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Penetration performance of double-ogive-nose projectiles
Accepted Manuscript Penetration performance of double-ogive-nose projectiles Jiancheng Liu, Aiguo Pi, Fenglei Huang PII:
S0734-743X(15)00081-0
DOI:
10.1016/j.ijimpeng.2015.05.003
Reference:
IE 2504
To appear in:
International Journal of Impact Engineering
Received Date: 25 January 2015 Revised Date:
17 April 2015
Accepted Date: 4 May 2015
Please cite this article as: Liu J, Pi A, Huang F, Penetration performance of double-ogive-nose projectiles, International Journal of Impact Engineering (2015), doi: 10.1016/j.ijimpeng.2015.05.003. 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.
ACCEPTED MANUSCRIPT
Penetration performance of double-ogive-nose projectiles
1 2
Jiancheng LIUa, Aiguo PIa*, Fenglei HUANGa
3 4
a)
State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, P. R. China
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*Corresponding author: Aiguo PI, State Key Laboratory of Explosion Science and Technology, Beijing Institute of
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Technology, Beijing 100081, P. R. China; E-mail: [email protected]; Tel: +86-10-68914087 Abstract
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The penetration ability of projectiles is closely related to their nose shape as this influences the high-velocity or ultrahigh-velocity
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kinetic-energy penetration of a rigid projectile. Based on classical cavity expansion theory and a double-ogive-nose scheme, this study
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set out to analyze the dependency relationship between the nose shape factor N* and the double-ogive characteristic parameters. This
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paper first discusses the effects of different nose shapes on the penetration performance of a projectile. The paper then proposes a
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double-ogive-nose penetration body scheme with a low penetration resistance. A penetration experiment with the double-ogive-nose
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projectile is set up and a Holmquist–Johnson–Cook concrete [1] model is described, together with its application to the prediction of
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the tendency of the depth of penetration. A comparison between the results of the experiment, simulation, and calculation for the ogive
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nose and double-ogive-nose projectiles proves that the analysis and the proposed method are both reasonable and feasible.
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Keywords
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Penetration; Double-ogive nose; Holmquist-Johnson-Cook (HJC) concrete model; Depth of penetration (DOP); Projectile
Highlights:
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A double-ogive-nose penetration body scheme with a low penetration resistance is proposed.
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The influence of the nose shape coefficient N* on the penetration resistance and DOP is discussed.
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The feasibility of the proposed method is validated through theoretical, experimental, and simulation results.
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ACCEPTED MANUSCRIPT
Radius of projectile in Fig. 2 and Fig. 4
a1
Rear radius of region 1 in Fig. 6
a2
Front radius of region 3 in Fig. 6
A
Coefficient of static resistance term
A1
Normalized cohesive strength
b
Projectile nose length
B
Coefficient of inertial resistance term
B1
Normalized pressure hardening coefficient
c
A constant given by reference [2]
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a
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Nomenclature
Caliber radius head
d
Projectile diameter
D
Damage parameter
D1
Constant in HJC model
D2
Constant in HJC model
EP
Young’s modulus of projectile
fc’
Unconfined compressive strength of concrete target
fs
Failure strain
F
Penetration resistance
TE D
Penetration resistance of region 1 Penetration resistance of region 2
AC C
F2
EP
F1
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CRH
F3
Penetration resistance of region 3
F’
Intermediate variable
k
k = dN * dx
k1
Constant in HJC model
k2
Constant in HJC model
k3
Constant in HJC model
ks
Slope of line segment CD in Fig. 4
K
Plastic volume modulus
2
ACCEPTED MANUSCRIPT Ke
Elastic bulk modulus
m
Mass of projectile
mnose
Mass of projectile nose Mass of CRH 4.5 ogive nose
N
Pressure hardening exponent
N*
Projectile nose factor
N 1*
Projectile nose factor of region 1
N 2*
Projectile nose factor of region 2
N 3*
Projectile nose factor of region 3
Pd
Final penetration depth
p
Loading on the concrete
pc
Pressure occurring in a uniaxial stress compression test
pl
Pressure of compaction
S
S = 72.0 f c’ − 0.5
S1
Radius of region 1
S2
Radius of region 2
EP
Normalized maximum strength that can be developed Maximum tensile hydrostatic pressure the material can withstand Normalized maximum tensile hydrostatic pressure
AC C
T*
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Normalized stress
T
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Maximum pressure reached prior to unloading
P*
Smax
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pmax
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mnose_CRH 4.5
v
Cavity expansion velocity
vs
Initial velocity
Vz
Instantaneous velocity
xc
X coordinate of point C in Fig. 4
xd
X coordinate of point D in Fig. 4
y
Nose generatrix equation
y’
Differential of y
yc
Y coordinate of point C in Fig. 4
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ACCEPTED MANUSCRIPT Y coordinate of point D in Fig. 4
yAC
Equation of curve AC
yDG
Equation of curve DG
y’AC
Differential of yAC
y’DG
Differential of yDG
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yd
Angle in Fig.6
ρ
Density of concrete target
ρP
Density of projectile
σ
Actual effective stress
σr
Stress of cavity expansion
σn
Normal stress on projectile surface
σ n1
Normal stress on region 1 in Fig. 4
σ n2
Normal stress on region 2 in Fig. 4
σ n3
Normal stress on region 3 in Fig. 4
σ*
Normalized effective stress
θ
Angle between projectile axial and nose generatix radius
θ0
Angle between projectile axial and nose generatix radius in Fig. 6
θ1
Angle between projectile axial and nose generatix radius in Fig. 6
θ2
Angle between projectile axial and nose generatix radius in Fig. 6
φ0
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Angle in Fig. 2
Angle between projectile axial and nose tip in Fig. 6
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φ1
EP
φ
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α
Angle between projectile axial and nose generatix extension in Fig. 6
γ
Variable of integration in Eq.(12)
η
Proportional coefficient in Eq.(11)
µ
Volume strain in Fig. 10
µc
Volumetric strain that occur in a uniaxial stress compression test
µl
Volume strain of compaction
∆µp
Equivalent plastic volumetric strain
µ max
Maximum volumetric strain reached prior to unloading
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µ
Modified volumetric strain
υP
Poisson’s ratio of projectile
ε0
Reference strain rate
∆ε p
Equivalent plastic strain
εf min
Amount of plastic strain before fracture
ε*
Dimensionless strain rate
1. Introduction
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Actual strain rate
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ε
Earth penetrating weapons (EPW) are an effective means of attacking hard and deeply buried targets. Therefore,
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the improvement of their penetration performance is currently the subject of considerable research. The penetration
26
ability of an EPW is commonly enhanced by taking one or more of three approaches, namely, increasing the impact
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velocity, raising the kinetic energy of the cross-section, and optimizing the nose structure of the penetrator. Under
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certain tactical conditions when the impact velocity, overall mass, and penetrator diameters are known, the key
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factor that influences the penetration performance of an EPW is its nose shape.
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Considerable research has been conducted to study the influence of the nose structure on the penetration
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ability of an EPW. From the viewpoint of the pertinent mechanics, Forrestal et al. [2, 3] created an empirical
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equation based on cavity expansion theory to estimate the penetration resistance of ogive-nose projectiles. By
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analyzing the nose force of a penetrator as a general y = f(x) form, Jones et al.[4] obtained a related dimensionless
34
constant, that is, the nose shape coefficient N*, and determined its relationship to the inertial resistance. Chen and
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Li [5] analyzed the coefficient N* of several different penetrators with different nose shapes, while Zhao et al. [6]
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introduced a relationship between N* and the velocity. Batra and Chen [7] discussed the effect of friction on the
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penetration.
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AC C
EP
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Based on a previously developed disk model [8], Yankelevsky [9] optimized the body shape of a penetrator by
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ACCEPTED MANUSCRIPT minimizing the instantaneous resistance force. An optimal shape was determined by using a single parameter that is
40
relevant to the velocity, overload, and the medium properties. Bunimovich and Yakunina [10, 11] and also
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Ostapenko and Ostapenko and Yakunina [12, 13] analytically determined the shape of a minimum-drag body and
42
the maximum depth of penetration. However, these works focused mainly on single-curve optimization. Only a
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limited amount of research has been applied to piecewise optimization.
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This paper presents the dependency relationship between the nose shape factor N* and the characteristic
45
parameters of a double-ogive nose. The effects of different double-ogive-nose projectile designs on the penetration
46
performance are discussed. A penetration body with a double-ogive nose, combined with a smaller penetration
47
resistance, is then introduced. Finally, the results of our experiments and the simulation data are presented.
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2. Design of double-ogive nose and related calculations
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2.1 Design of double-ogive nose based on cavity expansion theory
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Cavity expansion theory was originally established and developed for ductile metals [14]. Bishop et al. [15]
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first proposed spherical and cylindrical cavity expansion for the (quasi) static expression of metals. Motivated by
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this proposal, Hill [16] presented the dynamic expansion of the spherical cavity expression for metals. Spherical
53
cavity expansion theory was applied to concrete by Luk and Forrestal [17], which was simplified by fitting and
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obtaining the expression of the normal Cauchy stress component, as follows:
EP
AC C
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σn
v = A+ B ' ' 1/2 fc ( fc / ρ )
2
(1)
ρ
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where f c ' is the unconfined compressive strength of the concrete target,
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coefficients A and B are the resistance coefficients of the material obtained from the fitting. In the same manner as
58
for the resistance equation for index-hardening metal material, Eq. (1) consists of a static resistance related to Y and
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an inertial resistance related to the penetration velocity. Pi and Huang [14] discussed the ratio of the static
60
resistance to the inertial resistance during the process of penetrating 2024-O aluminum targets and concrete of a 6
is the density of the target, and
ACCEPTED MANUSCRIPT range of different strengths, as shown in Fig. 1. The results show that the ratio of static resistance increases
62
proportionally with the strength of the target in the resistance function. Moreover, the inertial resistance ratio
63
increases with the penetration velocity in the resistance function.
(a) Resistance of 2024-O aluminum target
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(b) Resistance of concrete target
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Fig. 1 Comparison between ratios of static term and inertial term in resistance function [14]
Fig. 2 Force on the cross-section of the tip
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Jones et al. [4] analyzed the normal penetration of a rigid axisymmetric projectile into a semi-infinite target.
