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Explosives with Lined Cavities
1.Rifle bullets ordinarily have muzzle velocities around 2000–3000 ft./sec., though with special devices projectiles have been shot from gun barrels up to 5000 ft./sec. F. Zwicky and F. Whipple have suggested using these jets as artificial meteors for controlled studies, since their velocities are about the same as the velocities of the slower meteors. Shaped charges are carried to the upper atmosphere by V‐2 rockets and exploded when the rocket has reached its highest point. Calculations indicate that these “meteors” should be observable from the ground.
2.Like most analogies this is far from perfect, for while a stream of water washes mud out of a mud bank the jet of metal does not wash or erode metal out of the target. Careful weighings have shown that a metal jet is captured by a metal target, which loses no weight except a very small amount at the front surface. The hole is produced by plastic flow of the target material in a radial direction.
3.Robert H. McLemore, “Formation penetrating with shaped explosive charges,” Oil Weekly July 8 (1946).
4.P. W. Bridgman has measured static pressures up to 100, 000 atmospheres, in spaces long and thick, and has obtained pressures estimated at 400, 000 atmospheres but was unable to make observations at the higher pressures.
5.Much pioneer work toward understanding this phenomenon was done by W. M. Evans and A. R. Ubbelohde under the auspices of the British Ministry of Supply.
6.The clearest and most revealing photographs the writers have seen were taken by L. B. Seely and J. C. Clark at the Aberdeen Ballistics Research Laboratory; the earliest photographs were taken by J. L. Tuck in England.
7.To show that bisects the angle in Fig. 12, consider a coordinate system having a constant velocity such that the origin moves from P to in unit time. In these coordinates a steady‐state condition exists in the region of the origin, with the liner flowing in along following a curved path and flowing out along PA. The curved path is caused by pressures on the liner from the detonation wave which have a constant distribution in these moving coordinates. The velocity of the liner passing through this region changes its direction but not its magnitude, since the pressure forces are everywhere perpendicular to the motion. Let and (parallel to PA) represent, respectively, the entering and emerging velocities of the liner in the moving system. These are equal in magnitude. Since the velocity of the moving system is the velocity of the collapsing liner in the stationary system is the vector sum Also, since the triangle is isosceles and since is parallel to PA, angle Therefore bisects the angle
8.L. L. Milne‐Thompson, Theoretical Hydrodynamics (Macmillan and Company, Ltd., London, 1938). For Bernoulli’s theorem see p. 10, 14; for free streamlines and jets, Chap. XI, expecially paragraph 11.43, where a discussion of two‐dimensional jets, is given, including a differential equation for the flow pattern.
9.Similar jets are formed under the extremely prosaic dropping of a sphere into water. An air‐filled cavity at first trails the sphere, symmetric about a vertical axis through the center of the sphere. After an instant the outward motion of the walls is checked, then reversed, and the walls collide on the axis, meeting on all sides. This causes vertical jets, both upwards and downwards. Cf. A. M. Worthington, A Study of Splash (Longmans, Green and Company, London, 1908); also a forthcoming article by Dr. David Gilbarg of the Naval Ordnance Laboratory.
10.I. S. Sokolnikoff, Mathematical Theory of Elasticity (McGraw‐Hill Book Company, New York, 1946), Chapter I.
11.Felix Hélie, Traite de balistique experimentale (Dumaine Paris, 1840), where an experimental law of direct proportionality between impact energy and hole (or crater) volume is asserted.
12.The theory presented in the next section was discovered independently by R. Hill, N. F. Mott, and D. C. Pack in England; earlier similar semiquantitative ideas had been advanced by Kistiakowsky, Messerly, and one of us.
13.It is probable that after the last jet particle strikes a relatively soft target material it will have sufficient residual momentum to open up the hole still deeper. This effect has been called “secondary penetration” to distinguish it from that given by Eq. (7). It helps to explain why deeper holes are produced in massive lead targets than in massive steel targets even though the lead targets have the higher densities.
14.In jets from conical liners a small amount of jet material at the rear of each jet travels slow enough to produce stresses lower than the yield strength in armor though higher than the yield strength in mild steel. Thus the penetration process may continue longer in mild steel than in armor. This phenomenon, together with the phenomenon of secondary penetration13 accounts for the fact that the total penetration into steel armor is a little less than the total penetration into mild steel and that the penetration into lead is greater than the total penetration into either of the steels.
15.The quantity J̄ and the integral both depend very slightly upon the density ρ of the target, since a different penetration in a different density target will change the average obtained for as well as the value of For both, the dependence upon ρ is so slight it will be neglected.
16.The process of ductile drawing of the jet due to the velocity gradient in it was first suggested because the increase in penetration proportional to for particle jets of case 3 did not appear to be rapid enough to account for the experimental observations.
17.Close to the charge (within 10 or 15 times the diameter of the base of the cone) air can be treated approximately like any other target having the same density. The front of the jet creates a very intense shock wave with an evacuated space behind it which reduces the air resistance on the rest of the particles in the jet to a negligible quantity.
18.This approximation is not serious because the forces acting upon the jet particles in this period are relatively small. The internal forces acting during the ductile drawing process change the velocities somewhat but not enough to seriously affect the rate at which the length of the jet changes, which is the quantity that now concerns us.
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