets
78
penetrating gelatin very well [29]. Compared with several useful penetration formulas
79
in previous work, no or less research has paid attention to the relationship between the
80
lift and drag coefficients exerted on a bullet and the angle of yaw in the gelatin
81
penetration. Roecker[25] gave the trend of drag coefficient
82
angle of yaw, while Sellier [26] derived the relationship based on external ballistics,
83
as follows,
with the varying of the
（2）
84 85
where
86
function of the relationship between the length of projectile and the caliber. However,
87
Sellier [26] made the remarks that Eq.(2) are only valid for small angles of yaw up to
88
is a constant within the observed range of velocities and
, and the integration of the differential equation above the value of
is a linear
never
6
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89
gives a real value. To the best of our knowledge, it is very difficult to find the
90
available correlation to describe the lift coefficient. The closest useful assumption is
91
the formula for rigid gyro-stabilized projectile in [30], which is defined as :
92
(3)
93
where
is a constant. Despite of limitations of Eq.(2) and little verification of Eq.(3)
94
in gelatin penetration, a motion model was established by K. Liu[28] based on Eq.(2)
95
and Eq.(3). The comparison of calculated results of K. Liu’s model [28] is
96
unsatisfactory with his experiment data.
97
In order to accurately predict the translational and yawing motion of rifle bullets
98
in gelatin penetration, we theoretically study the changing of the effective wetted area
99
of rifle bullets in the penetration progress and propose new frameworks for drag and
100
lift coefficients. Further, a new motion model for bullet penetrating gelatin is
101
established based on the frameworks and validated by comparison with the
102
experiment data and the previous model in Ref.[28]. The calculated results of new
103
frameworks for drag and lift coefficients are validated by simulated results and
104
compared with the older models found in Sellier [26] and Nestor [30]. The
105
comparative analysis is also made between results of new motion model and
106
simulated results in terms of penetration depth, deflection path, yaw angle, velocity,
107
lift force and drag force at different initial conditions. In our proposed motion model,
108
it is assumed that the bullet is not broken and its gravity and rotation are ignored in
109
the process of penetration. Because of the high velocity of rifle bullets, the influence 7
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110
of inertial component of resistance is only concerned. Fit parameters used in the
111
motion model are determined from one case of FEA simulation results.
112
2 Theory Basis
113
2.1 Yaw Motion
114
With the increasing of angle of yaw, the flow lies increasingly closer to one side
115
of the projectile surface. Thus, the calculation of the yawing motion of a projectile in
116
a dense medium can be applied for the whole penetration channel.
117 118
A functional form for the relationship of overturning moment to instantaneous yaw is derived by Roecker[21] as follows,
119 120
(4)
Where
121
[23]
122
where
is the overturning moment of inertia of the projectile,
123
diameter,
124
Length of the bullet is proportional to its diameter in the design of bullets. So it is
125
reasonable to replace
126
area
is the reference area,
of the bullet, then
is the overturning moment coefficient.
with the bullet length
with cross-sectional
(5)
The general solution of Eq.(4) can be simplified to
129 130
and
can be expressed as
127 128
is the reference
[22]
(6)
which will be used in the next section. Combined with Eq.(5), Equations (4) is 8
Page 8 of 33
131
expressed as functions of time ,
132
(7)
133
which is the overturning moment. Because of the rotation about its center of gravity in
134
the gelatin penetration, the base of the bullet (or the tip, if the bullet rotates in the
135
opposite direction) is forced into the medium at high speed. The resultant force is
136
applied away from the center of gravity, thus producing a torque that tends to oppose
137
the yaw, so the rotational motion around the transverse axis undergoes a damping
138
torque. A dimensionless damping coefficient
139
to enlarge Eq.(7)[26],
is introduced as a term of damping
140
(8)
141
It is considered to be appropriate for Eq.(8) to characterize the yaw motion of the rifle
142
bullet in gelatin penetration.
143
2.2 Drag Coefficient
144
It is well known that if full metal-jacketed and solid rifle bullets hit a soft
145
medium in stable flight with an initial angle of incidence, they cause a wound channel
146
which can be divided into three clearly-distinguishable phases, including narrow
147
channel phase, rapid overturning phase and damped oscillation phase. The drag force
148
experienced by bullets acts in the opposite direction of the bullet movement and can
149
be characterized by the drag coefficient [24], （9）
150 151
where
is the drag force, and
is the cross-sectional area of the projectile. 9
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152
In order to achieve a more realistic resistance coefficient and acting pressure, the
153
changing of the effective wetted area
of rifle bullet, which is projected onto a
154
plane perpendicular to the velocity vector, is introduced with the increasing of angle
155
of yaw. Moreover, we describe the influence of slenderness with the changing of yaw
156
angle to keep the equations manageable. The drag coefficient can be expressed as
157 158 159 160
(10)
where
is a constant, expressing the influences specific to head-shape of the bullet.
In the gelatin penetration of rifle bullets, the motion model of rifle bullets is complex with an initial angle of incidence
161 162
to. If Fig.1(a), and a drag coefficient
, where the results with
is adopted, a separation line can be seen in at
163
can be derived based on Eq.(10), (11)
,
164
where
165
reasonably assumed that
166
replacing
is an area detached ratio at
, and
is the caliber of the rifle bullet. It is
is constant within the observed range of velocity of rifle bullets. By
in Eq.(10) , the drag coefficient can be written as
167 168
(12)
2.2.1 Narrow Channel
169
(line-0: tangent line at the separation point when
170
point. Bold lines stands for the surface area in contact with gelatin or the effective wetted area.
171
Shadow area: the effective wetted area; )
172
line-1: tangent line at the separation
In the narrow channel, only a small part of the bullet's tip is in contact with the 10
Page 10 of 33
173
medium at the early stage. A large percentage of the bullet's surface is not in contact
174
with the medium and is hence subjected to virtually no forces, as shown in Fig1(b).
175
As the angle of yaw increases, the separation line moves progressively from the tip to
176
the end of the bullet, the surface in contact with gelatin increases, as shown in Fig1(b,
177
c, d). It leads to an increase in force and hence to an increase in the moment applied.
178
As a result, the angle of yaw increases and causes the overturning moment to increase
179
still further. It will come to an end until the entire surface of one side of the bullet is in
180
contact with the medium, marking the end of the narrow channel when
181
.
182 183
( : length of the bullet rifle;
If
，
(13)
:length of ogive head)
, shown in fig.2, Eq.(13) can be expressed as
184
(14)
185
At the end of the narrow channel, the maximum effective wetted area, indicated in
186
fig.1 (d), is approximately equal to
187 188
(15)
When
, Eq.(15) can be transformed into
189
(16)
190
Where the length to a diameter ratio
191
Eq.(16) can be simplified into
for rifle bullets, so
192 193
.
(17)
It is known that even if
is close to zero, the effective wetted area should be 11
Page 11 of 33
194
bigger than
195
expressed as
as show in Fig.1(b). Then,
in the narrow channel can be
196
(18)
197
where
is an increment with the increasing of yaw angle. It has been proved that
198
the drag coefficient is an even function of the yaw angle [26]. And the angle of yaw in
199
the narrow channel is small enough to suppose that
200
Then the effective wetted area in narrow channel can be calculated as
is proportional to
201 202
.
