then we superpose the component stresses of the last section multiplied by 4(1 − σ)W/F, and the component stresses here written down multiplied by −(1 − 2σ)W/2πF′, 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 traction across any plane parallel to the boundary is directed away from the place where W is supported, and its amount is 3W cos2θ / 2πr2. The corresponding displacement consists of
(1) a horizontal displacement radially outwards from the vertical through the origin of amount
| W (1 + σ) sin θ | ( cos θ − | 1 − 2σ | ), |
| 2πEr | 1 + cos θ |
(2) a vertical displacement downwards of amount
| W (1 + σ) | {2 (1 − σ) + cos2θ }. |
| 2πEr |
The effects produced by a system of loads on a solid bounded by a plane can be deduced.
The results for a solid body bounded by an infinite plane may be interpreted as giving the local effects of forces applied to a small part of the surface of a body. The results show that pressure is transmitted into a body from the boundary in such a way that the traction at a point on a section parallel to the boundary is the same at all points of any sphere which touches the boundary at the point of pressure, and that its amount at any point is inversely proportional to the square of the radius of this sphere, while its direction is that of a line drawn from the point of pressure to the point at which the traction is estimated. The transmission of force through a solid body indicated by this result was strikingly demonstrated in an attempt that was made to measure the lunar deflexion of gravity; it was found that the weight of the observer on the floor of the laboratory produced a disturbance of the instrument sufficient to disguise completely the effect which the instrument had been designed to measure (see G. H. Darwin, The Tides and Kindred Phenomena in the Solar System, London, 1898).
73. There is a corresponding theory of two-dimensional systems, that is to say, systems in which either the displacement is parallel to a fixed plane, or there is no traction across any plane of a system of parallel planes. This theory shows that, when pressure is applied at a point of the edge of a plate in any direction in the plane of the plate, the stress developed in the plate consists exclusively of radial pressure across any circle having the point of pressure as centre, and the magnitude of this pressure is the same at all points of any circle which touches the edge at the point of pressure, and its amount at any point is inversely proportional to the radius of this circle. This result leads to a number of interesting solutions of problems relating to plane systems; among these may be mentioned the problem of a circular plate strained by any forces applied at its edge.
74. The results stated in § 72 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 resultant 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 σ = 14, impinging directly with relative velocity v, the patches that come into contact are circles of radius
| ( | 45π | ) | 15 | ( | v | ) | 25 | r, |
| 256 | V |
where r is the radius of either, and V the velocity of longitudinal waves in a thin bar of the material. The duration of the impact is approximately
| (2.9432) ( | 2025π2 | ) | 1/5 | r | . |
| 512 | v1/5V4/5 |
For two steel spheres of the size of the earth impinging with a velocity of 1 cm. 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, and constitutes the reason for the adequacy of the statical theory here described.
75. 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 ρ, 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 110 gρa (3 − σ) / (1 − σ),
(2) radial tension of amount 110 gρ (r2/a) (3 − σ) / (1 − σ),
(3) uniform tension at right angles to the radius vector of amount
110 gρ (r2/a) (1 + 3σ) / (1 − σ),
where g is the value of gravity at the surface. The corresponding strains consist of
(1) uniform contraction of all lines of the body of amount
130 k−1gρa (3 − σ) / (1 − σ),
(2) radial extension of amount 110 k−1gρ (r2/a) (1 + σ) / (1 − σ),
(3) extension in any direction at right angles to the radius vector of amount
130 k−1gρ (r2/a) (1 + σ) / (1 − σ),
where k is the modulus of compression. The volume is diminished by the fraction gρa/5k of itself. The parts of the radii vectors within the sphere r = a {(3 − σ) / (3 + 3σ)}1/2 are contracted, and the parts without this sphere are extended. The application of the above results to the state of the interior of the earth involves a neglect of the caution emphasized in § 40, viz. that the strain determined by the solution must be small if the solution is to be accepted. In a body of the size and mass of the earth, and having a resistance to compression and a rigidity equal to those of steel, the radial contraction at the centre, as given by the above solution, would be nearly 13, and the radial extension at the surface nearly 16, and these fractions can by no means be regarded as “small.”
76. In a spherical shell of homogeneous isotropic material, of internal radius r1 and external radius r0, subjected to pressure p0 on the outer surface, and p1 on the inner surface, the stress at any point distant r from the centre consists of
(1) uniform tension in all directions of amount p1r13 − p0r03r03 − r13,
(2) radial pressure of amount p1 − p0r03 − r13 r03r13r3,
(3) tension in all directions at right angles to the radius vector of amount
| 12 | p1 − p0 | r03r13 | . | |
| r03 − r13 | r3 |
The corresponding strains consist of
(1) uniform extension of all lines of the body of amount
| 1 | p1r13 − p0r03 | , | |
| 3k | r03 − r13 |
(2) radial contraction of amount
| 1 | p1 − p0 | r03r13 | , | ||
| 2μ | r03 − r13 | r3 |
(3) extension in all directions at right angles to the radius vector of amount
| 1 | p1 − p0 | r03r13 | , | ||
| 4μ | r03 − r13 | r3 |
where μ is the modulus of rigidity of the material, = 12E / (1 + σ). The volume included between the two surfaces of the body is increased by the fraction p1r13 − p0r03)k(r03 − r13) of itself, and the volume within the inner surface is increased by the fraction
| 3 (p1 − p0) | r03 | + | p1r13 − p0r03 | |
| 4μ | r03 − r13 | k (r03 − r13) |
of itself. For a shell subject only to internal pressure p the greatest extension is the extension at right angles to the radius at the inner surface, and its amount is
| pr13 | ( | 1 | + | 1 | r03 | ); | |
| r03 − r13 | 3k | 4μ | r13 |
the greatest tension is the transverse tension at the inner surface, and its amount is p (12 r03 + r13) / (r03 − r13).
77. In the problem of a cylindrical shell under pressure a complication may arise from the effects of the ends; but when the ends are free from stress the solution is very simple. With notation similar to that in § 76 it can be shown that the stress at a distance r from the axis consists of
(1) uniform tension in all directions at right angles to the axis of amount
| p1r12 − p0r02 | , |
| r02 − r12 |
| (2) radial pressure of amount | p1 − p0 | r02r12 | , | |
| r02 − r12 | r2 |
(3) hoop tension numerically equal to this radial pressure.