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The force on the cross-section of the tip is shown in Fig. 2. The length of the nose is b and the radius of the shank
68
of the projectile is a. For all acceptable nose geometries, function y = y(x) with the boundary conditions y(0) = 0
69
and y(b) = a. The effects of friction in this analysis are assumed to be negligible for the whole projectile. The
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expression of resistance along the x axis, which is the direction of the projectile’s movement, is introduced. The 7
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equation can be expressed as follows: F = ∫ dF = 2π ∫ yy ′σ n (V z , φ )dx b
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(2)
0
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where σ n (V z , φ ) is the normal stress on the projectile surface. Luk and Forrestal [18] and Forrestal and Luk [19] adopted the normal velocity of the cavity surface as a
75
geometric function in discussing the penetration pressure. The normal component of the velocity acting on the
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projectile nose is given as follows:
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v = V z sin θ = V z cos φ
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where Vz is the instantaneous velocity in the direction of motion of the projectile. Considering the
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relationship between the normal stress in the cavity surface and the cavity expansion velocity, as given by the
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cavity expansion theory, Forrestal and Luk [19] derived Eq. (4) to express the relationship between the normal
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stress acting on the projectile surface and the penetration velocity.
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(3)
σ n (Vz ,φ ) = Afc' + Bρ (Vz cosφ )
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Based on the studies conducted by Jones et al. [4], without friction, we introduce the concept of the nose shape
84
factor N*, which is expressed as follows:
EP
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TE D
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N* =
2 a2
∫
b
0
yy ′3 dx 1 + y ′2
2
(4)
(5)
As shown in Eq. (5), N* is a dimensionless number and is only related to the geometrical nose shape. N*
87
quantitatively characterizes the nose shape and can be used as a quantity for evaluating the inertial resistance acting
88
on the projectile during penetration. Then, Eq. (2) can be written as
89
AC C
86
F = π a2 ( BN * ρVz2 + Afc' )
(6)
90
To simplify the formula A and B can be approximated using the tri-axial material data. The measured data has
91
shown that B varies over only a very small range for a wide variety of materials and can be approximated to 1.0. A
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can be set equal to S, which is a dimensionless parameter that modifies the compressive strength shown by
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ACCEPTED MANUSCRIPT reference [3], S = 72.0 f c’ − 0.5 as presented by Li and Chen [20]. With this, Eq. (6) becomes
F = π a 2 ( Sfc’ +N * ρVz 2 )
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z ≥ 4a
(7)
During crater formation, the projectile was shown to decelerate linearly so that the axial force acting on the projectile can be taken to be F = cz
97
0 ≤ z < 4a
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(8)
where z is the depth of the penetration from the surface of the target and c is a constant given by reference [3]. As
99
shown by Eq. (6) and (7), given the projectile diameter and initial velocity, the optimization of the penetration
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resistance is related only to the nose shape coefficient N*. In engineering practice, given the penetration stability
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and overall design limitation, armor-piercing or half armor-piercing penetrators are generally designed with an
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ogive nose; the caliber radius head (CRH) is generally between 3 and 5 to prevent the nose from becoming overly
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long [21]. In this study, the nose length within the scope of CRH 5 was studied to obtain the optimal scheme with a
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small coefficient N*. Then, the penetration depth was obtained by Forrestal [3], as follows:
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Pd =
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N * ρV12 m ln 1 + + 4a, 2π a2 ρ N * Sfc'
P > 4a
(9)
Where Pd is the final penetration depth, and V 1 is the velocity at the end of the crater given by reference [3] as
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V12 =
EP
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AC C
mvs2 − 4π a 2 Sf c' , where vs is the intimal velocity. m + 4π a3 N * ρ
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A CRH 3 projectile is selected and Eq. (5) is used to further compare the change in N * . To further discuss the
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change rules for parameter N*, a differential coefficient k which governs the change ratio of N* ( k = dN * dx ) is
110
introduced and a mathematical method can be used to explore the problem. N* and k are shown as curves only for
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an ogive nose and CRH > 0.5.
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ACCEPTED MANUSCRIPT 0.11
0.040
0.10 0.035
Ogive nose of CRH 3
0.08
0.030
0.07
0.025
0.06
k
N*
0.09
0.020
0.05 0.015
0.04 0.03
0.005
0.01 0.00 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.000 0.0
1.0
0.1
0.2
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0.010
Ogive nose of CRH 3 0.02
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
x/b
x/b
Fig. 3 (a) Relationship between N* and position
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Fig. 3 (b) Relationship between k and position
As shown in Fig. 3(a), the relationship between N* and the position on the nose are presented, with the
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x-coordinate signifying the dimensionless position on the nose, and the y-coordinate showing the integral value
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from the tip to this position, as given by Eq. (5). N* is a monotonously increasing function which increases quickly
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in the first half and then slowly in the second half. It indicates the change ratio of N* shown in Fig. 3(b) whereby k
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reaches a maximum when x/b = 0.208. Therefore, the region on both sides of x/b = 0.208 is selected to modify the
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nose. An ogive nose is usually adopted for a projectile because it is superior to a conical nose in terms of preventing
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ricochet. At the end of the nose, the ogive is tangential to the projectile shank, thus ensuring a greater filling mass
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and avoiding the stress concentration in the penetration process. only the middle of the projectile nose is optimal by
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linear transition. To attain better stability at the beginning of impact, the ogive part at the nose tip should be a little
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longer. Meanwhile, the tactical and technological aspects of the oblique and yaw angles should be considered, such
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that if a high impact angle adaptation is needed, the ogive at the tip should have a small diameter. For the ogive at
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the end of the nose, the ogive prior to optimization can be applied to avoid stress concentration in the penetration
124
process. The length of the middle region should generally be controlled such that it is no more than one third of the
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nose length. Then, the region where k > 0.027 is selected to optimize the design and minimize the increase in N* as
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far as possible. The modified nose is shown in Fig. 4, which also introduces the conical and ogive shapes;
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AEFHGDC encloses the modified region wherein ACE represents region 1, CEFD represents region 2, and DFHG
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129 130
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Fig. 4 Contrast between modified and original projectiles
For the modified projectile, ks is the slope of line segment CD; yAC and yDG are the curve equations of AC and
132
DG. According to the principle of the subsection integral, the nose coefficients N* can be added together to give the
133
value of N* for the three regions, as given by Eq. (10).
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* * * * N mod ified = N1 + N 2 + N3 =
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3 3 xd k (k x − k x + y ) b y 2 xc y AC y '3AC s s s D D DG y 'DG dx + dx + dx 2 ∫0 2 2 2 ∫ ∫ xc xd 1 + y ' a 1 + y 'AC 1 + ks DG
TE D
0.11
(10)
0.10 0.09
Ogive nose of CRH 3
0.08
N*
0.07 0.06
EP
0.05 0.04
Ogive nose of CRH 5
0.03
Double-ogive nose
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0.02 0.01
0.00 -0.4
135 136
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
x/b
Fig. 5 Relationship between N* of the three projectile types and nose position
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The significance of the double-ogive-nose projectile is to find a relatively superior geometrical shape between
138
CRH 3 and CRH 5 that can reduce the resistance during penetration. Fig. 5, in which the x-coordinate is the same
139
as in Fig. 4, presents the relationship between N* and the nose positions of the three types of noses. For the same
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length of projectile nose, the value of N* for the double-ogive-nose projectile is smaller than that for a typical ogive 11
ACCEPTED MANUSCRIPT 141
nose.
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2.2 Calculation analysis of penetration resistance For a double-ogive nose such as that shown in Fig. 6, the resistance is divided into three parts where region 1
144
at the tip is an ogive with radius S1, region 2 in the middle is a cone with a1 and a2 (distance from the central axis to
145
the two ends), and region 3 at the root is an ogive with radius S2.
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Fig. 6 Dimensions of double-ogive nose
To analyze the force on each part of a double-ogive-nose projectile in more detail, considering classical cavity
149
expansion theory, the integral equation is used to solve the stress that arises during the penetration. The force on
150
each unit is calculated, then integrated, and then iterated with time, so as to obtain the penetration depth more
151
accurately. Otherwise, it proves that the resistance of a double-ogive-nose projectile is smaller than a normal ogive
152
projectile. The force on the nose was discussed by Forrestal et al. [22] using both spherical and cylindrical
153
approximation. However, considering the uniformity with the force of an ogive-nose projectile, the spherical cavity
154
expansion stress is used for the cone part. The resistance of a double-ogive nose is obtained from Eq. (7).
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F = F1 + F2 + F3 155
φ1
θ1
φ0
θ0
= ∫ σ n1 ⋅ 2π S12 (sin φ − sin φ0 )cosφ dφ + (a22 − a12 )πσ n2 + ∫ σ n3 ⋅ 2π S22 (sin γ − sin θ2 )cos γ dγ
(11)
156
where σ n1 , σ n2 , and σ n3 are the normal stresses in regions 1, 2, and 3, respectively; F1, F2, and F3 are the
157
resistances
158
σn2 = Sfc' + ρ (Vz sinα ) , σn3 = Sfc' + ρ (Vz cosγ ) . φ and γ represent the integral variables shown in Figure 6.
of
regions 2
1,
2,
and
3,
respectively. 2
12
Then,
ρ = 2.5 g / cm 3 ,
σn1 = Sfc' + ρ (Vz cosφ ) , 2
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159
Fig. 7 Dimensions of CRH 3, double-ogive nose, and CRH 5 projectiles (mm)
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General ogive noses, that is, CRH 3 and CRH 5, as well as a double-ogive nose with a head length similar to
162
that of CRH 5, were selected as shown in Fig. 7. The three projectiles are all assumed to be rigid with a similar
163
mass of 450 kg and an initial velocity of 500 m/s and 1000 m/s. Head erosion during the penetration process was
164
ignored, and the unconfined compressive strength of the concrete target was 35 MPa.
