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:
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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
Attitude deflection of oblique perforation of concrete targets by a
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.
Oblique perforation；Penetration；Concrete target；Attitude deflection；Rigid
Over the past decades, the penetration of concrete targets by a rigid projectile has
been concerning continually in the field of impact and penetration, especially the
attitude deflection during the oblique perforation process [1-11]. When a rigid
projectile penetrates obliquely into a concrete target with finite thickness, the rigid
projectile will perforate the target along a deflected path due to the asymmetric
resistance, which is different from the normal perforation. Moreover, the well-known
spalling phenomenon resulting from the reflected tensile waves will occur on the rear
surface of target . 1
In order to describe such a complicated oblique perforation process, Chen et al.
 have developed a three-stage model, in which the perforation process is divided
into three sequent sub-stages, namely, the initial cratering, the tunneling and the shear
plugging sub-stages. The attitude deflection is characterized quantitatively by both the
attitude angle (i.e. the acute angle between the axis of the projectile and the normal
direction of the impacting surface of target) and the attitude deflection angle (i.e. the
increment of the attitude angle). However, in their model, the attitude deflection was
considered only in the initial cratering sub-stage, and hence the attitude angle remains
unchanged in both the tunneling and the shear plugging sub-stages, but which is not
the case in experimental observations and numerical simulations [6, 13-15].
In the oblique perforation experiments of the thick concrete targets, the residual
attitude angles (i.e. the attitude angle at the last time of perforation) are generally
larger than the initial attitude angles [16, 17]. While in the oblique perforation
experiments of the thin concrete targets, the residual attitude angles are found to be
less than the initial attitude angles [13, 14]. Moreover, the same phenomenon has also
been observed by the numerical simulations of oblique perforation of the thin concrete
targets [6, 15], which were investigated by incorporating a combined dynamic
constitutive model into the LS-DYNA code, based on the Holmquist-Johnson-Cook
(HJC)  and the Taylor-Chen-Kuszmaul (TCK)  models, with both the
compressive and the tensile damage of concrete are described simultaneously.
Consequently, it indicates that the directions of the attitude deflection in a thin target
are obviously different from that in a thick target, which, however, was rarely
The oblique perforation process in a thin concrete target is only composed of two
sub-stages, namely, the initial cratering and the shear plugging sub-stages . After it
deflects with an increasing attitude angle [4, 20, 21] during the initial cratering
sub-stage, the projectile has to deflect oppositely during the subsequent shear
plugging sub-stage to make the residual attitude angle less than the initial attitude 2
angle. Therefore, the disregard of the attitude deflection in the shear plugging
sub-stage makes the model developed by Chen et al.  fail to simulate the attitude
deflection in oblique perforation of thin concrete targets.
In addition, there are two more issues need to be justified. On the one hand,
Recht and Ipson [20, 22] investigated quantitatively the attitude deflection angle
during the initial cratering sub-stage with the initial momentum normal to the
penetration path, but their results fail to evaluate quantitatively the effects of the
initial impact velocity and the projectile geometry . Later, Chen et al.  obtained
the attitude deflection angle during the initial cratering sub-stage by analyzing the
kinetic energy consumption normal to the penetration path with two dimensionless
numbers, namely, the impact function I and the geometry function N [1, 23, 24].
However, although their results can describe quantitatively the effects of the initial
impact velocity and the initial attitude angle, only the attitude deflection angle at the
end time of the initial cratering sub-stage can be estimated, rather than the time
history of the attitude deflection angle during all sub-stages of perforation. On the
other hand, while the projectile enters the shear plugging sub-stage, a plug will be
formed between the projectile nose and the rear surface of the target. Referring to the
symmetric cone-shaped plug in normal perforation [25-27], Chen et al.  assumed
an asymmetric oblique-crossed cone-shaped plug in oblique perforation, with both of
their central axes being coincident with the direction of the projectile velocity.
However, Jena et al.  have pointed out that the shock waves formed at the moment
of impact will propagate spherically outward from the point of impact and finally
reflect at the nearest surface, so that the damaged area on the rear surface caused by
the reflected tensile waves is not in the direction of the projectile velocity in oblique
perforation. Moreover, according to the experimental results of oblique perforation of
thin concrete targets, the shapes of the damaged areas on the rear surface of targets
were symmetric circles when the attack angles (i.e. the acute angle between the axis
of the projectile and the direction of the projectile velocity) were negligible [7, 13]. It 3
indicates that the asymmetric oblique-crossed cone-shaped plug assumed by Chen et
al.  is questionable.