(19)
2.2.2 Rapid Overturning Phase
203
The second phase begins with the bullet yawing rapidly. The bullet undergoes a
204
very rapid deceleration. While the bullet continues to yaw beyond the point at which
205
it is perpendicular to its own direction of movement, the damping torque increasing
206
with the angle will be equal to the overturning moment at a specific point for the first
207
time, that is
208
equation related to the specific angle of yaw can be derived from Eq.(8) as follows,
, thereby marking the end of rapid overturning phase. The
209 210
(20)
By introducing Equations (5) and (6) above, Equation (20) is integrated to （21）
211 212
The solution is （22）
213
12
Page 12 of 33
214
which generally lies somewhere between 90°and 180°for rifle bullets.
215 216 217
When the line
is perpendicular to its own direction of movement, the
effective wetted area indicated in Fig.3(b) is approximately equal to
218 219
,
(23)
and the angle of yaw is
220
.
221
is supposed to be the maximum effective wetted area in the whole progress of
222
penetration. The bullet will continue to yaw beyond the point. Because of the abrupt
223
end of the bullet and a temporary cavity formed by the medium flowing away from
224
the bullet, the bottom is in no contact with the medium at certain angle of yaw, as
225
shown in Fig.3(c). It is supposed that the effective wetted area is symmetric about
226
when the slenderness is high and the change of
227
phase is proportional to
228
phase can be derived,
229
in rapid overturning
. Then the effective wetted area in rapid overturning
（24）
=
230
When related to the maximum effective wetted area and combined with
231
Equations (17), (18) and (24), it is reasonably assumed that the effective wetted area
232
in both narrow channel and rapid overturning phase can be expressed by one equation
233
as follows:
234
（25）
= 13
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235
Substituting Eq.(25) into Eq.(12) yields
236
(26）
237
where
238
2.2.3Damped oscillation phase
239
In the third phase, the bullet yaws mainly under the damping force and rocks
240
back and forth about its center of gravity with the tail forward. It is equivalent to the
241
gelatin penetration of projectiles with flat noses. Therefore the effective wetted area
242
will fluctuate around a fixed number. We chose the value of
243
second phase as the constant for the third phase，namely the point
244
245
. Then the drag coefficient in the third phase can be derived from
Equations (26) as follows,
246
(27)
247
when
248
where
249
2.3 Lift Coefficient
250
at the end of the
Lift force on a bullet acts in the direction perpendicular to the bullet velocity and 14
Page 14 of 33
251
can be characterized by lift coefficient,
252
(28)
253
where
is the actual lift force. Considering the effect of the slenderness of the rifle
254
bullet and that
255
the lift coefficient,
is an odd function of yaw angle, we propose a new equation for
256
(29)
257
where
258
3 Motion Modeling
259 260 261
and
: drag force.
are constant.
: lift force.
: overturning moment.
: damping moment.
: angle of yaw.
With Eq.(9) and Eq.(28), the drag and lift force model in the absolute coordinate system (seen in Fig.4) are transformed into
262
(30)
263
(31)
264
Combined with Equations (8), (30) and(31), a new motion model of rifle bullet in
265
gelatin penetration can be established as follows:
266
(32)
267 268
where
269
,
15
Page 15 of 33
270
(33)
271 272
,
,
.
Eq.(33) is integrated from Eq.(26) and Eq.(27).
273
Comparatively, the K. LIU’s motion model [28] is based on the older models for
274
the drag and lift coefficients referred to Sellier [26] and Nestor [30], Eq.(2) and Eq.(3),
275
while our new dynamic model in Equations (32) is based on the new proposed
276
frameworks for drag and lift coefficients in this paper.
277
4 Results and discussion
278
A Runge-Kutta solution of Eq.(32) is implemented to validate the proposed
279
equations for drag and lift coefficients and the new motion model based on the
280
experimental data and numeric results of FEA. The fit parameters in the new motion
281
model are identified by the weighted least square method based on one case of
282
simulation results, as listed in Table 1.
283
4.1 Comparison with Experiment results
284
4.1.1 Experiment set-up
285
In the experiments, 10% gelatin blocks (
) stored at 4
in
286
a refrigerator for 24h were impacted by 7.62mm rifle bullets by using a rifle fixed
287
with its muzzle 15m from the front face of the gelatin. The mass density of gelatin is
288
and parameters of rifle bullets were listed in Table.2. The speed of the 16
Page 16 of 33
289
bullet just before impacting the gelatin was measured with a double base optical
290
detector. Phantom V120 high-speed cameras with 16000 fps were used to record
291
motion of the bullets.
292
4.1.2 Comparison with Experiment results
293
For the 7.62mm rifle bullet at 636m/s with an initial yaw angle of
, the
294
process of gelatin penetration is illustrated in Fig.5. In combination with the results
295
plotted in Fig.6 and Tab.3, it can be concluded that no obvious instability of the bullet
296
is found from 0
297
gelatin, approximately 92.3mm, and the path of bullet deflects little. Furthermore, the
298
angle of yaw is increased from
299
phase is from 143.5us to 366.4us. The bullet becomes obviously unstable and its
300
yawing and deflection are accelerated during this time. In the third stage after 366.4us,
301
however, the yaw angle and deflection path continue to increase with a significant
302
deceleration. The comparison between the results of the new motion model and
303
experimental results indicates that conclusion of the new motion model based on the
304
new proposed frameworks for drag and lift coefficients is in great accordance with
305
experiment data in terms of the penetration depth, yaw angle and deflection path.
306
4.2 Comparison with Simulated Results of FEA
to 143.5 . A long narrow channel is formed in the front of the
to 9.5°. Time cost for the rapid overturning
307
Considering the limitation of experimental conditions and measuring error of
308
high-speed photography, it is difficult to get enough and accurate experiment data. It
309
is necessary to validate the accuracy of the new theoretical model by using FEA 17
Page 17 of 33
310
(Finite Element Analysis). A finite element model of 7.62mm rifle bullet penetrating
311
gelatin has been validated in the previous work [31]. So the critical value of the
312
element erosion strain was chosen as 0.9 here, the effect of which has been studied by
313
Wen [32]. Here we use the same finite element model to simulate the 10% gelatin
314
penetration of 7.62mm rifle bullet with an initial incidence angle of
315
velocity of 500m/s, 625m/s and 700m/s, respectively.
316
4.2.1 Drag coefficient
and impact
317
The drag force and velocity of projectiles are obtained from simulated results of
318
FEA directly and the drag coefficient of FEA are determined based on the Eq.(9).
319
Comparative study in drag coefficients is shown in Fig.7 among results of the new
320
proposed Eq.(33), FEA and Sellier's Eq.(2) with
321
obviously shown that impact velocity has no effect on the drag coefficient and the
322
drag coefficient increases with yaw angle, reaches the highest at an angle of yaw
323
between
324
calculation of the new proposed equation in the present paper is better agreement with
325
simulated results from the FEA. The reason for the oscillations of drag coefficients of
326
FEA is that finite elements between the bullet and gelatin with the rotation of rifle
327
bullets is contacted non-uniformly at high angles of yaw.
