Chapter Two

Introduction to Hollow

Charge

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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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57

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].

𝐅𝐢𝐠. 𝟓: 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 𝑖𝑛 ×

4 𝑖𝑛.× 0.5 𝑖𝑛.)[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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58

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 2𝛽 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 𝑉0 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 𝐴 moves to 𝐵 with a

velocity

𝑉1 = 𝑉0𝑐𝑜𝑠𝛽−𝛼

2/𝑠𝑖𝑛𝛽 (2-1)

A moving observer stationed at this moving junction would find any point 𝑃

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 𝑃 coming toward him with a velocity

𝑉2 = 𝑉0𝑐𝑜𝑠𝛽−𝛼

2/𝑡𝑎𝑛𝛽 + 𝑉0𝑠𝑖𝑛

𝛽−𝛼

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

𝑑𝑝/𝜌(𝑝) +1

2𝑈2 = 𝑐𝑜𝑛𝑠𝑡. (2-3)

Fig. 11:Formation of jet and slug from a cone

or wedge-shaped liner whose sides collapse

with constant velocity 𝑉0 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 𝑉0.

relating the pressure 𝑃 and the velocity 𝑈. If the liner is nearly incompressible

so that 𝑃 = 𝑃0 is constant, this reduces to the simpler and more familiar

equation; 𝑃 +1

2𝜌0𝑈

2 = 𝑐𝑜𝑛𝑠𝑡. (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,

𝑉2 , 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

𝑉 = 𝑉1 + 𝑉2 (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

𝑉𝑠 = 𝑉1 − 𝑉2 (2-6)

It may help to visualize the process to consider that: if the point 𝑃 (fixed in the

upper plane) (Fig. 11) travels to point 𝐵 (fixed in space) in unit time, the

material from the inner surface of the upper plane included between 𝑃𝐴 and

𝐴𝐵 moves into the jet, and the front of the jet moves to the right a distance

equal to 𝑃𝐴 + 𝐴𝐵 in the same time. Thus this material forms the very high

velocity jet that is responsible for the deep penetrations. Its velocity is

𝑉 = 𝑉0{𝑐𝑜𝑠

𝛽−𝛼

2

𝑠𝑖𝑛𝛽+

𝑐𝑜𝑠𝛽−𝛼

2

𝑡𝑎𝑛𝛽+ 𝑠𝑖𝑛

𝛽−𝛼

2} (2-7)

The material from the outer surface of each plane forms a slug and moves

with the relatively low velocity

𝑉s = 𝑉0{𝑐𝑜𝑠

1

2(𝛽−𝛼)

𝑠𝑖𝑛𝛽−

𝑐𝑜𝑠1

2(𝛽−𝛼)

𝑡𝑎𝑛𝛽− 𝑠𝑖𝑛

1

2(𝛽 − 𝛼)} (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 𝑚𝑗 be that part of 𝑚 going into the jet and 𝑚sbe

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62

that going into the slug. Then 𝑚 = 𝑚𝑗 + 𝑚𝑠 . Equating the horizontal

components of momentum before to those after passing the junction A in the

moving coordinate system of Fig. 13,

𝑚𝑉2𝑐𝑜𝑠𝛽 = 𝑚𝑠𝑉2 − 𝑚𝑗𝑉2 (2-9)

𝑚𝑗 = 𝑚/2(1 − 𝑐𝑜𝑠𝛽) , 𝑚𝑠 = 𝑚/2(1 + 𝑐𝑜𝑠𝛽)

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. 𝑂𝑃𝑃′ is the upper half of the original cone or

wedge. 𝐴𝑃, the collapsing section, is moving with a velocity 𝑉0 whose. Direction bisects

the angle 𝐴𝑃𝑃′ . The detonation wave (velocity 𝑈𝑑) will move from 𝑃 to 𝑃′ in unit time at

which time 𝑃′𝐵 will become the collapsing section. The junction 𝐴 will move to 𝐵 in unit

time at a velocity

𝑉1 = 𝑉0 𝑠𝑖𝑛𝜃/𝑠𝑖𝑛𝛽 = 𝑉0[𝑐𝑜𝑠1

2(𝛽 − 𝛼)/𝑠𝑖𝑛𝛽] since 𝜃 = 900 −

1

2(𝛽 − 𝛼)

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 𝑈𝑑 , we can even compute 𝑉0 from the fundamental relation

𝑈𝑑

𝑐𝑜𝑠𝛼=

𝑉0𝑐𝑜𝑠1

2 𝛽−𝛼

𝑠𝑖𝑛 𝛽−𝛼 (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

𝑉 = 𝑈𝑑𝑠𝑖𝑛 𝛽−𝛼

𝑐𝑜𝑠𝛼[𝑐𝑠𝑐𝛽 + 𝑐𝑡𝑛𝛽 + 𝑡𝑎𝑛

1

2 𝛽 − 𝛼 ] (2-11)

𝑉s = 𝑈𝑑𝑠𝑖𝑛 𝛽−𝛼

𝑐𝑜𝑠𝛼[𝑐𝑠𝑐𝛽 − 𝑐𝑡𝑛𝛽 − 𝑡𝑎𝑛

1

2 𝛽 − 𝛼 ] (2-12)

The jet velocity increases as the angle 𝛼 decreases, since 𝛽 also decreases.

With such a detonation wave the velocity approaches a maximum as 𝛼

approaches zero. From Eq. (2-11)

𝑉 = 2𝑈𝑑 when 𝛼 = 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, , 𝛽 = 𝛼 and the velocities of the jet and slug from Eqs.(2-7)-(2-8)

take the simple form,

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𝑉 = 𝑉0/𝑠𝑖𝑛𝛼 (1 + 𝑐𝑜𝑠𝛼) (2-13)

𝑉s = 𝑉0/𝑠𝑖𝑛𝛼 (1 − 𝑐𝑜𝑠𝛼) (2-14)

With wave fronts of this sort the velocity of the jet could be increased

indefinitely by decreasing the cone angle. However, as 𝛼 tends to zero, the

mass of the jet { 𝑚𝑗 = 𝑚/2(1 − 𝑐𝑜𝑠𝛼) } and the momentum of the jet

{𝑚𝑗𝑉 = (𝑚𝑉0/2)𝑠𝑖𝑛𝛼} 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 𝑉j and 𝑉s are not equal to the incoming liner velocity 𝑉

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].