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（a）vs = 500 m/s
（b）vs = 1000 m/s
Fig. 8 Deceleration history of three projectile types
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60
16 50
14 40
Pd /d
Pd /d
12 10
30
8 6
20
Ogive nose of projectile CRH 3 Ogive nose of projectile CRH 5 Double-ogive nose projectile
2
Ogive nose of projectile CRH 3 Ogive nose of projectile CRH 5 Double-ogive nose projectile
10
0
0
0
2
4
6
8
10
12
14
16
0
18
5
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4
10
15
20
25
30
t (ms)
t (ms)
（b）vs = 1000 m/s
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（a）vs = 500 m/s
Fig. 9 Dimensionless penetration depth history of three projectile types
The deceleration history of the three projectile types is shown in Fig. 8. The double-ogive-nose projectile has
166
the smallest deceleration peak, whereas the ogive nose projectile of CRH 3 has the largest peak. This phenomenon
167
proves that the penetration resistance of the double-ogive-nose type is smaller than that of the other two types [Fig.
168
8].
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The depth of penetration (DOP) history with its clearly demonstrated increment is shown in Fig. 9. The degree
170
of penetration at a low velocity is smaller than that at a high velocity. This phenomenon intuitively explains the
171
relationship between the nose shape factor N* and DOP and indirectly illustrates the influence of N* on the inertial
172
term. The calculated results prove that a double-ogive-nose projectile is subject to a small penetration resistance
173
and that the nose shape makes a major contribution to DOP.
174
3. Experimental Results and Simulation Analysis
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A penetration experiment with a double-ogive-nose projectile was performed by the authors to validate the
176
material model of simulation. Then, a numerical simulation parameter (failure strain) that can be used to predict the
177
penetration depth for different impact velocities and structures by simulation was determined from the measured
178
results.
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3.1 Material model used in simulation LS-DYNA3D was used for the simulation. The parameters were verified with the aid of experimental data.
181
The concrete was described by the Holmquist-Johnson-Cook (HJC) model [1] which is widely used for concrete. It
182
is similar to the Johnson–Cook model that is used for metals. The HJC model is commonly used to describe the
183
dynamic response of brittle material damage when that material is subjected to large strains, high strain rates, and
184
high pressures. The flow rule, consistency conditions, and strengthening rule are not strictly observed in this model.
185
Therefore, the model can describe the dynamic behavior leading to the penetration of a concrete target.
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186 187
190
EP
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The sectional state equation is used to describe the pressure and volume strain of the HJC model (Fig. 10). The first stage (OA) is linearly elastic at p < pc with
AC C
188
Fig. 10 Pressure and volume strain of HJC concrete model
p = Ke µ
(12)
191
in which K e = pc µ c is the elastic bulk modulus. p c and µ c are the pressure and volumetric strain, respectively,
192
which occur in a uniaxial stress compression test. The relationship between the pressure and volume strain are
193
linear at this stage.
194
The second stage (AB) is referred to as the transition stage and occurs at p c ≤ p ≤ pl . In this stage, the air
195
voids are gradually compressed out of the concrete, thus producing plastic volumetric strain. The loading is
196
described as 15
ACCEPTED MANUSCRIPT p = pl +
197
( pl − pc )( µ − µl )
(13)
µl − µc
where p l and µ l are the pressure and volume strain of compaction, respectively. The unloading in this stage
199
occurs along a modified path that is interpolated from the adjacent regions, as follows: p − pmax = (1 − F ’ ) K e + FK ( µ − µ max )
200
(14)
201
Furthermore, F = ( µmax − µc )
202
maximum pressure and volumetric strain, respectively, reached prior to unloading. The air voids in the concrete are
203
gradually eliminated, the structure is damaged, and a crack begins to appear at this stage.
205
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have been removed when the pressure reaches p l . This relationship is expressed as 2
p = k1 µ + k2 µ + k3 µ where
3
µ − µl 1 + µl
µ , is used so that the constants ( k 1 ,
The modified volumetric strain,
210
for a material with no voids, assuming that the concrete is fully crushed.
k 2 , and k 3 ) are equivalent to those used
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The yield surface of the HJC concrete model is given by
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(15)
(16)
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µ=
208
211
p max and µ max are the
The last stage (BC) defines the relationship for a fully dense concrete material from which all air voids
206 207
K is the plastic volume modulus.
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’
( µl − µc ) ， and
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198
σ * = A1 (1 − D ) + B1 P*N (1 + C ln ε * )
(17)
213
and σ * = σ f c’ is defined as the normalized effective stress,
214
( 0 < D < 1.0 ), P * = p f c' is the normalized stress, and ε * = ε ε 0 is the dimensionless strain rate (where
215
the actual strain rate and ε 0 =1 .0 s − 1 is the reference strain rate). A1 is the normalized cohesive strength, B1 is the
216
normalized pressure-hardening coefficient, N is the pressure-hardening exponent, C is the strain rate coefficient,
217
and Smax is the normalized maximum strength that can be developed.
218
σ
is the actual effective stress, D is the damage ε
is
Fracture damage is accumulated in a manner similar to that used in the Johnson–Cook fracture model [23]. 16
ACCEPTED MANUSCRIPT 219
The Johnson–Cook fracture model accumulates damage from the equivalent plastic strain. The model discussed
220
herein accumulates damage from both the equivalent plastic strain and plastic volumetric strain, and is expressed as D=∑
221
∆ε P + ∆µ P ε Pf + µ Pf
(18)
222
where ε pf + µ pf = D1 ( P * + T * ) , ∆ ε P , and ∆ µ P are the equivalent plastic strain and plastic volumetric strain,
223
respectively. D 1 and D2 are constants. T * = T f c' is the normalized maximum tensile hydrostatic pressure,
224
where T is the maximum tensile hydrostatic pressure the material can withstand.
225
strain that occurs before a fracture occurs, so as to prevent fractures from occurring under the influence of
226
low-magnitude tensile waves for which
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D2
min
is the amount of plastic
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εf
ε f = 0.01 .
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min
227
Table 1 lists the parameters for the concrete target used in the simulation. In view of the failure behavior
228
expressed by the eroding failure of a Lagrange mesh, the material failure strain fs is a critical parameter related to
229
DOP. fs = 0.38 is set in the simulation to prevent calculation errors.
230
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Table 1. Parameters for concrete target material
ρ (kg/m3)
E (GPa)
K (GPa)
A
B
C
N
fc (MPa)
2440
20.68
14.86
0.79
1.6
0.007
0.61
48
T (GPa)
ε0
εf
Smax
Pc (GPa)
µc
Pl (GPa)
µl
0.004
6
0.01
7.0
0.016
0.001
0.80
0.10
D1
D2
k1 (GPa)
k2 (GPa)
k3 (GPa)
fs
85
-171
208
0.38
EP
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0.04
min
1.0
231
For the experiment, the projectile was made of high-strength steel, which exhibits very little plastic
232
deformation or mass loss during penetration. Thus, the projectile can be defined as being a rigid body with a
233
density ρp = 7850 kg/m3. In the LS-DYNA program, the Young’s modulus Ep = 2.158 × 105 MPa and Poisson’s ratio
234
vp = 0.3 are also needed [24].
17
ACCEPTED MANUSCRIPT 3.2 Experimental results and finite element model
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235
236 237
Fig. 11 Models of concrete targets and projectiles used in the experiment
Experimental data for the structural parameters of the double-ogive-nose projectile and the target is listed in
239
Table 2. The finite element model is shown in Figure 11. To reduce the number of grids and improve the efficiency
240
of the calculation, a circular semi-infinite target was used in place of the square target used in the numerical
241
simulation. For the penetration experiment, the projectile was launched from a 155-mm gun, with caliber launching
242
technology so as to achieve the required velocity. The sabot was made of a lightweight and brittle material such that
243
its effects on the penetration ability could be ignored. The projectile penetrated the concrete vertically. The
244
dimensions of the concrete target used in the experiment were 3.6 m × 3.6 m × 5.4 m, respectively. The nose length
245
of the double-ogive-nose projectile was equivalent to a CRH 4.5 ogive nose. Then, the comparison in the
246
simulation was between CRH 3, CRH 4.5, and the experimental double-ogive-nose projectile described below.
EP
TE D
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238
247 NO.
DO1
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Table 2 Projectile and target data for double-ogive-nose projectile penetration experiment Mass
Strength of target
Projectile length
Projectile diameter
Initial velocity
DOP
(kg)
(MPa)
(m)
(m)
(m/s)
(m)
81.1
48
1.13
0.14
835
5.1
248
The experimental results show that the penetration depth of the double-ogive-nose projectile is 5.1 m. With the
249
verified numerical simulation parameters, Fig. 12 shows the change in the velocity and the contrast in the
250
dimensionless penetration depth between the numerical simulation and the experiment, which proves that the
251
selected parameters are reasonable and reliable. This group of parameters can be used to investigate the DOP rules 18
ACCEPTED MANUSCRIPT for projectiles with different nose shapes for a large range of impact velocities. 40
900
35
800 700
30
600
Simulation DOP Simulation velocity experiment DOP
20
500
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P/d
25
velocity (m/s)
252
400
15
300
10
200
5 0 0
3
6
9
12
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100
15
0 18
time (ms)
254
Fig. 12 Comparison of velocity between simulation and experiment
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255 256
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Fig. 13 Finite element models for three projectile types
To prove that the double-ogive-nose projectile can enhance the penetration ability, two projectile structures
258
similar to the double-ogive nose were selected to study the DOP rule with the previously selected simulation
259
parameters. Three types of nose projectiles were then selected for the penetration simulation shown in Fig. 13. The
260
first projectile has a common ogive nose with CRH 3; the second has a double-ogive nose, and the third is a CRH
261
4.5 ogive nose that has the same nose length as the double-ogive-nose projectile. The three projectiles were
262
assumed to have the same mass (with the aid of a filler), initial velocity, and target.
263
3.3 Results and discussion
264
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Four velocities were selected for the simulation. Table 3 compares the results of the theoretical calculation and 19
ACCEPTED MANUSCRIPT the numerical simulation, proving that the double-ogive nose has a better penetration ability than the equivalent
266
nose or CRH 3 nose. With the same initial velocity, the DOP of a double-ogive-nose projectile can be increased by
267
10% and 5%, relative to a common CRH 3 projectile and equivalent nose projectile, respectively. At a high velocity,
268
the DOP increment of a double-ogive nose is large but the growth rate is not obvious. For high-velocity penetration,
269
the nose shape has a significant influence on the DOP.