328
4.2.2 Lift coefficient
and
and
. It is
, and then decreases. Compared with the Sellier's solution,
329
Similar to the drag coefficient above, the lift coefficient of FEA can be
330
determined from the simulated results based on Eq.(28). A comparative analysis of lift 18
Page 18 of 33
331
coefficients is made among the results of the new proposed Eq.(29), FEA and Eq.(3)
332
given by Nestor[27] with
333
Fig.8 at different initial conditions. The results indicate that calculated results of the
334
new proposed Eq.(29) for lift coefficient have a better fit with that of FEA compared
335
with Nestor's solution. The reason for the oscillations of lift coefficients of FEA is
336
similar to that of drag coefficients at high angles of yaw.
337
4.2.3 Motion Model
. The lift coefficient behavior is illustrated in
338
Fig.9 exhibits the comparative analysis for the results of FEA and the new
339
motion model within the penetration depth of 30cm. It can be seen that calculated
340
results of the new motion model coincides well with the simulated results of FEA in
341
terms of penetration depth, deflection path, yaw angle, velocity, lift force and drag
342
force at different initial conditions. In Figure 9(e) and Figure 9(f), the drag force of
343
motion model is derived by using Eq.(9) and Eq.(33), and the lift force of motion
344
model is derived by using Eq.(28) and Eq.(29). While the drag force and lift force of
345
FEA are obtained directly from the simulated results. For lift force, it is indicated that
346
initially the two sets of results are very close to each other. However, after 220mm the
347
lift force in FEA decreases to a value near to zero more rapidly than that in our motion
348
model. But the difference for the impact velocity doesn’t exist.
349
5 Conclusion
350
The interaction of bullet with ballistic gelatin is a complex phenomenon.
351
Slenderness and head shape have a great influence on the motion of bullet. In order to 19
Page 19 of 33
352
characterize the behavior of rifle bullets in the gelatin penetration, the changing of the
353
effective wetted area of bullet is studied in the process of penetration and new
354
frameworks are proposed for drag and lift coefficients by introducing an area
355
detachment ratio and the effect of slenderness.
356
The motion model of rifle bullet based on the new frameworks is valid in the
357
ranges of penetration depth from 0cm to 30cm. The bullet's state of movement
358
calculated by our motion model at each point is almost consistent with the observed in
359
experiment and FEA in terms of penetration depth, deflection path and yaw angle. In
360
addition, the velocity, lift force and drag force according to the motion model are in
361
line with the simulated results of FEA. For drag and lift coefficients, new frameworks
362
can provide better predictions than was found in the literature. This may be significant
363
for research in wound ballistics.
364
Acknowledgments
365
This project are supported by National Natural Science Foundation of China
366
(Grant No.51575279 and No.11502119)
367
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[12] Allen WA, Mayfield EB, Morrison HL. Dynamics of a projectile penetrating sand. Journal of Applied Physics, 1957, 28(3):370-376.
396
[13] Hill R. Cavitation and the influence of headshape in attack of thick targets
397
by non-deforming projectiles†. Journal of the Mechanics & Physics of Solids, 1980,
398
28(s 5–6):249-263.
399 400 401 402 403 404
[14] Rosenberg Z, Dekel E., The deep penetration of concrete targets by rigid rods-revisited. International Journal of Protective Structures, 2010.1(1): 125-144 [15] Shan Y, Wu H, Huang F, Li J. On the Inertia term of projectile's penetration resistance. Advances in Materials Science & Engineering, 2013. [16] Peters CE, A mathematical-physical model of wound ballistics. Journal of Trauma, 1990,6(2):308-318.
405
[17] Peters CE, Dynamics of high-speed projectiles in tissue stimulants and
406
living tissue. Presentation given in the UTSI workshop on wound ballistics.
407
Tullahoma, Tennessee: The University of Tennessee Space Institute. July,
408
1986,p:1-69.
409
[18] Peters CE, Sebourn CL and Crowder HL, Wound ballistics of unstable
410
projectiles. Part I: projectile yaw growth and retardation. Journal of Trauma, 1996,
411
40(3S).
412 413 414
[19] Sturdivan LM, A mathematical model of penetration of chunky projectiles in a gelatin tissue simulant. Maryland, 1978. [20] Segletes SB, Modeling the penetration behavior of rigid spheres into 22
Page 22 of 33
415 416 417
ballistic gelatin. 2008, ART-TR-4393. [21] Roecker ET, Ricchiazzi AJ. Stability of penetrators in dense fluids. International Journal of Engineering Science, 1978, 16(11):917–920.
418
[22] Flis WJ. A Note on the Roecker-Ricchiazzi model of penetrator trajectory
419
instability, Proceedings of the 22nd International Symposium on Ballistics, Vol. 2,
420
DEStech Publications, Inc., Lancaster, PA, pp. 1287-1294, 2005.
421
[23] Weinacht P, Cooper GR. Analytical solutions for the linear and nonlinear
422
yawing motion in dense media. Proceedings of the 23nd International Symposium on
423
Ballistics, Vol. 2,pp:1339-1346, 2007
424 425 426 427 428 429 430 431 432 433 434 435
[24] Kneubuehl BP. Wound ballistics: basics and application. 4th ed. Berlin: Springer-Verlag, 2011 [25] Roecker ET. On yawing motion, velocity decay, and path deflection of projectiles penetrating cavitating media. 1986. [26] Sellier KG, Kneubuehl BP, Rufer R. Wound ballistics and the scientific background. Amsterdam: Elsevier,1994. [27] Mo GL, Wu ZL, Feng J, Liu K. Surface Pressure model of Rifle Bullets Penetrating into Ballistic Gelatin. Acta Armamentarii, 2014, 35(2):164-169. [28] Liu K, Wu ZL, Xu WH, Mo GL. A motion model for bullet penetrating gelatin. Explosion and Shock Waves, 2012,32(6): 616-622 [29] Liu K, Wu ZL, Xu WH, Mo GL. Research on Model Parameters of Bullet Penetrating Gelatin. Chinese Journal of High Pressure Physics, 2013, 27(5):677-684. 23
Page 23 of 33
436
[30] Nestor N. Theoretical study of the motion of a rigid gyro-stabilized
437
projectile into homogeneous dense media. Proceedings of the 24th International
438
Symposium of Ballistics, New Orlaeans,2008.
439
[31] Liu SS, Xu C, Chen AJ, Li HK. Effect of rifle bullet parameters on the
440
penetration into ballistic gelatin. Journal of Beijing Institute of Technology,
441
2015.24(4):487-493.
442
[32] Wen YK, Xu C, Wang HS, CHEN AJ, Batra RC. Impact of steel spheres on
443
ballistic gelatin at moderate velocity. International Journal of Impact Engineering,
444
2013,62(4):142-151.
445
24
Page 24 of 33
line-0
y’
d0
(x0,d0/2） 0
xz
xw
x1
x0
d
xq
v
x’
(a)
line-0
y’ ψ
0
x1
xz
xw
x0
(x1,-d1/2)
d
line-1
xq
v
x’
(b)
line-0
y’
0 d
x0
xz
xw x1
ψ v
xq
x’
(c)
(x1,-d1/2)
line-0
line-1
y’
ψ
d1 xw 0 x1
x’
xq
x0 v
xz
(d)
d
line-1
(x1,-d1/2)
line-0 y
446 447
0
x
Fig.1the motion of rifle bullet in gelatin penetration
448
H l0
449 450
Fig.2 structure of the rifle bullet
451 25
Page 25 of 33
x’ y ’
M1 ψ o
v (a) x’
M2
M1
A
lAB
ψ o
y ’
v
(b) B x’
M1
M2
ψ
M2
o y
o
v
y ’
x
(c)
452 453
Fig.3 the motion of rifle bullet in rapid overturning phase in gelatin penetration.
