532 



NATURE 



[February 15, 19 12 



an hydraulic press, but it differs in degree because of its 

 great intensity and of its extremely short duration, and 

 these characteristics, as we shall see, have a marked 

 influence on the effects which it produces. 



The first part of the problem, that is, the calculation of 

 the pressure in tons or pounds, is based on the familiar 

 principles of mechanics which were first precisely stated in 

 Newton's laws of motion. The cause of the pressure is 

 The rapid change of motion of the colliding bodies which 

 occurs when they come into contact, and, according to 

 Newton's second law, the force is simply proportional to 

 the rate at which this change is effected. The rate of 

 change may be measured in terms of energy and distance 

 or in terms of momentum and time. Thus a hammer 

 head moving at a rate of i6 feet per second, and weigh- 

 ing I lb., possesses 4 foot-lbs. of energy, because its 

 velocity could have been acquired by falling freely through 

 4 feet. If it strikes a nail and drives it one-eighth of an 

 inch, the energy which was generated by the weight of 

 I lb. acting through 4 feet is destroyed in 1/400 part of 

 that distance, and the force necessary to effect this change 

 of motion is 400 times as great — say, 400 lbs. The same 

 effect would be produced by a 4-lb. hammer striking with 

 the velocity which would be acquired by falling through 

 \ foot, namely, 8 feet per second. Regarding the same 

 instance from the point of view of momentum, the i-lb. 

 hammer would take half a second to fall 4 feet, and the 

 quantity of motion or " momentum," reckoned as the 

 product of the force acting into the time required to 

 generate it, would be one-half of a pound-second unit. 

 While driving the nail in, the hammer covers a distance 

 of \ inch with a velocity which starts at 16 feet per second 

 and drops to zero. To cover the distance of \ inch with 

 the average velocity of 8 feet per second takes i/8oo of a 

 second, which is 1/400 of the time (i second) which it 

 takes the weight of the hammer head (a force of i lb.) 

 to generate its motion. Thus the pressure required for the 

 rapid stoppage is, as before, 400 lbs. 



We may t?ke another instance essentially similar to the 

 hammer and nail, but differing greatly as regards scale. 

 A i4-inch armour-piercing shell weighs about 1400 lbs., 

 and when moving at 1800 feet per second possesses about 

 31,000 foot-tons of energy, or about 15,000,000 times as 

 much as our hammer head. Such a shell would just pierce 

 a plate of wrought iron i\ feet thick, and the average 

 force which must be exerted to pull it up in that distance, 

 which is, of course, the pressure which it exerts on the 

 plate, is 30.000 divided by 2^, or about 12,000 tons. This 

 is equivalent to some 80 tons on the square inch. 



When a hammer strikes a nail, the force acting during 

 the blow is practically constant, and the average value 

 obtained as above by dividing the energy by the distance 

 moved, or the momentum by the time taken, is equal to 

 the actual force exerted throughout the impact. In many 

 cases, however, this force is not constant, and it is then 

 necessary to divide the course of the impact into short 

 intervals either of space or of time, calculate the change 

 of energy or momentum in each, and add the result. A 

 familiar instance is that of two billiard balls. We may 

 suppose one ball to strike the other full with a velocity 

 of 16 feet per second, which corresponds to a fairly hard 

 stroke. It simplifies the consideration of the problem if 

 instead of one ball moving and the other at rest we sup- 

 pose them to be travelling in opposite directions with equal 

 velocities of 8 feet per second. At the instant when the 

 balls first touch there is no pressure between them, but as 

 they continue to approach each flattens the other at the 

 point of contact. The balls no longer touch at a point, 

 but over a circular area which rapidly increases in 

 diameter. Corresponding to any given amount of flatten- 

 ing or distance of approach, there is, of course, a definite 

 pressure, which might be measured by actually squeezing 

 the balls together under known forces and measuring the 

 corresponding amount of approach. Or the relation 

 between pressure and distance could be calculated, as was 

 done by Hertz. The area of the curve connecting pressure 

 and distance up to any point gives the number of foot- 

 pounds of energy destroyed. When this is just equal to 

 the original energy of the balls they will have been re- 

 duced to rest, and in the case supposed the distance of 

 approach is then 14/1000 of an inch, and the total pressure 



NO. 2207, VOL. 88] 



between them 1300 lbs. This pressure is distributed ov 

 the circle of contact, which is one-sixth of an inch 

 diameter, and the average intensity of the pressure is . 