270
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Table 3 Contrast between theoretical calculation and numerical simulation
Velocity (m/s)
Theoretical
Simulation
Calculation
DOP (m)
DOP (m) 2.248
500
2.454
Ogive nose of CRH 4.5
500
2.255
Ogive nose of CRH 3
500
2.174
Double-ogive nose
800
4.971
Ogive nose of CRH 4.5
800
4.652
Ogive nose of CRH 3
800
4.443
Double-ogive nose
1000
6.890
Ogive nose of CRH 4.5
1000
6.423
Ogive nose of CRH 3
1000
6.120
Double-ogive nose
1200
Ogive nose of CRH 4.5
1200
Ogive nose of CRH 3
1200
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Double-ogive nose
8000
2000
12.9
8.82
9.47
--
--
4.837
2.66
11.9
6.86
4.507
3.22
4.10
--
4.358
1.95
--
--
6.893
0.04
12.6
7.27
6.490
0.57
4.95
--
6.171
0.83
--
--
8.917
9.082
1.85
13.4
7.16
8.321
8.593
3.17
5.87
--
7.860
8.043
2.28
--
--
10000
Deceleration(g)
3000
8.39
(%)
1.986
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Deceleration(g)
4000
(%)
Equivalent Nose
--
vs=500m/s, double-ogive nose
5000
to CRH3 (%)
3.73
vs=500m/s, CRH 4.5 nose
6000
Deviation
DOP Increment to
7.43
vs=500m/s, CRH 3 nose
7000
DOP Increment
2.099
EP
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Relative
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Projectile
Numerical
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Impact
9000
vs=800m/s, CRH 3 nose
8000
vs=800m/s, CRH 4.5 nose vs=800m/s, double-ogive nose
7000 6000 5000 4000 3000 2000
1000
1000 0 0
1
2
3
4
5
6
7
8
9
10
11
12
13
0
14
0
time(ms)
2
4
6
8
10
time(ms)
(a) vs = 500 m/s
(b) vs = 800 m/s
20
12
14
16
18
ACCEPTED MANUSCRIPT 14000
12000
vs=1200m/s, CRH 3 nose
vs=1000m/s, CRH 3 nose
12000
vs=1000m/s, CRH 4.5 nose vs=1000m/s, double-ogive nose
Deceleration(g)
Deceleration(g)
10000
8000
6000
vs=1200m/s, CRH 4.5 nose vs=1200m/s, double-ogive nose
10000 8000 6000
4000
2000
2000
0 0
2
4
6
8
10
12
14
16
18
0
20
0
time(ms)
2
4
6
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4000
8
10
12
14
16
18
20
22
time(ms)
(d) vs = 1200 m/s
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(c) vs = 1000 m/s
Fig. 14 Deceleration time-history curves for three projectiles with different velocities
The deceleration time-history curves under different conditions are shown in Fig. 14. The double-ogive-nose
273
projectile has a smaller peak deceleration value and a wider pulse than the other two projectiles. It indicates that the
274
penetration resistance of a double-ogive nose is small and that the penetration time is long, which indicates that the
275
double-ogive-nose projectile has a better penetration ability.
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80 70
50
CRH 3 simulation CRH 4.5 simulation Double-ogive nose simulation Jones optimal simulation [4] Double-ogive experiment CRH 3 calculation CRH 4.5 calculation Double-ogive calculation Jones optimal calculation [4]
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Pd /d
60
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90
40 30
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20 10
0 400
276 277
500
600
700
800
900
1000
1100
1200
1300
1400
v(m/s)
Fig. 15 Dimensionless DOP with different penetration velocities
278
Fig. 15 shows the relationship between the dimensionless DOP and the impact velocity. The figure also shows
279
the simulated and calculated results for four types of projectiles, and the results of an experiment with a
280
double-ogive-nose projectile. It indicates that the penetration depth as determined by the theory-based calculation,
21
ACCEPTED MANUSCRIPT numerical simulation, and experiment are identical, which again proves the correctness of the theory-based
282
calculation. The penetration data obtained with impact velocities of between 400 m/s and 1400 m/s can be
283
predicted and the penetration ability of different projectile structures can be compared by applying the theory model.
284
With low-velocity penetration, the advantage of Jones [4] and double-ogive-nose projectile’s penetration ability is
285
not obvious. However, with high-velocity penetration, the double-ogive-nose projectile offers significant
286
advantages in terms of DOP over an ogive-nose projectile of the same mass. The DOP increment of the
287
double-ogive nose is 0.60 m over the CRH 4.5 ogive nose and 1.06 m more than that of the CRH 3 nose for an
288
initial velocity of 1200 m/s. This proves that nose optimization plays an essential role in reducing the penetration
289
resistance and enhancing DOP, especially for high-velocity penetration.
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1.0
y/a
0.8 0.6 0.4 0.0 0.0
0.1
290 291
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0.2 0.2
0.3
0.5
0.6
N*=0.0553 N*=0.0465 N*=0.0608 N*=0.0720
0.7
0.9
0.8
1.0
x/b
Fig. 16 Four types of nose shape outline
292
EP
Tab. 4 Contrast of four types of nose shape
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Conical nose
293
0.4
Cone Nose Jones Nose [4] Double-ogive Nose Ogive Nose CRH 4.5
Jones optimal nose
Double-ogive nose
CRH 4.5 Ogive nose
N*
0.0533
0.0465
0.0608
0.0720
mnose (kg)
11.58
14.12
16.44
18.87
mnose/mnose_CRH4.5
0.61
0.75
0.87
1
294
Four types of nose shape outline curves and their N* are shown in Fig. 16. The nose shape factor N* was
295
calculated for each of these. It can be seen that the N* of the Jones optimal nose exhibits the minimum value, that
296
the conical nose is second only to the Jones projectile, and that the N* of the double-ogive nose is smaller than that
22
ACCEPTED MANUSCRIPT of the CHR 4.5 ogive nose and close to that of the conical nose. The N* contrast between the double-ogive nose
298
projectile and Jones optimal projectile proves the correctness of the results in Fig. 15, which indicates that the
299
penetration ability of the Jones optimal projectile is better than that of the double-ogive nose projectile, as
300
determined by theory calculation and simulation. However, from the viewpoint of practical engineering
301
application, because of the limitation of the overall design platform, the penetrator should increase the quality of
302
the payload in the limit space as much as possible, as shown in Table. 4. Therefore, it is not common to choose the
303
conical nose shape in that it has a low-quality utilization ratio. However, for most engineering applications, an
304
ogive-nose projectile is widely used [2, 3, 21, 25-37]. The optimization described in this paper is based on the ogive
305
nose, and the N* of a double-ogive nose is close to that of the a conical nose. The mass utilization ratio is 41.97%
306
more than that of a conical nose, and 16.43% more than the Jones optimal nose.
307
4. Conclusion
309
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Based on cavity expansion theory, projectiles with different nose shapes were examined and the influences on
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297
their penetration ability were analyzed. Consequently, a double-ogive-nose structure is proposed. The influence of the nose shape coefficient N* on the penetration resistance and DOP was discussed based on
311
the results of the calculation. The resistance consists of the inertia resistance and the static resistance. The former is
312
related to the penetration velocity and the latter is associated with the material performance. As the velocity
313
increases, the percentage of the inertia item increases in the resistance function and thus affects the DOP with
314
high-velocity penetration.
316
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315
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310
For a given nose length, the double-ogive-nose projectile has a smaller nose shape coefficient N* than that of an ogive nose, and a larger mass than the Jones optimal nose.
317
A set of consistent target parameters was verified through experiment, and these parameters were used to
318
simulate the penetration performance of the three projectile types. The results were found to be consistent with the
23
ACCEPTED MANUSCRIPT 319
theoretical calculation, again proving the correctness of the theoretical model.
320
Acknowledgements
321
This work was supported by the National Natural Science Foundation of China under Grant (No.11221202) and the National Natural Science Foundation of China (No.11202029, No.11390362).
323
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322
Fourteenth International Symposium on Ballistics, Quebec, Canada, 1993.
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[2] Forrestal MJ, Frew DJ, Hanchak SJ, Brar NS, Penetration of grout and concrete targets with ogive-nose steel [3] Forrestal MJ, Altman BS, Cargile JD, Hanchak SJ, An empirical equation for penetration depth of ogive-nose projectiles into concrete targets, Int J Imp Eng, 15 (1994) 395-405. (1998) 413-417.
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[4] Jones SE, Rule WK, Jerome DM, Klug RT, On the optimal nose geometry for a rigid penetrator, Comput Mech, 22 [5] Chen XW, Li QM, Deep penetration of a non-deformable projectile with different geometrical characteristics, Int J Imp Eng, 27 (2002) 619-637.
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thermoviscoplastic target, Acta Mech, 106 (1994) 87-105.
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[12] Ostapenko NA, Yakunina GE, Least-drag bodies moving in media subject to the locality hypothesis, Fluid Dynamics, 27 (1992) 71-80.
[13] Ostapenko NA, Yakunina GY, The shape of slender three-dimensional bodies with maximum depth of penetration into dense media, Pmm Journal of Applied Mathematics and Mechanics, 63 (1999) 953-967. [14] Pi AG, Huang FL, Dynamic behavior of a slender projectile on oblique penetrating into concrete target, Explosion and Shock Waves, 27 (2007) 331-338. [15] Bishop RF, Hill R, Mott NF, The theory of indentation and hardness tests, Proc Phys Soc, 57 (1945) 147-159. [16] Hill R, Cavitation and the influence of headshape in attack of thick targets by non-deforming projectiles, J Mech Phys Solids, 28 (1980) 249-263. [17] Forrestal MJ, Cargile JD, Tzou RDY, Penetration of concrete targets, in:
Proceedings of the 1993 ASME Winter
Annual Meeting, November 28, 1993 - December 3, 1993, Publ by ASME, New Orleans, LA, USA, 1993, pp. 9-16. [18] Luk VK, Forrestal MJ, Penetration into semi-infinite reinforced-concrete targets with spherical and ogival nose projectiles, Int J Imp Eng, 6 (1987) 291-301.