454
moment.
: overturning
: damping moment.
455 y M2
D
o ψ
x v
L M1
456 457
Fig.4 Force acting on the rifle bullet.
458
26
Page 26 of 33
459 460
Fig.5 Progress of 7.62mm penetrating gelatin
461 462
Fig.6(a) Penetration depth versus time
463 464
Fig.6(b) Yaw angle versus penetration depth
27
Page 27 of 33
465 466
Fig.6(c) Path deflection in y direction versus penetration depth
467
Fig.6 Diagrams of 7.62mm rifle bullet in 10% gelatin penetration with initial conditions. Velocity:
468
636m/s. Angle of incidence:
. Initial angle of yaw:
.
469
470 471
(a) drag coefficient vs. angle of yaw at v = 500m/s
28
Page 28 of 33
472 473
(b) drag coefficient vs. angle of yaw at v = 625m/s
474 475
(c) drag coefficient vs. angle of yaw at v = 700m/s
476
Fig.7 drag coefficient vs. angle of yaw different impact velocity according to FEA, new proposed
477
Eq.(33) and Sellier's Eq.(2).
478
29
Page 29 of 33
479 480
(a) lift coefficient vs. angle of yaw at v = 500m/s
481
482 483
(b) lift coefficient vs. angle of yaw at v = 625m/s
484 485
(c) lift coefficient vs. angle of yaw at v = 700m/s 30
Page 30 of 33
486
Fig.8 lift coefficient vs. angle of yaw for different impact velocity according to FEA, new
487
proposed Eq.(29) and Nestor's Eq.(3).
488
489 490
(a) Penetration depth versus Time
491 492
(b)Path deflection versus Penetration depth
493 494
(c) Yaw angle versus Penetration depth
31
Page 31 of 33
495 496
(d) Penetration velocity versus Penetration depth
497 498
(e) Drag force versus Penetration depth
499 500
(f) Lift force versus Penetration depth
501
Fig.9 Comparative analysis of numerical results and theoretical results in the 10% gelatin
502 503
penetration of 7.62mm rifle bullet with an initial incidence angle of
and the impact velocity of
500m/s, 600m/s and 700m/s
504 505
32
Page 32 of 33
506
Table.1 Fit parameters for present motion model Caliber(mm) 7.62
0.4025
507
508
0.14
0.58
0.1713
0.58
1.78
Table.2 Structure parameters of rifle bullets Caliber(mm)
m/g
d/mm
7.62
7.90
7.92
Jq/(
)
l/mm
H/mm
26.8
16
366.9
Tab.3 computed results of the new motion model at typical moment
Typical moment
Instable moment
Penetration
Deflection
Angle of
depth(mm)
path(mm)
yaw(°)
143.5
92.3
-0.63
-9.5
366.4
212.3
-13.5
-111.2
t( )
Beginning of the damped oscillation phase
509
33
Page 33 of 33
S0734-743X(16)30031-8 http://dx.doi.org/doi: 10.1016/j.ijimpeng.2016.02.003 IE 2647
To appear in:
International Journal of Impact Engineering
Received date: Revised date: Accepted date:
25-9-2015 29-1-2016 8-2-2016
Please cite this article as: Liu Susu, Xu Cheng, Wen Yaoke, Zhang Xiaoyun, A new motion model of rifle bullet penetration into ballistic gelatin, International Journal of Impact Engineering (2016), http://dx.doi.org/doi: 10.1016/j.ijimpeng.2016.02.003. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.
1
A new motion model of rifle bullet penetration into ballistic gelatin
2
LIU Susu，XU Cheng, WEN Yaoke, ZHANG Xiaoyun
3
(School of Mechanical Engineering, Nanjing University of Science & Technology, Nanjing
4
210094,China )
5 6 7 8 9 10
Corresponding author: Xu Cheng Tel: +8613016962934
E-mail: [email protected]
Liu Susu Tel: +8613951760693
E-mail: [email protected]
Highlights
11
New frameworks for drag and lift coefficients are proposed.
12
A penetration model for rifle bullets penetrating ballistic gelatin is established.
13 14 15
The motion model is well consistent with the penetration tests and simulated results.
16
Abstract: An accurate description of the motion of bullets in ballistic gelatin
17
penetration can only be given if a corresponding mathematical model is derived. In
18
this paper, change of the effective wetted area of the bullet is studied well with the
19
increasing of angle of yaw in the penetration process. By introducing an area detached
20
ratio and the influence of slenderness, a novel framework is proposed for drag and lift 1
Page 1 of 33
21
coefficients. Further, a new motion model of rifle bullet is established based on the
22
new frameworks and validated by comparison with the results from experiment data
23
and FEA. The comparative analysis shows that results of the new motion model have
24
a better fit with experiment data than that of the traditional models in previous
25
literatures and the proposed framework for drag and lift coefficients is better than the
26
traditional ones in literatures by comparison with the numerical results. In addition,
27
the calculation of the new motion model are in great accordance with FEA in terms of
28
penetration depth, deflection path, yaw angle, velocity, lift force and drag force at
29
different initial conditions. Benefitting from the motion model based on the new
30
frameworks for drag and lift coefficients, the behavior of rifle bullet in gelatin
31
penetration can be characterized accurately, the prediction of the distribution of
32
energy deposited along the penetration trajectory and the potential for incapacitation
33
of rifle bullets may become possible.
34
Keyword: penetration, drag and lift coefficient, ballistic gelatin, motion model Notations , ,
positive constant
diameter of the bullet
angle of yaw
drag force
maximum angle of yaw in the reference diameter narrow channel angle of yaw in damped oscillation
the length to a diameter ratio
2
Page 2 of 33
phase ratio of length of bullet to yawing angular velocity length of ogive head yawing angular acceleration
drag force
angle of incidence
H
length of ogive head of bullet
media density
symbol of function
area detachment ratio
transverse moment of inertia
cross-sectional area of rifle bullet
length of rifle bullet
effective wetted area at
lift force tangent line at the separation
the effective wetted area of bullet
line-0,line-1 point
the maximum effective wetted area overturning moment. of bullet in narrow channel the maximum effective wetted area damping moment of bullet in second phase constant used in the Lift coefficient
increment of wetted area
lift coefficient
instantaneous penetration
drag coefficient
reference area
a constant, expressing the t
time after impact
influences specific to head-shape of
3
Page 3 of 33
the bullet an drag coefficient at
critical velocity
damping coefficient
instantaneous velocity displacement of bullet in X
symbol of function and Y coordinate velocity of bullet in X and Y overturning moment coefficient direction Acceleration of bullet in X and a detached diameter Y direction
35
1 Introduction
36
By now, ballistic gelatin (hereafter referred to as gelatin) has become popular as
37
a tissue simulant in wound ballistics [1-4] after a large number of experimental tests
38
were carried out in different countries. That is because the gelatin is homogeneous
39
and presents the same physical characteristics block after block. Besides, it is
40
transparent so that any changes inside can be recorded by high-speed movies. In
41
addition, its retarding properties are similar to those of skeletal muscle, especially 10%
42
gelatin [5-11]. Therefore, it is significant to characterize the behavior of a bullet
43
during gelatin penetration and understand the mechanisms acting between bullets and
44
bionic material.