The purpose of this paper is therefore to develop a perforation model to describe
the attitude deflection in oblique perforation of concrete targets, in which the inertia
moment of the projectile is introduced, and the attitude deflection during the shear
plugging sub-stage is taken into account together with introducing a corresponding
deflection mechanism. Moreover, a new classification of the target thickness is
presented, and the medium concrete target that has the medium thickness is first
proposed. This paper is divided into four sections as follows. After this brief
introduction, in Section 2, the perforation model is developed, which the three
sub-stages as well as the attitude deflections for different classes of target thickness
are analyzed respectively and the determination method of the classes is presented. In
Section 3, the numerical results by using the perforation model are presented and
compared with a series of the previous experimental data and simulated results,
moreover, the variations of the attitude deflection angle versus different initial
conditions are investigated and the numerical results of the medium concrete targets
are discussed. Finally, some conclusions are drawn out in Section 4.
2. Perforation model
In the analytical two-dimensional model, a rigid projectile moves in the ballistic
plane (i.e. the plane determined by the axis of the projectile and the normal direction
of the impacting surface of the target), which is subjected to the oblique target
resistance. For the sake of simplicity, the concrete target is assumed to be
homogeneous with the homogenized reinforcements and aggregates. As shown in
Fig.1, there are usually three kinds of angles in describing such an attitude deflection
process, namely, the attitude angle β, the attack angle δ and the oblique angle θ (i.e.
the difference between the attitude angle and the attack angle). In this study, we focus
mainly on the attitude angle β and define the attitude deflection angle ∆β = β - β0, in
which β is the current attitude angle and β0 is the initial attitude angle. In addition, the 4
self-rotation of the projectile around its own axis is not taken into account and the
attack angle δ is taken to be zero.
Fig.1. The attitude angle β, the attack angle δ and the oblique angle θ.
The other geometric parameters of the projectile and the target are shown in
Fig.2, where H is the target thickness; d, h, L and lC are the diameter, the nose length,
the body length and the distance between the nose tip and the mass centroid of the
projectile, respectively; v0 is the initial impact velocity.
Fig.2. Geometric parameters of the rigid projectile and the concrete target.
As aforementioned, the total oblique perforation process of the concrete target is
divided into three sub-stages, namely, the initial cratering, the tunneling and the shear
plugging sub-stages. The projectile does not deflect during the tunneling sub-stage
because of the symmetric lateral resistance, but it will deflect in both the initial
cratering and the shear plugging sub-stages with opposite directions. Therefore, as
Fig.3, the attitude deflection angle is positive during the initial cratering sub-stage but negative during the shear plugging sub-stage.
Fig.3 Attitude deflection during three perforation sub-stages.
To describe the motion of the projectile during the oblique perforation, in the
ballistic plane, three Euler rectangular coordinate systems (x1, y1), (x2, y2) and (x3, y3)
are established for the initial cratering, the tunneling and the shear plugging
sub-stages, respectively. Each of the three Euler rectangular coordinate systems is
determined at the initial time of the corresponding sub-stage, with the origin located at
the projectile nose and the xi-axis (i = 1, 2, 3) is along the positive axis of the
projectile and (xi, yi) forms a right hand system. Moreover, the motion and the stress
analysis of the projectile at the initial time of each perforation sub-stage is shown in
Fig.4, where Fni and Fti (i = 1, 2, 3) are the axial resistance and the lateral resistance, 6
respectively, and V0i and B0i (i = 1, 2, 3) are the initial projectile velocity and the initial
attitude angle of each sub-stage, respectively.
Fig.4 Motion and stress analysis of the projectile at the initial time of each
perforation sub-stage. The differential equations of motion of the projectile in each sub-stage are
d 2 xi m 2 = − Fni dt
d 2 βi = M C ( Fti ) dt 2
where i = 1, 2, 3 represent the initial cratering, the tunneling and the shear plugging
sub-stages, respectively, m is the mass of the projectile, xi represents the trajectory
distance during each sub-stage, MC(Fti) is the torque of Fti to the projectile centroid,
and JC is the inertia moment passing through the projectile centroid and normal to the
2.1 Analyses of three sub-stages in oblique perforation
2.1.1 Initial cratering sub-stage 7
In the initial cratering sub-stage, the axial resistance Fn1 is given by an empirical
formula according to the reference , and it satisfies the relationship of Eq. (2b)
with the lateral resistance Ft1 . According to the stress analysis of the projectile
Fig.4, MC(Ft1), the torque of Ft1 to the projectile centroid, is obtained as
Fn1 = c1 x1 ( t )
Ft1 = Fn1 sin β 0
M C ( Ft1 ) = Ft1lC
where c1 is the resistance constant in the initial cratering sub-stage.