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ACCEPTED MANUSCRIPT [19] Forrestal MJ, Luk VK, Penetration into soil targets, Int J Imp Eng, 12 (1992) 427-444. [20] Li QM, Chen XW, Dimensionless formulae for penetration depth of concrete target impacted by a non-deformable projectile, Int J Imp Eng, 28 (2003) 93-116. [21] Goldsmith W, Non-ideal projectile impact on targets, Int J Imp Eng, 22 (1999) 95-395. [22] Forrestal MJ, Okajima K, Luk VK, Penetration of 6061-T651 Aluminum Targets With Rigid Long Rods, J Appl Mech, 55 (1988) 755-760. temperatures and pressures, Engineering Fracture Mechanics, 21 (1985) 31-48.
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[23] Johnson GR, Cook WH, Fracture characteristics of three metals subjected to various strains, strain rates, [24] LS-DYNA Keyword User's Manual, Livermore Software Technology Corporation, 2007.
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material projectile penetrating concrete target, Transactions of Tianjin University, 14 (2008) 430-433.
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[32] Booker PM, Cargile JD, Kistler BL, La Saponara V, Investigation on the response of segmented concrete targets to projectile impacts, Int J Imp Eng, 36 (2009) 926-939.
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[34] Unosson M, Nilsson L, Projectile penetration and perforation of high performance concrete: experimental results
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and macroscopic modelling, Int J Imp Eng, 32 (2006) 1068-1085. [35] Warren TL, Fossum AF, Frew DJ, Penetration into low-strength (23MPa) concrete: target characterization and simulations, Int J Imp Eng, 30 (2004) 477-503. [36] Forrestal MJ, Frew DJ, Hickerson JP, Rohwer TA, Penetration of concrete targets with deceleration-time
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measurements, Int J Imp Eng, 28 (2003) 479-497. [37] Frew DJ, Hanchak SJ, Green ML, Forrestal MJ, Penetration of concrete targets with ogive-nose steel rods, Int J Imp Eng, 21 (1998) 489-497.
25
S0734-743X(15)00081-0
DOI:
10.1016/j.ijimpeng.2015.05.003
Reference:
IE 2504
To appear in:
International Journal of Impact Engineering
Received Date: 25 January 2015 Revised Date:
17 April 2015
Accepted Date: 4 May 2015
Please cite this article as: Liu J, Pi A, Huang F, Penetration performance of double-ogive-nose projectiles, International Journal of Impact Engineering (2015), doi: 10.1016/j.ijimpeng.2015.05.003. 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.
ACCEPTED MANUSCRIPT
Penetration performance of double-ogive-nose projectiles
1 2
Jiancheng LIUa, Aiguo PIa*, Fenglei HUANGa
3 4
a)
State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, P. R. China
6
*Corresponding author: Aiguo PI, State Key Laboratory of Explosion Science and Technology, Beijing Institute of
7
Technology, Beijing 100081, P. R. China; E-mail: [email protected]; Tel: +86-10-68914087 Abstract
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8
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5
The penetration ability of projectiles is closely related to their nose shape as this influences the high-velocity or ultrahigh-velocity
10
kinetic-energy penetration of a rigid projectile. Based on classical cavity expansion theory and a double-ogive-nose scheme, this study
11
set out to analyze the dependency relationship between the nose shape factor N* and the double-ogive characteristic parameters. This
12
paper first discusses the effects of different nose shapes on the penetration performance of a projectile. The paper then proposes a
13
double-ogive-nose penetration body scheme with a low penetration resistance. A penetration experiment with the double-ogive-nose
14
projectile is set up and a Holmquist–Johnson–Cook concrete [1] model is described, together with its application to the prediction of
15
the tendency of the depth of penetration. A comparison between the results of the experiment, simulation, and calculation for the ogive
16
nose and double-ogive-nose projectiles proves that the analysis and the proposed method are both reasonable and feasible.
17
Keywords
19
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18
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Penetration; Double-ogive nose; Holmquist-Johnson-Cook (HJC) concrete model; Depth of penetration (DOP); Projectile
Highlights:
20
A double-ogive-nose penetration body scheme with a low penetration resistance is proposed.
21
The influence of the nose shape coefficient N* on the penetration resistance and DOP is discussed.
22
The feasibility of the proposed method is validated through theoretical, experimental, and simulation results.
1
ACCEPTED MANUSCRIPT
Radius of projectile in Fig. 2 and Fig. 4
a1
Rear radius of region 1 in Fig. 6
a2
Front radius of region 3 in Fig. 6
A
Coefficient of static resistance term
A1
Normalized cohesive strength
b
Projectile nose length
B
Coefficient of inertial resistance term
B1
Normalized pressure hardening coefficient
c
A constant given by reference [2]
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a
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Nomenclature
Caliber radius head
d
Projectile diameter
D
Damage parameter
D1
Constant in HJC model
D2
Constant in HJC model
EP
Young’s modulus of projectile
fc’
Unconfined compressive strength of concrete target
fs
Failure strain
F
Penetration resistance
TE D
Penetration resistance of region 1 Penetration resistance of region 2
AC C
F2
EP
F1
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CRH
F3
Penetration resistance of region 3
F’
Intermediate variable
k
k = dN * dx
k1
Constant in HJC model
k2
Constant in HJC model
k3
Constant in HJC model
ks
Slope of line segment CD in Fig. 4
K
Plastic volume modulus
2
ACCEPTED MANUSCRIPT Ke
Elastic bulk modulus
m
Mass of projectile
mnose
Mass of projectile nose Mass of CRH 4.5 ogive nose
N
Pressure hardening exponent
N*
Projectile nose factor
N 1*
Projectile nose factor of region 1
N 2*
Projectile nose factor of region 2
N 3*
Projectile nose factor of region 3
Pd
Final penetration depth
p
Loading on the concrete
pc
Pressure occurring in a uniaxial stress compression test
pl
Pressure of compaction
S
S = 72.0 f c’ − 0.5
S1
Radius of region 1
S2
Radius of region 2
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Normalized maximum strength that can be developed Maximum tensile hydrostatic pressure the material can withstand Normalized maximum tensile hydrostatic pressure
AC C
T*
TE D
Normalized stress
T
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Maximum pressure reached prior to unloading
P*
Smax
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pmax
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mnose_CRH 4.5
v
Cavity expansion velocity
vs
Initial velocity
Vz
Instantaneous velocity
xc
X coordinate of point C in Fig. 4
xd
X coordinate of point D in Fig. 4
y
Nose generatrix equation
y’
Differential of y
yc
Y coordinate of point C in Fig. 4
3
ACCEPTED MANUSCRIPT Y coordinate of point D in Fig. 4
yAC
Equation of curve AC
yDG
Equation of curve DG
y’AC
Differential of yAC
y’DG
Differential of yDG
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yd
Angle in Fig.6
ρ
Density of concrete target
ρP
Density of projectile
σ
Actual effective stress
σr
Stress of cavity expansion
σn
Normal stress on projectile surface
σ n1
Normal stress on region 1 in Fig. 4
σ n2
Normal stress on region 2 in Fig. 4
σ n3
Normal stress on region 3 in Fig. 4
σ*
Normalized effective stress
θ
Angle between projectile axial and nose generatix radius
θ0
Angle between projectile axial and nose generatix radius in Fig. 6
θ1
Angle between projectile axial and nose generatix radius in Fig. 6
θ2
Angle between projectile axial and nose generatix radius in Fig. 6
φ0
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Angle in Fig. 2
Angle between projectile axial and nose tip in Fig. 6
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φ1
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φ
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α
Angle between projectile axial and nose generatix extension in Fig. 6
γ
Variable of integration in Eq.(12)
η
Proportional coefficient in Eq.(11)
µ
Volume strain in Fig. 10
µc
Volumetric strain that occur in a uniaxial stress compression test
µl
Volume strain of compaction
∆µp
Equivalent plastic volumetric strain
µ max
Maximum volumetric strain reached prior to unloading
4
ACCEPTED MANUSCRIPT
µ
Modified volumetric strain
υP
Poisson’s ratio of projectile
ε0
Reference strain rate
∆ε p
Equivalent plastic strain
εf min
Amount of plastic strain before fracture
ε*
Dimensionless strain rate
1. Introduction
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23
Actual strain rate
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ε
Earth penetrating weapons (EPW) are an effective means of attacking hard and deeply buried targets. Therefore,
25
the improvement of their penetration performance is currently the subject of considerable research. The penetration
26
ability of an EPW is commonly enhanced by taking one or more of three approaches, namely, increasing the impact
27
velocity, raising the kinetic energy of the cross-section, and optimizing the nose structure of the penetrator. Under
28
certain tactical conditions when the impact velocity, overall mass, and penetrator diameters are known, the key
29
factor that influences the penetration performance of an EPW is its nose shape.
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Considerable research has been conducted to study the influence of the nose structure on the penetration
31
ability of an EPW. From the viewpoint of the pertinent mechanics, Forrestal et al. [2, 3] created an empirical
32
equation based on cavity expansion theory to estimate the penetration resistance of ogive-nose projectiles. By
33
analyzing the nose force of a penetrator as a general y = f(x) form, Jones et al.[4] obtained a related dimensionless
34
constant, that is, the nose shape coefficient N*, and determined its relationship to the inertial resistance. Chen and
35
Li [5] analyzed the coefficient N* of several different penetrators with different nose shapes, while Zhao et al. [6]
36
introduced a relationship between N* and the velocity. Batra and Chen [7] discussed the effect of friction on the
37
penetration.
38
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Based on a previously developed disk model [8], Yankelevsky [9] optimized the body shape of a penetrator by
5
ACCEPTED MANUSCRIPT minimizing the instantaneous resistance force. An optimal shape was determined by using a single parameter that is
40
relevant to the velocity, overload, and the medium properties. Bunimovich and Yakunina [10, 11] and also
41
Ostapenko and Ostapenko and Yakunina [12, 13] analytically determined the shape of a minimum-drag body and
42
the maximum depth of penetration. However, these works focused mainly on single-curve optimization. Only a
43
limited amount of research has been applied to piecewise optimization.