45
In the study of ballistic penetration problems, one of major concerns is the
46
penetration resistance and its mathematical model. Allen[12] proposed a general 4
Page 4 of 33
47
model, composing of an inertia component, a viscous component and the natural
48
strength component of the target material, as follows: （1）
49 50
where
,
and
are positive constants. In some practical analysis, only the main
51
components are selected in mathematical models of penetration to simplify the
52
penetration problem. And the selection mainly depends on a critical velocity
53
an abrupt transition is believed to occur [12]. At a very low velocity
54
of material strength is greater than others and the resistance of the bullet is assumed to be
55
a constant [13]. While the major concern is the inertial component at a very high velocity
56
[14,15]. Some empirical models also consider the influence of more than one
57
component [16,17,18]. As to the gelatin penetration, Sturdivan[19] presents a general
58
mathematical model by considering viscous and inertia components jointly. Further,
59
Segletes[20] introduced a rate-based strength to bridge the gap between pure viscous
60
and pure strength-based velocity retardation models. And the mathematical model
61
used by Peters[16,17,18] is a special case of Segletes’.
, at which
, the influence
62
The yawing motion of projectiles in dense media is strikingly different from
63
free-flight yawing motion in air and has been investigated previously from the
64
perspective of both theoretical and experimental results [21-26]. Roecker [21]
65
proposed one important result, which formed the current theoretical foundation for
66
characterizing yawing behavior of projectile in dense media. Flis[22] obtained an
67
analytical solution to the nonlinear form of the governing equation. It can provide a 5
Page 5 of 33
68
relatively simple analytical form for Roecker’s numerical results and extend
69
Roecker’s analytical solution of the linear governing equation. Furthermore,
70
Weinacht[23] presented a complete set of analytical solutions for the linear and
71
nonlinear yaw growth of a projectile impacting and traversing dense media.
72
Mo [27] presented a surface pressure model to predict the translational and
73
yawing motion of rifle bullets in gelatin penetration. The resultant force and moment
74
on a projectile were achieved through numerical integration of surface pressure, and
75
the trace of a projectile was calculated by solving spatial motion equations deduced
76
from the mass center motion equations and Euler equations. K. LIU [28] established a
77
two-dimensional motion model and investigated the model parameters of rifle bullets
78
penetrating gelatin very well [29]. Compared with several useful penetration formulas
79
in previous work, no or less research has paid attention to the relationship between the
80
lift and drag coefficients exerted on a bullet and the angle of yaw in the gelatin
81
penetration. Roecker[25] gave the trend of drag coefficient
82
angle of yaw, while Sellier [26] derived the relationship based on external ballistics,
83
as follows,
with the varying of the
（2）
84 85
where
86
function of the relationship between the length of projectile and the caliber. However,
87
Sellier [26] made the remarks that Eq.(2) are only valid for small angles of yaw up to
88
is a constant within the observed range of velocities and
, and the integration of the differential equation above the value of
is a linear
never
6
Page 6 of 33
89
gives a real value. To the best of our knowledge, it is very difficult to find the
90
available correlation to describe the lift coefficient. The closest useful assumption is
91
the formula for rigid gyro-stabilized projectile in [30], which is defined as :
92
(3)
93
where
is a constant. Despite of limitations of Eq.(2) and little verification of Eq.(3)
94
in gelatin penetration, a motion model was established by K. Liu[28] based on Eq.(2)
95
and Eq.(3). The comparison of calculated results of K. Liu’s model [28] is
96
unsatisfactory with his experiment data.
97
In order to accurately predict the translational and yawing motion of rifle bullets
98
in gelatin penetration, we theoretically study the changing of the effective wetted area
99
of rifle bullets in the penetration progress and propose new frameworks for drag and
100
lift coefficients. Further, a new motion model for bullet penetrating gelatin is
101
established based on the frameworks and validated by comparison with the
102
experiment data and the previous model in Ref.[28]. The calculated results of new
103
frameworks for drag and lift coefficients are validated by simulated results and
104
compared with the older models found in Sellier [26] and Nestor [30]. The
105
comparative analysis is also made between results of new motion model and
106
simulated results in terms of penetration depth, deflection path, yaw angle, velocity,
107
lift force and drag force at different initial conditions. In our proposed motion model,
108
it is assumed that the bullet is not broken and its gravity and rotation are ignored in
109
the process of penetration. Because of the high velocity of rifle bullets, the influence 7
Page 7 of 33
110
of inertial component of resistance is only concerned. Fit parameters used in the
111
motion model are determined from one case of FEA simulation results.
112
2 Theory Basis
113
2.1 Yaw Motion
114
With the increasing of angle of yaw, the flow lies increasingly closer to one side
115
of the projectile surface. Thus, the calculation of the yawing motion of a projectile in
116
a dense medium can be applied for the whole penetration channel.
117 118
A functional form for the relationship of overturning moment to instantaneous yaw is derived by Roecker[21] as follows,
119 120
(4)
Where
121
[23]
122
where
is the overturning moment of inertia of the projectile,
123
diameter,
124
Length of the bullet is proportional to its diameter in the design of bullets. So it is
125
reasonable to replace
126
area
is the reference area,
of the bullet, then
is the overturning moment coefficient.
with the bullet length
with cross-sectional
(5)
The general solution of Eq.(4) can be simplified to
129 130
and
can be expressed as
127 128
is the reference
[22]
(6)
which will be used in the next section. Combined with Eq.(5), Equations (4) is 8
Page 8 of 33
131
expressed as functions of time ,
132
(7)
133
which is the overturning moment. Because of the rotation about its center of gravity in
134
the gelatin penetration, the base of the bullet (or the tip, if the bullet rotates in the
135
opposite direction) is forced into the medium at high speed. The resultant force is
136
applied away from the center of gravity, thus producing a torque that tends to oppose
137
the yaw, so the rotational motion around the transverse axis undergoes a damping
138
torque. A dimensionless damping coefficient
139
to enlarge Eq.(7)[26],
is introduced as a term of damping
140
(8)
141
It is considered to be appropriate for Eq.(8) to characterize the yaw motion of the rifle
142
bullet in gelatin penetration.
143
2.2 Drag Coefficient
144
It is well known that if full metal-jacketed and solid rifle bullets hit a soft
145
medium in stable flight with an initial angle of incidence, they cause a wound channel
146
which can be divided into three clearly-distinguishable phases, including narrow
147
channel phase, rapid overturning phase and damped oscillation phase. The drag force
148
experienced by bullets acts in the opposite direction of the bullet movement and can
149
be characterized by the drag coefficient [24], （9）
150 151
where
is the drag force, and
is the cross-sectional area of the projectile. 9
Page 9 of 33
152
In order to achieve a more realistic resistance coefficient and acting pressure, the
153
changing of the effective wetted area
of rifle bullet, which is projected onto a
154
plane perpendicular to the velocity vector, is introduced with the increasing of angle
155
of yaw. Moreover, we describe the influence of slenderness with the changing of yaw
156
angle to keep the equations manageable. The drag coefficient can be expressed as
157 158 159 160
(10)
where
is a constant, expressing the influences specific to head-shape of the bullet.
