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55
CHAPTER TWO
Introduction to Hollow Charge
For over a century mining engineers has known that some of the force
of an explosive charge could be concentrated on a small area by hollowing out
the charge opposite to the area ("hollow charge principle"). The earliest
known reference to this is 1792; a popular account appeared in von Nostrand's
Magazine in 1884. The effect became known as the Munroe effect in England
and the United States, and as the Neumann effect in Germany, in honor of
later workers (1885, 1911). Early in the recent war a large number of new
weapons, based on an important modification of this principle, appeared
almost simultaneously in the armed forces of the major combatants. It had
been discovered that by lining the hollow with a thin metal liner it became
possible to perforate armor plate, concrete walls, or other structures with a
surprisingly small weight of explosive charge. Very often these new weapons
took the form of a projectile with the lined hollow charge in the nose. Such
projectiles had an advantage over the ordinary artillery projectiles in that their
ability to perforate was practically independent of their striking velocity. In
fact, hand placed and statically detonated hollow charges are at least as
effective in perforating armor as those used in projectiles which detonate on
impact [fig.1].
Fig. 1: Cross section of the head of a U. S. Army bazooka, showing conical steel liner in
the shaped charge.
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When the booster is detonated, it starts an explosive wave which travels
parallel with the axis of the charge and collapses the steel cone starting from
its apex. The collapsing cone forms a long thin jet of steel that travels to the
right in the figure along the extension of the cone axis at speeds up to 30,000
ft./sec These high speed jets force their way through armor plate in much the
same way as a stream of water from a powerful fire hose would force its way
through a bank of soft mud. The pressures produced by the jet traveling at
these high velocities are so much greater than the ultimate strength of the
armor plate that the strength of the armor plays a negligible role in retarding
the penetration.
Many different kinds of liners were used in combat weapons.
Hemispheres, paraboloids, pear shapes, and trumpet shapes were all tried.
None of them appeared to provide remarkable advantages over any of the
others, though differences in performance did result from the different shapes.
2.1 Effects of Hollow Charge on Target
Fig. 2: A typical shaped charge with conical
steel liner for Use in static experiments. This
charge contains 115 g of Pentolite which has a
maximum diameter of 15
8 in. The steel liner
is 0.025 in. thick.
Fig.3: Photograph of a solid steel cylinder (3.25 in. in diameter,
7 in. in length) which has been shot and then sawed in half to
show the nature of the hole produced. A cross-sectioned replica
of the charge that produced this hole is shown in the position
that the real charge occupied before it was detonated. The real
charge contained 0.25 lb. of Pentolite. The cavity liner was a
steel cone (0.025 in. thick, 1.63-in. base diameter) [34,35].
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Fig.4: Photograph of a solid lead cylinder (4 in. in diameter,
9.5 in. in length) which has been treated the same as the steel
cylinder in Fig. 3. The steel slug from the liner can be seen
embedded in the lead about 5 in. from the bottom of the
cylinder[34,35].
π π’π . π: Photograph of a stack of plates treated like the
cylinders of Figs. 3 and 4. The plates are alternately steel and
lead; the top plate is steel, the next is lead, the third is steel,
etc. All of these plates were originally the same size (4 ππ Γ
4 ππ.Γ 0.5 ππ.)[34,35].
Fig.6: Photograph of a disk of lead (8 in. diameter, 2 in.
thick) which was placed 18 in. from a charge like that in Fig.
3)[34,35].
Fig. 7: Photograph of a steel cylinder, like that in Fig. 3, cut in
half after being shot with a charge from which the steel cone
had been removed. This unlined charge was detonated in
contact with the cylinder because it produces its deepest holes
at zero standoff, whereas lined charges produce their deepest
holes at a larger standoff [34, 35].
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Fig.8: Photograph of a steel cylinder that has been cut in half
after being blasted by a solid charge. This charge contained
more explosive than the others, since it' was cast in the same
mold without a cavity [34, 35].
