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Analytical models for penetration mechanics: A Review
Accepted Manuscript
ANALYTICAL MODELS FOR PENETRATION MECHANICS: A REVIEW Charles E. Anderson Jr. PII: DOI: Reference:
S0734-743X(16)31029-6 10.1016/j.ijimpeng.2017.03.018 IE 2878
To appear in:
International Journal of Impact Engineering
Received date: Revised date: Accepted date:
8 December 2016 22 March 2017 22 March 2017
Please cite this article as: Charles E. Anderson Jr. , ANALYTICAL MODELS FOR PENETRATION MECHANICS: A REVIEW, International Journal of Impact Engineering (2017), doi: 10.1016/j.ijimpeng.2017.03.018
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Highlights Analytical penetration models, including assumptions and results, are reviewed.
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Applicability and limitations of penetration models are highlighted.
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ANALYTICAL MODELS FOR PENETRATION MECHANICS: A REVIEW
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Charles E. Anderson, Jr.*
CEA Consulting, San Antonio, TX 78240 USA
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ABSTRACT
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A review of analytical penetration models has been conducted and summarized. Models include the Poncelet equation, hydrodynamic theory, modified hydrodynamic theory, Recht-Ipson, TateAlekseevskii, cavity expansion, Ravid-Bodner, Walker-Anderson, and similitude modeling. These models describe, depending upon assumptions, rigid-body penetration, eroding penetration, steady-state and transient penetration, and perforation. Model assumptions are highlighted, and examples are provided of model predictions against experimental data. The manuscript has 30 figures, many of which compare model results to experimental data; 59 reference citations are included.
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Keywords: penetration mechanics; analytical modeling; plastic-flow fields
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* Email: [email protected]
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NOTATION model constants
vc
critical velocity, Eq. (33)
A(r,)
vector potential, Eqs. (50) and (59)
vtr
transition velocity, Eq. (34)
a
projectile radius (Fig. 14), Eq. (39)
V
impact velocity
B, b
model constants
V
nondimensional velocity, Eq. (65)
Bmax
maximum hardness at bottom of crater, Eq. (16)
Vr
residual velocity
c
bar wave speed
V50
ballistic limit
co
bulk sound speed
W
plastic work
D
projectile diameter
Ws
work in shear, Eq. (20)
Dij
rate of deformation tensor, Eq. (62)
Wt
plastic work in target, Eq. (24)
d
hole diameter
Y
flow stress (strength)
eˆ r , eˆ , eˆ
unit vectors in r, , and directions
Yd
dynamic flow stress, Eq. (47)
E
Young’s modulus
z
direction of penetration
Ed
deformation energy, Eq. (19)
(p/t)1/2, Eq. (65)
F, Fn, Ft
Force, normal force, tangential force
~c
(1 + h/Rc), Eq. (61)
G
shear modulus
,
axial and radial extent of plastic flow field (Ravid-Bodner model)
s
Eq. (43)
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distance between projectile nose to target rear (back) surface
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h
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A, a
K
bulk modulus
cRc
extent of plastic flow field (WalkerAnderson model)
k
shock-particle velocity constant
blending parameter, Eq. (60)
L
projectile length
s
Eq. (43)
M, Mp
projectile mass
,
hardening exponents, Eq. (47)
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mass of L/D 1 projectile, Eq. (16)
, crit
strain; critical strain to failure
mp
plug mass, Eq. (17)
p
plastic strain rate
P
penetration depth
strain hardening coeff., Eq. (47)
Pc
penetration depth of a L/D = 1 projectile, Eq. (16)
dynamic effect parameter, Eq. (38)
Pr
pressure
instantaneous projectile length
p
Lambert-Jonas exponent, Eq. (26)
μ
(t /p)1/2, Eq. (32a)
R
projectile radius
f
friction coefficient, Eq. (40)
Rc
penetration channel (crater) radius
Poisson’s ratio
Rt
target resistance
~ R
(c – 1)Rc, Eq. (60)
s
extent of plastic zone in projectile
T
plate thickness
t
time
u
penetration velocity
uback
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m
density
t
dynamic yield strength
strength difference, Eq. (14)
stress
efs
effective flow stress
n
normal stress, Eq. (39)
target rear surface velocity
t
tangential stress
udebris
velocity of eroded projectile debris, Eq. (37)
xz
target shear stress, Eqs. (48-49)
us
shock velocity
Yt
target strength
projectile velocity
Yp
projectile strength
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Subscripts: p: projectile; t: target
1.0 Introduction
Several good articles exist in the literature that review penetration, perforation, and other aspects of terminal ballistics. Backman and Goldsmith [1] present an extensive survey (278 references) of the interaction of projectiles and targets. They looked at semi-infinite targets, thin plate penetration and perforation, and intermediate and thick targets, covering the entire velocity range. Jonas and Zukas [2] reviewed available analytical methods for the study of projectile-armor interactions, with an emphasis on numerical simulations (167 references), focused on the velocity range of 0.5 to 2.0 km/s. A special 4
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issue of the International Journal of Engineering Science [3] was dedicated to penetration mechanics, covering experimental investigations, analytical modeling, and numerical modeling. Anderson and Bodner [4] reviewed analytical and numerical modeling of ballistic impact (88 references). Goldsmith [5] provides an extensive review of non-ideal projectile impact on targets (367 references). The above articles concentrate, in varying detail, on experimental procedures/results, data trends, analytical modeling, numerical modeling, and the mechanics of projectile-target interactions.
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There has been considerable progress in analytical modeling since the publication of Refs. [1-4]. This article provides a review of analytical models that have been developed to describe the process of penetration into metallic targets, subject to several constraints. Firstly, the model has to provide significant insight into the mechanics of penetration. This constraint eliminates empirical models that require experimental data to fit an assumed functional form. It is noted that empirical models can be very useful, but their range of validity is limited by the experimental data used to determine the model parameters. Secondly, the model provided a major advance, and was not a slight modification of an existing model. Any errors of omission are those solely the author, either because the model did not meet the selection criteria, or simply because we were unaware of the model.
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In today’s world of large-scale numerical simulations, it can be asked: “Why is there a need for analytical models?” There are several answers to this question. If an analytic model can be developed that captures the essence of the phenomenology, then understanding has been demonstrated, and the essential and relevant physics/mechanics have been applied/incorporated. This fundamental understanding leads to improved designs. Further, analytical models are typically hundreds to thousands times faster in execution than numerical simulations. Thus, analytical models provide tools for predictions, rapid design studies, first-order optimization, and risk assessments. For example, a typical risk assessment might run thousands (maybe millions) of Monte Carlo scenarios, which would be prohibitively time-consuming for numerical simulations, but which are well suited for analytical models.
2.0 Poncelet Equation
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With the above as background, the arrangement of the paper is largely chronological by model development. The focus is on metallic targets.
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Jean-Victor Poncelet (1788–1867) was a French engineer and mathematician. He developed an ordinary differential equation to describe penetration of rigid projectiles. Newton’s second law gives the projectile deceleration as a result of a resisting force:
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dv F A Bv 2 dt
(1)
The A is a static target resistance term, and the Bv2 term states that the resisting force is proportional to the square of the velocity. In traditional fluid mechanics, this v2 term is usually called a drag term. Since the projectile is non-deforming, the cross-sectional area remains constant, and Eq. (1) can be written as:
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pL
dv a bv 2 dt
(2)
(L is the effective length of the projectile if the cross-section is not a constant.) Equation (2) can be integrated to find the total depth of penetration (P) by using the chain rule of differentiation, dv/dt = (dv/dz)(dz/dt) = v(dv/dz):
p
P
0
vdv
dz V a bv L0
Thus,
P p bV 2 ln 1 L 2b a
2
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(3)
(4)
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(A more general case, with a resisting force proportional to the velocity as well as the velocity squared, in addition to the constant term, can be written. Subsequent integration of the equation results in an arctangent term in addition to the logarithmic term. This form of the resisting force is generally not used.)
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Forrestal and Piekutowski [6] measured the depth of penetration for maraging steel projectiles with an ogival nose penetrating 6061-T6 aluminum as a function of impact velocity. The depths of penetration, normalized by the length of the projectile, are plotted in Fig. 1. The Rockwell C hardness of the projectiles varied between Rc38 and Rc53, depending upon the specific alloying and heat treatment. A least-squares regression fit through the experimental data gives the values of the parameters a and b, given by the solid line in Fig. 1 (several points, above 1500 m/s, that fall below the general trend were ignored in the regression analysis; they will be discussed later).
2 P pV L 2a
(5)
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For small values of bV 2/a, the natural logarithm can be expanded to give:
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which is shown as the dashed line in Fig. 1. Thus, the depth of penetration is proportional to the square of the impact velocity. The resistive force, Eq. (2), is approximately a constant, independent of the impact velocity, since the parameter b does not enter into Eq. (5). The normalized penetration for blunt-nose tungsten-alloy projectiles into a 7000 series aluminum target is shown in Fig. 2. The experimental data are from two data sets generated by researchers at the Ernst-Mach-Institut [7-8]. A least-squares regression was conducted on all data below 0.7 km/s; the result is shown with the solid line. However, if four data points that are circled are deleted from the analysis, the resulting fit is shown with the dashed line. The fitted standard error is 45% less for the dashed curve than for the solid curve. Because we know that at sufficiently high impact velocities that the projectile will begin to deform, this Poncelet analysis is suggestive that the operative mechanics (that is, the assumption of rigid-body penetration) is not true for these four omitted data points. Note 6
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that without experimental evidence, it can only be said that this analysis is suggestive. Although not very satisfying, an alternative explanation is that the Poncelet parameters change during penetration. However, a similar analysis of Poncelet fits to the data for a completely different set of experiments, described in Section 6.3, also show that a better fit to the experimental data is achieved if some of the higher velocity data points are not included. In these experiments, recovered projectiles of the deleted data points show considerable deformation.
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One last comment is worth mentioning before proceeding. The small argument expansion, given by Eq. (5), is only valid to about 0.3 km/s in Fig. 2, given the values of a and b (shown in the figure). Thus, for this projectile-target combination, the assumption of a constant resisting force is only valid to about 0.3 km/s; at higher impact velocities, a “drag” resistive force must also be included to reproduce the experimental data.
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We will see that the Poncelet equation will manifest itself several times in the development of penetration models.
3.0 Hydrodynamic Penetration and Modified Theory of Hydrodynamic Penetration 3.1 Hydrodynamic Theory
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The hydrodynamic theory of penetration emerged during WWII and was applied to shaped-charge jet penetration. The theory was developed by Birkhoff, MacDougall, Pugh, and Taylor [9]; the authors acknowledged in a footnote that independent work of Hill, Mott and Pack in England led to the same results. The operative equation is derived from conservation of momentum, assuming that the strengths and viscosity of target and penetrator materials can be neglected, that is, the problem can be treated by hydrodynamics. Thus, the momentum equation is:
v 1 1 v 2 v v Pr t 2
(6)
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where is the vector operator. With the assumption that the jet and target materials are incompressible, the term on the right-hand side can be written as Pr . A further simplification is
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done by using a coordinate transformation, as shown in Fig. 3. It is assumed that the jet moves at a constant velocity v, and that the penetration velocity, u, is also a constant; thus the projectile-target interface does not move in Fig. 3b. The momentum equation, Eq. (6), then becomes
v v
P 1 v 2 r 2
(7)
with the assumptions of hydrodynamic and incompressible materials under steady-state conditions. The dot product of both sides of the equation by the vector velocity gives: 7
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P 1 v v 2 r 2 since
v v is perpendicular to v.
0
(8)
Thus, we have that the gradient of the term in the parenthesis
Along the projectile-target centerline, Eq. (8) becomes:
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is perpendicular to v. The term within the parenthesis is usually called Bernoulli’s equation. It is important to note that while the units of the terms within the parenthesis in Eq. (8) have units of specific energy, Eq. (8) was derived from the momentum equation.1
1 2 Pr 1 2 v v Pr 0 z 2 z 2
(9)
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Equation (9) is integrated from the back of the projectile to the projectile-target interface, and then from the interface into the target. The pressure at the back of the projectile is zero, and the target is assumed to be semi-infinite (the pressure at infinity is zero). Using the nomenclature from Fig. 3b, so that the velocity at the projectile-target interface is zero, and the target velocity at infinity is –u, integration of Eq. (9) gives:
1 1 p v u 2 t u 2 2 2
(10)
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This is the hydrodynamic, steady-state, incompressible model for penetration. Solving for the penetration velocity gives:
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u
v 1 t / p
(11)
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Since penetration is steady state2, the time of penetration is:
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L vu
(12)
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and the penetration depth is P = ut. After rearranging, the normalized depth of penetration is:
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P u p / t L v u
(13)
Most elementary physics textbooks derive Bernoulli’s equation using conservation of energy methods.
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Specifically, the shock phase of penetration is neglected, that is, steady state is reached instantaneously; and there is not a terminal phase of penetration, i.e., penetration stops as soon as the last particle of the jet has struck the target.
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3.2 Modified Hydrodynamic Theory
1 1 p v u 2 t u 2 2 2
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Shaped-charge jets have a velocity gradient. Thus, the jet can be partitioned into a series of connected jets with different velocities. The hydrodynamic theory is then applied to each section of the jet. This procedure overpredicted the final depth of penetration. Eichelberger [10] reasoned that at sufficiently low velocities, particularly into strong (steel) targets, that target strength effects could not be neglected. He modified Eq. (10): (14)
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where = Yt – Yp represents the resistance to plastic deformation. The penetration velocity, Eq. (11), is modified by the addition of to the right-hand side of Eq. (14). A more general expression for the penetration velocity will be given when the Tate model is described. Eichelberger determined that was 1 to 3 times the uniaxial yield strength of the target material (as the jet is typically copper, Yp is very small compared to Yt).
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Allen and Rogers [11] applied the modified hydrodynamic theory to high-velocity impacts of six different rod projectile materials (gold, lead, copper, tin, aluminum, and magnesium) into 7075-T6 aluminum. Instead of , they used the symbol t, which they called the dynamic yield strength of a solid target relative to a fluid jet. They found that t 1.89 GPa, or 3.9 times the yield strength of 7075-T6 aluminum (0.48 GPa). But they also determined that t had to be written as a function of impact velocity to reproduce the final depth of penetration.
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These modifications to the hydrodynamic theory apply to relatively weak projectiles. The assumption is that the projectile is completely consumed, that is, no projectile material remains at the bottom of the penetration channel. This assumption is true only for high velocity impacts and/or weak projectiles. Additionally, the above modifications apply to relatively long projectiles where steady-state penetration is applicable.
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Christman and Gehring [12] described the four phases of high velocity penetration: (I) a transient phase, (II) the primary phase, (III) the secondary phase, and (IV) a recovery phase. Although the concept was correct, their original figure is highly distorted. These phases of penetration are shown in Fig. 4 for an L/D 20 tungsten-alloy projectile impacting a steel target at 3.0 km/s. In Fig. 4, the phases are renamed: (I) shock phase, (II) steady-state phase, (III) terminal (transient) phase, and (IV) recovery (elastic rebound of the bottom of the penetration channel). The relative time duration of phases II and III depend on the aspect ratio of the projectile. The shock phase persists for a very short time (that depends upon the projectile diameter), and penetration during this phase is typically included in the steady-state phase. Christman and Gehring then made a very interesting modification to the hydrodynamic theory that accounted for terminal transient effects; thus, their modification is applicable to small-aspect-ratio projectiles. They separated penetration into a primary (steady-state) phase and a secondary (transient) phase. They stated penetration for the transient phase could be approximately by the penetration of an L/D = 1 projectile. Therefore, for high impact velocities, total penetration is given by: 9
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1
p 2 P ( L D) Pc t
(15)
p Pc 0.13 t
1
3
1 2 mv 2 Bmax
1
3
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where penetration of the primary (steady-state) phase is given by a rod that is one diameter shorter than its initial length. Then they added the crater depth, Pc, obtained for a L/D = 1 rod. Christman and Gehring correlated Pc with experimental data:
(16)
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where Bmax is the maximum hardness at the bottom of the crater (to account for strain hardening of target material). The correlation of Eqs. (15) and (16) with the measured penetration depths is shown in Fig. 5 for rods with aspect ratios of 1 to 25 at impact velocities of 2.0 – 6.7 km/s, for aluminum and steel projectiles impacting aluminum and steel targets (only a subset of the data in Ref. [12] have been plotted).
4.0 Conservation of Energy and Momentum Model
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4.1 Recht-Ipson Model
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Up to this point, the models have considered the targets to be semi-infinite, although assumptions have been invoked by various investigators in using these models to account for finite-thick targets. Recht and Ipson [13] developed a model to estimate the residual velocity of chunky projectiles (L/D ~ 1) impacting relatively thin plates using conservation of momentum and energy considerations.
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The authors considered impact at ordnance velocities where the projectile stays intact, although it can deform (mushroom) slightly. An experimental observation for impacts against thin plates is that a plug is ejected and is attached to the projectile. Application of the conservation of momentum gives:
M pV M p m p Vr
(17)
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where the large M and small m refer to the masses of the projectile and plug, respectively Application of conservation of energy gives:
1 1 M pV 2 M p m p Vr2 Ws Ed 2 2
(18)
where Ws is the energy used to shear the plug and Ed is the energy associated with deformation and heating (plastic work). Ed is simply the difference between the initial and final kinetic energies, which, using Eq. (17) is:
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1 m p Ed M pV 2 2 M p mp
(19)
The energy loss due to shearing the plug is estimated from Eq. (18) by finding the minimum velocity that gives a residual velocity of zero. This minimum velocity is V50. With Vr = 0, and substituting Eq. (19) into Eq. (18), the shear energy is calculated:
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1 M p Ws M pV502 2 M p m p
(20)
Substituting Eqs. (19-20) into Eq. (18) and solving for the residual velocity gives: Mp Vr M m p p
2 V V502
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2
(21)
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d T 1 t p D L 2
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2
V502
1
2
(22)
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Vr
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Assuming that the projectile is nominally a right-circular cylinder, then Eq. (21) can be rewritten in terms of the geometric parameters, with d/D being the ratio of plug diameter to projectile diameter, and T/L the ratio of plate thickness to projectile length:
The authors note that Eq. (22) is valid for T/L < 1/2 and T/D < 1/2.
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Results for three sets of experiments for steel cylinders into mild steel plate are shown in Fig. 6. At relatively low impact velocities, the hard steel fragment does not deform. The authors argue that the effect of the ratio d/D is small, and since d is not usually known, they set the ratio to 1.0. This statement is true only for T/L = 0.15. As the impact velocity increases and the plate thickness increases, the stresses at the projectile-target interface become sufficiently large to deform (mushroom) the projectile. To reproduce the analytical results in Ref. [13], the authors must have assumed that d/D increases with impact velocity; that is, the projectile mushrooms and pushes out a plug with a larger diameter than the original projectile. To reproduce the Recht-Ipson results for T/L = 0.29 and 0.44, it was assumed that d/D goes from 1.0 to approximately 1.4 linearly with impact velocity as the velocity goes from V50 to approximately 950 m/s. The dashed lines nominally replicate the analytical results of Fig. 3 in Ref. [13]. Recht and Ipson [13] examined oblique perforation of thin plates and perforation of thick plates by cylinders. Equation (21) must be modified and additional assumptions and experimental data are required to make analytical predictions. Thus, some of the elegance of the normal impact against a thin plate is lost. Recht and Ipson also examined the case of an armor-piercing (AP) bullet perforating a thick plate [13]. In this case, no plug is ejected; and there is no deformation of the bullet. Thus, all kinetic energy loss is through plastic deformation of the target: 11
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1 1 M pV 2 M pVr2 Wt 2 2
(23)
If it is assumed that the plastic deformation is relatively independent of the impact velocity, then
Wt
1 M pV502 2
(24)
Vr V 2 2 1 V50 V50
1
2
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and after some rearranging, Eq. (23) becomes:
(25)
Experimental data for an AP bullet into a steel target is shown in Fig.7, along with the analytical prediction given by Eq. (25).