In the gelatin penetration of rifle bullets, the motion model of rifle bullets is complex with an initial angle of incidence
161 162
to. If Fig.1(a), and a drag coefficient
, where the results with
is adopted, a separation line can be seen in at
163
can be derived based on Eq.(10), (11)
,
164
where
165
reasonably assumed that
166
replacing
is an area detached ratio at
, and
is the caliber of the rifle bullet. It is
is constant within the observed range of velocity of rifle bullets. By
in Eq.(10) , the drag coefficient can be written as
167 168
(12)
2.2.1 Narrow Channel
169
(line-0: tangent line at the separation point when
170
point. Bold lines stands for the surface area in contact with gelatin or the effective wetted area.
171
Shadow area: the effective wetted area; )
172
line-1: tangent line at the separation
In the narrow channel, only a small part of the bullet's tip is in contact with the 10
Page 10 of 33
173
medium at the early stage. A large percentage of the bullet's surface is not in contact
174
with the medium and is hence subjected to virtually no forces, as shown in Fig1(b).
175
As the angle of yaw increases, the separation line moves progressively from the tip to
176
the end of the bullet, the surface in contact with gelatin increases, as shown in Fig1(b,
177
c, d). It leads to an increase in force and hence to an increase in the moment applied.
178
As a result, the angle of yaw increases and causes the overturning moment to increase
179
still further. It will come to an end until the entire surface of one side of the bullet is in
180
contact with the medium, marking the end of the narrow channel when
181
.
182 183
( : length of the bullet rifle;
If
，
(13)
:length of ogive head)
, shown in fig.2, Eq.(13) can be expressed as
184
(14)
185
At the end of the narrow channel, the maximum effective wetted area, indicated in
186
fig.1 (d), is approximately equal to
187 188
(15)
When
, Eq.(15) can be transformed into
189
(16)
190
Where the length to a diameter ratio
191
Eq.(16) can be simplified into
for rifle bullets, so
192 193
.
(17)
It is known that even if
is close to zero, the effective wetted area should be 11
Page 11 of 33
194
bigger than
195
expressed as
as show in Fig.1(b). Then,
in the narrow channel can be
196
(18)
197
where
is an increment with the increasing of yaw angle. It has been proved that
198
the drag coefficient is an even function of the yaw angle [26]. And the angle of yaw in
199
the narrow channel is small enough to suppose that
200
Then the effective wetted area in narrow channel can be calculated as
is proportional to
201 202
.
(19)
2.2.2 Rapid Overturning Phase
203
The second phase begins with the bullet yawing rapidly. The bullet undergoes a
204
very rapid deceleration. While the bullet continues to yaw beyond the point at which
205
it is perpendicular to its own direction of movement, the damping torque increasing
206
with the angle will be equal to the overturning moment at a specific point for the first
207
time, that is
208
equation related to the specific angle of yaw can be derived from Eq.(8) as follows,
, thereby marking the end of rapid overturning phase. The
209 210
(20)
By introducing Equations (5) and (6) above, Equation (20) is integrated to （21）
211 212
The solution is （22）
213
12
Page 12 of 33
214
which generally lies somewhere between 90°and 180°for rifle bullets.
215 216 217
When the line
is perpendicular to its own direction of movement, the
effective wetted area indicated in Fig.3(b) is approximately equal to
218 219
,
(23)
and the angle of yaw is
220
.
221
is supposed to be the maximum effective wetted area in the whole progress of
222
penetration. The bullet will continue to yaw beyond the point. Because of the abrupt
223
end of the bullet and a temporary cavity formed by the medium flowing away from
224
the bullet, the bottom is in no contact with the medium at certain angle of yaw, as
225
shown in Fig.3(c). It is supposed that the effective wetted area is symmetric about
226
when the slenderness is high and the change of
227
phase is proportional to
228
phase can be derived,
229
in rapid overturning
. Then the effective wetted area in rapid overturning
（24）
=
230
When related to the maximum effective wetted area and combined with
231
Equations (17), (18) and (24), it is reasonably assumed that the effective wetted area
232
in both narrow channel and rapid overturning phase can be expressed by one equation
233
as follows:
234
（25）
= 13
Page 13 of 33
235
Substituting Eq.(25) into Eq.(12) yields
236
(26）
237
where
238
2.2.3Damped oscillation phase
239
In the third phase, the bullet yaws mainly under the damping force and rocks
240
back and forth about its center of gravity with the tail forward. It is equivalent to the
241
gelatin penetration of projectiles with flat noses. Therefore the effective wetted area
242
will fluctuate around a fixed number. We chose the value of
243
second phase as the constant for the third phase，namely the point
244
245
. Then the drag coefficient in the third phase can be derived from
Equations (26) as follows,
246
(27)
247
when
248
where
249
2.3 Lift Coefficient
250
at the end of the
Lift force on a bullet acts in the direction perpendicular to the bullet velocity and 14
Page 14 of 33
251
can be characterized by lift coefficient,
252
(28)
253
where
is the actual lift force. Considering the effect of the slenderness of the rifle
254
bullet and that
255
the lift coefficient,
is an odd function of yaw angle, we propose a new equation for
256
(29)
257
where
258
3 Motion Modeling
259 260 261
and
: drag force.
are constant.
: lift force.
: overturning moment.
: damping moment.
: angle of yaw.
With Eq.(9) and Eq.(28), the drag and lift force model in the absolute coordinate system (seen in Fig.4) are transformed into
262
(30)
263
(31)
264
Combined with Equations (8), (30) and(31), a new motion model of rifle bullet in
265
gelatin penetration can be established as follows:
266
(32)
267 268
where
269
,
15
Page 15 of 33
270
(33)
271 272
,
,
.
Eq.(33) is integrated from Eq.(26) and Eq.(27).
273
Comparatively, the K. LIU’s motion model [28] is based on the older models for
274
the drag and lift coefficients referred to Sellier [26] and Nestor [30], Eq.(2) and Eq.(3),
275
while our new dynamic model in Equations (32) is based on the new proposed
276
frameworks for drag and lift coefficients in this paper.
277
4 Results and discussion
278
A Runge-Kutta solution of Eq.(32) is implemented to validate the proposed
279
equations for drag and lift coefficients and the new motion model based on the
280
experimental data and numeric results of FEA. The fit parameters in the new motion
281
model are identified by the weighted least square method based on one case of
282
simulation results, as listed in Table 1.
283
4.1 Comparison with Experiment results
284
4.1.1 Experiment set-up
285
In the experiments, 10% gelatin blocks (
) stored at 4
in
286
a refrigerator for 24h were impacted by 7.62mm rifle bullets by using a rifle fixed
287
with its muzzle 15m from the front face of the gelatin. The mass density of gelatin is
288
and parameters of rifle bullets were listed in Table.2. The speed of the 16
Page 16 of 33
289
bullet just before impacting the gelatin was measured with a double base optical
290
detector. Phantom V120 high-speed cameras with 16000 fps were used to record
291
motion of the bullets.
292
4.1.2 Comparison with Experiment results
293
For the 7.62mm rifle bullet at 636m/s with an initial yaw angle of
, the
294
process of gelatin penetration is illustrated in Fig.5. In combination with the results
295
plotted in Fig.6 and Tab.3, it can be concluded that no obvious instability of the bullet
296
is found from 0
297
gelatin, approximately 92.3mm, and the path of bullet deflects little. Furthermore, the
298
angle of yaw is increased from
299
phase is from 143.5us to 366.4us. The bullet becomes obviously unstable and its
300
yawing and deflection are accelerated during this time. In the third stage after 366.4us,
301
however, the yaw angle and deflection path continue to increase with a significant
302
deceleration. The comparison between the results of the new motion model and
303
experimental results indicates that conclusion of the new motion model based on the
304
new proposed frameworks for drag and lift coefficients is in great accordance with
305
experiment data in terms of the penetration depth, yaw angle and deflection path.