2.2-The Birkhoff et Al. Theory
An elementary mathematical discussion can be given of the formation
of jets from conical and wedge-shaped liners [36]. Consider a charge having a
cross section as shown in Fig. 9. In the conical case the charge is obtained by
rotating the cross section about the dashed axis of symmetry. The wedge-
shaped case is obtained by supposing the cross section to be the same for an
indefinite distance perpendicular to the cross section or, equivalently, to be
confined between rigid walls at both ends. We first consider the case of a
wedge.
The mechanism of liner collapse is illustrated in Fig. 10. Starting from
the booster a detonation wave has traveled to the right in the figure and is in
the process of collapsing the liner. This wave has produced pressures so great
that the strength of the liner may be neglected and the material may be treated
as a non-viscous fluid. We assume that, after the walls have received an
original impulse from the detonation wave, the pressure on all sides of the
liner quickly equalizes and the walls continue to collapse inward with no
appreciable change in velocity. Because of the finite time required for the
wave to travel from the apex of the liner to the base, the angle 2π½ between the
moving walls is larger than that between the walls of the original liner.
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Actually the effect of the detonation pressures acting for a very short
distance is to give the liner a velocity π0 which bisects the angle between the
perpendicular to the original liner surface, and the perpendicular to the
collapsing liner surface.
Fig.9: Hollow charge with wedge-shaped liner ready to be set off with an
electric detonator.
Fig.10: Hollow charge with wedge-shaped liner in the process of exploding. Detonation
wave which was started by the detonator and booster has passed over most of the liner
which is in the process of collapsing.
The walls of the collapsing liner are two planes moving inward, as shown in
Fig. 11. From Fig. 12 the junction of these planes around π΄ moves to π΅ with a
velocity
π1 = π0πππ π½βπΌ
2/π πππ½ (2-1)
A moving observer stationed at this moving junction would find any point π
in the upper plane moving with a velocity equal to the vector difference
between the velocity of the walls and the velocity of the junction. Thus he
would see the point π coming toward him with a velocity
π2 = π0πππ π½βπΌ
2/π‘πππ½ + π0π ππ
π½βπΌ
2 (2-2)
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Furthermore, as shown qualitatively by x-ray pictures and as shown in Fig. 13,
he will see a "jet" moving off to the right and a "slug" moving to the left.
We now come to the crucial point of the discussion. As viewed by our
observer, the whole process appears to be unchanged by the lapse of time. In
hydro-dynamical language it appears to be a "steady motion" from which it
follows that we can use Bernoulli's equation
ππ/π(π) +1
2π2 = ππππ π‘. (2-3)
Fig. 11:Formation of jet and slug from a cone
or wedge-shaped liner whose sides collapse
with constant velocity π0 as a result of the
explosion of a charge that was in contact with
the outer surface. The solid lines show
conditions at an early instant of time, and the
dotted lines show conditions after the walls
have moved a distance equal to the velocity π0.
relating the pressure π and the velocity π. If the liner is nearly incompressible
so that π = π0 is constant, this reduces to the simpler and more familiar
equation; π +1
2π0π
2 = ππππ π‘. (2-4)
In either case the pressure at any point in the fluid determines the velocity of
the fluid at that point. Assume that the liner moves away from the exploded
gases so fast that the pressure on its surface is very low and hence the
pressures are constant on all of the surfaces of the collapsing liner. This is a
well-known situation, and the boundary streamlines at constant pressure
(hence velocity) are called "free streamlines." Hence, as viewed by the
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observer, the jet and slug will appear to recede with exactly the same speed,
π2 , as the walls appear to approach; this is shown in Fig. 13. In particular,
during collapse the jet and slug will have exactly the same length. This is
observed experimentally.