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4.2 Conservation of Energy
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A remark is in order concerning the use of conservation of energy in penetration modeling. It is difficult to account for all the various mechanisms that dissipate energy. Further, the proportion of energy dissipation (energy transfer mechanisms) changes with impact velocity. In particular, as the impact velocity increases, projectile kinetic energy is transferred to the target in terms of target kinetic energy and elastic compression energy; this compression energy is dissipated by plastic work at later times [14]. Walker [15] showed that it is the transfer of this energy at the time of penetration that defines the forces on the projectile, and for an energy rate balance to be successful, it must include transfer of energy stored in the target as elastic compression. Although this is done automatically within numerical simulations, this would seem to be intractable within the context of an analytical model.
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In summary, conservation of energy can be useful in analytical modeling, but generally only over a limited velocity range and/or projectile-target combination. For example, Recht and Ipson [13] made assumptions on energy dissipation, but then commented on the restrictions and range of applicability.
5.0 Similitude Analysis
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Similitude analysis, sometimes called scale modeling, is a procedure for casting results in a nondimensional form; for example, see Ref. [16]. Equation (25) is an example of a relationship between a normalized residual velocity and a normalized impact velocity. Often, nondimensionalization allows what would seem to be disparate data to be collapse into a single response curve. One example for penetration mechanics will be given here, and then another will be described later. An experimental study examined the ballistic performance of 17 different projectiles (a tungsten alloy and six different steels with different heat treats) into two different armor steel targets (different hardnesses) [17]. A subset of the data is shown in Fig. 8. The Lambert-Jonas equation [18] provides for a general expression for the residual velocity as a function of impact velocity: 12
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a V p V p 50 Vr 0
1p
V V50 V V50
(26)
Dividing Eq. (26) by V50 gives: V p Vr a 1 V50 V50
1p
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A least-squares regression analysis is used to determine fit parameters a, p, and V50. Note that if the parameter p is equal to 2.0, Eq. (26) is nominally an expression for the conservation of energy [similar to Eqs. (23-25) above], but including the parameters a and p allow for a more general (better) curve fit to experimental data.
(27)
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This suggests that the experimental data should be plotted as Vr/V50 versus V/V50. The data in Fig. 8 are plotted using these nondimensional quantities in Fig. 9. Besides the experimental data in Fig. 8, twelve other projectile-target combinations are also plotted as the light gray hexagons in Fig. 9. All the experimental data collapse to a single curve; the expression for the curve is given in the figure.
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Although nondimensionalization, using the correct choice of parameters, permitted collapse of the experimental data, it does not necessarily explicitly allow one to determine the operative mechanics. Nevertheless, the results of Fig. 9 demonstrate that a substantial amount of basic information concerning projectile and target material response (including deformation and failure) is contained in V50.
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6.1 Eroding Penetration
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6.0 Tate-Alekseevski Model
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In contrast to shaped-charge jets, projectiles (rods) decelerate while penetrating a target. Independently, Tate [19] and Alekseevski [20] proposed a modified Bernoulli equation:
1 1 p v u 2 Y p t u 2 Rt 2 2
(28)
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where Yp is considered to be the dynamic flow stress of the projectile and Rt is the target resistance to penetration. Separating the rigid potion of the rod from that part that is undergoing plastic deformation (mushrooming), the force acting to decelerate the rigid portion of the rod of length is given by:
p R p2
dv R p2 Y p dt
Yp dv dt p 13
(29)
(30)
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The initial length of the rod is L, but since the rigid portion of the rod is moving faster than the penetration velocity, the rod is getting shorter with time:
d v u dt
(31)
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Thus, the three coupled equations, (28), (30), and (31) are solved simultaneously to determine the time history of v, u, and . Equation (28) can be rearranged to solve for the instantaneous penetration velocity in terms of the projectile velocity (which is changing with time) and the material characteristics of the projectile and target:
v v2 A 1 2
t p u
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1/ 2
p t
2Rt Y p 1 2
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u
A
v Rt Y p 2 v
(32a)
t
p t
(32b)
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The solution for u in Eq. (14) is given by Eq. (32) with Yp 0, and Rt replaced by . From Eq. (32), it is seen that there is a critical velocity, vc, were u = 0:
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2Rt Yp vc p
1/ 2
(33)
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That is, when the rod velocity drops to the critical velocity, there is no further penetration. We will examine procedures for estimating Yp and Rt after the next couple of subsections.
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6.2 Rigid-Body Penetration and Transition from Rigid-Body to Eroding Penetration
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In the previous subsection, Rt > Yp and there is erosion of the rod. Tate examined the case where Yp > Rt [21].3 For this situation, there exists a transition velocity, vtr, at and below which the rod no long erodes, that is, penetration occurs in rigid-body mode. For rigid-body penetration, u = v. The transition velocity is found from Eq. (32a) by equating u and v vtr, and solving for vtr:
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Notwithstanding that the model described in Section 2.5.1 is usually called the Tate-Alekseevki model, Addison Tate extensively explored the modified Bernoulli equation, including rigid-body penetration, transition of rigid-body to eroding penetration, and procedures for estimating Yp and Rt. Thus, the coupled equations (28), (30), and (31) can truly be called Tate’s model.
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2 Y p Rt v tr t
1/ 2
(34)
When penetrating as a rigid projectile, the stress on the end of the rod is given by the right-hand side of Eq. (28). Thus, the deceleration is:
(35)
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dv 1 1 2 t v Rt dt pL 2
Equation (35) has the same form as Eq. (2), Poncelet’s equation, with a = Rt, and b = t /2. Thus, Eq. (4) is rewritten as:
P p t V 2 ln 1 L t 2 Rt
(36)
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Normalized penetration versus impact velocity is shown in Fig. 10 for a steel (7.9 g/cm3) projectile penetrating an aluminum (2.7 g/cm3) target. A variety of hypothetical Yp–Rt combinations are shown. Since Yp > Rt for all the examples shown, penetration at low impact velocities is initially in the rigid-body mode. But at some point, given by Eq. (34), the rod begins to erode. The X’s in the figure mark the transition velocity from rigid-body to eroding penetration.
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Experimental data from Forrestal et al. [22-23], are plotted in Fig. 11. The results for three nose shapes are plotted: hemispherical, conical, and an ogive. T-200 and C-300 are types of maraging steel. A least-squares regression analysis was conducted to find Rt in Eq. (36); the resulting fit is shown in Fig. 11 with Rt = 1.62 GPa. With Yp = 3.0 GPa, vtr is 0.95 km/s, and if Yp = 4.0, vtr is 1.28 km/s. Regardless, the Tate-Poncelet equation cannot reproduce the conical- and ogival-nose data at the higher impact velocities. Nose shape can result in deeper rigid-body penetration than can be accounted for in a simple theory.4
6.3 Rigid-Body, Eroding, and Secondary Penetration
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The projectile will begin to erode, for the case of Yp > Rt, when the velocity is sufficiently high that the penetration pressure is larger than the strength of the projectile. As the impact velocity increases, it can be seen in Fig. 10 that P/L approaches the hydrodynamic limit from above, except for when Yp is only slightly larger than Rt.5 Although the transition velocity for rigid-body to eroding penetration is given by Eq. (34), this is overly simplistic. Depending upon the ductility of the projectile, substantial mushrooming can occur at
4
The two-term Poncelet equation, Eq. (4), can be used to fit the whole range of experimental data, but the fit parameters do not have any significant relationship/interpretation to physical parameters. 5
This is in contrast to Rt > Yp, where P/L approaches the hydrodynamic limit from below as the impact velocity increases.
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the projectile-target interface, which shortens the rod, prior to failure of projectile material (erosion). Further, as the projectile material is made stronger, to resist deformation and maintain rigid-body penetration, the projectile material tends to become more brittle, which means that small strains lead to material failure and erosion. The results for three strong steel projectiles with an ogival nose into 6061-T6 aluminum are shown in Fig. 12. The experimental data are from Ref. [24]. The solid curve is a fit to the rigid-portion of the P/L results using Eq. (4), with a = 1.956 GPa and b = 0.0518 g/cm3. Note that although the Poncelet equation provides a good overall fit to the experimental data, all the low velocity results tend to lie above the curve fit.
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It is observed that the decrease in penetration efficiency (P/L) is not gradual as indicated in Fig. 10. Vertical lines in Fig. 12 denote the approximate impact velocity where the various projectiles begin to erode. The onset of erosion occurs over a relatively narrow velocity range; this is particularly evident for the VAR 4340 projectile, where this transition velocity interval is approximately 75 m/s. The AerMet 100 appears to be deforming at about 1.81 m/s, but unfortunately, the next data point is at 2.04 km/s. However, similar data using spherical-nose projectiles also show a very narrow velocity interval for the transition from rigid-body to eroding penetration [25].
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Experimental data for a flat-nosed, tungsten-alloy projectile into 7020 aluminum (t = 2.8 g/cm3) are shown in Fig. 13 [7-8]. Tungsten alloy is more ductile than the maraging steels used in the experiments described above, and thus, will plastically deform considerably more before failure. Only the data to an impact velocity of approximately 0.54 km/s were used for the Tate-Poncelet curve fit; at impact velocities between approximately 0.5 and 0.7 km/s, it is believed that the projectile is mushrooming.6 By 0.8 km/s, the projectile begins to erode, and there is a significant decrease in normalized penetration for a small change in impact velocity.
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The velocity of the erosion debris, udebris, is (2u – v), as shown in Fig. 3. The direction of this velocity depends upon the ratio of target to projectile density. Ignoring strength effects, if p << t, then u 0, and udebris –v. If p = t, then udebris = 0; and if p > t, then udebris is into the target. Then, if
1 2 p u debris Rt 2
(37)
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there is additional penetration, called secondary penetration [11].
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The penetration velocity u can be estimated from the Tate model if Yp and Rt are known. Given the uncertainties in Yp and Rt, it is estimated that secondary penetration will occur between 1.25 and 1.40 km/s (Yp = 2.0 GPa, and Rt between 2.15 GPa and 2.35 GPa). The vertical dotted lines in Fig. 13 denote where secondary penetration begins. Of course, if udebris is only a little greater than Rt, one would not expect substantial secondary penetration. But as the impact velocity increases, secondary penetration
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Wickert [8] shows photographs of recovered projectiles that impacted the aluminum target at 0.540 km/s and 0.664 km/s. The projectile had not deformed at the lower impact velocity, but there is deformation at the higher impact velocity.
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keeps the P/L response from asymptotically approaching the hydrodynamic limit; instead, P/L significantly exceeds the hydrodynamic limit as shown in Fig. 13.
6.4 Estimates for Yp and Rt
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Tate realized that for the modified hydrodynamic model to be useful, he needed a methodology for estimating Yp and Rt. Two articles summarized his efforts [26-27]. The term Yp is associated with the deceleration of the projectile, and is estimated from the flow stress, Yp, by:
Y p 1 Yp
(38)
where is a constant independent of velocity and accounts for dynamic material effects.
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Rt is the resistance to plastic deformation of the target flow. Various methods have been proposed to estimate Rt. In particular, cavity expansion theory has been used [28]. The concept is to assume an incompressible or compressible plastic region and an elastic region. A cavity from zero radius to a radius R is opened quasi-statically. A similarity solution is obtained; the solution leads to a plastic region and an elastic region, and allows calculation of the stress at the interfaces. Spherical cavity expansion or cylindrical cavity expansion can be assumed. The various solutions are shown in Table 1. If the material is assumed to be incompressible, then 0.5, as indicated in the table.
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Tate conducted a cavity expansion on a flow field inspired by the magnetic flow lines in a solenoid (the impact velocity is sufficiently high that hydrodynamic flow is established, with a cavity radius larger than that of the projectile) Assuming that the target material is incompressible, that the J2 flow law applies, and when yielding, the material is perfectly plastic (no account for rate effects or microstructural features, etc.), he derived a different expression for Rt, as shown in the third row of Table 1. Note that this solution depends upon the value of in Eq. (38). After analyzing experimental data, Tate determined that = 0.7 was an appropriate value to account for dynamic effects [27].
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As an example, consider that the target is an armor-like steel, with Yp = 1.0 GPa, t = 0.3 (t = 0.5 for an incompressible material), and Et = 200 GPa. The values calculated for Rt are shown in Table 2. Note that the resistance to plastic deformation is 3 to 6 times the flow stress of the target material; the target material resists the opening of a cavity (or penetration channel) due to self-confinement. When comparing model predictions to experimental data, Tate’s solenoidal model provides a much more realistic value for Rt [29].
Table 1. Quasi-static Cavity Expansion Solutions Spherical
2 Yt Et 1 ln 3 1 t Yt 2 Et 2 Yt 1 ln 3 3 Yt
Rt Incompressible
Cylindrical
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Rt
2 Yt 3
3E t 1 ln 21 t Yt
3E t 2 Yt 1 ln 3 3 Yt
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Rt
Compressible
2 Yt 3
Et 1 ln 31 t Yt
Rt
Yt
3Et 1 ln 6 1 t Yt 3
2 2 0.7 0.57 Et 2 Et Yt ln Rt Yt ln Yt 4 e Yt 3 3
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Tate Solenoidal Model
Table 2. Estimates for Rt for an Armor-Like Steel Spherical
Cylindrical
Incompressible
3.93 GPa
3.32 GPa
Compressible
3.74 GPa
3.15 GPa
5.44 GPa
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Tate Solenoidal Model
A problem, though, is that Rt is not a material property, although it does depend upon material properties. Anderson, Littlefield, and Walker [14] showed that Rt also depends upon the impact velocity, just as Allen and Rogers showed that t was a function of impact velocity [11].
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6.5 Summary of the Tate Model
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The Tate model is a one-dimensional model that predicts the time history of penetration, including projectile deceleration. The model provides insights into the dependence of penetration on material properties (for example, density and strength). It predicts a critical velocity below which no (additional) penetration occurs. It also predicts rigid-body penetration, and a threshold velocity for the transition from rigid-body penetration to eroding penetration. Tate provided engineering estimates for evaluating Yp and Rt; the primary weakness of the model is obtaining a priori accurate estimates for Rt, which depends upon the impact velocity as well as material properties.
7.0 Rigid-Body Penetration Revisited
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7.1 Nose Shape and Dynamic Cavity Expansion Forrestal and his co-authors examined rigid-body penetration. Although primarily interested in penetration into geologic materials, they were uncertain about the accuracy of their model because of scatter in experimental data, inherent in brittle materials. Therefore, for model validation, they decided to study penetration into 6061-T6 aluminum. In their work, they calculated the force on the projectile nose, examining different nose shapes such as hemispherical, conical, and an ogive. An example for calculating the force on a hemispherical-nose projectile will be shown here; the procedure for other nose shapes is found in Ref. [22]. 18
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Assuming the direction of penetration is in the z-direction, the incremental normal force (dFn) on the hemispherical nose, see Fig. 14, is given by:
dFn 2 R a n v, d
R a sin
(39)
dFt 2 R a f n v, d
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where n(Vz,) represents the normal stress on the projectile nose, which is a function of the penetration velocity and . They also assume that there is frictional (tangential) force (dFt) proportional to the normal stress, t f n , then (40)
To obtain the component of the normal and tangential forces in the z-direction, dFn and dFt are multiplied by cos and sin, respectively. Integrating over the hemisphere gives the retarding force:
0
n v, sin 2 2 f sin 2 d
It now is required to calculate n(v,).
(41)
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/2
Fz a 2
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The authors used dynamic cavity expansion [30], including strain hardening and rate effects, to calculate the force on the projectile nose. Similar to quasi-static cavity expansion, there is an incompressible or compressible plastic region and an elastic region. The difference is that in dynamic cavity expansion, the cavity is opened at a constant velocity. A similarity solution leads to a plastic boundary, an elastic boundary (plastic, elastic, and undisturbed regions); and stresses at the interfaces.
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The math becomes more complicated for compressible materials, and for materials where a more realistic constitutive response is assumed, for example, strain hardening and strain rate effects, e.g., Ref. [31-32]. Here, only the incompressible spherical dynamic cavity result is shown for an elastic-perfectly plastic material:
2 Et 3 2 2Yt t v As Bv 2 1 ln 3 3Yt 2
(42)
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The first term, As, is the quasi-static spherical cavity expansion term (Table 1). Note that the stress (or resistance to penetration) includes strength and inertial effects. Also note that the resistance is proportional to the square of the cavity (expansion) velocity. The stress normal to the projectile nose surface is then given by Eq. (41) using the component of the velocity normal to the surface, i.e., vcos. Inserting Eq. (42) into Eq. (41) and integrating gives:
f 3 t f 2 1 v a 2 s s v 2 Fz a 2 As 1 2 4 4
(43)
Note that the force has the form of the Poncelet equation with the coefficients calculated from physical properties. The final depth of penetration, following a similar procedure as done for Eqs. (1-4) gives: 19
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sV 2 M P ln 1 s a 2 s
(44)
where M is the projectile mass. The authors found that the total depth of penetration was relatively insensitive to friction coefficients between 0.02 and 0.1 for the hemispherical-nose projectile, but predictions for the conical and ogival nose projectiles were more sensitive to values of f [22].
M p a 2 L
2 p a 3 3
Thus, Eq. (44) can be rewritten as:
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p sV 2 P ln 1 L 2a / 3 s s
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Using the nomenclature of Forrestal et al. [23], the mass of the projectile, where L is the length of the shank and a is the spherical nose radius, so that the total projectile length is (L+a), is given by: (45)
(46)
Their work perhaps culminated in an article by Warren and Forrestal that considered strain hardening and strain-rate sensitivity on the penetration of aluminum targets with spherical-nosed rods [32]. They considered the constitutive relationship:
Yd
M
E E Y Y o d
(47)
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Yd
Yd Y 0
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where the exponents and , and the proportionality constant , are determined from fitting experimental stress-strain data. The dot symbolizes the strain rate. The solutions for incompressible and compressible spherical cavity expansion7, with and without rate effects, as a function of impact velocity are shown in Fig. 15.
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Experimental data from [22-23] are plotted in Fig. 15. It is seen that the compressible model with rate effects provides the best overall prediction of penetration, specifically going through the experimental data at the higher impact velocities.
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7.2 Frictional Effects
Initially, Forrestal and colleagues were concerned about friction effects between the projectile and the penetration cavity wall, so they included frictional forces in their model, e.g., Eq. (40). However, Warren and Forrestal [32] had to neglect frictional forces in order for the analytical model to reproduce the experimental data, Fig. 15. The reason is described in the paragraphs below.