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This paper presents the dependency relationship between the nose shape factor N* and the characteristic
45
parameters of a double-ogive nose. The effects of different double-ogive-nose projectile designs on the penetration
46
performance are discussed. A penetration body with a double-ogive nose, combined with a smaller penetration
47
resistance, is then introduced. Finally, the results of our experiments and the simulation data are presented.
48
2. Design of double-ogive nose and related calculations
49
2.1 Design of double-ogive nose based on cavity expansion theory
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Cavity expansion theory was originally established and developed for ductile metals [14]. Bishop et al. [15]
51
first proposed spherical and cylindrical cavity expansion for the (quasi) static expression of metals. Motivated by
52
this proposal, Hill [16] presented the dynamic expansion of the spherical cavity expression for metals. Spherical
53
cavity expansion theory was applied to concrete by Luk and Forrestal [17], which was simplified by fitting and
54
obtaining the expression of the normal Cauchy stress component, as follows:
EP
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50
σn
v = A+ B ' ' 1/2 fc ( fc / ρ )
2
(1)
ρ
56
where f c ' is the unconfined compressive strength of the concrete target,
57
coefficients A and B are the resistance coefficients of the material obtained from the fitting. In the same manner as
58
for the resistance equation for index-hardening metal material, Eq. (1) consists of a static resistance related to Y and
59
an inertial resistance related to the penetration velocity. Pi and Huang [14] discussed the ratio of the static
60
resistance to the inertial resistance during the process of penetrating 2024-O aluminum targets and concrete of a 6
is the density of the target, and
ACCEPTED MANUSCRIPT range of different strengths, as shown in Fig. 1. The results show that the ratio of static resistance increases
62
proportionally with the strength of the target in the resistance function. Moreover, the inertial resistance ratio
63
increases with the penetration velocity in the resistance function.
(a) Resistance of 2024-O aluminum target
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(b) Resistance of concrete target
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Fig. 1 Comparison between ratios of static term and inertial term in resistance function [14]
Fig. 2 Force on the cross-section of the tip
66
Jones et al. [4] analyzed the normal penetration of a rigid axisymmetric projectile into a semi-infinite target.
67
The force on the cross-section of the tip is shown in Fig. 2. The length of the nose is b and the radius of the shank
68
of the projectile is a. For all acceptable nose geometries, function y = y(x) with the boundary conditions y(0) = 0
69
and y(b) = a. The effects of friction in this analysis are assumed to be negligible for the whole projectile. The
70
expression of resistance along the x axis, which is the direction of the projectile’s movement, is introduced. The 7
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equation can be expressed as follows: F = ∫ dF = 2π ∫ yy ′σ n (V z , φ )dx b
72
(2)
0
73
where σ n (V z , φ ) is the normal stress on the projectile surface. Luk and Forrestal [18] and Forrestal and Luk [19] adopted the normal velocity of the cavity surface as a
75
geometric function in discussing the penetration pressure. The normal component of the velocity acting on the
76
projectile nose is given as follows:
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v = V z sin θ = V z cos φ
78
where Vz is the instantaneous velocity in the direction of motion of the projectile. Considering the
79
relationship between the normal stress in the cavity surface and the cavity expansion velocity, as given by the
80
cavity expansion theory, Forrestal and Luk [19] derived Eq. (4) to express the relationship between the normal
81
stress acting on the projectile surface and the penetration velocity.
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(3)
σ n (Vz ,φ ) = Afc' + Bρ (Vz cosφ )
83
Based on the studies conducted by Jones et al. [4], without friction, we introduce the concept of the nose shape
84
factor N*, which is expressed as follows:
EP
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N* =
2 a2
∫
b
0
yy ′3 dx 1 + y ′2
2
(4)
(5)
As shown in Eq. (5), N* is a dimensionless number and is only related to the geometrical nose shape. N*
87
quantitatively characterizes the nose shape and can be used as a quantity for evaluating the inertial resistance acting
88
on the projectile during penetration. Then, Eq. (2) can be written as
89
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86
F = π a2 ( BN * ρVz2 + Afc' )
(6)
90
To simplify the formula A and B can be approximated using the tri-axial material data. The measured data has
91
shown that B varies over only a very small range for a wide variety of materials and can be approximated to 1.0. A
92
can be set equal to S, which is a dimensionless parameter that modifies the compressive strength shown by
8
ACCEPTED MANUSCRIPT reference [3], S = 72.0 f c’ − 0.5 as presented by Li and Chen [20]. With this, Eq. (6) becomes
F = π a 2 ( Sfc’ +N * ρVz 2 )
94 95 96
z ≥ 4a
(7)
During crater formation, the projectile was shown to decelerate linearly so that the axial force acting on the projectile can be taken to be F = cz
97
0 ≤ z < 4a
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(8)
where z is the depth of the penetration from the surface of the target and c is a constant given by reference [3]. As
99
shown by Eq. (6) and (7), given the projectile diameter and initial velocity, the optimization of the penetration
100
resistance is related only to the nose shape coefficient N*. In engineering practice, given the penetration stability
101
and overall design limitation, armor-piercing or half armor-piercing penetrators are generally designed with an
102
ogive nose; the caliber radius head (CRH) is generally between 3 and 5 to prevent the nose from becoming overly
103
long [21]. In this study, the nose length within the scope of CRH 5 was studied to obtain the optimal scheme with a
104
small coefficient N*. Then, the penetration depth was obtained by Forrestal [3], as follows:
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Pd =
105
N * ρV12 m ln 1 + + 4a, 2π a2 ρ N * Sfc'
P > 4a
(9)
Where Pd is the final penetration depth, and V 1 is the velocity at the end of the crater given by reference [3] as
107
V12 =
EP
106
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mvs2 − 4π a 2 Sf c' , where vs is the intimal velocity. m + 4π a3 N * ρ
108
A CRH 3 projectile is selected and Eq. (5) is used to further compare the change in N * . To further discuss the
109
change rules for parameter N*, a differential coefficient k which governs the change ratio of N* ( k = dN * dx ) is
110
introduced and a mathematical method can be used to explore the problem. N* and k are shown as curves only for
111
an ogive nose and CRH > 0.5.
9
ACCEPTED MANUSCRIPT 0.11
0.040
0.10 0.035
Ogive nose of CRH 3
0.08
0.030
0.07
0.025
0.06
k
N*
0.09
0.020
0.05 0.015
0.04 0.03
0.005
0.01 0.00 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.000 0.0
1.0
0.1
0.2
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0.010
Ogive nose of CRH 3 0.02
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
x/b
x/b
Fig. 3 (a) Relationship between N* and position
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Fig. 3 (b) Relationship between k and position
As shown in Fig. 3(a), the relationship between N* and the position on the nose are presented, with the
113
x-coordinate signifying the dimensionless position on the nose, and the y-coordinate showing the integral value
114
from the tip to this position, as given by Eq. (5). N* is a monotonously increasing function which increases quickly
115
in the first half and then slowly in the second half. It indicates the change ratio of N* shown in Fig. 3(b) whereby k
116
reaches a maximum when x/b = 0.208. Therefore, the region on both sides of x/b = 0.208 is selected to modify the
117
nose. An ogive nose is usually adopted for a projectile because it is superior to a conical nose in terms of preventing
118
ricochet. At the end of the nose, the ogive is tangential to the projectile shank, thus ensuring a greater filling mass
119
and avoiding the stress concentration in the penetration process. only the middle of the projectile nose is optimal by
120
linear transition. To attain better stability at the beginning of impact, the ogive part at the nose tip should be a little
121
longer. Meanwhile, the tactical and technological aspects of the oblique and yaw angles should be considered, such
122
that if a high impact angle adaptation is needed, the ogive at the tip should have a small diameter. For the ogive at
123
the end of the nose, the ogive prior to optimization can be applied to avoid stress concentration in the penetration
124
process. The length of the middle region should generally be controlled such that it is no more than one third of the
125
nose length. Then, the region where k > 0.027 is selected to optimize the design and minimize the increase in N* as
126
far as possible. The modified nose is shown in Fig. 4, which also introduces the conical and ogive shapes;
127
AEFHGDC encloses the modified region wherein ACE represents region 1, CEFD represents region 2, and DFHG
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10
ACCEPTED MANUSCRIPT represents region 3. ABG represents the shape of CRH 5 and ODG represents the shape of CRH 3.
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129 130
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Fig. 4 Contrast between modified and original projectiles
For the modified projectile, ks is the slope of line segment CD; yAC and yDG are the curve equations of AC and
132
DG. According to the principle of the subsection integral, the nose coefficients N* can be added together to give the
133
value of N* for the three regions, as given by Eq. (10).
134
* * * * N mod ified = N1 + N 2 + N3 =
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3 3 xd k (k x − k x + y ) b y 2 xc y AC y '3AC s s s D D DG y 'DG dx + dx + dx 2 ∫0 2 2 2 ∫ ∫ xc xd 1 + y ' a 1 + y 'AC 1 + ks DG
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0.11
(10)
0.10 0.09
Ogive nose of CRH 3
0.08
N*
0.07 0.06
EP
0.05 0.04
Ogive nose of CRH 5
0.03
Double-ogive nose
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0.02 0.01
0.00 -0.4
135 136
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
x/b
Fig. 5 Relationship between N* of the three projectile types and nose position
137
The significance of the double-ogive-nose projectile is to find a relatively superior geometrical shape between
138
CRH 3 and CRH 5 that can reduce the resistance during penetration. Fig. 5, in which the x-coordinate is the same
139
as in Fig. 4, presents the relationship between N* and the nose positions of the three types of noses. For the same
140
length of projectile nose, the value of N* for the double-ogive-nose projectile is smaller than that for a typical ogive 11
ACCEPTED MANUSCRIPT 141
nose.
142
2.2 Calculation analysis of penetration resistance For a double-ogive nose such as that shown in Fig. 6, the resistance is divided into three parts where region 1
144
at the tip is an ogive with radius S1, region 2 in the middle is a cone with a1 and a2 (distance from the central axis to
145
the two ends), and region 3 at the root is an ogive with radius S2.