 tons per square inch. The distribution, however, is n. 

 uniform, the pressure at the centre being \\ times tl 

 average. The balls are then like compressed springs, th' 

 original energy of motion having been completely tran 

 formed into strain energy in their substance. The reasc 

 of the high intensity of pressure developed is that tl. 

 strain energy is concentrated into a very small volume ■ 

 ivory near to the point of contact. The balls then be^; 

 to separate, and the whole process of compression is go; 

 through in reverse order, the strain energy being tran- 

 formed back into energy of motion by the pressuf'-. 

 Finally, the balls rebound unstrained, with nearly the 

 velocity with which they approached. 



If for the ivory balls we substituted hollow balls of steel 

 having the same mass, the pressure produced by the blow 

 would be greater, because the steel is much more rigid 

 than ivory, and gives less under a given force. Thus the 

 distance of approach is less, the circle of contact smaller, 

 and the maximum intensity of pressure much greater. It 

 reaches 280 tons per square inch averaged over the surface 

 of contact. .Such a pressure could only be sustained with- 

 out permanent effect by a very hard steel. Ordinan,' mild 

 steel would begin to flow when the pressure passed about 

 100 tons, a permanent flat would be left by the blow, and 

 the balls would rebound with less velocity than that of 

 approach. The theory the results of which I have given 

 does not, of course, apply to such a case, as it depends 

 on the assumption of perfect elasticity. 



It is rather remarkable that materials can sustain with- 

 out injury such large pressures as are produced by these 

 blows. Mild steel balls are not crushed perceptibly until 

 the pressure reaches 100 tons per square inch, yet a short 

 column of the same steel would be crushed by a pressure 

 of 30 tons per square inch. One reason is the extremely 

 short duration of the pressure — it has no time to produce 

 much effect. The other is the fact that in the blow it is 

 accompanied by large lateral pressures exerted by the 

 metal surrounding the area of contact. Pressure equal in 

 all directions, such as is exerted by the water at the bottom 

 of a deep ocean, produces generally no permanent effect 

 on solids or liquids. To produce breakage or permanent 

 deformation there must be difference of pressure in different 

 directions, and the most important, if not the only, factor 

 determining whether such breakage or deformation shall 

 occur is the amount of the difference. If, for example, 

 our column of mild steel, which in the absence of lateral 

 support begins to crush at 30 tons, were surrounded by a 

 jacket exerting a radial pressure of 30 tons, it is probable 

 that the end pressure might be increased to 60 tons without 

 any movement occurring. In the impact of balls the 

 metal surrounding the point of contact, by resisting the 

 lateral expansion of the compressed part, sets up radial 

 pressure of this kind. It can be shown, in fact, that th 

 lateral pressure at the centre of the circle of contac: 

 corresponding to a maximum normal pressure of 100 ton? 

 per square inch, is 75 tons per square inch, leaving 25 tons 

 effective for producing deformation or breakage. 



These calculations of pressure are based on theory, and 

 it may be asked what direct experimental evidence we have 

 that the theory is correct. It is not, of course, possible 

 actually to measure the pressures over the minute circle 

 of contact between the balls, nor is it possible accurately 

 to measure the amount of the flattening. We can, how- 

 ever, pursue the calculation a little further, and determine 

 the time during which the balls are in contact from the 

 moment when they first touch to the moment at w-hich 

 they separate on the rebound. In the case of billiard balls 

 moving with a relative velocity' of 16 feet per second, this 

 time is 1/4000 of a second. A precisely similar calcula- 

 tion can be made for balls of steel or other metal, and it is 

 not difficult to measure in the laboratory the time during 

 which such balls remain in contact. The method is of 

 considerable use in connection with impact problems, and 

 it consists in making the two balls, by their contact, close 

 a galvanometer circuit in which there is also a battery and 

 resistance. A certain quantity of electricity, which i- 

 simply proportional to the time of contact, then pass^- 

 through the galvanometer and produces a proportionat 