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The modeling work of Forrestal and colleagues largely focused on using spherical cavity expansion.
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Hill [33], in revisiting work done during WWII, makes the following statement as he outlined the assumptions for his work: “The frictional component can be disregarded because of surface melting.” Tate made an estimate of temperature effects during rod penetration [34]. He found that thermal conduction is significant only very close to the projectile-target interface (during the timeframe of penetration). Further, that when distances are scaled relative to the crater diameter, the temperature distribution is independent of the impact velocity, and the temperature approaches the melting temperature in a small region which is of the same order of size as the conduction zone. Wilkins [35] did not initially include frictional forces in the Lagrangian hydrocode HEMP [36], and had good agreement with experiments. When he added friction, the calculated results became worse.
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Lastly, Camacho and Ortiz [37] performed detailed finite element simulations of some of the Forrestal et al., experiments using a new adaptive meshing technique and a constitutive material law that included strain hardening, rate-dependent plasticity, heat conduction, and thermal-mechanical coupling. The simulations showed a very thin melted layer in the target next to the projectile that resulted in a nearly frictionless interface. Thus, it is concluded that friction in penetration mechanics is a term added to an analytical model to improve agreement with experiments, whose sole justification is that the friction coefficient is on the order of 0.01-0.2, and thus appears reasonable. But, when an accurate constitutive model is used, there is no need to include friction for ballistic penetration modeling.
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7.3 Applicability of Cavity Expansion for Rigid-Body Penetration
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Warren [38] examines rigid-body penetration for an elastic, strain-hardening aluminum using quasistatic spherical cavity expansion and dynamic spherical cavity expansion solutions to obtain the force on an ogival projectile. Warren compares the depth of penetration as a function of penetration velocity against a number of different data sets. He concludes that quasi-static target strength dominates penetration resistance at the lower impact velocities (where target inertial effects have been ignored), but that at higher velocities, target inertial effects must be included to match the experimental data. In effect, although more simplistic than the analysis in Ref. [38], the quasi-static solution is similar to Eq. (5), whereas, the dynamic solution is similar to Eq. (4). Warren concludes: “Final depth of penetration predictions…that include target inertia are found to be in excellent agreement with all of the experimental penetration data for all striking velocities, projectile geometries and ogive-nose shapes, projectiles densities, and material properties” *38+. He further notes: “Additionally, it is observed that target inertia must eventually be included at some particular striking velocity that is…dependent on the projectile geometry, nose shape…for the spherical cavity-expansion approximation model to accurately predict final depths of penetration.” *38]. Rosenberg and Dekel [39] and Rubin et al. [40], take issue with the applicability of the cavity expansion approximation applied to rigid-body penetration. They state that the resisting stress on a rigid projectile is constant, and that deceleration during penetration is independent of velocity [39]. They argue that a constant penetration channel diameter implies that the deceleration is a constant [39], and they support their argument by examining data from Piekutowski et al. [24]. They also note that Hill [33] states that a penetration channel with a constant diameter, which is equal to that of the projectile, is an indication for a constant resisting stress on the projectile [39]. Using experimental data 21
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from Ref. [24], Rosenberg and Dekel calculate an average deceleration, a V 2 / 2P , where V is the impact velocity and P is the final depth of penetration. They then show excellent agreement with experimental data by plotting P V 2 /( 2a) , albeit the value for a was obtained by analyzing the experimental data.
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Rubin et al. [40], develop a model for the drag force using what they believe to be a more realistic flow field in the target. They find that the force is a constant below a critical impact velocity, called the separation velocity Vs, which is what Hill calls the cavitation velocity [33].8 Since the force is a constant, then the deceleration is a constant. They calculate the resisting force that is larger than that calculated from cavity expansion. Thus, Refs. [39-40] state that the one-dimensional cavity expansion solutions, whether cylindrical or spherical, do not adequately represent the two-dimensional flow field around the penetrating projectile, and therefore, the cavity expansion solutions predict a value for the resisting stress that is too low. In consequence, they conclude that Warren needed to include target inertia to increase the resisting stress as the impact velocity increased.
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Warren responds *41+ that Hill’s comment concerns a projectile that is moving at constant velocity, which is not the case here because the projectile is decelerating due to target resistance; further, section 4 of Hill’s paper *33+ states “that if the velocity is not constant, then the deceleration of the projectile will have an effect on the target resistance” *41+. Warren then shows experimental data (with the tunneling region the same diameter as the projectile so that there is no cavitation) where the deceleration is not constant. Warren concludes that “when the strength term overshadows the inertial term…the target inertia term can be neglected in the penetration model. However, when the strength does not overshadow the target inertia effects, then a target inertia term must be included” *41+.
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Having laid out these arguments, it is observed that Warren has modeled the penetration process using an accurate constitutive description of the target material. On the other hand, the analytical model of Rubin et al., cannot explicitly account for the influence of compressibility, strain hardening, strain rate, and temperature of the target material, and must rely on numerical simulations for the influence of these effects [40]. Further, for accurate replication of the P versus V response, the deceleration parameter a must be determined from experimental data.
8.0 Dynamic Plasticity and the Ravid-Bodner Model
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The next major advance in analytical penetration modeling was the introduction of a multi-stage penetration/perforation model, developed by Ravid and Bodner [42]. The authors proposed five stages of penetration/perforation for a rigid projectile, as shown in Fig. 16: 1) penetration, 2) bulge formation, 3) bulge advancement, 4) plug formation and exit, and 5) projectile exit.
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At the critical velocity, the penetration channel begins to open up (and is no longer the same diameter as the projectile), penetration resistance increases, and target inertia must be included to determine the deceleration force on the projectile. Vs depends upon nose shaped, target material, etc.
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The authors established plastic flow fields within the target. As shown in Fig. 17, there are three zones where target material is moving (there is no target motion in zone IV). Velocity distributions are assigned to each zone, subject to compatibility and continuity conditions, and then the plastic work rates (including strain-rate effects) are computed. Work hardening is considered indirectly by taking the flow stress to be the average value, with respect to plastic work, over the full strain range. The radial and axial extent of the plastic zone fields (R and R, where R is the projectile radius) are solved for by minimizing the plastic work rate. Projectile deceleration is computed from an energy rate balance.
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Stage I (the penetration stage) continues until the plastic flow field reaches the rear of the target, at which point the target begins to form a spherical bulge, Fig. 18. Uplifting of target material at the surface ceases, and all target motion is in the direction of penetration. Target mass motion, combined with the incompressibility condition of plastic flow, determines the plastic flow velocity field (thus, no minimization procedures are required).
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At some point, the bulge no longer expands radially (the end of Stage II), and instead, the bulge advances in the direction of penetration, Stage III, Fig. 19. The geometry is determined by the plastic flow incompressibility condition.
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The target can possibly fail during Stages II or III. Ravid and Bodner considered a number of possible failure mechanisms, Stage IV. In their approach, it is possible to estimate the strains in the zones and between zone boundaries, which allows them to consider the formation of adiabatic shear bands, brittle failure in shear, conventional plug formation, and ductile failure. The length and shape of the plug depends upon the failure mode; the specific failure mode is target material dependent.
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An example of the model is shown in Fig. 20. A 7.62-mm armor-piercing (AP) bullet impacted a 12mm-thick steel plate at 855 m/s. The force on the projectile, the projectile velocity, and bullet displacement are plotted in the figure. The various stages of penetration are also shown in the figure. The model predicted a plug velocity of 424 m/s and a bullet velocity of 364 m/s. The experimental exit velocity was 345 45 m/s.
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Ravid, Bodner, and Holcman [43] extended the concept of dynamic plastic flow fields to high velocity impact (greater than 1 km/s). The analysis considered impact shock generation, plastic flow of the target, and mushrooming of the projectile for normal impact by a rod. The analysis continues until the shock effects in the rod become relatively unimportant (a few microseconds after impact). This model would be the first stage of a postulated seven-stage model, which was never fully developed.
9.0 Walker-Anderson Model
9.1 Semi-infinite Targets The momentum equation along the projectile-target centerline is given by: u z 1 u z zz 2 xz 0 t 2 z z x 2
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Walker and Anderson integrated this equation along the projectile-target centerline [44]:
p
z1
zp
u z dz t t
p
zi
zi
u z dz t t
u z 1 dz p u z2 t 2
zi
zi
zp
1 t u z2 2
zz
zi
zp
2
zp
xz dz 0 x
u z 1 1 xz dz p u 2 v 2 t u 2 2 dz 0 zp t 2 2 x
(49a)
(49b)
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zp
where zp and zi denote the positions of the rear of the projectile and the projectile-target interface, respectively; the target is assumed to be semi-infinite in the original model. To integrate Eq. (49b) explicitly, three major assumptions were invoked: (1) the velocity profile along the centerline in the projectile and target are specified (so that the first two integrals can be integrated); 2) the rear of the projectile is decelerated by elastic waves (similar to Tate); and 3) the shear behavior of the target material is specified (so that the gradient of the shear stress can be evaluated).
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The velocity profiles were motivated by the results of numerical simulations [29]. The velocity profile in the projectile is assumed to be bi-linear, see Fig. 21. The velocity field in the target is derived from a vector potential A:
ur Ar , 2 c2 1
M
c Rc 2 1 sin r
1 A sin eˆr 1 rAeˆ r sin r r
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v Ar , eˆ
(50)
(51)
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where Rc is the radius of the penetration channel. Thus,
vr
u c2 Rc2 2 1 cos 2 c 1 r
(52a)
u sin 1
(52b)
v z v r cos v sin
(53)
v
2 c
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The velocity in the z-direction is given by:
which gives the expression for the velocity field in the target, along the centerline, in Fig. 21. The velocity profiles are shown in Fig. 21, along with a comparison to results of a numerical simulation. In the figure, the symbol s represents the extent of plastic flow in the projectile (from the projectile-target interface); and cRc is the extent of the plastic flow field in the target. The specific comparison is for a tungsten-alloy projectile that impacted a steel target at 1.5 km/s (the projectiles has decelerated to approximately 1.4 km/s at the time that the velocity profile was plotted). 24
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Assuming a von Mises yield surface and that the shear stress is proportional to the rate of 7 deformation (rigid plasticity), the integral of the gradient of the shear stress gives Yt ln c [44]. 6 Taking the time derivatives of the velocity profiles in Fig. 21, and assembling the various terms in Eq. (49b), the centerline momentum balance becomes:
c 1 2 Rc u d v u s2 t c p c 1 dt s 2 ( c 1) 2
1 7 1 p ( v u ) 2 t u 2 Yt ln c 2 3 2
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p v ( L s) u p s t Rc
(54)
where the dot over a symbol implies the time derivative. Deceleration of the rear of the projectile is given by:
p
v u s 1 p ( L s) c c
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(55)
where it is assumed that the decelerating elastic waves run between the tail and the plastic zone in the projectile (a material nonlinearity) at a bar wave speed of c, (Ep /p)1/2. Lastly, the length of the projectile shortens with time:
M
L v u
(56)
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Equations (54-56) are solved simultaneously with an equation that provides for the extent of the plastic flow field (c), which is estimated by dynamic compressible cylindrical cavity expansion [44]:
u 2 c2 K t t u 2 c2 1 t 2Gt
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u2 1 t Yt
u K t K o 1 k co
K t t u 2
(57a)
(57b)
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Compressibility of the target is reflected in c. Equation (57b) is an heuristic expression to represent the stiffening of the bulk modulus with increasing penetration velocity [44], where the shock velocity is related to the particle (penetration) velocity by us = co + ku, and Ko = oco2. As the penetration velocity increases, c decreases as shown in Fig. 22. This prediction was confirmed by numerical simulations in Ref. [14]. As already stated, c is dependent on the penetration velocity, and not the initial impact velocity. Therefore, c changes as the penetration velocity changes. The elastic energy stored in compression is extremely important as describe in Section 4.2; c reflects this compression. For example, the dashed curve in Fig. 22 represents c for rigid-body penetration into aluminum. As the penetration velocity decreases, c becomes larger, effectively accounting for less energy being stored in elastic compression, and more energy being directly dissipated in target plastic work. 25
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For eroding long-rod penetration, the penetration velocity is approximately constant during the primary phase of penetration, so c is approximately a constant. It then increases during the terminal phase when there is rapid deceleration of the remaining projectile.
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It is interesting to note that Chocron, Anderson, and Walker [45] compared the results for compressible cylindrical cavity expansion and compressible spherical cavity expansion to compute c. They found that the use of the compressible cylindrical model, using values for the material flow stresses consistent with the constitutive behavior of the materials, gave better agreement with experimental data at impact velocities below 2 km/s. (Cavity expansion is used to provide an estimate of the extent of plastic flow in the target. This is in contrast to the discussion in Section 7 where cavity expansion is used to calculate directly the force on the projectile. The hemispherical flow field in the Walker-Anderson model is a good approximation of the applicable mechanics. Any error is estimating the extent of plastic zone in the target using cavity expansion is somewhat mitigated by the fact it enters the Walker-Anderson model as ln(c).)
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The Walker-Anderson model uses material properties for the projectile and target; the only empirical input is the crater radius Rc as a function of impact velocity [44]. Examples of the model, compared to experimental data, are shown in Fig. 23; other examples are provided in Ref. [44]. Although comparison with experimental data is excellent, there is a discrepancy for eroding penetration that has not been resolved. As already discussed, the projectile is decelerated by elastic waves. However, when the length of the projectile has decreased to approximately one projectile diameter, the projectile material is within the plastically deforming region, and the material is “sees” the high pressures associated with the penetration front (for example, see Ref. [47]). This high pressure decelerates the remaining projectile sooner than calculated by the analytical model, as shown in Fig. 24 where the nose and tail velocities are plotted versus time. The result for the Tate model is also shown for comparison. For this specific example, the Walker-Anderson model, the Tate model, and the simulation have the same total depth of penetration. Notwithstanding this discrepancy in the current model, the Walker-Anderson model provides a reasonable estimate of the total depth of penetration for L/D = 1 projectiles (as shown in Ref. [44]), probably due to the fact that the target mechanics (the plastic flow zone) is being treated correctly combined with the robustness of the conservation of momentum approach.
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In the limit where s0, Young’s modulus of the projectile becomes large (c), and the threedimensional terms are removed (Rc0), Eqs. (54-56) reduce to Tate’s model, Eqs. (28), (30), and (31), but with an explicit expression for Rt:
Rt
7 Yt ln c 3
It is noted that p in Eq. (55) is the dynamic flow stress of the projectile material; whereas, Tate heuristically accounted for dynamic effects by adjusting the parameter in Eq. (38). More will be discussed about the appropriate value for Yt in Section 9.3.
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9.2 Finite Targets
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Initially, the Walker-Anderson model was extended to finite-thick targets by combining the failure modes of the Ravid-Bodner model with the Walker-Anderson model [48]. In this approach, the WalkerAnderson model is considered operative until the presence of the target rear surface begins to influence projectile penetration, which was assumed to occur when the plastic zone reached the effected target rear surface covering the projected area of the projectile (R2). The modified model predicted bulging, failure of the target, and an estimate of the residual velocity as a function of impact velocity, including the ballistic limit velocity. Then, in 1999, Walker extended the Walker-Anderson model to include explicitly bulging [49] and failure [50]. When the flow field reaches the rear surface of the target, the flow field is achieved through a multiplicative blending of the potentials for hemispherical flow (Adeep), Eq. (50), and shear flow (Ashear):
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(59)
where determines how much of the potential is deep versus shear. The velocity flow fields are found by taking the curl of Eq. (59), as was done for Eqs. (50-51). Details are provided in Ref. [49]. The volume ~ of the material in the truncated hemisphere with radius R and height h is V h 2 R~ h / 3 [51]. The
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~3 : weighting is determined by dividing this volume by the original volume of the hemisphere, (2 / 3)R
3 h 2 R
2
1 h 2 R
3
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~ ~ In practice, h is the distance between the projectile nose and the target rear surface, and
(60) ~ R c 1Rc .
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equals 1 until the plastic zone reaches the rear surface of the target. As the distance between the projectile nose and the target rear surface decreases, decreases until it reaches zero at h = 0, and the velocity field is pure shear.
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After considerable math, and a few minor approximations for simplification, modifications for the back surface is effected by replacing the last terms in Eq. (54):
7 7 1 1 2 t u 2 Yt ln c by t u uback Yt ln ~c 3 3 2 2
(61)
~
where c = 1 + (distance to back surface)/Rc. The change in the target resistance term reflects the fact that there is less target material being plastically deformed. The velocity of the rear (back) surface is obtained explicitly from the flow fields, and integrating the velocity over time provides the position of the rear surface. A comparison of the Walker-Anderson model with bulging to a numerical simulation is shown in Fig. 25. Since the flow field is known, it is possible to evaluate the equivalent plastic strain rate at any point of the target: 27
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p
1 u u j Dij i 2 x j xi
2 Dij Dij 3
(62)
p
4 Rc 3 r
2
u r
Integration over time provides the equivalent plastic strain.
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where Dij is the strain rate tensor, and the ui’s and xj’s are the velocity and coordinate directions, respectively. However, Eq. (62) is difficult to evaluate for a general location within the target. It is straightforward, however, to evaluate the equivalent plastic strain rate along the projectile-target centerline where the expression for the flow field is algebraically simple [50]:
(63)
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The strains were calculated a specified locations along the back surface [50], and it was found that the maximum equivalent plastic strain occurs at a distance of approximately two projectile radii from the centerline, and not on the centerline. Nevertheless, the simplest failure criterion that can be posed is to invoke target failure (that is, perforation) when the centerline strain at the rear surface exceeds some critical strain, crit. It is necessary to use an experimental data point to determine crit, such as the ballistic limit velocity. Once determined, then crit is considered to be a constant for other targets that use the same material.
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The model is compared to two experiments in Fig. 26. Both experiments are near the V50 limit for the specific target thickness (a tungsten-alloy projectile into an armor steel). For the thinner target, the impact velocity is just below V50 of 1250 m/s. The projectile has perforated the thicker target at the V50 impact velocity. Penetration, bulging, and perforation are well predicted by the analytical model.
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It was found that a critical strain to failure calculated in this manner permits accurate predictions of residual velocities for relatively ductile targets. However, this failure criterion is too simplistic for very hard targets that fail by brittle shear or strain localization [50].
9.3 Similitude Analysis Revisited and the Effective Flow Stress
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Normalized penetration as a function of impact velocity is shown for five sets of data in Fig. 27 [46]. The projectiles were flat-nosed with an aspect ratio (L/D) of 10. There is a tungsten alloy (WA) projectile into an armor steel, a maraging (W8) steel, and a mild steel (Steel 52). There is also a moderately strong steel projectile into a mild steel (Steel 37/52) and into an armor steel. A nonlinear regression was used to fit each individual set of experimental data to an equation with the form: a2 P a1 L 1 10 a3 ( a4 V )
(64)
where the ai’s are determined from the regression analysis; the solid lines represent the curve fits [53]. The data are replotted in Fig. 28 using nondimensional variables [54]. The ratio of the projectile to target densities () is used to account for density differences in normalized penetration. A nondimensional velocity is defined, pV 2/efs, where efs is the effective flow stress. pV 2 is proportional 28
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to the penetration pressure, and efs is related to the strength of the target. Thus, the nondimensional velocity is a ratio of penetration pressure to target resistance. Similitude theory does not specify the relationship between the nondimensional variables, and it was found that better correlation was obtained by also including the ratio of the densities on the abscissa, raised to a small power. The exponents were determined by adjusting them to provide a relatively tightly collapsed data set.9
0.2
a2 P a1 L 1 10 a3 *a4 V
pV 2 0.14 V efs
The regression coefficients are shown in Fig. 28.