306
4.2 Comparison with Simulated Results of FEA
to 143.5 . A long narrow channel is formed in the front of the
to 9.5°. Time cost for the rapid overturning
307
Considering the limitation of experimental conditions and measuring error of
308
high-speed photography, it is difficult to get enough and accurate experiment data. It
309
is necessary to validate the accuracy of the new theoretical model by using FEA 17
Page 17 of 33
310
(Finite Element Analysis). A finite element model of 7.62mm rifle bullet penetrating
311
gelatin has been validated in the previous work [31]. So the critical value of the
312
element erosion strain was chosen as 0.9 here, the effect of which has been studied by
313
Wen [32]. Here we use the same finite element model to simulate the 10% gelatin
314
penetration of 7.62mm rifle bullet with an initial incidence angle of
315
velocity of 500m/s, 625m/s and 700m/s, respectively.
316
4.2.1 Drag coefficient
and impact
317
The drag force and velocity of projectiles are obtained from simulated results of
318
FEA directly and the drag coefficient of FEA are determined based on the Eq.(9).
319
Comparative study in drag coefficients is shown in Fig.7 among results of the new
320
proposed Eq.(33), FEA and Sellier's Eq.(2) with
321
obviously shown that impact velocity has no effect on the drag coefficient and the
322
drag coefficient increases with yaw angle, reaches the highest at an angle of yaw
323
between
324
calculation of the new proposed equation in the present paper is better agreement with
325
simulated results from the FEA. The reason for the oscillations of drag coefficients of
326
FEA is that finite elements between the bullet and gelatin with the rotation of rifle
327
bullets is contacted non-uniformly at high angles of yaw.
328
4.2.2 Lift coefficient
and
and
. It is
, and then decreases. Compared with the Sellier's solution,
329
Similar to the drag coefficient above, the lift coefficient of FEA can be
330
determined from the simulated results based on Eq.(28). A comparative analysis of lift 18
Page 18 of 33
331
coefficients is made among the results of the new proposed Eq.(29), FEA and Eq.(3)
332
given by Nestor[27] with
333
Fig.8 at different initial conditions. The results indicate that calculated results of the
334
new proposed Eq.(29) for lift coefficient have a better fit with that of FEA compared
335
with Nestor's solution. The reason for the oscillations of lift coefficients of FEA is
336
similar to that of drag coefficients at high angles of yaw.
337
4.2.3 Motion Model
. The lift coefficient behavior is illustrated in
338
Fig.9 exhibits the comparative analysis for the results of FEA and the new
339
motion model within the penetration depth of 30cm. It can be seen that calculated
340
results of the new motion model coincides well with the simulated results of FEA in
341
terms of penetration depth, deflection path, yaw angle, velocity, lift force and drag
342
force at different initial conditions. In Figure 9(e) and Figure 9(f), the drag force of
343
motion model is derived by using Eq.(9) and Eq.(33), and the lift force of motion
344
model is derived by using Eq.(28) and Eq.(29). While the drag force and lift force of
345
FEA are obtained directly from the simulated results. For lift force, it is indicated that
346
initially the two sets of results are very close to each other. However, after 220mm the
347
lift force in FEA decreases to a value near to zero more rapidly than that in our motion
348
model. But the difference for the impact velocity doesn’t exist.
349
5 Conclusion
350
The interaction of bullet with ballistic gelatin is a complex phenomenon.
351
Slenderness and head shape have a great influence on the motion of bullet. In order to 19
Page 19 of 33
352
characterize the behavior of rifle bullets in the gelatin penetration, the changing of the
353
effective wetted area of bullet is studied in the process of penetration and new
354
frameworks are proposed for drag and lift coefficients by introducing an area
355
detachment ratio and the effect of slenderness.
356
The motion model of rifle bullet based on the new frameworks is valid in the
357
ranges of penetration depth from 0cm to 30cm. The bullet's state of movement
358
calculated by our motion model at each point is almost consistent with the observed in
359
experiment and FEA in terms of penetration depth, deflection path and yaw angle. In
360
addition, the velocity, lift force and drag force according to the motion model are in
361
line with the simulated results of FEA. For drag and lift coefficients, new frameworks
362
can provide better predictions than was found in the literature. This may be significant
363
for research in wound ballistics.
364
Acknowledgments
365
This project are supported by National Natural Science Foundation of China
366
(Grant No.51575279 and No.11502119)
367
References
368 369 370 371 372
[1] Maiden N. Historical overview of wound ballistics research. Forensic Science Medicine & Pathology, 2009, 5(2):85-89. [2] Nicholas NC, Welsch JR. Institute for non-lethal defense technologies report: ballistic gelatin. ADA446543, 2004. [3] Grimal Q, Gama BA, Naili S, Watzky A, Gllespie JW. Finite element study of 20
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high-speed blunt impact on thorax: linear elastic considerations. International Journal
374
of Impact Engineering, 2004, 30(6):665-683.
375
[4] Swain MV, Kieser DC, Shah S, JA Kieser. Projectile penetration into ballistic
376
gelatin. Journal of the Mechanical Behavior of Biomedical Materials, 2014,
377
29(6):385-392.
378 379
[5] Jussila J. Preparing ballistic gelatin--review and proposal for a standard method. Forensic Science International, 2004, 141:91-98.
380
[6] Cronin DS, Falzon C. Characterization of 10% ballistic gelatin to evaluate
381
temperature, aging and strain rate effects. Experimental Mechanics, 2011,
382
51(7):1197-1206.
383 384 385 386
[7] Salisbury CP, Cronin DS. Mechanical properties of ballistic gelatin at high deformation rates. Experimental Mechanics, 2009, 49(6):829-840. [8] Jiwoon K, Ghatu S. Compressive strain rate sensitivity of ballistic gelatin. Journal of Biomechanics, 2009, 43(3):420-425.
387
[9] Moy P, Foster M, Gunnarsson CA, Weerasooriya T. Loading rate effect on
388
tensile failure behavior of gelatins under mode I. In: SEM annual conference on
389
experimental and applied mechanics 2010. p.15-23. Indianapoli, USA.
390 391 392 393
[10] Aihaiti M, Hemley RJ. Equation of state of ballistic gelatin(II). 55048-EG.1.Carnegie Institute of Washington, DC; 2011. [11] Parker NG, Povey MJW. Ultrasonic study of the gelation of gelatin: phase diagram, hysteresis and kinetics. Food Hydrocolloids 2012, 26(1):99-107. 21
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[12] Allen WA, Mayfield EB, Morrison HL. Dynamics of a projectile penetrating sand. Journal of Applied Physics, 1957, 28(3):370-376.
396
[13] Hill R. Cavitation and the influence of headshape in attack of thick targets
397
by non-deforming projectiles†. Journal of the Mechanics & Physics of Solids, 1980,
398
28(s 5–6):249-263.