At the initial time of this sub-stage t = 0, the initial trajectory distance X01 = x1(0)
= 0, the initial projectile velocity V01 = v1(0) = v0, the initial angular velocity Ω01 =
ω1(0) = 0 and the initial attitude angle B01 = β1(0) =β0. Substituting Eq.(2) into Eq.(1)
leads to the initial-value problems for x1(t) and β1(t), which, under the aforementioned
initial conditions, gives the solutions as x1 ( t ) =
v1 ( t ) =
β1 ( t ) = −
c V01 sin 1 t c1 m m
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
m sin B01 + B01 c1
At the end time of this stage t = T1, we denote the end trajectory distance, the end
projectile velocity and the end attitude angle of the first stage as X1 = x1(T1), V1 = v1(T1)
and Β1 = β1(T1), respectively. Moreover, we define the first attitude deflection angle as
∆β1 = Β1 − B01. From Eqs.(3)-(5), it reaches
c V01 sin 1 T1 c1 m m 8
c V1 = V01 cos 1 T1 m
ml V ∆β1 = C 01 JC
2 V1 V1 m sin B01 arccos − 1− c1 V01 V01
In these equations, the end time of the first stage T1 and the resistance constant c1 are
unknown yet and need to be determined in the following.
The initial cratering is also divided into the complete and the incomplete cases.
For a target thick enough, the projectile will accomplish a complete initial cratering
sub-stage, making the trajectory distance reach up to X1 that can be determined by the
following empirical formula [1, 24]
X 1 = kd
where k = 0.707 + h/d is a dimensionless coefficient. However, in the case of
incomplete initial cratering, more details of other sub-stages are necessary to
2.1.2 Tunneling sub-stage
The projectile enters the tunneling sub-stage only after the complete initial
cratering. In other words, if the projectile is only able to accomplish an incomplete
initial cratering, it will skip over the tunneling sub-stage and enter the shear plugging
sub-stage directly. In the tunneling sub-stage, the lateral resistance is symmetric,
which leads to Ft2 = 0 and MC(Ft2) = 0. Thus, the projectile moves forward in a
straight line under the axial resistance Fn2 that can be estimated by the following
empirical formula [2, 29]
Fn 2 =
2 πd 2 R + N ∗ ρ ( v2 ( t ) ) 4
where R and ρ are the target strength parameter and density of the concrete target, N*
is the geometric factor of the projectile nose. 9
At the initial time of the tunneling sub-stage t = T1, which is also the end time of
the initial cratering sub-stage, the initial trajectory distance of the second sub-stage
X02 = x2(T1) = 0 and the projectile velocity V02 = v2(T1) = V1. Similarly as done for the
first sub-stage, at the end time of this sub-stage t = T2, we define the end trajectory
distance, the end projectile velocity and the end attitude angle of the second sub-stage
as X2 = x2(T2), V2 = v2(T2) and Β2 = β2(T2) =Β1, respectively. Obviously, the second
attitude deflection angle ∆β2 = Β2 − Β1 = 0. Substituting Eq.(10) into Eq.(1) for the
second sub-stage and then integrating the resulted differential equation, X2 can be
derived as X2 =
R + N ∗ ρV12 2m ln π d 2 ρ N ∗ R + N ∗ ρV22
The tunnel formed in this stage will impede the lateral motion of the projectile
and the attitude deflection in the shear plugging sub-stage in a way, which is called
the clamping mechanism of the tunneling sub-stage. If the tunnel is long enough to
impede the lateral motion of the projectile completely, the attitude deflection in the
shear plugging sub-stage can be negligible, which is called here as the complete
clamping mechanism. Contrarily, if the tunnel is not long enough, the attitude
deflection of the projectile in the shear plugging sub-stage will depend on the tunnel
length, which is called as the incomplete clamping mechanism.
2.1.3 Shear plugging sub-stage
According to the experimental observation of symmetric damaged areas on the
rear surfaces of concrete targets, as shown in Fig.5 [7, 13], a symmetric cone-shaped
plug is assumed in this paper, whose central axis is coincident with the normal
direction of the target. The cone crater formed in the shear plugging sub-stage is
shown as Fig.6(a), which is different from the asymmetric oblique-crossed cone crater
proposed by Chen et al. (see Fig.6(b)) .
229 230 231
Fig.5 Photographs of damaged areas on the rear surfaces of concrete targets in oblique perforation experiments [7, 13].
233 234 235
Fig.6 (a) Symmetric cone crater proposed in this study; (b) Asymmetric oblique-crossed cone crater .
The geometric parameters of the symmetric cone crater are shown in Fig.7,
where H* is the plug’s thickness, the cone slope angle α (i.e. the acute angle between
the central axis and the conical edge of the plug) is taken to be 66.1°, and the
shear surface area is As = πl(R + r). Using the relations l2 = (R−r)2 + H*2, r =
d/2·sec(β0+∆β1) and R = r + H*tanα, there is
H* AS cos α = π dH * sec ( β 0 + ∆ β1 ) + tan α d
Fig.7 Geometric parameters of the symmetric cone crater.