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The form of the equation used to fit the collapsed data is similar to that of Eq. (64), but now it is P/L as a function of V [54]:
p t
(65)
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Key to this analysis is the selection of the effective flow stress, efs. efs was determined by conducting a parametric study on P/L as a function of velocity for various values of efs. The valued selected for efs was the one that minimized the root mean square error on P/L, see Fig. 29 for an example [53]. This was done for each of the experimental data sets. efs for Steel 37, Steel 52, and German armor steel were found to be 0.797 GPa, 0.858 GPa, and 1.41 GPa, respectively.
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The target flow stress used in the Walker-Anderson model (Yt), which is a single value, must somehow encompass the volumetric effect of the target flow stress that is a function of strain, strain rate, and temperature; and which varies with time and location in the target. Adiabatic stress strain curves10, using the Johnson-Cook constitutive model [55], are shown for two materials at various strain rates in Fig. 30: a mild steel (A36) and 4340 steel hardened to Rockwell C 30. A36 is similar to the German mild steels 37 and 52. The long horizontal dashed lines represent efs for Steel 37 and 52, determined by the minimization procedure. The dashed-dotted line represents an average equivalent stress for A36 with some strain hardening; this dashed-dotted line is only 8% lower than the calculated efs for Steel 37.
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4340 steel, hardened to Rc30, is often used as an experimental surrogate for armor steel. The average equivalent stress at a strain rate of 103 s-1 is 1.2 GPa. This is the value used for Yt for one of the steels in Fig. 23. efs for German armor steel was determined to be 1.4 GPa, which is the value for Yt used for the stronger steel in Fig. 23. Thus, Yt in the Walker-Anderson model and efs are equivalent; Yt, i.e., the effective flow stress, encompasses an averaged volumetric flow stress for the target material. And it is seen that an estimate for Yt can be made from a stress-strain curve.
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It is interesting to note that the strength of the projectile is not included in the analysis. This is because the penetration pressures greatly exceeds the flow stress of the projectile material for these projectile-target combinations and impact velocities. 10
Plastic work is converted to temperature, which then results in thermal softening.
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In fact, Walker developed a procedure for estimating the target flow stress [56] using Johnson-Cook constitutive parameters. The strain field in the target can be calculated, and the plastic strain rate can be determined directly from the velocity field. If no thermal softening is present, the resulting equations can be integrated explicitly; if thermal softening is taken into account, then the target resistance term in Eq. (58) must be solved in the presence of thermal softening [56].
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10.0 Summary
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This article has reviewed analytical penetration models for metallic targets. The models range in applicability from relatively low impact velocities (rigid-body penetration) to very high-velocity impacts (hydrodynamic approximation). It was seen that the oldest of the models, represented by the Poncelet equation, Eq. (1), is derived naturally from the Tate model. The Poncelet equation, where the force is proportional to a static term and a velocity-squared term, was also derived from calculating the decelerating force on the projectile using dynamic cavity expansion. The objective of all of these models is to replicate experimental data using real material properties while minimizing empirical factors. Exploring the applicability and accuracy of the models has led to insights as well as limitations.
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Backman and Goldsmith [1] highlighted eight areas that merited further investigations to improve modeling of penetration mechanics. It is instructive to comment on these to see what progress has been made, and where there remain outstanding issues and requirements for further research.
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1. The development of more rational bases of specifying modeling parameters. Model parameters should be specified beforehand, i.e., they should not be determined from the experimental data that they are supposed to predict. Specifically, Backman and Goldsmith state that the next two topics can contribute to less empirical procedures. I will comment after item number 3.
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2. The improved description of interface phenomena (e.g., the formation of the penetration channel, interface phenomena involving oblique impact, nonzero flight orientations, deforming projectiles).
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3. The improved analytical modeling of force contributions (a need to extend modeling to thicker targets, and to include dishing and bulging of target elements, plug displacement, etc.).
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In general, the models described above do not use empirical fits for model parameters; instead, procedures have been developed to estimate model parameters from independent laboratory measurements, such as density, flow stress, etc. A major advance, particularly for penetration and perforation of thick targets, was the inclusion of dynamic plasticity, i.e., the description of the plastic flow field(s). Thus, considerable progress has been made in these three areas for normal impact of an unyawed projectile. 4. Effects of obliquity. Typically, the line-of-sight thickness is used to account for target obliquity. This is a reasonable approximation. However, at large obliquities, particularly for thick targets, the presence of the free surface results in asymmetric loading on the projectile nose, which can lead to bending and possibly fracture of the projectile. Although this can be modeled quite 30
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nicely in three-dimensional numerical simulations, this is currently beyond the ability of firstprinciples analytical models.
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5. Effects of flight orientation of the projectile. It is well-known that yaw and pitch can dramatically affect penetration response, which is amplified by target obliquity, as demonstrated by Anderson, Behner, and Hohler [57]. Although analytical modeling of some of the initial conditions can be performed, the complete projectile-target interaction is probably not analytically tractable. However, approximations can be made. An example is a paper by Walker, Anderson, and Goodlin [58] that combines several analytical expressions (based on experimental data) for predicting the depth of penetration as a function of velocity, L/D ratio, and impact inclination.
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6. Improved constitutive representations. Forrestal, Warren, and colleagues explicitly use the stress-strain behavior of the target material, including work hardening and strain-rate effects, in their modeling. In the Tate and Walker-Anderson models, a single value for the projectile flow stress, including strain-rate effects, accurately predicts deceleration of the projectile. For the Walker-Anderson model, a single value for the target flow stress is used, but it was shown that this value is related to a volumetric average of the target stress that includes work hardening, strain-rate effects, and thermal softening. Backman and Goldsmith’s comment also had to do with materials other than metals. The focus of this article was metallic targets; ceramics, glasses, fabrics, composites, etc., are beyond the scope of this article.
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7. Thermo-mechanical coupling and heat dissipation. Aside from the concept of the effective flow stress in the Walker-Anderson model, thermo-mechanical coupling, including localization, is probably beyond the capability of analytical models without some empiricism. However, thermo-mechanical coupling is readily modeled in numerical simulations, although the issue of different length scales of various phenomena remains an issue. 8. Failure criteria. Prediction of failure is the really tough problem. See the paragraph below.
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Unresolved at this time are accurate predictions for the transition from rigid-body to eroding penetration, which depends upon projectile material properties, projectile nose shape, and target strength, although some interesting work has been done by Segletes [59]. Another area of uncertainty is target failure, which depends upon the target material and loading conditions. Thus, it is failure (whether projectile and/or target) that is difficult, if not impossible, to predict a priori. This is also true of numerical simulations. It may be that some empiricism will always be required to calibrate a model for accurate predictions. Another area that requires further work is multi-material layered targets. The stress field precedes the penetration front, thus, the next layer will affect the rate of penetration. This effect will increase as the difference in material properties of the two materials increases. This is true for a metallic layered target, and the effect may become even larger when other materials such as ceramics, fabrics, or composites are used. It was observed that when models are based on conservation of momentum, such as the Tate and Walker-Anderson models, they tend to be robust and applicable to a wide range of materials and impact 31
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velocities. Models based on conservation of energy typically have limited applicability because energy sinks/dissipation change with impact velocity and materials. This article focused on analytical models for metallic targets. An article is planned on reviewing analytical models for non-metallic targets, such as glasses, ceramics, yarns and composites.
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11.0 ACKNOWLEDGEMENTS The author is very grateful for the interactions over the years with most (but not all) of the authors referenced in this article.
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29. C. E. Anderson, Jr. and James D. Walker, “An examination of long-rod penetration,” Int. J. Impact Engng., 11(4): 481-501, 1991. 30. H. G. Hopkins, “Dynamic expansion of spherical cavities in metals,” Progress in Solid Mechanics, Vol. 1 (I. Sneddon and R. Hill, Eds.), North-Holland, NY, pp. 85-164, 1960. 31. V. K. Luk, M. J. Forrestal, and D. E. Amos, “Dynamic spherical cavity expansion of strain-hardening materials,” J. Appl. Mech., 58: 1-6, 1991.
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32. T. L. Warren and M. J. Forrestal, “Effects of strain hardening and strain-rate sensitivity on the penetration of aluminum targets with spherical-nosed rods,” Int. J. Solids Structures, 35(28-29): 3737-3753, 1998. 33. R. Hill, “Cavitation and the influence of headshape in attack of thick targets by non-deforming projectiles,” J. Mech. Phys. Solids, 28: 249-263, 1980.
35. M. Wilkins, personal communication.
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34. A. Tate, “A theoretical estimate of temperature effects during rod penetration,” Proc. 9th Int. Symp. Ballistics, 2: 307-314, Shriverham, UK, 1986.
36. M. L. Wilkins, “Calculations of elastic-plastic flow,” in Methods of Computational Physics, Vol. 3 (B. Adler, S. Fernback, and M. Rotenberg, Eds.), Academic Press, NY, NY, 1964.
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37. G.T. Camacho and M. Ortiz, “Adaptive Lagrangian modeling of ballistic penetration of metallic targets,” Comput. Methods Appl. Mech. Eng., 142: 269-301, 1997. 38. T. L. Warren, “The effect of target inertia on the penetration of aluminum targets by rigid ogivenosed long rods,” Int. J. Impact Engng., 91: 6-13, 2016.
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39. Z. Rosenberg and E. Dekel, “A comment on ‘The effect of target inertia on the penetration of aluminum targets by rigid ogive-nosed long rods’ by T. L. Warren,” Letter to the Editor, Int. J. Impact Engng., 93: 231-233, 2016. 40. M. B. Rubin, R. Kositski, and Z. Rosenberg, “Essential physics of target inertia in penetration problems missed by cavity expansion models,” Int. J. Impact Engng., 98: 97-104, 2016.
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41. T. L. Warren, “Response to: ‘A comment on ‘The effect of target inertia on the penetration of aluminum targets by rigid ogive-nosed long rods by T. L. Warren’ by Z. Rosenberg and E. Dekel,” Letter to the Editor, Int. J. Impact Engng., 93: 234-235, 2016.
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42. M. Ravid and S. R. Bodner, “Dynamic perforation of viscoplastic plates by rigid projectiles,” Int. J. Engng. Sci., 21(6): 577-591, 1983. 43. M. Ravid, S. R. Bodner, and I. Holcman, “Analysis of very high speed impact,” Int. J. Engng. Sci., 25(4): 473-482, 1987. 44. J. D. Walker and C. E. Anderson, Jr., “A time-dependent model for long-rod penetration,” Int. J. Impact Engng., 16(1): 19-48, 1995.
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45. S. Chocron, C. E. Anderson, Jr., and J. D. Walker, “Long-rod penetration: cylindrical vs. spherical cavity expansion for the extent of plastic flow,” Proc 17th Symp. Ballistics, (C. Van Niekerk, Ed.), 3: 319-326, South African Ballistics Organisation, Morleta Park, South African, 1998. 46. C. E. Anderson, Jr. and D. L. Orphal, “Analysis of the terminal phase of penetration,” Int. J. Impact Engng., 29: 69-80, 2003.
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47. M. Ravid, S. R. Bodner, J. D. Walker, S. Chocron, C. E. Anderson, Jr., and J. P. Riegel, III, “Modification of the Walker-Anderson penetration model to include exit failure modes and fragmentation,” Proc. 17th Int. Symp. Ballistics (C. Van Niekerk, Ed.), 3: 267-274, South African Ballistics Organisation, Morleta Park, South African, 1998. 48. J. D. Walker, “An analytic velocity field for back surface bulging,” Proc. 18th Int. Symp. Ballistics (W. G. Reinecke, Ed.), 2: 1239-1246, Technomic Publishing Company, Lancaster, PA, 1999.
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49. S. Chocron, C. E. Anderson, Jr., and J. D. Walker, “A consistent flow approach to model penetration and failure of finite-thickness metallic targets,” Proc. 18th Int. Symp. Ballistics (W. G. Reinecke, Ed.), 2: 761-768, Technomic Publishing Company, Lancaster, PA, 1999. 50. V. Hohler and A. J. Stilp, compiled in: C. E. Anderson, Jr., B. L. Morris, and D. L. Littlefield, “A penetration mechanics database,” SwRI Report 3593/001, Southwest Research Institute, San Antonio TX, 1992.
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51. Standard Mathematical Tables, 20th Edtion (S. M. Selby, Ed-in-Chief), CRC Press, Cleveland, OH, 1972.
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52. C. E. Anderson, Jr., V. Hohler, J. D. Walker, and A. J. Stilp, “Time-resolved penetration of long rods into steel targets,” Int. J. Impact Engng., 16(1): 1-18, 1995.
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53. J.P. Riegel, III and C. E. Anderson, Jr., “Target effective flow stress calibrated using the WalkerAnderson penetration model,” Proc. 28th Int. Symp. Ballistics, DESTech Publications, Inc., 2: 12421253, 2014. 54. C. E. Anderson, Jr. and J.P. Riegel III, “A penetration model based on experimental data,” Int. J. Impact Engng., 80: 24-35, 2015.
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55. G. R. Johnson and W. H. Cook, “A constitutive model and data for metals subjected to large strain, high strain rates, and high temperatures,” Proc. 7th Int. Symp. Ballistics, pp. 541-548, The Hague, The Netherlands, April 19-23 1983. 56. J. D. Walker, S. Chocron, C. E. Anderson, Jr., J. P. Riegel III, D.S. Riha, J. M. McFarland, M. Moore, G. Willden, M. Murphy, and B. Abbott, “Survivability modeling in DARPA's adaptive vehicle make (AVM) program,” Proc. 27th Int. Symp. Ballistics, 2: 967-979, DEStech Publications, Lancaster, PA, 2013. 57. C. E. Anderson, Jr., T. Behner, and V. Hohler, “Penetration efficiency as a function of target obliquity and projectile pitch,” J. Appl. Mech., 80: 031801-1/11, 2013.
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58. J. D. Walker, C. E. Anderson, Jr., and D. L. Goodlin, “Tungsten into steel penetration including velocity, L/D, and impact inclination effects,” Proc. 19th Int. Ballistics Symp. (I. R. Crewther, Ed.), 3: 1133-1140, Interlaken, Switzerland, 2001.
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59. S. B. Segletes, “The erosion transition of tungsten-alloy long rods into aluminum targets,” Int. J. Impact Engng., 44: 2168-2191, 2007.
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Figure 1. Normalized penetration for very hard steel projectiles into 6061-T6 aluminum.
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Data from Ref. [6].
Figure 2. Normalized penetration for tungsten-alloy projectiles into a structural grade aluminum. Date are from Refs. [7-8].
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(a) Jet penetrating a target
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(b) Coordinate transformation.
Figure 3. Schematic of an incompressible, hydrodynamic jet penetrating an incompressible
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hydrodynamic target.
Figure 4. Phases of penetration for a L/D 20 projectile impacting at 3.0 km/s. 38
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Figure 5. Measured penetration vs. application of Eqs. (15-16);
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the figure is reproduced from Ref. [12].
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Figure 6. Comparison of Recht-Ipson model for chunky projectiles into thin plates
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with experimental data. Data from Ref. [13].
Figure 7. Comparison of Recht-Ipson model for an armor-piercing bullet against a thick target. Data from Ref. [13].
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Figure 8. Residual velocity vs. impact velocity for various types of projectiles into two
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hardnesses of armor steel [17].
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Figure 9. Normalized residual velocity vs. normalized impact velocity for data in Fig. 8 [17]
Figure 10. Normalized penetration vs. impact velocity for various Yp – Rt
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combinations for a steel projectile penetrating an aluminum target.
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Figure 11. Comparison of Tate-Poncelet equation with experimental data for steel projectiles
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penetrating 6061-T6 aluminum targets. The dashed line is given by Eq. (36). Data are from Refs. [24-25].
Figure 12. Normalized penetration for hard steel ogive projectiles into 6061-T6 aluminum
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depicting the rapid transition from rigid-body to eroding penetration. Data from Ref. [24].
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Figure 13. Normalized penetration for tungsten-alloy projectiles into 7020 aluminum targets.
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Data are from Ref. [7-8].
Figure 14. Rigid projectile geometry with hemispherical nose.
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Figure 15. Normalized penetration as a function of impact velocity for various dynamic
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spherical cavity solutions and experimental data. Data from Refs. [22-23].
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Figure 16. Multi-stage penetration/perforation model by Ravid and Bodner [42].
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Figure 17. Ravid-Bodner penetraton model, Stage I (from Ref. [42]).
Figure 18. Bulge formation, Stage II (from Ref. [42]).
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Figure 19. Bulge advancement, Stage III (from Ref. [42]).
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Figure 20. Example of the Ravid-Bodner model: a 7.62-mm AP bullet impacting a 12-mmthick steel plate at 855 m/s.
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Projectile
( zi s) z zi z p z ( zi s)
Target 2 1
Rc r ( z ) c Rc r ( z ) c Rc
(a) Centerline velocity profiles.
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u R c c u z ( z ) c2 1 r ( z ) 0
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vu ( z zi ) u u z ( z) s v
(b) Comparision of analytic approximation and numerical simulation
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Figure 21. Centerline velocity profiles of Walker-Anderson model [44].
Figure 22. c as a function of penetration velocity for steel and aluminum targets.
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Figure 23. Results for Walker-Anderson model for L/D 10 tungsten-alloy and steel projectiles
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penetrating two different armor steels. Data are from Ref. [50].
Figure 24. Comparison of nose and tail velocities vs. time for an L/D = 10 tungstenalloy projectile into an armor steel target at an impact velocity of 1.5 km/s.
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Figure 25. Comparison of Walker-Anderson model with bulging: red lines denote model
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predictions superimposed on a numerical simulation.
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(b) T = 4.95 cm; V = 1700 m/s.
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(a) T = 2.90 cm; V = 1241 m/s.
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Figure 26. Bulging and failure of the Walker-Anderson model. Data from Ref. [52].
Figure 27. Normalized penetration for five sets of projectile-target combinations as a function of impact velocity. Data from Ref. [50].
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Figure 28. Normalized penetration vs. a nondimensional impact velocity [54].
Figure 29. Root mean square error as t is varied for the steel projectile steel 37/52 target.
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Figure 30. Equivalent stress vs. equivalent plastic strain for two materials.
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ANALYTICAL MODELS FOR PENETRATION MECHANICS: A REVIEW Charles E. Anderson Jr. PII: DOI: Reference:
S0734-743X(16)31029-6 10.1016/j.ijimpeng.2017.03.018 IE 2878
To appear in:
International Journal of Impact Engineering
Received date: Revised date: Accepted date:
8 December 2016 22 March 2017 22 March 2017
Please cite this article as: Charles E. Anderson Jr. , ANALYTICAL MODELS FOR PENETRATION MECHANICS: A REVIEW, International Journal of Impact Engineering (2017), doi: 10.1016/j.ijimpeng.2017.03.018
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.