Going back to the stationary system of coordinates, it is seen that the
forward jet (traveling to the right in Fig. 13) has a velocity given by
π = π1 + π2 (2-5)
while the backward jet or "slug" (traveling to the left in the moving system of
Fig. 13) actually has a small velocity to the right given by
ππ = π1 β π2 (2-6)
It may help to visualize the process to consider that: if the point π (fixed in the
upper plane) (Fig. 11) travels to point π΅ (fixed in space) in unit time, the
material from the inner surface of the upper plane included between ππ΄ and
π΄π΅ moves into the jet, and the front of the jet moves to the right a distance
equal to ππ΄ + π΄π΅ in the same time. Thus this material forms the very high
velocity jet that is responsible for the deep penetrations. Its velocity is
π = π0{πππ
π½βπΌ
2
π πππ½+
πππ π½βπΌ
2
π‘πππ½+ π ππ
π½βπΌ
2} (2-7)
The material from the outer surface of each plane forms a slug and moves
with the relatively low velocity
πs = π0{πππ
1
2(π½βπΌ)
π πππ½β
πππ 1
2(π½βπΌ)
π‘πππ½β π ππ
1
2(π½ β πΌ)} (2-8)
The principle of the conservation of momentum determines the manner with
which the material of the collapsing planes divides between the jet and the
slug. Let m be the total mass per unit length of the two collapsing planes
approaching the junction. Let ππ be that part of π going into the jet and πsbe
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that going into the slug. Then π = ππ + ππ . Equating the horizontal
components of momentum before to those after passing the junction A in the
moving coordinate system of Fig. 13,
ππ2πππ π½ = ππ π2 β πππ2 (2-9)
ππ = π/2(1 β πππ π½) , ππ = π/2(1 + πππ π½)
According to this simple picture, the velocities of the jet and slug and
their cross-sectional thickness are constant.
Fig.12: Geometry of the collapse process. πππβ² is the upper half of the original cone or
wedge. π΄π, the collapsing section, is moving with a velocity π0 whose. Direction bisects
the angle π΄ππβ² . The detonation wave (velocity ππ) will move from π to πβ² in unit time at
which time πβ²π΅ will become the collapsing section. The junction π΄ will move to π΅ in unit
time at a velocity
π1 = π0 π πππ/π πππ½ = π0[πππ 1
2(π½ β πΌ)/π πππ½] since π = 900 β
1
2(π½ β πΌ)
The case of a conical liner may be treated in the same way. However, in
this case the walls converge on the axis from all sides. The moving observer
must travel at the same rate as in the case of the wedge. However, in order for
the process to appear stationary to him the total mass per unit distance along
the axis must be constant. This is only approximately true in the case of a liner
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of constant thickness; to be exactly true; the liner thickness would have to be
inversely proportional to the distance from the apex.
In the case of a plane detonation wave traveling parallel to the axis with
constant speed ππ , we can even compute π0 from the fundamental relation
ππ
πππ πΌ=
π0πππ 1
2 π½βπΌ
π ππ π½βπΌ (2-10)
which follows by pure geometry from Fig. 12.
Fig. 13: Formation of jet and slug by a cone or wedge-shaped liner shown in Fig.11 from
the point of view of an observer stationed at the moving junction A.
This replaces formulas (2-7)-(2-8) by
π = πππ ππ π½βπΌ
πππ πΌ[ππ ππ½ + ππ‘ππ½ + π‘ππ
1
2 π½ β πΌ ] (2-11)
πs = πππ ππ π½βπΌ
πππ πΌ[ππ ππ½ β ππ‘ππ½ β π‘ππ
1
2 π½ β πΌ ] (2-12)
The jet velocity increases as the angle πΌ decreases, since π½ also decreases.
With such a detonation wave the velocity approaches a maximum as πΌ
approaches zero. From Eq. (2-11)
π = 2ππ when πΌ = 0
and the jet velocity cannot exceed twice the detonation velocity.
