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256 THE SAND-BLAST [386
begin by supposing the bars to be of the same length, material, and section, and before impact to be moving with equal and opposite velocities v. At impact, the impinging faces are reduced to rest, and remain at rest so long as the bars are in contact at all. This condition of rest is propagated in each bar as a wave moving with a velocity a, characteristic of the material. In such a progressive wave there is a general relation between the particle-velocity (estimated relatively to the parts outside the wave) and the compression (e), viz., that the velocity,, is equal to ae. In the present case the relative particle-velocity is v, so that v = ae. The limit of the strength of the material is reached when e has a certain value, and from this the greatest value of v (half the original relative velocity) which the bars can bear is immediately inferred.
But the importance of the conclusion depends upon an extension now to be considered. It will be seen that the length of the bars does not enter into the 'question. . Neither does the equality of the lengths. However short one of them may be, we may contemplate an interval after first impact so short that the wave will not have reached the further end, and then the argument remains unaffected. However short one of the impinging bars, the above calculated relative velocity is the highest which the material can bear without undergoing disruption.
As more closely related to practice, the case of two spheres of radii r, ?-', impinging directly with relative velocity v, is worthy of consideration. According to ordinary elastic theory the only remaining data of the problem are the densities p, p, and the elasticities. The latter may be taken to be the Young's moduli q, q', and the Poisson's ratios, a, a-', of which the two last are purely numerical. The same may be said of the ratios q'jq, p'/p, and r'jr. So far as dimensional quantities are concerned, any maximum strain e may be regarded as a function of r, v, q, and p. The two last can occur only in the combination q/p, since strain is of no dimensions. Moreover, q/p = a?, where a is a velocity. Regarding e as a function of r, v, and a, we see that v and a can occur only as the ratio vfa, and that r cannot appear at all. The maximum strain then is independent of the linear scale; and if the rupture depends only on the. maximum strain, it is as likely to occur with small spheres as with large ones. The most interesting case occurs when one sphere is very large relatively to the other, as when a grain of sand impinges upon a glass surface. If the velocity of impact be given, the glass is as likely to be broken by a small grain as by a much larger one. It may be remarked that this conclusion would be upset if rupture depends upon the duration of a strain as well as upon its magnitude.
1 The general argument from dynamical similarity that the maximum strain during impact is independent of linear scale, is, of course, not limited to the case of spheres, which has been chosen merely for convenience of statement.