In view of von-Mises failure criterion, the plug occurs once its shear stress τf
reaches the failure stress of concrete with τ f = f C
force of the plug Fτ′ = τ f As ,where As is the shear surface area. As the critical force Fs′
is the projection of Fτ′ along the axis of plug and satisfies Fs′ = Fτ′·cosα, the Eq. (12b)
can be obtained as Fs′ =
3 . At that moment, the shear
1 f C AS cos α 3
By using the geometric relations as shown in Fig.8 as well as the Newton’s third law,
the axial resistance acted on the projectile is Fns = Fs·sec(β0+∆β1) = Fs′·sec(β0+∆β1).
Therefore, with Eq.(
12b), at the initial time of the shear plugging sub-stage t = T2, the axial resistance Fns
can be written as
1 H* f C πdH * sec ( β 0 + ∆ β1 ) + tan α sec ( β 0 + ∆ β1 ) d 3
Fig.8 The critical force Fs′of the plug at the initial time of the shear plugging
If it enters the shear plugging sub-stage after an incomplete initial cratering, the
projectile will deflect again with the opposite direction of the initial cratering
sub-stage. Referring to Chen et al. , in this sub-stage, with the motion of the
projectile, the axial resistance Fn3 can be assumed to decrease linearly from Fns to zero
as shown by Eq.(14a), and there exists a relationship between the lateral resistance Ft3
and the axial resistance Fn3 as shown by Eq.(14b). It is assumed that the acting point
of Ft3 moves gradually to the projectile centroid from the nose tip and the torque arm l
= lC – x3(t) decreases linearly from lC. Therefore, the axial resistance Fn3 and the
lateral resistance Ft3 are given as
Fn3 = Fns − c2 x3 ( t )
Ft3 = −Fn3 sin ( β0 + ∆β1 )
M C ( Ft3 ) = Ft3 lC − x3 ( t )
where c2 is the resistance constant in the shear plugging sub-stage.
At the initial time of the shear plugging sub-stage t = T2, the initial trajectory
distance X03 = x3(T2) = 0, the initial projectile velocity V03 = v3(T2) = V2, the initial 13
angular velocity Ω03 = ω3(T2) = 0 and the initial attitude angle B03 = β3(T2) = Β2 = Β1
= β0 + ∆β1. Substituting Eq.(14) into Eq.(1) leads to the initial-value problems for x3(t)
and β3(t), respectively. Solving the initial-value problems under the initial conditions,
the trajectory distance x3(t), the projectile velocity v3(t), the attitude angle β3(t), the
angular velocity ω3(t) = dβ3(t)/dt and the acceleration a3(t) = dv3(t)/dt of the projectile
during the third sub-stage can be obtained as
m Fns (t −T2 ) x3 ( t ) = V03 − e 4 c 2 c 2 2
v3 ( t ) =
m Fns (t −T2 ) − V03 e 4c2 2c2
sin Β 03 β3 ( t ) = − JC
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
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
A1 D1 α1 ( t −T2 ) A2 D2 −α1 ( t −T2 ) t − T2 ) 2 ( − 2 e + D3 2 e 2 α1 α1
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.
The shear plugging sub-stage is finished at the time t = T3 when the plug
separates from the projectile, and from then on, there is no longer action between
them (the re-contact after separation is not taken into account) and the acceleration
of the projectile vanishes. Similarly, we denote the end trajectory distance, the end
projectile velocity, the end angular velocity, the end attitude angle and the end
attitude deflection angle of the third sub-stage as X3 = x3(T3), V3 = v3(T3), Ω3 =
ω3(T3), Β3 = β3(T3) and ∆β3 = Β3 − Β03, respectively. Thus, from Eqs.(15)-(18), we
m Fns (T3 −T2 ) X 3 = V03 − e 4 c 2 c 2 2
m Fns −(T3 −T2 ) − V03 + e 4 c 2 c 2 2
m Fns (T3 −T2 ) − V03 e 4 2 c c 2 2
m Fns (T3 −T2 ) − V03 e 4c2 2c2
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
mV032 Fns2 − + FnslC c2 (T3 −T2 ) 2 2c2 m − c2 m
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
Ω3 = −
where the resistance constant c2 is unknown and needs to be determined as well.
As the analyses presented above are focused on each sub-stage independently, the
continuous conditions between the sub-stages are still required. However, referring to
Chen et al. , the oblique perforations of a thin target and a thick target are
composed of different sub-stages. Therefore, for the concrete targets with different
thickness, the continuous conditions are also different and their oblique perforation
processes should be considered respectively.
Chen et al.  have divided the target thickness into two classes according to the
trajectory distance in the initial cratering sub-stage, namely, the thin and the thick
concrete targets. However, It is insufficient while the attitude deflection in the shear
plugging sub-stage as well as the clamping mechanism of the tunneling sub-stage are
taken into account. Therefore, this paper classifies the target thickness into three
classes, among which the medium target between the thin and the thick targets is
newly proposed, and the attitude deflection angle ∆βr will be investigated respectively
for each class with their individual characteristics and continuous conditions.