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Highlights Analytical penetration models, including assumptions and results, are reviewed.
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Applicability and limitations of penetration models are highlighted.
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ANALYTICAL MODELS FOR PENETRATION MECHANICS: A REVIEW
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Charles E. Anderson, Jr.*
CEA Consulting, San Antonio, TX 78240 USA
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ABSTRACT
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A review of analytical penetration models has been conducted and summarized. Models include the Poncelet equation, hydrodynamic theory, modified hydrodynamic theory, Recht-Ipson, TateAlekseevskii, cavity expansion, Ravid-Bodner, Walker-Anderson, and similitude modeling. These models describe, depending upon assumptions, rigid-body penetration, eroding penetration, steady-state and transient penetration, and perforation. Model assumptions are highlighted, and examples are provided of model predictions against experimental data. The manuscript has 30 figures, many of which compare model results to experimental data; 59 reference citations are included.
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Keywords: penetration mechanics; analytical modeling; plastic-flow fields
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* Email: [email protected]
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NOTATION model constants
vc
critical velocity, Eq. (33)
A(r,)
vector potential, Eqs. (50) and (59)
vtr
transition velocity, Eq. (34)
a
projectile radius (Fig. 14), Eq. (39)
V
impact velocity
B, b
model constants
V
nondimensional velocity, Eq. (65)
Bmax
maximum hardness at bottom of crater, Eq. (16)
Vr
residual velocity
c
bar wave speed
V50
ballistic limit
co
bulk sound speed
W
plastic work
D
projectile diameter
Ws
work in shear, Eq. (20)
Dij
rate of deformation tensor, Eq. (62)
Wt
plastic work in target, Eq. (24)
d
hole diameter
Y
flow stress (strength)
eˆ r , eˆ , eˆ
unit vectors in r, , and directions
Yd
dynamic flow stress, Eq. (47)
E
Young’s modulus
z
direction of penetration
Ed
deformation energy, Eq. (19)
(p/t)1/2, Eq. (65)
F, Fn, Ft
Force, normal force, tangential force
~c
(1 + h/Rc), Eq. (61)
G
shear modulus
,
axial and radial extent of plastic flow field (Ravid-Bodner model)
s
Eq. (43)
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distance between projectile nose to target rear (back) surface
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h
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A, a
K
bulk modulus
cRc
extent of plastic flow field (WalkerAnderson model)
k
shock-particle velocity constant
blending parameter, Eq. (60)
L
projectile length
s
Eq. (43)
M, Mp
projectile mass
,
hardening exponents, Eq. (47)
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mass of L/D 1 projectile, Eq. (16)
, crit
strain; critical strain to failure
mp
plug mass, Eq. (17)
p
plastic strain rate
P
penetration depth
strain hardening coeff., Eq. (47)
Pc
penetration depth of a L/D = 1 projectile, Eq. (16)
dynamic effect parameter, Eq. (38)
Pr
pressure
instantaneous projectile length
p
Lambert-Jonas exponent, Eq. (26)
μ
(t /p)1/2, Eq. (32a)
R
projectile radius
f
friction coefficient, Eq. (40)
Rc
penetration channel (crater) radius
Poisson’s ratio
Rt
target resistance
~ R
(c – 1)Rc, Eq. (60)
s
extent of plastic zone in projectile
T
plate thickness
t
time
u
penetration velocity
uback
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m
density
t
dynamic yield strength
strength difference, Eq. (14)
stress
efs
effective flow stress
n
normal stress, Eq. (39)
target rear surface velocity
t
tangential stress
udebris
velocity of eroded projectile debris, Eq. (37)
xz
target shear stress, Eqs. (48-49)
us
shock velocity
Yt
target strength
projectile velocity
Yp
projectile strength
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Subscripts: p: projectile; t: target
1.0 Introduction
Several good articles exist in the literature that review penetration, perforation, and other aspects of terminal ballistics. Backman and Goldsmith [1] present an extensive survey (278 references) of the interaction of projectiles and targets. They looked at semi-infinite targets, thin plate penetration and perforation, and intermediate and thick targets, covering the entire velocity range. Jonas and Zukas [2] reviewed available analytical methods for the study of projectile-armor interactions, with an emphasis on numerical simulations (167 references), focused on the velocity range of 0.5 to 2.0 km/s. A special 4
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issue of the International Journal of Engineering Science [3] was dedicated to penetration mechanics, covering experimental investigations, analytical modeling, and numerical modeling. Anderson and Bodner [4] reviewed analytical and numerical modeling of ballistic impact (88 references). Goldsmith [5] provides an extensive review of non-ideal projectile impact on targets (367 references). The above articles concentrate, in varying detail, on experimental procedures/results, data trends, analytical modeling, numerical modeling, and the mechanics of projectile-target interactions.
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There has been considerable progress in analytical modeling since the publication of Refs. [1-4]. This article provides a review of analytical models that have been developed to describe the process of penetration into metallic targets, subject to several constraints. Firstly, the model has to provide significant insight into the mechanics of penetration. This constraint eliminates empirical models that require experimental data to fit an assumed functional form. It is noted that empirical models can be very useful, but their range of validity is limited by the experimental data used to determine the model parameters. Secondly, the model provided a major advance, and was not a slight modification of an existing model. Any errors of omission are those solely the author, either because the model did not meet the selection criteria, or simply because we were unaware of the model.
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In today’s world of large-scale numerical simulations, it can be asked: “Why is there a need for analytical models?” There are several answers to this question. If an analytic model can be developed that captures the essence of the phenomenology, then understanding has been demonstrated, and the essential and relevant physics/mechanics have been applied/incorporated. This fundamental understanding leads to improved designs. Further, analytical models are typically hundreds to thousands times faster in execution than numerical simulations. Thus, analytical models provide tools for predictions, rapid design studies, first-order optimization, and risk assessments. For example, a typical risk assessment might run thousands (maybe millions) of Monte Carlo scenarios, which would be prohibitively time-consuming for numerical simulations, but which are well suited for analytical models.
2.0 Poncelet Equation
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With the above as background, the arrangement of the paper is largely chronological by model development. The focus is on metallic targets.
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Jean-Victor Poncelet (1788–1867) was a French engineer and mathematician. He developed an ordinary differential equation to describe penetration of rigid projectiles. Newton’s second law gives the projectile deceleration as a result of a resisting force:
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dv F A Bv 2 dt
(1)
The A is a static target resistance term, and the Bv2 term states that the resisting force is proportional to the square of the velocity. In traditional fluid mechanics, this v2 term is usually called a drag term. Since the projectile is non-deforming, the cross-sectional area remains constant, and Eq. (1) can be written as:
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pL
dv a bv 2 dt
(2)
(L is the effective length of the projectile if the cross-section is not a constant.) Equation (2) can be integrated to find the total depth of penetration (P) by using the chain rule of differentiation, dv/dt = (dv/dz)(dz/dt) = v(dv/dz):
p
P
0
vdv
dz V a bv L0
Thus,
P p bV 2 ln 1 L 2b a
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(3)
(4)
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(A more general case, with a resisting force proportional to the velocity as well as the velocity squared, in addition to the constant term, can be written. Subsequent integration of the equation results in an arctangent term in addition to the logarithmic term. This form of the resisting force is generally not used.)
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Forrestal and Piekutowski [6] measured the depth of penetration for maraging steel projectiles with an ogival nose penetrating 6061-T6 aluminum as a function of impact velocity. The depths of penetration, normalized by the length of the projectile, are plotted in Fig. 1. The Rockwell C hardness of the projectiles varied between Rc38 and Rc53, depending upon the specific alloying and heat treatment. A least-squares regression fit through the experimental data gives the values of the parameters a and b, given by the solid line in Fig. 1 (several points, above 1500 m/s, that fall below the general trend were ignored in the regression analysis; they will be discussed later).
2 P pV L 2a
(5)
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For small values of bV 2/a, the natural logarithm can be expanded to give:
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which is shown as the dashed line in Fig. 1. Thus, the depth of penetration is proportional to the square of the impact velocity. The resistive force, Eq. (2), is approximately a constant, independent of the impact velocity, since the parameter b does not enter into Eq. (5). The normalized penetration for blunt-nose tungsten-alloy projectiles into a 7000 series aluminum target is shown in Fig. 2. The experimental data are from two data sets generated by researchers at the Ernst-Mach-Institut [7-8]. A least-squares regression was conducted on all data below 0.7 km/s; the result is shown with the solid line. However, if four data points that are circled are deleted from the analysis, the resulting fit is shown with the dashed line. The fitted standard error is 45% less for the dashed curve than for the solid curve. Because we know that at sufficiently high impact velocities that the projectile will begin to deform, this Poncelet analysis is suggestive that the operative mechanics (that is, the assumption of rigid-body penetration) is not true for these four omitted data points. Note 6
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that without experimental evidence, it can only be said that this analysis is suggestive. Although not very satisfying, an alternative explanation is that the Poncelet parameters change during penetration. However, a similar analysis of Poncelet fits to the data for a completely different set of experiments, described in Section 6.3, also show that a better fit to the experimental data is achieved if some of the higher velocity data points are not included. In these experiments, recovered projectiles of the deleted data points show considerable deformation.
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One last comment is worth mentioning before proceeding. The small argument expansion, given by Eq. (5), is only valid to about 0.3 km/s in Fig. 2, given the values of a and b (shown in the figure). Thus, for this projectile-target combination, the assumption of a constant resisting force is only valid to about 0.3 km/s; at higher impact velocities, a “drag” resistive force must also be included to reproduce the experimental data.
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We will see that the Poncelet equation will manifest itself several times in the development of penetration models.
3.0 Hydrodynamic Penetration and Modified Theory of Hydrodynamic Penetration 3.1 Hydrodynamic Theory
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The hydrodynamic theory of penetration emerged during WWII and was applied to shaped-charge jet penetration. The theory was developed by Birkhoff, MacDougall, Pugh, and Taylor [9]; the authors acknowledged in a footnote that independent work of Hill, Mott and Pack in England led to the same results. The operative equation is derived from conservation of momentum, assuming that the strengths and viscosity of target and penetrator materials can be neglected, that is, the problem can be treated by hydrodynamics. Thus, the momentum equation is:
v 1 1 v 2 v v Pr t 2
(6)
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where is the vector operator. With the assumption that the jet and target materials are incompressible, the term on the right-hand side can be written as Pr . A further simplification is
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done by using a coordinate transformation, as shown in Fig. 3. It is assumed that the jet moves at a constant velocity v, and that the penetration velocity, u, is also a constant; thus the projectile-target interface does not move in Fig. 3b. The momentum equation, Eq. (6), then becomes
v v
P 1 v 2 r 2
(7)
with the assumptions of hydrodynamic and incompressible materials under steady-state conditions. The dot product of both sides of the equation by the vector velocity gives: 7
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P 1 v v 2 r 2 since
v v is perpendicular to v.
0
(8)
Thus, we have that the gradient of the term in the parenthesis
Along the projectile-target centerline, Eq. (8) becomes:
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is perpendicular to v. The term within the parenthesis is usually called Bernoulli’s equation. It is important to note that while the units of the terms within the parenthesis in Eq. (8) have units of specific energy, Eq. (8) was derived from the momentum equation.1
1 2 Pr 1 2 v v Pr 0 z 2 z 2
(9)
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Equation (9) is integrated from the back of the projectile to the projectile-target interface, and then from the interface into the target. The pressure at the back of the projectile is zero, and the target is assumed to be semi-infinite (the pressure at infinity is zero). Using the nomenclature from Fig. 3b, so that the velocity at the projectile-target interface is zero, and the target velocity at infinity is –u, integration of Eq. (9) gives:
1 1 p v u 2 t u 2 2 2
(10)
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This is the hydrodynamic, steady-state, incompressible model for penetration. Solving for the penetration velocity gives:
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u
v 1 t / p
(11)
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Since penetration is steady state2, the time of penetration is:
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L vu
(12)
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and the penetration depth is P = ut. After rearranging, the normalized depth of penetration is:
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P u p / t L v u
(13)
Most elementary physics textbooks derive Bernoulli’s equation using conservation of energy methods.
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Specifically, the shock phase of penetration is neglected, that is, steady state is reached instantaneously; and there is not a terminal phase of penetration, i.e., penetration stops as soon as the last particle of the jet has struck the target.
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3.2 Modified Hydrodynamic Theory
1 1 p v u 2 t u 2 2 2
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Shaped-charge jets have a velocity gradient. Thus, the jet can be partitioned into a series of connected jets with different velocities. The hydrodynamic theory is then applied to each section of the jet. This procedure overpredicted the final depth of penetration. Eichelberger [10] reasoned that at sufficiently low velocities, particularly into strong (steel) targets, that target strength effects could not be neglected. He modified Eq. (10): (14)
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where = Yt – Yp represents the resistance to plastic deformation. The penetration velocity, Eq. (11), is modified by the addition of to the right-hand side of Eq. (14). A more general expression for the penetration velocity will be given when the Tate model is described. Eichelberger determined that was 1 to 3 times the uniaxial yield strength of the target material (as the jet is typically copper, Yp is very small compared to Yt).
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Allen and Rogers [11] applied the modified hydrodynamic theory to high-velocity impacts of six different rod projectile materials (gold, lead, copper, tin, aluminum, and magnesium) into 7075-T6 aluminum. Instead of , they used the symbol t, which they called the dynamic yield strength of a solid target relative to a fluid jet. They found that t 1.89 GPa, or 3.9 times the yield strength of 7075-T6 aluminum (0.48 GPa). But they also determined that t had to be written as a function of impact velocity to reproduce the final depth of penetration.
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These modifications to the hydrodynamic theory apply to relatively weak projectiles. The assumption is that the projectile is completely consumed, that is, no projectile material remains at the bottom of the penetration channel. This assumption is true only for high velocity impacts and/or weak projectiles. Additionally, the above modifications apply to relatively long projectiles where steady-state penetration is applicable.
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Christman and Gehring [12] described the four phases of high velocity penetration: (I) a transient phase, (II) the primary phase, (III) the secondary phase, and (IV) a recovery phase. Although the concept was correct, their original figure is highly distorted. These phases of penetration are shown in Fig. 4 for an L/D 20 tungsten-alloy projectile impacting a steel target at 3.0 km/s. In Fig. 4, the phases are renamed: (I) shock phase, (II) steady-state phase, (III) terminal (transient) phase, and (IV) recovery (elastic rebound of the bottom of the penetration channel). The relative time duration of phases II and III depend on the aspect ratio of the projectile. The shock phase persists for a very short time (that depends upon the projectile diameter), and penetration during this phase is typically included in the steady-state phase. Christman and Gehring then made a very interesting modification to the hydrodynamic theory that accounted for terminal transient effects; thus, their modification is applicable to small-aspect-ratio projectiles. They separated penetration into a primary (steady-state) phase and a secondary (transient) phase. They stated penetration for the transient phase could be approximately by the penetration of an L/D = 1 projectile. Therefore, for high impact velocities, total penetration is given by: 9
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1
p 2 P ( L D) Pc t
(15)
p Pc 0.13 t
1
3
1 2 mv 2 Bmax
1
3
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where penetration of the primary (steady-state) phase is given by a rod that is one diameter shorter than its initial length. Then they added the crater depth, Pc, obtained for a L/D = 1 rod. Christman and Gehring correlated Pc with experimental data:
(16)
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where Bmax is the maximum hardness at the bottom of the crater (to account for strain hardening of target material). The correlation of Eqs. (15) and (16) with the measured penetration depths is shown in Fig. 5 for rods with aspect ratios of 1 to 25 at impact velocities of 2.0 – 6.7 km/s, for aluminum and steel projectiles impacting aluminum and steel targets (only a subset of the data in Ref. [12] have been plotted).
4.0 Conservation of Energy and Momentum Model
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4.1 Recht-Ipson Model
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Up to this point, the models have considered the targets to be semi-infinite, although assumptions have been invoked by various investigators in using these models to account for finite-thick targets. Recht and Ipson [13] developed a model to estimate the residual velocity of chunky projectiles (L/D ~ 1) impacting relatively thin plates using conservation of momentum and energy considerations.
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The authors considered impact at ordnance velocities where the projectile stays intact, although it can deform (mushroom) slightly. An experimental observation for impacts against thin plates is that a plug is ejected and is attached to the projectile. Application of the conservation of momentum gives:
M pV M p m p Vr
(17)
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where the large M and small m refer to the masses of the projectile and plug, respectively Application of conservation of energy gives:
1 1 M pV 2 M p m p Vr2 Ws Ed 2 2
(18)
where Ws is the energy used to shear the plug and Ed is the energy associated with deformation and heating (plastic work). Ed is simply the difference between the initial and final kinetic energies, which, using Eq. (17) is:
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1 m p Ed M pV 2 2 M p mp
(19)
The energy loss due to shearing the plug is estimated from Eq. (18) by finding the minimum velocity that gives a residual velocity of zero. This minimum velocity is V50. With Vr = 0, and substituting Eq. (19) into Eq. (18), the shear energy is calculated:
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1 M p Ws M pV502 2 M p m p
(20)
Substituting Eqs. (19-20) into Eq. (18) and solving for the residual velocity gives: Mp Vr M m p p
2 V V502
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(21)
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d T 1 t p D L 2
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2
V502
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2
(22)
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Assuming that the projectile is nominally a right-circular cylinder, then Eq. (21) can be rewritten in terms of the geometric parameters, with d/D being the ratio of plug diameter to projectile diameter, and T/L the ratio of plate thickness to projectile length:
The authors note that Eq. (22) is valid for T/L < 1/2 and T/D < 1/2.
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Results for three sets of experiments for steel cylinders into mild steel plate are shown in Fig. 6. At relatively low impact velocities, the hard steel fragment does not deform. The authors argue that the effect of the ratio d/D is small, and since d is not usually known, they set the ratio to 1.0. This statement is true only for T/L = 0.15. As the impact velocity increases and the plate thickness increases, the stresses at the projectile-target interface become sufficiently large to deform (mushroom) the projectile. To reproduce the analytical results in Ref. [13], the authors must have assumed that d/D increases with impact velocity; that is, the projectile mushrooms and pushes out a plug with a larger diameter than the original projectile. To reproduce the Recht-Ipson results for T/L = 0.29 and 0.44, it was assumed that d/D goes from 1.0 to approximately 1.4 linearly with impact velocity as the velocity goes from V50 to approximately 950 m/s. The dashed lines nominally replicate the analytical results of Fig. 3 in Ref. [13]. Recht and Ipson [13] examined oblique perforation of thin plates and perforation of thick plates by cylinders. Equation (21) must be modified and additional assumptions and experimental data are required to make analytical predictions. Thus, some of the elegance of the normal impact against a thin plate is lost. Recht and Ipson also examined the case of an armor-piercing (AP) bullet perforating a thick plate [13]. In this case, no plug is ejected; and there is no deformation of the bullet. Thus, all kinetic energy loss is through plastic deformation of the target: 11
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1 1 M pV 2 M pVr2 Wt 2 2
(23)
If it is assumed that the plastic deformation is relatively independent of the impact velocity, then
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1 M pV502 2
(24)
Vr V 2 2 1 V50 V50
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and after some rearranging, Eq. (23) becomes:
(25)
Experimental data for an AP bullet into a steel target is shown in Fig.7, along with the analytical prediction given by Eq. (25).