399 400 401 402 403 404
[14] Rosenberg Z, Dekel E., The deep penetration of concrete targets by rigid rods-revisited. International Journal of Protective Structures, 2010.1(1): 125-144 [15] Shan Y, Wu H, Huang F, Li J. On the Inertia term of projectile's penetration resistance. Advances in Materials Science & Engineering, 2013. [16] Peters CE, A mathematical-physical model of wound ballistics. Journal of Trauma, 1990,6(2):308-318.
405
[17] Peters CE, Dynamics of high-speed projectiles in tissue stimulants and
406
living tissue. Presentation given in the UTSI workshop on wound ballistics.
407
Tullahoma, Tennessee: The University of Tennessee Space Institute. July,
408
1986,p:1-69.
409
[18] Peters CE, Sebourn CL and Crowder HL, Wound ballistics of unstable
410
projectiles. Part I: projectile yaw growth and retardation. Journal of Trauma, 1996,
411
40(3S).
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[19] Sturdivan LM, A mathematical model of penetration of chunky projectiles in a gelatin tissue simulant. Maryland, 1978. [20] Segletes SB, Modeling the penetration behavior of rigid spheres into 22
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ballistic gelatin. 2008, ART-TR-4393. [21] Roecker ET, Ricchiazzi AJ. Stability of penetrators in dense fluids. International Journal of Engineering Science, 1978, 16(11):917–920.
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[22] Flis WJ. A Note on the Roecker-Ricchiazzi model of penetrator trajectory
419
instability, Proceedings of the 22nd International Symposium on Ballistics, Vol. 2,
420
DEStech Publications, Inc., Lancaster, PA, pp. 1287-1294, 2005.
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[23] Weinacht P, Cooper GR. Analytical solutions for the linear and nonlinear
422
yawing motion in dense media. Proceedings of the 23nd International Symposium on
423
Ballistics, Vol. 2,pp:1339-1346, 2007
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[24] Kneubuehl BP. Wound ballistics: basics and application. 4th ed. Berlin: Springer-Verlag, 2011 [25] Roecker ET. On yawing motion, velocity decay, and path deflection of projectiles penetrating cavitating media. 1986. [26] Sellier KG, Kneubuehl BP, Rufer R. Wound ballistics and the scientific background. Amsterdam: Elsevier,1994. [27] Mo GL, Wu ZL, Feng J, Liu K. Surface Pressure model of Rifle Bullets Penetrating into Ballistic Gelatin. Acta Armamentarii, 2014, 35(2):164-169. [28] Liu K, Wu ZL, Xu WH, Mo GL. A motion model for bullet penetrating gelatin. Explosion and Shock Waves, 2012,32(6): 616-622 [29] Liu K, Wu ZL, Xu WH, Mo GL. Research on Model Parameters of Bullet Penetrating Gelatin. Chinese Journal of High Pressure Physics, 2013, 27(5):677-684. 23
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[30] Nestor N. Theoretical study of the motion of a rigid gyro-stabilized
437
projectile into homogeneous dense media. Proceedings of the 24th International
438
Symposium of Ballistics, New Orlaeans,2008.
439
[31] Liu SS, Xu C, Chen AJ, Li HK. Effect of rifle bullet parameters on the
440
penetration into ballistic gelatin. Journal of Beijing Institute of Technology,
441
2015.24(4):487-493.
442
[32] Wen YK, Xu C, Wang HS, CHEN AJ, Batra RC. Impact of steel spheres on
443
ballistic gelatin at moderate velocity. International Journal of Impact Engineering,
444
2013,62(4):142-151.
445
24
Page 24 of 33
line-0
y’
d0
(x0,d0/2） 0
xz
xw
x1
x0
d
xq
v
x’
(a)
line-0
y’ ψ
0
x1
xz
xw
x0
(x1,-d1/2)
d
line-1
xq
v
x’
(b)
line-0
y’
0 d
x0
xz
xw x1
ψ v
xq
x’
(c)
(x1,-d1/2)
line-0
line-1
y’
ψ
d1 xw 0 x1
x’
xq
x0 v
xz
(d)
d
line-1
(x1,-d1/2)
line-0 y
446 447
0
x
Fig.1the motion of rifle bullet in gelatin penetration
448
H l0
449 450
Fig.2 structure of the rifle bullet
451 25
Page 25 of 33
x’ y ’
M1 ψ o
v (a) x’
M2
M1
A
lAB
ψ o
y ’
v
(b) B x’
M1
M2
ψ
M2
o y
o
v
y ’
x
(c)
452 453
Fig.3 the motion of rifle bullet in rapid overturning phase in gelatin penetration.
454
moment.
: overturning
: damping moment.
455 y M2
D
o ψ
x v
L M1
456 457
Fig.4 Force acting on the rifle bullet.
458
26
Page 26 of 33
459 460
Fig.5 Progress of 7.62mm penetrating gelatin
461 462
Fig.6(a) Penetration depth versus time
463 464
Fig.6(b) Yaw angle versus penetration depth
27
Page 27 of 33
465 466
Fig.6(c) Path deflection in y direction versus penetration depth
467
Fig.6 Diagrams of 7.62mm rifle bullet in 10% gelatin penetration with initial conditions. Velocity:
468
636m/s. Angle of incidence:
. Initial angle of yaw:
.
469
470 471
(a) drag coefficient vs. angle of yaw at v = 500m/s
28
Page 28 of 33
472 473
(b) drag coefficient vs. angle of yaw at v = 625m/s
474 475
(c) drag coefficient vs. angle of yaw at v = 700m/s
476
Fig.7 drag coefficient vs. angle of yaw different impact velocity according to FEA, new proposed
477
Eq.(33) and Sellier's Eq.(2).
478
29
Page 29 of 33
479 480
(a) lift coefficient vs. angle of yaw at v = 500m/s
481
482 483
(b) lift coefficient vs. angle of yaw at v = 625m/s
484 485
(c) lift coefficient vs. angle of yaw at v = 700m/s 30
Page 30 of 33
486
Fig.8 lift coefficient vs. angle of yaw for different impact velocity according to FEA, new
487
proposed Eq.(29) and Nestor's Eq.(3).
488
489 490
(a) Penetration depth versus Time
491 492
(b)Path deflection versus Penetration depth
493 494
(c) Yaw angle versus Penetration depth
31
Page 31 of 33
495 496
(d) Penetration velocity versus Penetration depth
497 498
(e) Drag force versus Penetration depth
499 500
(f) Lift force versus Penetration depth
501
Fig.9 Comparative analysis of numerical results and theoretical results in the 10% gelatin
502 503
penetration of 7.62mm rifle bullet with an initial incidence angle of
and the impact velocity of
500m/s, 600m/s and 700m/s
504 505
32
Page 32 of 33
506
Table.1 Fit parameters for present motion model Caliber(mm) 7.62
0.4025
507
508
0.14
0.58
0.1713
0.58
1.78
Table.2 Structure parameters of rifle bullets Caliber(mm)
m/g
d/mm
7.62
7.90
7.92
Jq/(
)
l/mm
H/mm
26.8
16
366.9
Tab.3 computed results of the new motion model at typical moment
Typical moment
Instable moment
Penetration
Deflection
Angle of
depth(mm)
path(mm)
yaw(°)
143.5
92.3
-0.63
-9.5
366.4
212.3
-13.5
-111.2
t( )
Beginning of the damped oscillation phase
509
33
Page 33 of 33