WERRMAw༞༞ I This gives the value of ▲ at (x', y', z') in terms of the given surface-displacements u, v, w. 145. Determination of the displacements. We may now determine u, v, w at (x', y', z'). Let L, M, N denote the functions L = [[ dødy, M = [[%, dady, N= ff ddy......(10), which are finite, continuous and one-valued within the solid; then the value of ▲ at (x', y', z') is given by the equation and the equations for u, v, w are three such as (11), ..(12). Now L, M, N are the potentials of distributions of densities u, v, w on the surface, and therefore L, M, N, & all satisfy Laplace's equation. Also the surface-value of u is 1 ƏL for 2п дг this is the density of the distribution whose potential is L. Thus we may take 1 ƏL 1 λ+μ дф 2πλ+3μ θα + 2п дг We shall devote the next seven articles to the discussion and generalisation of a particular example, returning in art. 153 to the problem of determining the displacements when the surfacetractions are given. To find ▲ from these, we have to calculate two sets of surfacetractions. Let A', wi, w, ws' be the dilatation and rotations corresponding to any system of displacements u', v', w', and let F, G', H' be the corresponding surface-tractions. Then, if z be positive within the medium, the boundary-conditions are, by (15) of art. 29, F" = 2μ + 2μπή, Əz 2μ - 2μπί, .(6). 0 H': == The system F., G., H, is obtained by putting and we get = The system Lo, M., No is obtained by putting u' = §。, v' = no, w'. We get 4μ 2R-1 0 Δ': = Lo = 2μ λ+ 3μ θαλα 2μ This gives the value of ▲ at (x, y, z) in terms of the given surface-displacements u, v, w. 145. Determination of the displacements. We may now determine u, v, w at (x', y', z'). Let L, M, N denote the functions L= ff dødy, M= - ff. dady, N = ffTM, dady......(10), which are finite, continuous and one-valued within the solid; then the value of ▲ at (x', y', z') is given by the equation and the equations for u, v, w are three such as Now L, M, N are the potentials of distributions of densities u, v, w on the surface, and therefore L, M, N, & all satisfy Laplace's equation. Also the surface-value of u is 1 ƏL for this is the density of the distribution whose potential is L. Thus We shall devote the next seven articles to the discussion and generalisation of a particular example, returning in art. 153 to the problem of determining the displacements when the surfacetractions are given. 146. Particular Example. The simplest example of these formulæ will be found by To fix ideas suppose the bounding plane horizontal, and the axis z drawn vertically downwards from a point in the plane. Then this example will correspond to the case when part of the bounding plane is vertically depressed, and the remainder held fixed. (It is Now is the potential of a distribution of matter on the surface, and the simplest example we can take is that of a single mass dm distributed over a small area do at the origin. convenient to take this dm.) We shall shew hereafter that dm is a constant multiple of the force required to depress the part of the surface near the origin. Suppose then that dm r .(15), where r is the distance from the origin to any point of the solid. Since the only (x, y, z) that occurs is the origin, we may suppress the accents on (x', y', z') and write If dm be regarded as a small finite quantity the depression near the origin is very great, and we must regard the origin as excluded from the part of the solid whose deformation we investigate. The problem is that of a considerable depression near a single point, and the above formulæ shew how to find the displacements at a distance from the point. 147. Elementary Discussion of Particular Example. Simple Solutions of First Type. On account of its importance we shall consider this solution à priori. It can be readily verified that the displacements where r is the distance of the point (x, y, z) from the origin, satisfy the general equations of equilibrium, when there is no bodily force, at all points not indefinitely near the origin. This is M. Boussinesq's first type of simple solutions of these equations. Now these expressions can be written where r is the distance of (x, y, z) from a given point. If the above expressions be multiplied by any quantity independent of x, y, z we still have a solution, and the sum of any number of such solutions is a solution, and therefore is a solution, r being the distance of (x, y, z) from the point (x'y' ́) on the plane z = 0, and p1 any function of x', y'. Now we may regard p1 as the surface-density of a distribution of matter on the plane z = 0, and then ffprda'dy' is the "direct potential” of this distribution at (x, y, z), and, since Var = 2/r, Vap.rdx'dy' is the "inverse potential” (i.e. the ordinary gravitation potential) of this distribution. 148. Solid bounded by Infinite Plane. Purely Normal Surface Displacement. We shall suppose the solid bounded by the plane z=0, and seek the distribution of surface-traction which would produce the above system of displacements. It corresponds to purely normal displacement of a part of the bounding surface, the remainder being kept fixed. |