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4.2 Conservation of Energy
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A remark is in order concerning the use of conservation of energy in penetration modeling. It is difficult to account for all the various mechanisms that dissipate energy. Further, the proportion of energy dissipation (energy transfer mechanisms) changes with impact velocity. In particular, as the impact velocity increases, projectile kinetic energy is transferred to the target in terms of target kinetic energy and elastic compression energy; this compression energy is dissipated by plastic work at later times [14]. Walker [15] showed that it is the transfer of this energy at the time of penetration that defines the forces on the projectile, and for an energy rate balance to be successful, it must include transfer of energy stored in the target as elastic compression. Although this is done automatically within numerical simulations, this would seem to be intractable within the context of an analytical model.
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In summary, conservation of energy can be useful in analytical modeling, but generally only over a limited velocity range and/or projectile-target combination. For example, Recht and Ipson [13] made assumptions on energy dissipation, but then commented on the restrictions and range of applicability.
5.0 Similitude Analysis
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Similitude analysis, sometimes called scale modeling, is a procedure for casting results in a nondimensional form; for example, see Ref. [16]. Equation (25) is an example of a relationship between a normalized residual velocity and a normalized impact velocity. Often, nondimensionalization allows what would seem to be disparate data to be collapse into a single response curve. One example for penetration mechanics will be given here, and then another will be described later. An experimental study examined the ballistic performance of 17 different projectiles (a tungsten alloy and six different steels with different heat treats) into two different armor steel targets (different hardnesses) [17]. A subset of the data is shown in Fig. 8. The Lambert-Jonas equation [18] provides for a general expression for the residual velocity as a function of impact velocity: 12
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a V p V p 50 Vr 0
1p
V V50 V V50
(26)
Dividing Eq. (26) by V50 gives: V p Vr a 1 V50 V50
1p
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A least-squares regression analysis is used to determine fit parameters a, p, and V50. Note that if the parameter p is equal to 2.0, Eq. (26) is nominally an expression for the conservation of energy [similar to Eqs. (23-25) above], but including the parameters a and p allow for a more general (better) curve fit to experimental data.
(27)
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This suggests that the experimental data should be plotted as Vr/V50 versus V/V50. The data in Fig. 8 are plotted using these nondimensional quantities in Fig. 9. Besides the experimental data in Fig. 8, twelve other projectile-target combinations are also plotted as the light gray hexagons in Fig. 9. All the experimental data collapse to a single curve; the expression for the curve is given in the figure.
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Although nondimensionalization, using the correct choice of parameters, permitted collapse of the experimental data, it does not necessarily explicitly allow one to determine the operative mechanics. Nevertheless, the results of Fig. 9 demonstrate that a substantial amount of basic information concerning projectile and target material response (including deformation and failure) is contained in V50.
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6.1 Eroding Penetration
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6.0 Tate-Alekseevski Model
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In contrast to shaped-charge jets, projectiles (rods) decelerate while penetrating a target. Independently, Tate [19] and Alekseevski [20] proposed a modified Bernoulli equation:
1 1 p v u 2 Y p t u 2 Rt 2 2
(28)
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where Yp is considered to be the dynamic flow stress of the projectile and Rt is the target resistance to penetration. Separating the rigid potion of the rod from that part that is undergoing plastic deformation (mushrooming), the force acting to decelerate the rigid portion of the rod of length is given by:
p R p2
dv R p2 Y p dt
Yp dv dt p 13
(29)
(30)
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The initial length of the rod is L, but since the rigid portion of the rod is moving faster than the penetration velocity, the rod is getting shorter with time:
d v u dt
(31)
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Thus, the three coupled equations, (28), (30), and (31) are solved simultaneously to determine the time history of v, u, and . Equation (28) can be rearranged to solve for the instantaneous penetration velocity in terms of the projectile velocity (which is changing with time) and the material characteristics of the projectile and target:
v v2 A 1 2
t p u
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1/ 2
p t
2Rt Y p 1 2
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u
A
v Rt Y p 2 v
(32a)
t
p t
(32b)
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The solution for u in Eq. (14) is given by Eq. (32) with Yp 0, and Rt replaced by . From Eq. (32), it is seen that there is a critical velocity, vc, were u = 0:
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2Rt Yp vc p
1/ 2
(33)
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That is, when the rod velocity drops to the critical velocity, there is no further penetration. We will examine procedures for estimating Yp and Rt after the next couple of subsections.
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6.2 Rigid-Body Penetration and Transition from Rigid-Body to Eroding Penetration
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In the previous subsection, Rt > Yp and there is erosion of the rod. Tate examined the case where Yp > Rt [21].3 For this situation, there exists a transition velocity, vtr, at and below which the rod no long erodes, that is, penetration occurs in rigid-body mode. For rigid-body penetration, u = v. The transition velocity is found from Eq. (32a) by equating u and v vtr, and solving for vtr:
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Notwithstanding that the model described in Section 2.5.1 is usually called the Tate-Alekseevki model, Addison Tate extensively explored the modified Bernoulli equation, including rigid-body penetration, transition of rigid-body to eroding penetration, and procedures for estimating Yp and Rt. Thus, the coupled equations (28), (30), and (31) can truly be called Tate’s model.
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2 Y p Rt v tr t
1/ 2
(34)
When penetrating as a rigid projectile, the stress on the end of the rod is given by the right-hand side of Eq. (28). Thus, the deceleration is:
(35)
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dv 1 1 2 t v Rt dt pL 2
Equation (35) has the same form as Eq. (2), Poncelet’s equation, with a = Rt, and b = t /2. Thus, Eq. (4) is rewritten as:
P p t V 2 ln 1 L t 2 Rt
(36)
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Normalized penetration versus impact velocity is shown in Fig. 10 for a steel (7.9 g/cm3) projectile penetrating an aluminum (2.7 g/cm3) target. A variety of hypothetical Yp–Rt combinations are shown. Since Yp > Rt for all the examples shown, penetration at low impact velocities is initially in the rigid-body mode. But at some point, given by Eq. (34), the rod begins to erode. The X’s in the figure mark the transition velocity from rigid-body to eroding penetration.
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Experimental data from Forrestal et al. [22-23], are plotted in Fig. 11. The results for three nose shapes are plotted: hemispherical, conical, and an ogive. T-200 and C-300 are types of maraging steel. A least-squares regression analysis was conducted to find Rt in Eq. (36); the resulting fit is shown in Fig. 11 with Rt = 1.62 GPa. With Yp = 3.0 GPa, vtr is 0.95 km/s, and if Yp = 4.0, vtr is 1.28 km/s. Regardless, the Tate-Poncelet equation cannot reproduce the conical- and ogival-nose data at the higher impact velocities. Nose shape can result in deeper rigid-body penetration than can be accounted for in a simple theory.4
6.3 Rigid-Body, Eroding, and Secondary Penetration
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The projectile will begin to erode, for the case of Yp > Rt, when the velocity is sufficiently high that the penetration pressure is larger than the strength of the projectile. As the impact velocity increases, it can be seen in Fig. 10 that P/L approaches the hydrodynamic limit from above, except for when Yp is only slightly larger than Rt.5 Although the transition velocity for rigid-body to eroding penetration is given by Eq. (34), this is overly simplistic. Depending upon the ductility of the projectile, substantial mushrooming can occur at
4
The two-term Poncelet equation, Eq. (4), can be used to fit the whole range of experimental data, but the fit parameters do not have any significant relationship/interpretation to physical parameters. 5
This is in contrast to Rt > Yp, where P/L approaches the hydrodynamic limit from below as the impact velocity increases.
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the projectile-target interface, which shortens the rod, prior to failure of projectile material (erosion). Further, as the projectile material is made stronger, to resist deformation and maintain rigid-body penetration, the projectile material tends to become more brittle, which means that small strains lead to material failure and erosion. The results for three strong steel projectiles with an ogival nose into 6061-T6 aluminum are shown in Fig. 12. The experimental data are from Ref. [24]. The solid curve is a fit to the rigid-portion of the P/L results using Eq. (4), with a = 1.956 GPa and b = 0.0518 g/cm3. Note that although the Poncelet equation provides a good overall fit to the experimental data, all the low velocity results tend to lie above the curve fit.
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It is observed that the decrease in penetration efficiency (P/L) is not gradual as indicated in Fig. 10. Vertical lines in Fig. 12 denote the approximate impact velocity where the various projectiles begin to erode. The onset of erosion occurs over a relatively narrow velocity range; this is particularly evident for the VAR 4340 projectile, where this transition velocity interval is approximately 75 m/s. The AerMet 100 appears to be deforming at about 1.81 m/s, but unfortunately, the next data point is at 2.04 km/s. However, similar data using spherical-nose projectiles also show a very narrow velocity interval for the transition from rigid-body to eroding penetration [25].
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Experimental data for a flat-nosed, tungsten-alloy projectile into 7020 aluminum (t = 2.8 g/cm3) are shown in Fig. 13 [7-8]. Tungsten alloy is more ductile than the maraging steels used in the experiments described above, and thus, will plastically deform considerably more before failure. Only the data to an impact velocity of approximately 0.54 km/s were used for the Tate-Poncelet curve fit; at impact velocities between approximately 0.5 and 0.7 km/s, it is believed that the projectile is mushrooming.6 By 0.8 km/s, the projectile begins to erode, and there is a significant decrease in normalized penetration for a small change in impact velocity.
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The velocity of the erosion debris, udebris, is (2u – v), as shown in Fig. 3. The direction of this velocity depends upon the ratio of target to projectile density. Ignoring strength effects, if p << t, then u 0, and udebris –v. If p = t, then udebris = 0; and if p > t, then udebris is into the target. Then, if
1 2 p u debris Rt 2
(37)
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there is additional penetration, called secondary penetration [11].
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The penetration velocity u can be estimated from the Tate model if Yp and Rt are known. Given the uncertainties in Yp and Rt, it is estimated that secondary penetration will occur between 1.25 and 1.40 km/s (Yp = 2.0 GPa, and Rt between 2.15 GPa and 2.35 GPa). The vertical dotted lines in Fig. 13 denote where secondary penetration begins. Of course, if udebris is only a little greater than Rt, one would not expect substantial secondary penetration. But as the impact velocity increases, secondary penetration
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Wickert [8] shows photographs of recovered projectiles that impacted the aluminum target at 0.540 km/s and 0.664 km/s. The projectile had not deformed at the lower impact velocity, but there is deformation at the higher impact velocity.
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keeps the P/L response from asymptotically approaching the hydrodynamic limit; instead, P/L significantly exceeds the hydrodynamic limit as shown in Fig. 13.
6.4 Estimates for Yp and Rt
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Tate realized that for the modified hydrodynamic model to be useful, he needed a methodology for estimating Yp and Rt. Two articles summarized his efforts [26-27]. The term Yp is associated with the deceleration of the projectile, and is estimated from the flow stress, Yp, by:
Y p 1 Yp
(38)
where is a constant independent of velocity and accounts for dynamic material effects.
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Rt is the resistance to plastic deformation of the target flow. Various methods have been proposed to estimate Rt. In particular, cavity expansion theory has been used [28]. The concept is to assume an incompressible or compressible plastic region and an elastic region. A cavity from zero radius to a radius R is opened quasi-statically. A similarity solution is obtained; the solution leads to a plastic region and an elastic region, and allows calculation of the stress at the interfaces. Spherical cavity expansion or cylindrical cavity expansion can be assumed. The various solutions are shown in Table 1. If the material is assumed to be incompressible, then 0.5, as indicated in the table.
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Tate conducted a cavity expansion on a flow field inspired by the magnetic flow lines in a solenoid (the impact velocity is sufficiently high that hydrodynamic flow is established, with a cavity radius larger than that of the projectile) Assuming that the target material is incompressible, that the J2 flow law applies, and when yielding, the material is perfectly plastic (no account for rate effects or microstructural features, etc.), he derived a different expression for Rt, as shown in the third row of Table 1. Note that this solution depends upon the value of in Eq. (38). After analyzing experimental data, Tate determined that = 0.7 was an appropriate value to account for dynamic effects [27].
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As an example, consider that the target is an armor-like steel, with Yp = 1.0 GPa, t = 0.3 (t = 0.5 for an incompressible material), and Et = 200 GPa. The values calculated for Rt are shown in Table 2. Note that the resistance to plastic deformation is 3 to 6 times the flow stress of the target material; the target material resists the opening of a cavity (or penetration channel) due to self-confinement. When comparing model predictions to experimental data, Tate’s solenoidal model provides a much more realistic value for Rt [29].
Table 1. Quasi-static Cavity Expansion Solutions Spherical
2 Yt Et 1 ln 3 1 t Yt 2 Et 2 Yt 1 ln 3 3 Yt
Rt Incompressible
Cylindrical
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Rt
2 Yt 3
3E t 1 ln 21 t Yt
3E t 2 Yt 1 ln 3 3 Yt
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Rt
Compressible
2 Yt 3
Et 1 ln 31 t Yt
Rt
Yt
3Et 1 ln 6 1 t Yt 3
2 2 0.7 0.57 Et 2 Et Yt ln Rt Yt ln Yt 4 e Yt 3 3
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Tate Solenoidal Model
Table 2. Estimates for Rt for an Armor-Like Steel Spherical
Cylindrical
Incompressible
3.93 GPa
3.32 GPa
Compressible
3.74 GPa
3.15 GPa
5.44 GPa
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Tate Solenoidal Model
A problem, though, is that Rt is not a material property, although it does depend upon material properties. Anderson, Littlefield, and Walker [14] showed that Rt also depends upon the impact velocity, just as Allen and Rogers showed that t was a function of impact velocity [11].
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6.5 Summary of the Tate Model
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The Tate model is a one-dimensional model that predicts the time history of penetration, including projectile deceleration. The model provides insights into the dependence of penetration on material properties (for example, density and strength). It predicts a critical velocity below which no (additional) penetration occurs. It also predicts rigid-body penetration, and a threshold velocity for the transition from rigid-body penetration to eroding penetration. Tate provided engineering estimates for evaluating Yp and Rt; the primary weakness of the model is obtaining a priori accurate estimates for Rt, which depends upon the impact velocity as well as material properties.
7.0 Rigid-Body Penetration Revisited
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7.1 Nose Shape and Dynamic Cavity Expansion Forrestal and his co-authors examined rigid-body penetration. Although primarily interested in penetration into geologic materials, they were uncertain about the accuracy of their model because of scatter in experimental data, inherent in brittle materials. Therefore, for model validation, they decided to study penetration into 6061-T6 aluminum. In their work, they calculated the force on the projectile nose, examining different nose shapes such as hemispherical, conical, and an ogive. An example for calculating the force on a hemispherical-nose projectile will be shown here; the procedure for other nose shapes is found in Ref. [22]. 18
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Assuming the direction of penetration is in the z-direction, the incremental normal force (dFn) on the hemispherical nose, see Fig. 14, is given by:
dFn 2 R a n v, d
R a sin
(39)
dFt 2 R a f n v, d
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where n(Vz,) represents the normal stress on the projectile nose, which is a function of the penetration velocity and . They also assume that there is frictional (tangential) force (dFt) proportional to the normal stress, t f n , then (40)
To obtain the component of the normal and tangential forces in the z-direction, dFn and dFt are multiplied by cos and sin, respectively. Integrating over the hemisphere gives the retarding force:
0
n v, sin 2 2 f sin 2 d
It now is required to calculate n(v,).
(41)
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/2
Fz a 2
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The authors used dynamic cavity expansion [30], including strain hardening and rate effects, to calculate the force on the projectile nose. Similar to quasi-static cavity expansion, there is an incompressible or compressible plastic region and an elastic region. The difference is that in dynamic cavity expansion, the cavity is opened at a constant velocity. A similarity solution leads to a plastic boundary, an elastic boundary (plastic, elastic, and undisturbed regions); and stresses at the interfaces.
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The math becomes more complicated for compressible materials, and for materials where a more realistic constitutive response is assumed, for example, strain hardening and strain rate effects, e.g., Ref. [31-32]. Here, only the incompressible spherical dynamic cavity result is shown for an elastic-perfectly plastic material:
2 Et 3 2 2Yt t v As Bv 2 1 ln 3 3Yt 2
(42)
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The first term, As, is the quasi-static spherical cavity expansion term (Table 1). Note that the stress (or resistance to penetration) includes strength and inertial effects. Also note that the resistance is proportional to the square of the cavity (expansion) velocity. The stress normal to the projectile nose surface is then given by Eq. (41) using the component of the velocity normal to the surface, i.e., vcos. Inserting Eq. (42) into Eq. (41) and integrating gives:
f 3 t f 2 1 v a 2 s s v 2 Fz a 2 As 1 2 4 4
(43)
Note that the force has the form of the Poncelet equation with the coefficients calculated from physical properties. The final depth of penetration, following a similar procedure as done for Eqs. (1-4) gives: 19
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sV 2 M P ln 1 s a 2 s
(44)
where M is the projectile mass. The authors found that the total depth of penetration was relatively insensitive to friction coefficients between 0.02 and 0.1 for the hemispherical-nose projectile, but predictions for the conical and ogival nose projectiles were more sensitive to values of f [22].
M p a 2 L
2 p a 3 3
Thus, Eq. (44) can be rewritten as:
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p sV 2 P ln 1 L 2a / 3 s s
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Using the nomenclature of Forrestal et al. [23], the mass of the projectile, where L is the length of the shank and a is the spherical nose radius, so that the total projectile length is (L+a), is given by: (45)
(46)
Their work perhaps culminated in an article by Warren and Forrestal that considered strain hardening and strain-rate sensitivity on the penetration of aluminum targets with spherical-nosed rods [32]. They considered the constitutive relationship:
Yd
M
E E Y Y o d
(47)
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Yd
Yd Y 0
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where the exponents and , and the proportionality constant , are determined from fitting experimental stress-strain data. The dot symbolizes the strain rate. The solutions for incompressible and compressible spherical cavity expansion7, with and without rate effects, as a function of impact velocity are shown in Fig. 15.
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Experimental data from [22-23] are plotted in Fig. 15. It is seen that the compressible model with rate effects provides the best overall prediction of penetration, specifically going through the experimental data at the higher impact velocities.
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7.2 Frictional Effects
Initially, Forrestal and colleagues were concerned about friction effects between the projectile and the penetration cavity wall, so they included frictional forces in their model, e.g., Eq. (40). However, Warren and Forrestal [32] had to neglect frictional forces in order for the analytical model to reproduce the experimental data, Fig. 15. The reason is described in the paragraphs below.
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The modeling work of Forrestal and colleagues largely focused on using spherical cavity expansion.
