ELASTIC when the body is suspended from one or more points in a horizontal plane its volume is increased by dv—Wh/3k, where W is the weight of the body, and h the depth of its centre of gravity below the plane ; when the body is supported by upward vertical pressures at one or more points in a horizontal plane the volume is diminished by - ov = 'Vh'/3k, where h' is the height of the centre of gravity above the plane ; if the body is a cylinder, of length l and section A, standing with its base on a smooth horizontal plane, its length is shortened by an amount -5Z=WZ/2EA; if the same cylinder lies on the plane with its generators horizontal, its length is increased by an amount 8l=aWh'/EA. 21. In recent years important results have been found by considering the effects produced in an elastic solid by forces applied at isolated points. Taking the case of a single force F applied at a point in the interior, it can be shown that the stress at a distance r from the point consists of (1) a radial pressure of amount 2-0- F cos 0 1 - 0- 47r r2 ’ (2) tension in all directions at right angles to the radius of amount 1-20- F cosfl 2(1 — <t) 47t r2 ’ (3) shearing stress acting along the radius dr on the surface of the cone d = const, and acting along the meridian d9 on the surface of the sphere r=const, of amount 1 - 2o- F sin 0 2(1 - o) 47r r2 ’ where 0 is the angle between the radius vector r and the line of action of F. The line marked T in Fig. 19 shows the direction of this shearing stress on the spherical surface. Thus the principal stresses are in and perpendicular to the meridian plane, and the direction of one of those in the meridian plane is inclined to the radius vector r at an angle The corresponding displacement at any point is compounded of a radial Fig. 19. displacement of amount l + o- F cos# 2(1 — o-) 4^E r and a displacement parallel to the line of action of F of amount (3-4o-)(l + o-) F_ 1. 2(1 - o-) 47tE r The effects of forces applied at different points and in different directions can be obtained by summation, and the effect of continuously distributed forces can be obtained by integration. 22. The stress system considered in the last section is equivalent, on the plane through the origin at right angles to the line of action of F, to a pressure of magnitude at the origin and a radial shearing stress of amount --I--— 2 , and, 2(l-0-)47rr ’ ’ by the application of this system of tractions Fig. 20. to a solid bounded by a plane, the displacement just described would be produced. There is also another (Fig. 20) stress system for a solid
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so bounded which is equivalent, on the same plane, to a pressure at the origin, and a radial shearing stress proportional to 1/r2, but these are in the ratio 227t : r~2, instead of being in the ratio 47t(1 - cr) : (1 - 2o-)r~ . The second stress system consists of (1) radial pressure FV-2, (2) tension in the meridian plane across the radius vector of amount FV-2 cos 0/(1 + cos 0), (3) tension across the meridian plane of amount FV-2/(l +cos 0), (4) shearing stress as in the last section of amount FV-2 sin 0/(1 + cos 0), and the stress across the plane boundary consists of a pressure of magnitude 27tF' and a radial shearing stress of amount FV-2. If then we superpose the component stresses of the last section multiplied by 4(1 - 0-)'W/F, and the component stresses here written down multiplied by - (1 - 20-)'W'/27^F,, the stress on the plane boundary will reduce to a single pressure W at the origin. We shall thus obtain the stress system at any point due to such a force applied at one point of the boundary. In the stress system thus arrived at the stress across any plane parallel to the boundary is directed away from the place where W is supported, and its amount is 3Wcos20/27rr2. The corresponding displacement consists of (1) a horizontal displacement radially outwards from the vertical through the origin of amount W(1 + 0-) sin 0 1-2027rEr
COS0 - l+cos0y>
(2) a vertical displacement downwards of amount W-O+rfe}. The effects produced by a system of loads on a solid bounded by a plane can be deduced. 23. The results stated in the last section have been applied to give an account of the nature of the actions concerned in the impact of two solid bodies. The dissipation of energy involved in the impact is neglected, and the pressure between the bodies at any instant during the impact is equal to the rate of destruction of momentum of either along the normal to the plane of contact drawn towards the interior of the other. It has been shown that in general the bodies come into contact over a small area bounded by an ellipse, and remain in contact for a time which varies inversely as the fifth root of the initial relative velocity. For equal spheres of the same material, with o- = I, impinging directly with relative velocity v, the patches that come into contact are circles of radius /757ri/v i V256; ) r, where r is the radius of either, and Y is the velocity of longitudinal waves in a thin bar of the material. The duration of the impact is approximately 2 1/5
r (2-9432) 3757r 128 ) v^W6’ For two steel spheres of the size of the earth impinging with a velocity of 10 mm. per second the duration of the impact would be about twenty-seven hours. The fact that the duration of impact is, for moderate velocities, a considerable multiple of the time taken by a wave of compression to travel through either of two impinging bodies has been ascertained experimentally,1 and constitutes the reason for the adequacy of the statical theory here described. 24. Spheres and Cylinders. — Simple results can be found for spherical and cylindrical bodies strained by radial forces. For a sphere of radius a, and of homogeneous isotropic material of density p, strained by the mutual gravitation of its parts, the stress at a distance r from the centre consists of (1) uniform hydrostatic pressure of amount ^gpa{3 - o-)/(l - o’), (2) radial tension of amount ^gpir^/a)^ - o-)/(l - a), 1 Cf. Auerbach in Winkelmann’s Handbuch der Physik, i. 303. Breslau, 1891.