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146 147
Fig. 6 Dimensions of double-ogive nose
To analyze the force on each part of a double-ogive-nose projectile in more detail, considering classical cavity
149
expansion theory, the integral equation is used to solve the stress that arises during the penetration. The force on
150
each unit is calculated, then integrated, and then iterated with time, so as to obtain the penetration depth more
151
accurately. Otherwise, it proves that the resistance of a double-ogive-nose projectile is smaller than a normal ogive
152
projectile. The force on the nose was discussed by Forrestal et al. [22] using both spherical and cylindrical
153
approximation. However, considering the uniformity with the force of an ogive-nose projectile, the spherical cavity
154
expansion stress is used for the cone part. The resistance of a double-ogive nose is obtained from Eq. (7).
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F = F1 + F2 + F3 155
φ1
θ1
φ0
θ0
= ∫ σ n1 ⋅ 2π S12 (sin φ − sin φ0 )cosφ dφ + (a22 − a12 )πσ n2 + ∫ σ n3 ⋅ 2π S22 (sin γ − sin θ2 )cos γ dγ
(11)
156
where σ n1 , σ n2 , and σ n3 are the normal stresses in regions 1, 2, and 3, respectively; F1, F2, and F3 are the
157
resistances
158
σn2 = Sfc' + ρ (Vz sinα ) , σn3 = Sfc' + ρ (Vz cosγ ) . φ and γ represent the integral variables shown in Figure 6.
of
regions 2
1,
2,
and
3,
respectively. 2
12
Then,
ρ = 2.5 g / cm 3 ,
σn1 = Sfc' + ρ (Vz cosφ ) , 2
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159
Fig. 7 Dimensions of CRH 3, double-ogive nose, and CRH 5 projectiles (mm)
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General ogive noses, that is, CRH 3 and CRH 5, as well as a double-ogive nose with a head length similar to
162
that of CRH 5, were selected as shown in Fig. 7. The three projectiles are all assumed to be rigid with a similar
163
mass of 450 kg and an initial velocity of 500 m/s and 1000 m/s. Head erosion during the penetration process was
164
ignored, and the unconfined compressive strength of the concrete target was 35 MPa.
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（a）vs = 500 m/s
（b）vs = 1000 m/s
Fig. 8 Deceleration history of three projectile types
13
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60
16 50
14 40
Pd /d
Pd /d
12 10
30
8 6
20
Ogive nose of projectile CRH 3 Ogive nose of projectile CRH 5 Double-ogive nose projectile
2
Ogive nose of projectile CRH 3 Ogive nose of projectile CRH 5 Double-ogive nose projectile
10
0
0
0
2
4
6
8
10
12
14
16
0
18
5
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10
15
20
25
30
t (ms)
t (ms)
（b）vs = 1000 m/s
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（a）vs = 500 m/s
Fig. 9 Dimensionless penetration depth history of three projectile types
The deceleration history of the three projectile types is shown in Fig. 8. The double-ogive-nose projectile has
166
the smallest deceleration peak, whereas the ogive nose projectile of CRH 3 has the largest peak. This phenomenon
167
proves that the penetration resistance of the double-ogive-nose type is smaller than that of the other two types [Fig.
168
8].
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The depth of penetration (DOP) history with its clearly demonstrated increment is shown in Fig. 9. The degree
170
of penetration at a low velocity is smaller than that at a high velocity. This phenomenon intuitively explains the
171
relationship between the nose shape factor N* and DOP and indirectly illustrates the influence of N* on the inertial
172
term. The calculated results prove that a double-ogive-nose projectile is subject to a small penetration resistance
173
and that the nose shape makes a major contribution to DOP.
174
3. Experimental Results and Simulation Analysis
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175
A penetration experiment with a double-ogive-nose projectile was performed by the authors to validate the
176
material model of simulation. Then, a numerical simulation parameter (failure strain) that can be used to predict the
177
penetration depth for different impact velocities and structures by simulation was determined from the measured
178
results.
14
ACCEPTED MANUSCRIPT 179
3.1 Material model used in simulation LS-DYNA3D was used for the simulation. The parameters were verified with the aid of experimental data.
181
The concrete was described by the Holmquist-Johnson-Cook (HJC) model [1] which is widely used for concrete. It
182
is similar to the Johnson–Cook model that is used for metals. The HJC model is commonly used to describe the
183
dynamic response of brittle material damage when that material is subjected to large strains, high strain rates, and
184
high pressures. The flow rule, consistency conditions, and strengthening rule are not strictly observed in this model.
185
Therefore, the model can describe the dynamic behavior leading to the penetration of a concrete target.
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186 187
190
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The sectional state equation is used to describe the pressure and volume strain of the HJC model (Fig. 10). The first stage (OA) is linearly elastic at p < pc with
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188
Fig. 10 Pressure and volume strain of HJC concrete model
p = Ke µ
(12)
191
in which K e = pc µ c is the elastic bulk modulus. p c and µ c are the pressure and volumetric strain, respectively,
192
which occur in a uniaxial stress compression test. The relationship between the pressure and volume strain are
193
linear at this stage.
194
The second stage (AB) is referred to as the transition stage and occurs at p c ≤ p ≤ pl . In this stage, the air
195
voids are gradually compressed out of the concrete, thus producing plastic volumetric strain. The loading is
196
described as 15
ACCEPTED MANUSCRIPT p = pl +
197
( pl − pc )( µ − µl )
(13)
µl − µc
where p l and µ l are the pressure and volume strain of compaction, respectively. The unloading in this stage
199
occurs along a modified path that is interpolated from the adjacent regions, as follows: p − pmax = (1 − F ’ ) K e + FK ( µ − µ max )
200
(14)
201
Furthermore, F = ( µmax − µc )
202
maximum pressure and volumetric strain, respectively, reached prior to unloading. The air voids in the concrete are
203
gradually eliminated, the structure is damaged, and a crack begins to appear at this stage.
205
SC
have been removed when the pressure reaches p l . This relationship is expressed as 2
p = k1 µ + k2 µ + k3 µ where
3
µ − µl 1 + µl
µ , is used so that the constants ( k 1 ,
The modified volumetric strain,
210
for a material with no voids, assuming that the concrete is fully crushed.
k 2 , and k 3 ) are equivalent to those used
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209
The yield surface of the HJC concrete model is given by
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212
(15)
(16)
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µ=
208
211
p max and µ max are the
The last stage (BC) defines the relationship for a fully dense concrete material from which all air voids
206 207
K is the plastic volume modulus.
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( µl − µc ) ， and
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σ * = A1 (1 − D ) + B1 P*N (1 + C ln ε * )
(17)
213
and σ * = σ f c’ is defined as the normalized effective stress,
214
( 0 < D < 1.0 ), P * = p f c' is the normalized stress, and ε * = ε ε 0 is the dimensionless strain rate (where
215
the actual strain rate and ε 0 =1 .0 s − 1 is the reference strain rate). A1 is the normalized cohesive strength, B1 is the
216
normalized pressure-hardening coefficient, N is the pressure-hardening exponent, C is the strain rate coefficient,
217
and Smax is the normalized maximum strength that can be developed.
218
σ
is the actual effective stress, D is the damage ε
is
Fracture damage is accumulated in a manner similar to that used in the Johnson–Cook fracture model [23]. 16
ACCEPTED MANUSCRIPT 219
The Johnson–Cook fracture model accumulates damage from the equivalent plastic strain. The model discussed
220
herein accumulates damage from both the equivalent plastic strain and plastic volumetric strain, and is expressed as D=∑
221
∆ε P + ∆µ P ε Pf + µ Pf
(18)
222
where ε pf + µ pf = D1 ( P * + T * ) , ∆ ε P , and ∆ µ P are the equivalent plastic strain and plastic volumetric strain,
223
respectively. D 1 and D2 are constants. T * = T f c' is the normalized maximum tensile hydrostatic pressure,
224
where T is the maximum tensile hydrostatic pressure the material can withstand.
225
strain that occurs before a fracture occurs, so as to prevent fractures from occurring under the influence of
226
low-magnitude tensile waves for which
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D2
min
is the amount of plastic
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εf
ε f = 0.01 .
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Table 1 lists the parameters for the concrete target used in the simulation. In view of the failure behavior
228
expressed by the eroding failure of a Lagrange mesh, the material failure strain fs is a critical parameter related to
229
DOP. fs = 0.38 is set in the simulation to prevent calculation errors.
230
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Table 1. Parameters for concrete target material
ρ (kg/m3)
E (GPa)
K (GPa)
A
B
C
N
fc (MPa)
2440
20.68
14.86
0.79
1.6
0.007
0.61
48
T (GPa)
ε0
εf
Smax
Pc (GPa)
µc
Pl (GPa)
µl
0.004
6
0.01
7.0
0.016
0.001
0.80
0.10
D1
D2
k1 (GPa)
k2 (GPa)
k3 (GPa)
fs
85
-171
208
0.38
EP
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0.04
min
1.0
231
For the experiment, the projectile was made of high-strength steel, which exhibits very little plastic
232
deformation or mass loss during penetration. Thus, the projectile can be defined as being a rigid body with a
233
density ρp = 7850 kg/m3. In the LS-DYNA program, the Young’s modulus Ep = 2.158 × 105 MPa and Poisson’s ratio
234
vp = 0.3 are also needed [24].
17
ACCEPTED MANUSCRIPT 3.2 Experimental results and finite element model
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236 237
Fig. 11 Models of concrete targets and projectiles used in the experiment
Experimental data for the structural parameters of the double-ogive-nose projectile and the target is listed in
239
Table 2. The finite element model is shown in Figure 11. To reduce the number of grids and improve the efficiency
240
of the calculation, a circular semi-infinite target was used in place of the square target used in the numerical
241
simulation. For the penetration experiment, the projectile was launched from a 155-mm gun, with caliber launching
242
technology so as to achieve the required velocity. The sabot was made of a lightweight and brittle material such that
243
its effects on the penetration ability could be ignored. The projectile penetrated the concrete vertically. The
244
dimensions of the concrete target used in the experiment were 3.6 m × 3.6 m × 5.4 m, respectively. The nose length
245
of the double-ogive-nose projectile was equivalent to a CRH 4.5 ogive nose. Then, the comparison in the
246
simulation was between CRH 3, CRH 4.5, and the experimental double-ogive-nose projectile described below.