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Hill [33], in revisiting work done during WWII, makes the following statement as he outlined the assumptions for his work: “The frictional component can be disregarded because of surface melting.” Tate made an estimate of temperature effects during rod penetration [34]. He found that thermal conduction is significant only very close to the projectile-target interface (during the timeframe of penetration). Further, that when distances are scaled relative to the crater diameter, the temperature distribution is independent of the impact velocity, and the temperature approaches the melting temperature in a small region which is of the same order of size as the conduction zone. Wilkins [35] did not initially include frictional forces in the Lagrangian hydrocode HEMP [36], and had good agreement with experiments. When he added friction, the calculated results became worse.
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Lastly, Camacho and Ortiz [37] performed detailed finite element simulations of some of the Forrestal et al., experiments using a new adaptive meshing technique and a constitutive material law that included strain hardening, rate-dependent plasticity, heat conduction, and thermal-mechanical coupling. The simulations showed a very thin melted layer in the target next to the projectile that resulted in a nearly frictionless interface. Thus, it is concluded that friction in penetration mechanics is a term added to an analytical model to improve agreement with experiments, whose sole justification is that the friction coefficient is on the order of 0.01-0.2, and thus appears reasonable. But, when an accurate constitutive model is used, there is no need to include friction for ballistic penetration modeling.
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7.3 Applicability of Cavity Expansion for Rigid-Body Penetration
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Warren [38] examines rigid-body penetration for an elastic, strain-hardening aluminum using quasistatic spherical cavity expansion and dynamic spherical cavity expansion solutions to obtain the force on an ogival projectile. Warren compares the depth of penetration as a function of penetration velocity against a number of different data sets. He concludes that quasi-static target strength dominates penetration resistance at the lower impact velocities (where target inertial effects have been ignored), but that at higher velocities, target inertial effects must be included to match the experimental data. In effect, although more simplistic than the analysis in Ref. [38], the quasi-static solution is similar to Eq. (5), whereas, the dynamic solution is similar to Eq. (4). Warren concludes: “Final depth of penetration predictions…that include target inertia are found to be in excellent agreement with all of the experimental penetration data for all striking velocities, projectile geometries and ogive-nose shapes, projectiles densities, and material properties” *38+. He further notes: “Additionally, it is observed that target inertia must eventually be included at some particular striking velocity that is…dependent on the projectile geometry, nose shape…for the spherical cavity-expansion approximation model to accurately predict final depths of penetration.” *38]. Rosenberg and Dekel [39] and Rubin et al. [40], take issue with the applicability of the cavity expansion approximation applied to rigid-body penetration. They state that the resisting stress on a rigid projectile is constant, and that deceleration during penetration is independent of velocity [39]. They argue that a constant penetration channel diameter implies that the deceleration is a constant [39], and they support their argument by examining data from Piekutowski et al. [24]. They also note that Hill [33] states that a penetration channel with a constant diameter, which is equal to that of the projectile, is an indication for a constant resisting stress on the projectile [39]. Using experimental data 21
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from Ref. [24], Rosenberg and Dekel calculate an average deceleration, a V 2 / 2P , where V is the impact velocity and P is the final depth of penetration. They then show excellent agreement with experimental data by plotting P V 2 /( 2a) , albeit the value for a was obtained by analyzing the experimental data.
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Rubin et al. [40], develop a model for the drag force using what they believe to be a more realistic flow field in the target. They find that the force is a constant below a critical impact velocity, called the separation velocity Vs, which is what Hill calls the cavitation velocity [33].8 Since the force is a constant, then the deceleration is a constant. They calculate the resisting force that is larger than that calculated from cavity expansion. Thus, Refs. [39-40] state that the one-dimensional cavity expansion solutions, whether cylindrical or spherical, do not adequately represent the two-dimensional flow field around the penetrating projectile, and therefore, the cavity expansion solutions predict a value for the resisting stress that is too low. In consequence, they conclude that Warren needed to include target inertia to increase the resisting stress as the impact velocity increased.
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Warren responds *41+ that Hill’s comment concerns a projectile that is moving at constant velocity, which is not the case here because the projectile is decelerating due to target resistance; further, section 4 of Hill’s paper *33+ states “that if the velocity is not constant, then the deceleration of the projectile will have an effect on the target resistance” *41+. Warren then shows experimental data (with the tunneling region the same diameter as the projectile so that there is no cavitation) where the deceleration is not constant. Warren concludes that “when the strength term overshadows the inertial term…the target inertia term can be neglected in the penetration model. However, when the strength does not overshadow the target inertia effects, then a target inertia term must be included” *41+.
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Having laid out these arguments, it is observed that Warren has modeled the penetration process using an accurate constitutive description of the target material. On the other hand, the analytical model of Rubin et al., cannot explicitly account for the influence of compressibility, strain hardening, strain rate, and temperature of the target material, and must rely on numerical simulations for the influence of these effects [40]. Further, for accurate replication of the P versus V response, the deceleration parameter a must be determined from experimental data.
8.0 Dynamic Plasticity and the Ravid-Bodner Model
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The next major advance in analytical penetration modeling was the introduction of a multi-stage penetration/perforation model, developed by Ravid and Bodner [42]. The authors proposed five stages of penetration/perforation for a rigid projectile, as shown in Fig. 16: 1) penetration, 2) bulge formation, 3) bulge advancement, 4) plug formation and exit, and 5) projectile exit.
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At the critical velocity, the penetration channel begins to open up (and is no longer the same diameter as the projectile), penetration resistance increases, and target inertia must be included to determine the deceleration force on the projectile. Vs depends upon nose shaped, target material, etc.
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The authors established plastic flow fields within the target. As shown in Fig. 17, there are three zones where target material is moving (there is no target motion in zone IV). Velocity distributions are assigned to each zone, subject to compatibility and continuity conditions, and then the plastic work rates (including strain-rate effects) are computed. Work hardening is considered indirectly by taking the flow stress to be the average value, with respect to plastic work, over the full strain range. The radial and axial extent of the plastic zone fields (R and R, where R is the projectile radius) are solved for by minimizing the plastic work rate. Projectile deceleration is computed from an energy rate balance.
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Stage I (the penetration stage) continues until the plastic flow field reaches the rear of the target, at which point the target begins to form a spherical bulge, Fig. 18. Uplifting of target material at the surface ceases, and all target motion is in the direction of penetration. Target mass motion, combined with the incompressibility condition of plastic flow, determines the plastic flow velocity field (thus, no minimization procedures are required).
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At some point, the bulge no longer expands radially (the end of Stage II), and instead, the bulge advances in the direction of penetration, Stage III, Fig. 19. The geometry is determined by the plastic flow incompressibility condition.
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The target can possibly fail during Stages II or III. Ravid and Bodner considered a number of possible failure mechanisms, Stage IV. In their approach, it is possible to estimate the strains in the zones and between zone boundaries, which allows them to consider the formation of adiabatic shear bands, brittle failure in shear, conventional plug formation, and ductile failure. The length and shape of the plug depends upon the failure mode; the specific failure mode is target material dependent.
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An example of the model is shown in Fig. 20. A 7.62-mm armor-piercing (AP) bullet impacted a 12mm-thick steel plate at 855 m/s. The force on the projectile, the projectile velocity, and bullet displacement are plotted in the figure. The various stages of penetration are also shown in the figure. The model predicted a plug velocity of 424 m/s and a bullet velocity of 364 m/s. The experimental exit velocity was 345 45 m/s.
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Ravid, Bodner, and Holcman [43] extended the concept of dynamic plastic flow fields to high velocity impact (greater than 1 km/s). The analysis considered impact shock generation, plastic flow of the target, and mushrooming of the projectile for normal impact by a rod. The analysis continues until the shock effects in the rod become relatively unimportant (a few microseconds after impact). This model would be the first stage of a postulated seven-stage model, which was never fully developed.
9.0 Walker-Anderson Model
9.1 Semi-infinite Targets The momentum equation along the projectile-target centerline is given by: u z 1 u z zz 2 xz 0 t 2 z z x 2
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Walker and Anderson integrated this equation along the projectile-target centerline [44]:
p
z1
zp
u z dz t t
p
zi
zi
u z dz t t
u z 1 dz p u z2 t 2
zi
zi
zp
1 t u z2 2
zz
zi
zp
2
zp
xz dz 0 x
u z 1 1 xz dz p u 2 v 2 t u 2 2 dz 0 zp t 2 2 x
(49a)
(49b)
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zp
where zp and zi denote the positions of the rear of the projectile and the projectile-target interface, respectively; the target is assumed to be semi-infinite in the original model. To integrate Eq. (49b) explicitly, three major assumptions were invoked: (1) the velocity profile along the centerline in the projectile and target are specified (so that the first two integrals can be integrated); 2) the rear of the projectile is decelerated by elastic waves (similar to Tate); and 3) the shear behavior of the target material is specified (so that the gradient of the shear stress can be evaluated).
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The velocity profiles were motivated by the results of numerical simulations [29]. The velocity profile in the projectile is assumed to be bi-linear, see Fig. 21. The velocity field in the target is derived from a vector potential A:
ur Ar , 2 c2 1
M
c Rc 2 1 sin r
1 A sin eˆr 1 rAeˆ r sin r r
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v Ar , eˆ
(50)
(51)
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where Rc is the radius of the penetration channel. Thus,
vr
u c2 Rc2 2 1 cos 2 c 1 r
(52a)
u sin 1
(52b)
v z v r cos v sin
(53)
v
2 c
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The velocity in the z-direction is given by:
which gives the expression for the velocity field in the target, along the centerline, in Fig. 21. The velocity profiles are shown in Fig. 21, along with a comparison to results of a numerical simulation. In the figure, the symbol s represents the extent of plastic flow in the projectile (from the projectile-target interface); and cRc is the extent of the plastic flow field in the target. The specific comparison is for a tungsten-alloy projectile that impacted a steel target at 1.5 km/s (the projectiles has decelerated to approximately 1.4 km/s at the time that the velocity profile was plotted). 24
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Assuming a von Mises yield surface and that the shear stress is proportional to the rate of 7 deformation (rigid plasticity), the integral of the gradient of the shear stress gives Yt ln c [44]. 6 Taking the time derivatives of the velocity profiles in Fig. 21, and assembling the various terms in Eq. (49b), the centerline momentum balance becomes:
c 1 2 Rc u d v u s2 t c p c 1 dt s 2 ( c 1) 2
1 7 1 p ( v u ) 2 t u 2 Yt ln c 2 3 2
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p v ( L s) u p s t Rc
(54)
where the dot over a symbol implies the time derivative. Deceleration of the rear of the projectile is given by:
p
v u s 1 p ( L s) c c
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(55)
where it is assumed that the decelerating elastic waves run between the tail and the plastic zone in the projectile (a material nonlinearity) at a bar wave speed of c, (Ep /p)1/2. Lastly, the length of the projectile shortens with time:
M
L v u
(56)
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Equations (54-56) are solved simultaneously with an equation that provides for the extent of the plastic flow field (c), which is estimated by dynamic compressible cylindrical cavity expansion [44]:
u 2 c2 K t t u 2 c2 1 t 2Gt
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u2 1 t Yt
u K t K o 1 k co
K t t u 2
(57a)
(57b)
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Compressibility of the target is reflected in c. Equation (57b) is an heuristic expression to represent the stiffening of the bulk modulus with increasing penetration velocity [44], where the shock velocity is related to the particle (penetration) velocity by us = co + ku, and Ko = oco2. As the penetration velocity increases, c decreases as shown in Fig. 22. This prediction was confirmed by numerical simulations in Ref. [14]. As already stated, c is dependent on the penetration velocity, and not the initial impact velocity. Therefore, c changes as the penetration velocity changes. The elastic energy stored in compression is extremely important as describe in Section 4.2; c reflects this compression. For example, the dashed curve in Fig. 22 represents c for rigid-body penetration into aluminum. As the penetration velocity decreases, c becomes larger, effectively accounting for less energy being stored in elastic compression, and more energy being directly dissipated in target plastic work. 25
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For eroding long-rod penetration, the penetration velocity is approximately constant during the primary phase of penetration, so c is approximately a constant. It then increases during the terminal phase when there is rapid deceleration of the remaining projectile.
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It is interesting to note that Chocron, Anderson, and Walker [45] compared the results for compressible cylindrical cavity expansion and compressible spherical cavity expansion to compute c. They found that the use of the compressible cylindrical model, using values for the material flow stresses consistent with the constitutive behavior of the materials, gave better agreement with experimental data at impact velocities below 2 km/s. (Cavity expansion is used to provide an estimate of the extent of plastic flow in the target. This is in contrast to the discussion in Section 7 where cavity expansion is used to calculate directly the force on the projectile. The hemispherical flow field in the Walker-Anderson model is a good approximation of the applicable mechanics. Any error is estimating the extent of plastic zone in the target using cavity expansion is somewhat mitigated by the fact it enters the Walker-Anderson model as ln(c).)
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The Walker-Anderson model uses material properties for the projectile and target; the only empirical input is the crater radius Rc as a function of impact velocity [44]. Examples of the model, compared to experimental data, are shown in Fig. 23; other examples are provided in Ref. [44]. Although comparison with experimental data is excellent, there is a discrepancy for eroding penetration that has not been resolved. As already discussed, the projectile is decelerated by elastic waves. However, when the length of the projectile has decreased to approximately one projectile diameter, the projectile material is within the plastically deforming region, and the material is “sees” the high pressures associated with the penetration front (for example, see Ref. [47]). This high pressure decelerates the remaining projectile sooner than calculated by the analytical model, as shown in Fig. 24 where the nose and tail velocities are plotted versus time. The result for the Tate model is also shown for comparison. For this specific example, the Walker-Anderson model, the Tate model, and the simulation have the same total depth of penetration. Notwithstanding this discrepancy in the current model, the Walker-Anderson model provides a reasonable estimate of the total depth of penetration for L/D = 1 projectiles (as shown in Ref. [44]), probably due to the fact that the target mechanics (the plastic flow zone) is being treated correctly combined with the robustness of the conservation of momentum approach.
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In the limit where s0, Young’s modulus of the projectile becomes large (c), and the threedimensional terms are removed (Rc0), Eqs. (54-56) reduce to Tate’s model, Eqs. (28), (30), and (31), but with an explicit expression for Rt:
Rt
7 Yt ln c 3
It is noted that p in Eq. (55) is the dynamic flow stress of the projectile material; whereas, Tate heuristically accounted for dynamic effects by adjusting the parameter in Eq. (38). More will be discussed about the appropriate value for Yt in Section 9.3.
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9.2 Finite Targets
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Initially, the Walker-Anderson model was extended to finite-thick targets by combining the failure modes of the Ravid-Bodner model with the Walker-Anderson model [48]. In this approach, the WalkerAnderson model is considered operative until the presence of the target rear surface begins to influence projectile penetration, which was assumed to occur when the plastic zone reached the effected target rear surface covering the projected area of the projectile (R2). The modified model predicted bulging, failure of the target, and an estimate of the residual velocity as a function of impact velocity, including the ballistic limit velocity. Then, in 1999, Walker extended the Walker-Anderson model to include explicitly bulging [49] and failure [50]. When the flow field reaches the rear surface of the target, the flow field is achieved through a multiplicative blending of the potentials for hemispherical flow (Adeep), Eq. (50), and shear flow (Ashear):
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1 Ar, θ Adeep Ashear eˆ
(59)
where determines how much of the potential is deep versus shear. The velocity flow fields are found by taking the curl of Eq. (59), as was done for Eqs. (50-51). Details are provided in Ref. [49]. The volume ~ of the material in the truncated hemisphere with radius R and height h is V h 2 R~ h / 3 [51]. The
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~3 : weighting is determined by dividing this volume by the original volume of the hemisphere, (2 / 3)R
3 h 2 R
2
1 h 2 R
3
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~ ~ In practice, h is the distance between the projectile nose and the target rear surface, and
(60) ~ R c 1Rc .
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equals 1 until the plastic zone reaches the rear surface of the target. As the distance between the projectile nose and the target rear surface decreases, decreases until it reaches zero at h = 0, and the velocity field is pure shear.
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After considerable math, and a few minor approximations for simplification, modifications for the back surface is effected by replacing the last terms in Eq. (54):
7 7 1 1 2 t u 2 Yt ln c by t u uback Yt ln ~c 3 3 2 2
(61)
~
where c = 1 + (distance to back surface)/Rc. The change in the target resistance term reflects the fact that there is less target material being plastically deformed. The velocity of the rear (back) surface is obtained explicitly from the flow fields, and integrating the velocity over time provides the position of the rear surface. A comparison of the Walker-Anderson model with bulging to a numerical simulation is shown in Fig. 25. Since the flow field is known, it is possible to evaluate the equivalent plastic strain rate at any point of the target: 27
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p
1 u u j Dij i 2 x j xi
2 Dij Dij 3
(62)
p
4 Rc 3 r
2
u r
Integration over time provides the equivalent plastic strain.
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where Dij is the strain rate tensor, and the ui’s and xj’s are the velocity and coordinate directions, respectively. However, Eq. (62) is difficult to evaluate for a general location within the target. It is straightforward, however, to evaluate the equivalent plastic strain rate along the projectile-target centerline where the expression for the flow field is algebraically simple [50]:
(63)
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The strains were calculated a specified locations along the back surface [50], and it was found that the maximum equivalent plastic strain occurs at a distance of approximately two projectile radii from the centerline, and not on the centerline. Nevertheless, the simplest failure criterion that can be posed is to invoke target failure (that is, perforation) when the centerline strain at the rear surface exceeds some critical strain, crit. It is necessary to use an experimental data point to determine crit, such as the ballistic limit velocity. Once determined, then crit is considered to be a constant for other targets that use the same material.
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The model is compared to two experiments in Fig. 26. Both experiments are near the V50 limit for the specific target thickness (a tungsten-alloy projectile into an armor steel). For the thinner target, the impact velocity is just below V50 of 1250 m/s. The projectile has perforated the thicker target at the V50 impact velocity. Penetration, bulging, and perforation are well predicted by the analytical model.
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It was found that a critical strain to failure calculated in this manner permits accurate predictions of residual velocities for relatively ductile targets. However, this failure criterion is too simplistic for very hard targets that fail by brittle shear or strain localization [50].
9.3 Similitude Analysis Revisited and the Effective Flow Stress
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Normalized penetration as a function of impact velocity is shown for five sets of data in Fig. 27 [46]. The projectiles were flat-nosed with an aspect ratio (L/D) of 10. There is a tungsten alloy (WA) projectile into an armor steel, a maraging (W8) steel, and a mild steel (Steel 52). There is also a moderately strong steel projectile into a mild steel (Steel 37/52) and into an armor steel. A nonlinear regression was used to fit each individual set of experimental data to an equation with the form: a2 P a1 L 1 10 a3 ( a4 V )
(64)
where the ai’s are determined from the regression analysis; the solid lines represent the curve fits [53]. The data are replotted in Fig. 28 using nondimensional variables [54]. The ratio of the projectile to target densities () is used to account for density differences in normalized penetration. A nondimensional velocity is defined, pV 2/efs, where efs is the effective flow stress. pV 2 is proportional 28
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to the penetration pressure, and efs is related to the strength of the target. Thus, the nondimensional velocity is a ratio of penetration pressure to target resistance. Similitude theory does not specify the relationship between the nondimensional variables, and it was found that better correlation was obtained by also including the ratio of the densities on the abscissa, raised to a small power. The exponents were determined by adjusting them to provide a relatively tightly collapsed data set.9
0.2
a2 P a1 L 1 10 a3 *a4 V
pV 2 0.14 V efs
The regression coefficients are shown in Fig. 28.