In the hypothetical case of a conical wave front, moving perpendicular
to the surface of a conical liner so that it strikes all surfaces at the same
instant, , π½ = πΌ and the velocities of the jet and slug from Eqs.(2-7)-(2-8)
take the simple form,
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π = π0/π πππΌ (1 + πππ πΌ) (2-13)
πs = π0/π πππΌ (1 β πππ πΌ) (2-14)
With wave fronts of this sort the velocity of the jet could be increased
indefinitely by decreasing the cone angle. However, as πΌ tends to zero, the
mass of the jet { ππ = π/2(1 β πππ πΌ) } and the momentum of the jet
{πππ = (ππ0/2)π πππΌ} both tend to zero.
To sum up, our mathematical theory predicts jet and slug velocities as in (2-
7)-(2-8), (2-11)-(2-12), and masses as in (2-9) both for conical and wedge-
shaped liners.
Hirsch [37] extended the PER model to include non-steady effects at the
collision point (or formation point) by assuming that relative to this point. The
jet and slug velocities πj and πs are not equal to the incoming liner velocity π
as the PER theory assumed the Bernoulli equation.
Lampson [38] devised a method to account for the velocity gradient through
the liner thickness for conical liners. The existence of a velocity gradient
through the liner walls was first noted by Sterne [39] for collapsing cylindrical
shells.
Other studies involve modifications or extensions of the PER theory. Such as
Perez et al. [40], who modified PER to account for two-dimensional effects of
the flow near the collapse axis, and Hirsh [41] .Leidel provides an overview of
the PER model and analyzes annular cutting charges, or cookie cutter
warheads as first conceived by glass et al. Harrison [42] and Harrison and
Karpp[43] compare the simple one dimensional models to hydrocodes and
experimental data. Other shaped charge publications of interest include
Hornemann and Senf[44] on wide-angle cones, Van Thiel and Levatin
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[45],Trinks [46] for his shaped charge discussion and historical inferences,
Rostoker[47],Simon and Dipersio[48] ,and the USSR studies by Novikov[49]
, and Titov[50]. These papers relates to jet formation and collapse and relevant
topics pursuing the concepts reviewed so far. An excellent Soviet paper was
published by Ivanova and Rozantseva[51], which is very long and very
informative. It provides a detailed discussion and derivation of PER theory,
and simple penetration models including experimental data and numerous
flash radiographs or X-ray photographs of shaped charge jets. The USSR
models are discussed in more detail [52].
Excellent flash radiography data and analysis thereof is given by Breidenbach
[53] and by Heine-Geldern and Pugh [54]. Classical papers on jets and jet
formation are given by Harlow and Pracht[55] and on collapsing cylinders by
Koski et al.[56].
Zernow and Simon [57] invesigated the behaviour of jets from several
different shaped charge materials. Aseltine[58] studied the effect of several
different warhead asymmetries on the behaviour and performance of shaped
charge jets. Also, Panzarella And Longobardi[59] investigated conical shaped
charge liners that were intentionally deformed , and investigated deficiencies
in charge explosive casting. Merendino et al. [60] altered a shaped charge
liner, that is, inhibited the collapse of the liner, to obtain a massive
hypervelocity pellet in lieu of a jet. Other studies regarding shaped charge
analysis are given in the paper by Robinson [61, 62] and Lee [63].
Kolsky et al. [64] studied the formation and penetration of conical shaped
charge liners. Kolsky[65] continued his studies to include the formation and
penetration of hemispherical shaped charge liners. The formation and collapse
of hemispherical liners were also studied via hydro-codes by Kiwan and
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Arbuckle [66], Arbuckle et al. [67], Chou et al. [68], Aseltine et al. [69], Chou
et al. [70], and Lacetera and Walters [71].
Studied on the formation of hemispherical shaped charge liners were
presented by Lee [63], Singh [72], Shepherd [73], and Grace et al. [74]. A
closed form analytical expression (analogous to the PER theory for conical
liners) for the collapse and formation of hemispherical shaped charge liners
does not currently exist.
Vigil and Robinson [75] provided a theoretical model to analyze the three-
dimensional collapse, jetting, and penetration of a linear shaped charge. The
results were in agreement with the available experimental data.
More literature on hollow charge can be found in Refs. [76-87].