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247 NO.
DO1
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Table 2 Projectile and target data for double-ogive-nose projectile penetration experiment Mass
Strength of target
Projectile length
Projectile diameter
Initial velocity
DOP
(kg)
(MPa)
(m)
(m)
(m/s)
(m)
81.1
48
1.13
0.14
835
5.1
248
The experimental results show that the penetration depth of the double-ogive-nose projectile is 5.1 m. With the
249
verified numerical simulation parameters, Fig. 12 shows the change in the velocity and the contrast in the
250
dimensionless penetration depth between the numerical simulation and the experiment, which proves that the
251
selected parameters are reasonable and reliable. This group of parameters can be used to investigate the DOP rules 18
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900
35
800 700
30
600
Simulation DOP Simulation velocity experiment DOP
20
500
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P/d
25
velocity (m/s)
252
400
15
300
10
200
5 0 0
3
6
9
12
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100
15
0 18
time (ms)
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Fig. 12 Comparison of velocity between simulation and experiment
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255 256
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Fig. 13 Finite element models for three projectile types
To prove that the double-ogive-nose projectile can enhance the penetration ability, two projectile structures
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similar to the double-ogive nose were selected to study the DOP rule with the previously selected simulation
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parameters. Three types of nose projectiles were then selected for the penetration simulation shown in Fig. 13. The
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first projectile has a common ogive nose with CRH 3; the second has a double-ogive nose, and the third is a CRH
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4.5 ogive nose that has the same nose length as the double-ogive-nose projectile. The three projectiles were
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assumed to have the same mass (with the aid of a filler), initial velocity, and target.
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3.3 Results and discussion
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Four velocities were selected for the simulation. Table 3 compares the results of the theoretical calculation and 19
ACCEPTED MANUSCRIPT the numerical simulation, proving that the double-ogive nose has a better penetration ability than the equivalent
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nose or CRH 3 nose. With the same initial velocity, the DOP of a double-ogive-nose projectile can be increased by
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10% and 5%, relative to a common CRH 3 projectile and equivalent nose projectile, respectively. At a high velocity,
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the DOP increment of a double-ogive nose is large but the growth rate is not obvious. For high-velocity penetration,
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the nose shape has a significant influence on the DOP.
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Table 3 Contrast between theoretical calculation and numerical simulation
Velocity (m/s)
Theoretical
Simulation
Calculation
DOP (m)
DOP (m) 2.248
500
2.454
Ogive nose of CRH 4.5
500
2.255
Ogive nose of CRH 3
500
2.174
Double-ogive nose
800
4.971
Ogive nose of CRH 4.5
800
4.652
Ogive nose of CRH 3
800
4.443
Double-ogive nose
1000
6.890
Ogive nose of CRH 4.5
1000
6.423
Ogive nose of CRH 3
1000
6.120
Double-ogive nose
1200
Ogive nose of CRH 4.5
1200
Ogive nose of CRH 3
1200
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Double-ogive nose
8000
2000
12.9
8.82
9.47
--
--
4.837
2.66
11.9
6.86
4.507
3.22
4.10
--
4.358
1.95
--
--
6.893
0.04
12.6
7.27
6.490
0.57
4.95
--
6.171
0.83
--
--
8.917
9.082
1.85
13.4
7.16
8.321
8.593
3.17
5.87
--
7.860
8.043
2.28
--
--
10000
Deceleration(g)
3000
8.39
(%)
1.986
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Deceleration(g)
4000
(%)
Equivalent Nose
--
vs=500m/s, double-ogive nose
5000
to CRH3 (%)
3.73
vs=500m/s, CRH 4.5 nose
6000
Deviation
DOP Increment to
7.43
vs=500m/s, CRH 3 nose
7000
DOP Increment
2.099
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Relative
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Projectile
Numerical
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Impact
9000
vs=800m/s, CRH 3 nose
8000
vs=800m/s, CRH 4.5 nose vs=800m/s, double-ogive nose
7000 6000 5000 4000 3000 2000
1000
1000 0 0
1
2
3
4
5
6
7
8
9
10
11
12
13
0
14
0
time(ms)
2
4
6
8
10
time(ms)
(a) vs = 500 m/s
(b) vs = 800 m/s
20
12
14
16
18
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12000
vs=1200m/s, CRH 3 nose
vs=1000m/s, CRH 3 nose
12000
vs=1000m/s, CRH 4.5 nose vs=1000m/s, double-ogive nose
Deceleration(g)
Deceleration(g)
10000
8000
6000
vs=1200m/s, CRH 4.5 nose vs=1200m/s, double-ogive nose
10000 8000 6000
4000
2000
2000
0 0
2
4
6
8
10
12
14
16
18
0
20
0
time(ms)
2
4
6
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4000
8
10
12
14
16
18
20
22
time(ms)
(d) vs = 1200 m/s
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(c) vs = 1000 m/s
Fig. 14 Deceleration time-history curves for three projectiles with different velocities
The deceleration time-history curves under different conditions are shown in Fig. 14. The double-ogive-nose
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projectile has a smaller peak deceleration value and a wider pulse than the other two projectiles. It indicates that the
274
penetration resistance of a double-ogive nose is small and that the penetration time is long, which indicates that the
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double-ogive-nose projectile has a better penetration ability.
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80 70
50
CRH 3 simulation CRH 4.5 simulation Double-ogive nose simulation Jones optimal simulation [4] Double-ogive experiment CRH 3 calculation CRH 4.5 calculation Double-ogive calculation Jones optimal calculation [4]
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Pd /d
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90
40 30
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20 10
0 400
276 277
500
600
700
800
900
1000
1100
1200
1300
1400
v(m/s)
Fig. 15 Dimensionless DOP with different penetration velocities
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Fig. 15 shows the relationship between the dimensionless DOP and the impact velocity. The figure also shows
279
the simulated and calculated results for four types of projectiles, and the results of an experiment with a
280
double-ogive-nose projectile. It indicates that the penetration depth as determined by the theory-based calculation,
21
ACCEPTED MANUSCRIPT numerical simulation, and experiment are identical, which again proves the correctness of the theory-based
282
calculation. The penetration data obtained with impact velocities of between 400 m/s and 1400 m/s can be
283
predicted and the penetration ability of different projectile structures can be compared by applying the theory model.
284
With low-velocity penetration, the advantage of Jones [4] and double-ogive-nose projectile’s penetration ability is
285
not obvious. However, with high-velocity penetration, the double-ogive-nose projectile offers significant
286
advantages in terms of DOP over an ogive-nose projectile of the same mass. The DOP increment of the
287
double-ogive nose is 0.60 m over the CRH 4.5 ogive nose and 1.06 m more than that of the CRH 3 nose for an
288
initial velocity of 1200 m/s. This proves that nose optimization plays an essential role in reducing the penetration
289
resistance and enhancing DOP, especially for high-velocity penetration.
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1.0
y/a
0.8 0.6 0.4 0.0 0.0
0.1
290 291
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0.2 0.2
0.3
0.5
0.6
N*=0.0553 N*=0.0465 N*=0.0608 N*=0.0720
0.7
0.9
0.8
1.0
x/b
Fig. 16 Four types of nose shape outline
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Tab. 4 Contrast of four types of nose shape
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Conical nose
293
0.4
Cone Nose Jones Nose [4] Double-ogive Nose Ogive Nose CRH 4.5
Jones optimal nose
Double-ogive nose
CRH 4.5 Ogive nose
N*
0.0533
0.0465
0.0608
0.0720
mnose (kg)
11.58
14.12
16.44
18.87
mnose/mnose_CRH4.5
0.61
0.75
0.87
1
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Four types of nose shape outline curves and their N* are shown in Fig. 16. The nose shape factor N* was
295
calculated for each of these. It can be seen that the N* of the Jones optimal nose exhibits the minimum value, that
296
the conical nose is second only to the Jones projectile, and that the N* of the double-ogive nose is smaller than that
22
ACCEPTED MANUSCRIPT of the CHR 4.5 ogive nose and close to that of the conical nose. The N* contrast between the double-ogive nose
298
projectile and Jones optimal projectile proves the correctness of the results in Fig. 15, which indicates that the
299
penetration ability of the Jones optimal projectile is better than that of the double-ogive nose projectile, as
300
determined by theory calculation and simulation. However, from the viewpoint of practical engineering
301
application, because of the limitation of the overall design platform, the penetrator should increase the quality of
302
the payload in the limit space as much as possible, as shown in Table. 4. Therefore, it is not common to choose the
303
conical nose shape in that it has a low-quality utilization ratio. However, for most engineering applications, an
304
ogive-nose projectile is widely used [2, 3, 21, 25-37]. The optimization described in this paper is based on the ogive
305
nose, and the N* of a double-ogive nose is close to that of the a conical nose. The mass utilization ratio is 41.97%
306
more than that of a conical nose, and 16.43% more than the Jones optimal nose.
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4. Conclusion
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Based on cavity expansion theory, projectiles with different nose shapes were examined and the influences on
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their penetration ability were analyzed. Consequently, a double-ogive-nose structure is proposed. The influence of the nose shape coefficient N* on the penetration resistance and DOP was discussed based on
311
the results of the calculation. The resistance consists of the inertia resistance and the static resistance. The former is
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related to the penetration velocity and the latter is associated with the material performance. As the velocity
313
increases, the percentage of the inertia item increases in the resistance function and thus affects the DOP with
314
high-velocity penetration.
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For a given nose length, the double-ogive-nose projectile has a smaller nose shape coefficient N* than that of an ogive nose, and a larger mass than the Jones optimal nose.
317
A set of consistent target parameters was verified through experiment, and these parameters were used to
318
simulate the penetration performance of the three projectile types. The results were found to be consistent with the
23
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theoretical calculation, again proving the correctness of the theoretical model.
320
Acknowledgements
321
This work was supported by the National Natural Science Foundation of China under Grant (No.11221202) and the National Natural Science Foundation of China (No.11202029, No.11390362).
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