For a thin concrete target, referring to Chen et al. , its oblique perforation
consists of only two sub-stages, i.e. an incomplete initial cratering and a shear 15
plugging sub-stages. That is, the attitude deflection angle in a thin target is ∆βr = ∆β1 +
∆β3 (see Fig.10).
While the target thickness reaches up to a certain value, a completed initial
cratering sub-stage will be accomplished and a tunneling sub-stage will appear. At this
moment, we define that the target transforms into a medium one. Thus, the oblique
perforation of a medium concrete target is defined to be composed of three sub-stages,
i.e. a complete initial cratering, a tunneling and a shear plugging sub-stages, but
among which the tunnel length X2 is limited with just the incomplete clamping
mechanism. That is, the attitude deflection angle in a medium target is ∆βr = ∆β1 + ∆β3
(X2) (see Fig.11), in which the quantitative description of the incomplete clamping
mechanism of the tunneling sub-stage need to be dealt with.
When the tunnel length keeps increasing with the target thickness to a certain
value, the attitude deflection in the shear plugging sub-stage will be impeded
completely. At this moment, it is defined that the target transforms into a thick one.
Therefore, the oblique perforation of a thick concrete target is also comprised of three
sub-stages, i.e. a complete initial cratering, a tunneling and a shear plugging
sub-stages, among which the large enough tunnel length X2 with the complete
clamping mechanism leads to ∆β3 = 0. That is, the attitude deflection angle in a thick
target is ∆βr = ∆β1 (see
2.2 Analyses of concrete targets with different thickness
2.2.1 Thick target 16
Fig.9 Attitude deflection in oblique perforation of a thick concrete target.
The attitude deflection in oblique perforation of a thick concrete target is
depicted as shown in
Fig.9. At the end time of the initial cratering sub-stage t = T1, which is also the
initial time of the tunneling sub-stage, the axial resistance satisfies Fn1 = Fn2 at this
moment, thus we have
c1 X1 =
πd 2 R + N * ρV12 ) ( 4
Solving Eqs.(6)-(9) and (24) simultaneously, the end time T1, the end projectile
velocity V1 and the end attitude deflection angle ∆β1 of the initial cratering sub-stage
as well as the resistance constant c1 can be obtained as
V = 2 1
mv02 − ( πkd 3 4 ) R
m + ( πkd 3 4 ) N * ρ
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
From the geometric relation as shown in
Fig.9, the end trajectory distance in the tunneling sub-stage X2 and that in the
shear plugging sub-stage X3 satisfy, respectively,
X 2 = ( H − H * ) sec ( β 0 + ∆β1 ) − X 1
X 3 = H * sec ( β0 + ∆β1 )
Moreover, as the end time of the tunneling sub-stage is just the initial time of the
shear plugging sub-stage, the axial resistance satisfies Fn2 = Fn3 at this moment.
Substituting Eqs.(10) and (13) into them, one has
πd 2 1 H* ∗ 2 * R + N ρV2 ) = f C πdH sec ( β0 + ∆β1 ) + tan α sec ( β0 + ∆β1 ) ( 4 d 3
Therefore, X2 and X3, the end projectile velocities V2 and V3, the end time of the shear
plugging sub-stage T3, the resistance constant c2 and the plug thickness H* can be
obtained respectively by solving Eqs.(11), (19)-(21) and (29)-(31) simultaneously,
which the Eq.(11) is approximated simply by the first order Taylor expansion of the
2.2.2 Thin target
Fig.10 Attitude deflection in oblique perforation of a thin concrete target.
The attitude deflection in oblique perforation of a thin concrete target is shown in
Fig.10. Similarity, at the end time of the initial cratering sub-stage, which is also the
initial time of the shear plugging sub-stage, the axial resistance satisfies Fn1 = Fn3 at
this moment. Thus from Eqs.(2a) and (13), there is
c1 X1 =
1 H* f C πdH * sec ( β0 + ∆β1 ) + tan α sec ( β0 + ∆β1 ) d 3
Moreover, as shown in Fig.10, the end trajectory distance of the initial cratering
sub-stage X1 and that of the shear plugging sub-stage X3 can be written as
X1 = ( H − H* ) sec( β0 +∆β1 )
X 3 = H * sec ( β0 + ∆β1 + ∆β3 )
As the resistance constant c1 given by Eq.(28) is only determined by the
geometric and the material parameters of the projectile and the target, it is
independent of the trajectory distance during the initial cratering sub-stage and the
target thickness. Therefore, the Eq.(28) is also adapted for the incomplete initial
cratering in the oblique perforation of the thin concrete target . Solving Eqs.(6)-(8), 19
(19)-(23) and (32)-(34) simultaneously, the end time of the initial cratering sub-stage
T1 and the shear plugging sub-stage T3, the end trajectory distances X1 and X3, the end
projectile velocities V1 and V3, the end attitude deflection angles ∆β1 and ∆β3, the end
angular velocity Ω3, the resistance constant c2 and the plug thickness H* can be
2.2.3 Medium target
Fig.11 Attitude deflection in oblique perforation of a medium concrete target.