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The form of the equation used to fit the collapsed data is similar to that of Eq. (64), but now it is P/L as a function of V [54]:
p t
(65)
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Key to this analysis is the selection of the effective flow stress, efs. efs was determined by conducting a parametric study on P/L as a function of velocity for various values of efs. The valued selected for efs was the one that minimized the root mean square error on P/L, see Fig. 29 for an example [53]. This was done for each of the experimental data sets. efs for Steel 37, Steel 52, and German armor steel were found to be 0.797 GPa, 0.858 GPa, and 1.41 GPa, respectively.
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The target flow stress used in the Walker-Anderson model (Yt), which is a single value, must somehow encompass the volumetric effect of the target flow stress that is a function of strain, strain rate, and temperature; and which varies with time and location in the target. Adiabatic stress strain curves10, using the Johnson-Cook constitutive model [55], are shown for two materials at various strain rates in Fig. 30: a mild steel (A36) and 4340 steel hardened to Rockwell C 30. A36 is similar to the German mild steels 37 and 52. The long horizontal dashed lines represent efs for Steel 37 and 52, determined by the minimization procedure. The dashed-dotted line represents an average equivalent stress for A36 with some strain hardening; this dashed-dotted line is only 8% lower than the calculated efs for Steel 37.
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4340 steel, hardened to Rc30, is often used as an experimental surrogate for armor steel. The average equivalent stress at a strain rate of 103 s-1 is 1.2 GPa. This is the value used for Yt for one of the steels in Fig. 23. efs for German armor steel was determined to be 1.4 GPa, which is the value for Yt used for the stronger steel in Fig. 23. Thus, Yt in the Walker-Anderson model and efs are equivalent; Yt, i.e., the effective flow stress, encompasses an averaged volumetric flow stress for the target material. And it is seen that an estimate for Yt can be made from a stress-strain curve.
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It is interesting to note that the strength of the projectile is not included in the analysis. This is because the penetration pressures greatly exceeds the flow stress of the projectile material for these projectile-target combinations and impact velocities. 10
Plastic work is converted to temperature, which then results in thermal softening.
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In fact, Walker developed a procedure for estimating the target flow stress [56] using Johnson-Cook constitutive parameters. The strain field in the target can be calculated, and the plastic strain rate can be determined directly from the velocity field. If no thermal softening is present, the resulting equations can be integrated explicitly; if thermal softening is taken into account, then the target resistance term in Eq. (58) must be solved in the presence of thermal softening [56].
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10.0 Summary
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This article has reviewed analytical penetration models for metallic targets. The models range in applicability from relatively low impact velocities (rigid-body penetration) to very high-velocity impacts (hydrodynamic approximation). It was seen that the oldest of the models, represented by the Poncelet equation, Eq. (1), is derived naturally from the Tate model. The Poncelet equation, where the force is proportional to a static term and a velocity-squared term, was also derived from calculating the decelerating force on the projectile using dynamic cavity expansion. The objective of all of these models is to replicate experimental data using real material properties while minimizing empirical factors. Exploring the applicability and accuracy of the models has led to insights as well as limitations.
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Backman and Goldsmith [1] highlighted eight areas that merited further investigations to improve modeling of penetration mechanics. It is instructive to comment on these to see what progress has been made, and where there remain outstanding issues and requirements for further research.
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1. The development of more rational bases of specifying modeling parameters. Model parameters should be specified beforehand, i.e., they should not be determined from the experimental data that they are supposed to predict. Specifically, Backman and Goldsmith state that the next two topics can contribute to less empirical procedures. I will comment after item number 3.
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2. The improved description of interface phenomena (e.g., the formation of the penetration channel, interface phenomena involving oblique impact, nonzero flight orientations, deforming projectiles).
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3. The improved analytical modeling of force contributions (a need to extend modeling to thicker targets, and to include dishing and bulging of target elements, plug displacement, etc.).
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In general, the models described above do not use empirical fits for model parameters; instead, procedures have been developed to estimate model parameters from independent laboratory measurements, such as density, flow stress, etc. A major advance, particularly for penetration and perforation of thick targets, was the inclusion of dynamic plasticity, i.e., the description of the plastic flow field(s). Thus, considerable progress has been made in these three areas for normal impact of an unyawed projectile. 4. Effects of obliquity. Typically, the line-of-sight thickness is used to account for target obliquity. This is a reasonable approximation. However, at large obliquities, particularly for thick targets, the presence of the free surface results in asymmetric loading on the projectile nose, which can lead to bending and possibly fracture of the projectile. Although this can be modeled quite 30
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nicely in three-dimensional numerical simulations, this is currently beyond the ability of firstprinciples analytical models.
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5. Effects of flight orientation of the projectile. It is well-known that yaw and pitch can dramatically affect penetration response, which is amplified by target obliquity, as demonstrated by Anderson, Behner, and Hohler [57]. Although analytical modeling of some of the initial conditions can be performed, the complete projectile-target interaction is probably not analytically tractable. However, approximations can be made. An example is a paper by Walker, Anderson, and Goodlin [58] that combines several analytical expressions (based on experimental data) for predicting the depth of penetration as a function of velocity, L/D ratio, and impact inclination.
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6. Improved constitutive representations. Forrestal, Warren, and colleagues explicitly use the stress-strain behavior of the target material, including work hardening and strain-rate effects, in their modeling. In the Tate and Walker-Anderson models, a single value for the projectile flow stress, including strain-rate effects, accurately predicts deceleration of the projectile. For the Walker-Anderson model, a single value for the target flow stress is used, but it was shown that this value is related to a volumetric average of the target stress that includes work hardening, strain-rate effects, and thermal softening. Backman and Goldsmith’s comment also had to do with materials other than metals. The focus of this article was metallic targets; ceramics, glasses, fabrics, composites, etc., are beyond the scope of this article.
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7. Thermo-mechanical coupling and heat dissipation. Aside from the concept of the effective flow stress in the Walker-Anderson model, thermo-mechanical coupling, including localization, is probably beyond the capability of analytical models without some empiricism. However, thermo-mechanical coupling is readily modeled in numerical simulations, although the issue of different length scales of various phenomena remains an issue. 8. Failure criteria. Prediction of failure is the really tough problem. See the paragraph below.
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Unresolved at this time are accurate predictions for the transition from rigid-body to eroding penetration, which depends upon projectile material properties, projectile nose shape, and target strength, although some interesting work has been done by Segletes [59]. Another area of uncertainty is target failure, which depends upon the target material and loading conditions. Thus, it is failure (whether projectile and/or target) that is difficult, if not impossible, to predict a priori. This is also true of numerical simulations. It may be that some empiricism will always be required to calibrate a model for accurate predictions. Another area that requires further work is multi-material layered targets. The stress field precedes the penetration front, thus, the next layer will affect the rate of penetration. This effect will increase as the difference in material properties of the two materials increases. This is true for a metallic layered target, and the effect may become even larger when other materials such as ceramics, fabrics, or composites are used. It was observed that when models are based on conservation of momentum, such as the Tate and Walker-Anderson models, they tend to be robust and applicable to a wide range of materials and impact 31
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velocities. Models based on conservation of energy typically have limited applicability because energy sinks/dissipation change with impact velocity and materials. This article focused on analytical models for metallic targets. An article is planned on reviewing analytical models for non-metallic targets, such as glasses, ceramics, yarns and composites.
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11.0 ACKNOWLEDGEMENTS The author is very grateful for the interactions over the years with most (but not all) of the authors referenced in this article.
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25. M. J. Forrestal and A. J. Piekutowski, “Penetration experiments with 6061-T6511 aluminum targets and spherical-nose steel projectiles at striking velocities between 0.5 and 3.0 km/s,” Int. J. Impact Engng., 24(1): 57-67, 2000. 26. A. Tate, “Long rod penetration models—Part I. A flow field model for high speed long rod penetration,” Int. J. Mech. Sci., 28(8): 535-548, 1986. 27. A. Tate, “Long rod penetration models—Part II. Extensions to the hydrodynamic theory of penetration,” Int. J. Mech. Sci., 28(9): 599-612, 1986. 28. R. F. Bishop, R. Hill, and N. F. Mott, “The theory of indentation and hardness,” Proc. Royal Soc., 57(3): 147-159, 1945. 33
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29. C. E. Anderson, Jr. and James D. Walker, “An examination of long-rod penetration,” Int. J. Impact Engng., 11(4): 481-501, 1991. 30. H. G. Hopkins, “Dynamic expansion of spherical cavities in metals,” Progress in Solid Mechanics, Vol. 1 (I. Sneddon and R. Hill, Eds.), North-Holland, NY, pp. 85-164, 1960. 31. V. K. Luk, M. J. Forrestal, and D. E. Amos, “Dynamic spherical cavity expansion of strain-hardening materials,” J. Appl. Mech., 58: 1-6, 1991.
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32. T. L. Warren and M. J. Forrestal, “Effects of strain hardening and strain-rate sensitivity on the penetration of aluminum targets with spherical-nosed rods,” Int. J. Solids Structures, 35(28-29): 3737-3753, 1998. 33. R. Hill, “Cavitation and the influence of headshape in attack of thick targets by non-deforming projectiles,” J. Mech. Phys. Solids, 28: 249-263, 1980.
35. M. Wilkins, personal communication.
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34. A. Tate, “A theoretical estimate of temperature effects during rod penetration,” Proc. 9th Int. Symp. Ballistics, 2: 307-314, Shriverham, UK, 1986.
36. M. L. Wilkins, “Calculations of elastic-plastic flow,” in Methods of Computational Physics, Vol. 3 (B. Adler, S. Fernback, and M. Rotenberg, Eds.), Academic Press, NY, NY, 1964.
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37. G.T. Camacho and M. Ortiz, “Adaptive Lagrangian modeling of ballistic penetration of metallic targets,” Comput. Methods Appl. Mech. Eng., 142: 269-301, 1997. 38. T. L. Warren, “The effect of target inertia on the penetration of aluminum targets by rigid ogivenosed long rods,” Int. J. Impact Engng., 91: 6-13, 2016.
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39. Z. Rosenberg and E. Dekel, “A comment on ‘The effect of target inertia on the penetration of aluminum targets by rigid ogive-nosed long rods’ by T. L. Warren,” Letter to the Editor, Int. J. Impact Engng., 93: 231-233, 2016. 40. M. B. Rubin, R. Kositski, and Z. Rosenberg, “Essential physics of target inertia in penetration problems missed by cavity expansion models,” Int. J. Impact Engng., 98: 97-104, 2016.
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41. T. L. Warren, “Response to: ‘A comment on ‘The effect of target inertia on the penetration of aluminum targets by rigid ogive-nosed long rods by T. L. Warren’ by Z. Rosenberg and E. Dekel,” Letter to the Editor, Int. J. Impact Engng., 93: 234-235, 2016.
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42. M. Ravid and S. R. Bodner, “Dynamic perforation of viscoplastic plates by rigid projectiles,” Int. J. Engng. Sci., 21(6): 577-591, 1983. 43. M. Ravid, S. R. Bodner, and I. Holcman, “Analysis of very high speed impact,” Int. J. Engng. Sci., 25(4): 473-482, 1987. 44. J. D. Walker and C. E. Anderson, Jr., “A time-dependent model for long-rod penetration,” Int. J. Impact Engng., 16(1): 19-48, 1995.
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45. S. Chocron, C. E. Anderson, Jr., and J. D. Walker, “Long-rod penetration: cylindrical vs. spherical cavity expansion for the extent of plastic flow,” Proc 17th Symp. Ballistics, (C. Van Niekerk, Ed.), 3: 319-326, South African Ballistics Organisation, Morleta Park, South African, 1998. 46. C. E. Anderson, Jr. and D. L. Orphal, “Analysis of the terminal phase of penetration,” Int. J. Impact Engng., 29: 69-80, 2003.
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47. M. Ravid, S. R. Bodner, J. D. Walker, S. Chocron, C. E. Anderson, Jr., and J. P. Riegel, III, “Modification of the Walker-Anderson penetration model to include exit failure modes and fragmentation,” Proc. 17th Int. Symp. Ballistics (C. Van Niekerk, Ed.), 3: 267-274, South African Ballistics Organisation, Morleta Park, South African, 1998. 48. J. D. Walker, “An analytic velocity field for back surface bulging,” Proc. 18th Int. Symp. Ballistics (W. G. Reinecke, Ed.), 2: 1239-1246, Technomic Publishing Company, Lancaster, PA, 1999.
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49. S. Chocron, C. E. Anderson, Jr., and J. D. Walker, “A consistent flow approach to model penetration and failure of finite-thickness metallic targets,” Proc. 18th Int. Symp. Ballistics (W. G. Reinecke, Ed.), 2: 761-768, Technomic Publishing Company, Lancaster, PA, 1999. 50. V. Hohler and A. J. Stilp, compiled in: C. E. Anderson, Jr., B. L. Morris, and D. L. Littlefield, “A penetration mechanics database,” SwRI Report 3593/001, Southwest Research Institute, San Antonio TX, 1992.
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51. Standard Mathematical Tables, 20th Edtion (S. M. Selby, Ed-in-Chief), CRC Press, Cleveland, OH, 1972.
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52. C. E. Anderson, Jr., V. Hohler, J. D. Walker, and A. J. Stilp, “Time-resolved penetration of long rods into steel targets,” Int. J. Impact Engng., 16(1): 1-18, 1995.
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53. J.P. Riegel, III and C. E. Anderson, Jr., “Target effective flow stress calibrated using the WalkerAnderson penetration model,” Proc. 28th Int. Symp. Ballistics, DESTech Publications, Inc., 2: 12421253, 2014. 54. C. E. Anderson, Jr. and J.P. Riegel III, “A penetration model based on experimental data,” Int. J. Impact Engng., 80: 24-35, 2015.
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55. G. R. Johnson and W. H. Cook, “A constitutive model and data for metals subjected to large strain, high strain rates, and high temperatures,” Proc. 7th Int. Symp. Ballistics, pp. 541-548, The Hague, The Netherlands, April 19-23 1983. 56. J. D. Walker, S. Chocron, C. E. Anderson, Jr., J. P. Riegel III, D.S. Riha, J. M. McFarland, M. Moore, G. Willden, M. Murphy, and B. Abbott, “Survivability modeling in DARPA's adaptive vehicle make (AVM) program,” Proc. 27th Int. Symp. Ballistics, 2: 967-979, DEStech Publications, Lancaster, PA, 2013. 57. C. E. Anderson, Jr., T. Behner, and V. Hohler, “Penetration efficiency as a function of target obliquity and projectile pitch,” J. Appl. Mech., 80: 031801-1/11, 2013.
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58. J. D. Walker, C. E. Anderson, Jr., and D. L. Goodlin, “Tungsten into steel penetration including velocity, L/D, and impact inclination effects,” Proc. 19th Int. Ballistics Symp. (I. R. Crewther, Ed.), 3: 1133-1140, Interlaken, Switzerland, 2001.
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59. S. B. Segletes, “The erosion transition of tungsten-alloy long rods into aluminum targets,” Int. J. Impact Engng., 44: 2168-2191, 2007.
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Figure 1. Normalized penetration for very hard steel projectiles into 6061-T6 aluminum.
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Data from Ref. [6].
Figure 2. Normalized penetration for tungsten-alloy projectiles into a structural grade aluminum. Date are from Refs. [7-8].
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(a) Jet penetrating a target
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(b) Coordinate transformation.
Figure 3. Schematic of an incompressible, hydrodynamic jet penetrating an incompressible
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hydrodynamic target.
Figure 4. Phases of penetration for a L/D 20 projectile impacting at 3.0 km/s. 38
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Figure 5. Measured penetration vs. application of Eqs. (15-16);
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the figure is reproduced from Ref. [12].
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Figure 6. Comparison of Recht-Ipson model for chunky projectiles into thin plates
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with experimental data. Data from Ref. [13].
Figure 7. Comparison of Recht-Ipson model for an armor-piercing bullet against a thick target. Data from Ref. [13].
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Figure 8. Residual velocity vs. impact velocity for various types of projectiles into two
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hardnesses of armor steel [17].
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Figure 9. Normalized residual velocity vs. normalized impact velocity for data in Fig. 8 [17]
Figure 10. Normalized penetration vs. impact velocity for various Yp – Rt
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combinations for a steel projectile penetrating an aluminum target.
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Figure 11. Comparison of Tate-Poncelet equation with experimental data for steel projectiles
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penetrating 6061-T6 aluminum targets. The dashed line is given by Eq. (36). Data are from Refs. [24-25].
Figure 12. Normalized penetration for hard steel ogive projectiles into 6061-T6 aluminum
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depicting the rapid transition from rigid-body to eroding penetration. Data from Ref. [24].
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Figure 13. Normalized penetration for tungsten-alloy projectiles into 7020 aluminum targets.
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Data are from Ref. [7-8].
Figure 14. Rigid projectile geometry with hemispherical nose.
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Figure 15. Normalized penetration as a function of impact velocity for various dynamic
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spherical cavity solutions and experimental data. Data from Refs. [22-23].
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Figure 16. Multi-stage penetration/perforation model by Ravid and Bodner [42].
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Figure 17. Ravid-Bodner penetraton model, Stage I (from Ref. [42]).
Figure 18. Bulge formation, Stage II (from Ref. [42]).
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Figure 19. Bulge advancement, Stage III (from Ref. [42]).
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Figure 20. Example of the Ravid-Bodner model: a 7.62-mm AP bullet impacting a 12-mmthick steel plate at 855 m/s.
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Projectile
( zi s) z zi z p z ( zi s)
Target 2 1
Rc r ( z ) c Rc r ( z ) c Rc
(a) Centerline velocity profiles.
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u R c c u z ( z ) c2 1 r ( z ) 0
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vu ( z zi ) u u z ( z) s v
(b) Comparision of analytic approximation and numerical simulation
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Figure 21. Centerline velocity profiles of Walker-Anderson model [44].
Figure 22. c as a function of penetration velocity for steel and aluminum targets.
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Figure 23. Results for Walker-Anderson model for L/D 10 tungsten-alloy and steel projectiles
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penetrating two different armor steels. Data are from Ref. [50].
Figure 24. Comparison of nose and tail velocities vs. time for an L/D = 10 tungstenalloy projectile into an armor steel target at an impact velocity of 1.5 km/s.
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Figure 25. Comparison of Walker-Anderson model with bulging: red lines denote model
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predictions superimposed on a numerical simulation.
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(b) T = 4.95 cm; V = 1700 m/s.
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(a) T = 2.90 cm; V = 1241 m/s.
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Figure 26. Bulging and failure of the Walker-Anderson model. Data from Ref. [52].
Figure 27. Normalized penetration for five sets of projectile-target combinations as a function of impact velocity. Data from Ref. [50].
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Figure 28. Normalized penetration vs. a nondimensional impact velocity [54].
Figure 29. Root mean square error as t is varied for the steel projectile steel 37/52 target.
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Figure 30. Equivalent stress vs. equivalent plastic strain for two materials.
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