The attitude deflection in oblique perforation of a medium concrete target is
shown in Fig.11. The analysis of the complete initial cratering and the tunneling
sub-stages for the medium target is the same as that for the thick target, by which the
end projectile velocity V2 and the end trajectory distance X2 at the end time of the
tunneling sub-stage can be obtained. In order to determine the attitude deflection
angle and the angular velocity in the shear plugging sub-stage affected by the
tunneling sub-stage, an oblique perforation process of a fictitious thin concrete target
is assumed to consist of a complete initial cratering and a shear plugging sub-stages,
in which we denote the initial impact velocity as v0t and the projectile velocity at the
end time of the complete initial cratering sub-stage as V1t = V2. Thus, using Eq.(25),
v0t can be obtained as
v0 t = 1 + ( πkd 3 4 m ) N * ρ V22 + ( πkd 3 4m ) R 20
Moreover, to describe the effects of the tunnel length X2 on the attitude deflection
angle ∆β3, a coefficient δx is introduced as
v0 t X2 1 − v0 XL
where XL is the minimum tunnel length to make ∆β3 = 0. Empirically, XL = 2d for the
projectiles with the ratio of the length to the diameter larger than 4. Using Eqs.(22)
and (23), the attitude deflection angle ∆β3 and the end angular velocity Ω3 in the
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
c2 (T3 −T2 ) m
mV22 Fns2 − + FnslC 2 2c2 − c2 m
2 mV22 Fns2 (T3 − T2 ) − − + mlCV2 (T3 − T2 ) 2c2 2 2
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
As aforementioned, the tunneling sub-stage only follows a complete initial
cratering sub-stage. Thus the existence of the tunneling sub-stage can be used to
distinguish a thin concrete target from a medium or a thick concrete target. A critical
target thickness HL is then introduced as the minimum thickness of the target that
consists of a complete initial cratering and a shear plugging sub-stages but without a
tunneling sub-stage, as shown in Fig.12. Therefore, a concrete target with the
thickness H less than HL is defined as a thin concrete target, and, vice versa, either a
thick or a medium target.
Fig.12 Attitude deflection in the oblique perforation of a concrete target with the
critical thickness HL.
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*
At the end time of the complete initial cratering sub-stage, the end projectile velocity
V1L, the end attitude deflection angle ∆β1L and the resistance constant c1 can be
obtained from Eqs.(25), (27) and (28). Moreover, as the end time of the complete
initial cratering sub-stage is also the initial time of the shear plugging sub-stage, so
that Fn1 = Fn3 at this moment. By virtue of Eqs.(2a) and (13), ∆β1L and H*L can be
solved, and then the critical target thickness HL can be derived from Eq. (39).
As XL is the minimum tunnel length to impede completely the attitude deflection
in the shear plugging sub-stage (to make ∆β3 = 0), it can be used to distinguish a
medium concrete target from a thick target. Therefore, a concrete target with the
tunnel length X2 less than XL (0 < X2 < XL) belongs to a medium concrete target, while
a target with X2 ≥ XL belongs to a thick concrete target.
3. Numerical results and discussions
The perforation model is used and the numerical results are compared with the
experimental data of the oblique perforation of both thick and thin concrete targets, as
well as the simulated results of the oblique perforation of thin concrete targets [6, 14,
In the oblique perforation experiments of thick concrete targets reported by Fan
et al. , for the tested ogive-nose projectiles, the mass m = 3.64 kg, the diameter d
= 50 mm, the caliber-radius-head CRH = 3, and the inertia moment JC = 0.018 kg·m2.
For the tested thick targets, the compressive strength fc = 42 MPa, the density ρ =
2280 kg/m3, and the target thickness H = 250 mm, 300 mm, 400 mm, 550 mm,
respectively. With the same initial attitude angle β0 = 30°, the experimental data and
the numerical results of the residual velocity (i.e. the projectile velocity at the last
time of perforation) and the attitude deflection angle are shown in Fig.13.
Fig.13 The experimental data and the numerical results of the oblique perforation
experiments of thick concrete targets : (a) residual velocity; (b) attitude deflection
In the numerical simulations of the oblique perforation of thin concrete targets
that investigated by LS-DYNA software and reported by Liu et al. , the mass of the
numerical ogive-nose projectile is m = 100 kg, the diameter d = 189 mm, the length L
= 1050 mm, CRH = 3.3, and the inertia moment JC = 9.15 kg·m2. For the numerical 23
thin targets, the compressive strength fc = 30 MPa, the density ρ = 2500 kg/m3, and the
target thickness H = 180 mm. The initial attitude angle β0 varies from 15° to 45° and
the initial impact velocity v0 varies from 200 m/s to 500 m/s. The simulated results
and the numerical results of the attitude deflection angle are shown in Fig.14, which
the negative attitude deflection angles indicate the projectiles deflect with decreasing
472 473 474
Fig.14 The simulated results and the numerical results of the oblique perforation simulations of thin concrete targets .
In the two oblique perforation experiments of multilayered thin concrete targets
performed by Ma et al. , for the tested truncated-ogive-nose projectiles, the mass
m = 290 kg, the diameter d = 250 mm, CRH = 1.56, and the inertia moment JC =
34.53 kg·m2. For the tested thin targets, the compressive strength fc = 46 MPa, the
density ρ = 2500 kg/m3, the target thicknesses of the first layers are both 300mm
while others are 180 mm, and the target spacing is 3.5 m. The initial impact velocities
and the initial attitude angles of the two tests (numbered No.290-1 and No.290-2) are
v0 = 833 m/s, β0 = 15° and v0 = 688 m/s, β0 = 14.7°, respectively. There were no attack
angles when the projectiles impacted on the first layers of targets.
The ballistic deflections of the two tests recorded by a High Speed Photograph
System are shown in Fig.15 , in which the blue lines are the projectile axes when
they impacted on the first layers of targets, the purple and the red lines are the
connection of the impacting points on each layer of targets during the penetration 24
process. These observations indicate that the projectiles deflected with decreasing
attitude angle during the penetration process. As the perforation processes of the first
two or three layers of targets are less affected by the attack angles, only them are
calculated. Both the experimental data and the numerical results of the residual
velocity Vr and the attitude deflection angle ∆βr are listed in Table 1 and Table 2,
respectively, in which the errors are defined as ERr = |Vr,exp − Vr,num|/|Vr,exp|×100% and
ERa = |∆βr,exp − ∆βr,num|, respectively.
Fig.15 The ballistic deflections recorded in the oblique perforation experiments
of multilayered thin concrete targets .
Table 1. The experimental data and the numerical results of the residual velocity Vr in the oblique perforation
experiments of multilayered thin concrete targets.
Vr,exp /m·s-1 Vr,num/m·s-1
Table 2. The experimental data and the numerical results of the attitude deflection angle ∆βr in the oblique
perforation experiments of multilayered thin concrete targets.
As shown in Fig.13, the prediction of the perforation model is in good agreement
with the experimental data for the thick concrete targets, which indicates that the
inertia moment JC is effective in describing the attitude deflection during the initial
cratering sub-stage. Moreover, it can also be found from Fig.14, Table 1 and Table 2
that considering the attitude deflection during the shear plugging sub-stage and
introducing the corresponding deflection mechanism make it possible to describe the
phenomena in the oblique perforation of thin concrete targets, which the residual
attitude angle is less than the initial attitude angle and the penetration path of the
projectile tends to be the shortest path, i.e. along the normal direction of the target.
With the geometric and the material parameters of the numerical projectile (m =
100 kg) and the concrete target given by Liu et al. , the variations of the attitude
deflection angle ∆βr versus the dimensionless target thickness χ (= H/d), the initial
attitude angle β0 and the initial impact velocity v0 are investigated respectively. The
numerical results are shown in Fig.16-18, respectively. The signs of ∆βr represent the
deflection directions, i.e. ∆βr > 0 indicates that the projectile deflects with increasing
attitude angle while ∆βr < 0 indicates the decreasing attitude angle. Moreover, while
the signs of ∆β1 (> 0) and ∆β3 (< 0) both represent the deflection directions, the
increase of ∆β1 and the decrease of ∆β3 indicate a more violent attitude deflection in
the initial cratering sub-stage and the shear plugging sub-stage, respectively.
The variation of ∆βr versus χ for four initial attitude angles β0 = 10°, 15°, 20°, 25°
under the same initial impact velocity v0 = 500 m/s are shown in Fig.16. It can be seen 26
clearly that there exist two critical target thicknesses χL and χLT on each curve, cross
which the class of the target will transform and the attitude deflection angle will vary
versus χ in a different way. Both of them decrease with the increase of β0. Typically,
the two critical target thicknesses χL = 3.84, χLT = 5.74 for β0 = 15°. Moreover, it can
be found that, with the increase of χ, ∆βr decreases in the thin target, but increases in
the medium target and then keeps constant in the thick target.
In the medium target, the attitude deflection angle ∆β1 of the complete initial
cratering sub-stage remains unchanged with the variation of χ. However, the increase
of the tunnel length X2, resulting from the increase of χ, and the enhanced clamping
mechanism will alleviate the attitude deflection in the shear plugging sub-stage, so
that ∆β3 increases with χ. Therefore, the attitude deflection angle ∆βr increases with χ
in the medium target.
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.
The variation of ∆βr versus β0 for seven target thicknesses χ = 1.2, 1.6 (thin
targets), 4.3, 4.5, 4.8 (medium targets), 7.0 and 8.0 (thick targets) under the same
initial impact velocity v0 = 500 m/s are shown in Fig.17. It is found that with the
increase of β0, ∆βr decreases in the thin target and increases in the thick target, which
both indicate a more violent attitude deflection. But in the medium target, ∆βr
decreases first and then increases. Moreover, as shown in Fig.17, there exists a critical 27
initial attitude angle β0L on each curve for the medium targets, and, for β0 > β0L, the
medium target will transform into a thick target. Typically, the critical initial attitude
angle β0L = 47.5° for χ = 4.5.
In the medium target, the increased β0 results in the increase of both ∆β1 and X2 as
well as the decrease of ∆β3, however, the increased X2 leads contrarily to the increase
of ∆β3. When β0 is small and the clamping mechanism under a small X2 is not
significant, the variation of ∆β3 is dominated by β0. But with the increase of β0, the
increased X2 will gradually dominate the variation of ∆β3. Therefore, ∆β3 as well as
∆βr first decreases and then increases in the medium target. When β0 increases to β0L,
X2 becomes large enough such that ∆β3 increases to zero, and then the medium target
transforms into the thick one.
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.
The variation of ∆βr versus v0 for six target thicknesses χ = 2.2, 2.8 (thin targets),
5.1, 5.4 (medium targets), 7.0 and 8.0 (thick targets) under the same initial attitude
angle β0 = 15° are shown in Fig.18. It is found that with the increase of v0, ∆βr 28
increases in the thin target but decreases in the thick target, which both indicate that
the attitude deflection is alleviated. However, in the medium targets, the variation of
∆βr depends on the target thickness. For example, for a thinner medium target (χ =
5.1), ∆βr increases monotonously, while, for a thicker medium target (χ = 5.4), ∆βr
increases first and then decreases.
In the medium target, both ∆β1 and X2 will decrease with v0. Similarly, while the
increased v0 leads to the increase of ∆β3, the decreased X2 leads to the decrease of ∆β3
contrarily. For a thinner medium target (χ = 5.1), the negligible clamping mechanism
under a small X2 dominates the variation of ∆β3 with v0, so that the attitude deflection
angle ∆βr increases monotonously. However, for a thicker medium target (χ = 5.4), the
remarkable clamping mechanism under a larger X2 leads to that the variation of ∆β3 is
gradually changed into a X2-dominated one, so that ∆β3 as well as the attitude
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.
In a word, the attitude deflection in a medium concrete target is under the
deflection mechanisms of both the initial cratering and the shear plugging sub-stages
as well as the incomplete clamping mechanism of the tunneling sub-stage. The
quantitative description of the incomplete clamping mechanism of the tunneling
sub-stage can be realized by the introduced coefficient δx, and the variations of the
attitude deflection angle ∆βr versus different initial conditions can be predicted for a
A perforation model has been developed to predict the attitude deflection in the
oblique perforation of concrete targets by a rigid projectile, with broader applications,
and the numerical results are in good agreement with a series of experimental data and
simulated results. Some conclusions can be drawn out as follows.
(1) The inertial moment of a projectile is an important parameter in predicting the
time history of the attitude deflection angle during the initial cratering sub-stage,
by which, the perforation model can predict accurately the attitude deflection in
the oblique perforation of thick concrete targets.
(2) The shape revision of the plug formed on the rear surface of the target is
reasonable. Moreover, the introduced deflection mechanism in the shear
plugging sub-stage is of great significance in describing the attitude deflection in
the oblique perforation of thin concrete targets, by which, the phenomenon that
the residual attitude angle is less than the initial attitude angle can be predicted
(3) The incomplete clamping mechanism during the tunneling sub-stage can be
described quantitatively by an introduced coefficient δx, by which the attitude
deflection in the oblique perforation of medium concrete targets can be predicted
(4) The classification for the concrete targets in the oblique perforation is necessary
to predict well the attitude deflection angle, which can be realized by defining
the critical target thickness HL and the critical tunnel length XL. Thus, the targets
with the thickness less than HL belong to the thin targets, that with the thickness
larger than HL but the tunnel length larger than XL belong to the thick targets, and
the rest belong to the medium targets. 30
Acknowledgments This work was supported by the National Natural Science Foundation of China
[grant numbers 11521062].
Declarations of interest
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: