Page:Encyclopædia Britannica, Ninth Edition, v. 15.djvu/775

Rh MECHANICS 743 The physical condition is expressed by Hooke s Law in the form Cylin drical or pris matic wire, origin ally straight. Bending. Twist. Hence This expresses (ns in 267) the passage of simultaneous waves. They are now waves of condensation and extension, not of trans verse displacement. The nature of the interpretation of the equation is of the same general character as before, the speed being /E/^. DYNAMICS OF AN ELASTIC SOLID. 271. This subject, which is a very extensive and difficult one, and in its generality quite unsuitable for discussion here, has already been to some extent treated of under ELASTICITY. We therefore content ourselves with one or two examples, whose treatment is comparatively simple, while their applications are frequent and of considerable practical importance. 272. Even so restricted a problem as that of determin ing the form assumed by a wire or thin rod of homogeneous isotropic elastic material, under the action of given forces and couples, presents somewhat formidable difficulties un less in its unstrained state the wire be straight and truly cylindrical or prismatic. And, even with these limitations, the problem again becomes formidable if we introduce the consideration of non-isotropic material ; while, in any case, if the radius of curvature at each point is not very large in proportion to the thickness of the rod in the plane of bending, the problem is to no appreciable extent simplified by the limitation of form of the body. We will therefore give the comparatively simple case of the mere bending and twist of a homogeneous isotropic wire whose natural form is cylindrical or prismatic, the amounts of these from various sources being so small as to be superposable. Bending lengthens one set of lines of particles originally parallel to the axis of the wire and shortens others. Twist lengthens all but one such line, forming them into helices. The more detailed investigation, which we cannot give here, shows that there is one line of particles (the &quot; elastic central line &quot;) which passes through the centre of inertia of each transverse section, and which may be treated (under our present limitations) as rigorously unchanged in length. The mutual molecular action of the parts of the wire on opposite sides of any transverse section may of course be reduced to a force and a couple, and the force may be conveniently treated as passing through the centre of inertia of the section. Also the twist and curvature of the wire near this section obviously depend on the couple and not on the force. For the moment of the couple is in general finite, while that of the force (about any point in the corresponding element of the wire) is infinitesimal. 273. Let any two planes, at right angles to one another, be drawn through the elastic central line before distortion ; and let them be cut in lines PR and PS by a transverse section through a point P of the central line. Also let PT be a tangent to that line. Suppose a similar construction to be carried out for every point P of the central line. Then it is clear that the form of the distorted wire will be completely determined if we know the form assumed by the central line, and the positions taken by the lines PR and PS drawn from each point in it. In their new positions P T, P R , and P S will still form (in consequence of the limitations we have imposed) a rectangular system ; and the nature of the distortion will be clearly indicated by the change of position of this rectangular system as it passes from point to point of the distorted central line. The plane of rotation of P T is the osculating plane of the bending ; its rate of rotation in that plane per unit length of the central line is the amount of bending ; and the rate of rotation of the system P R , P S, about P T , per unit length of the central line, is the rate of twist. Suppose P to move with unit velocity along the distorted central line, and let p, a, r be the angular velocities of the system about P R, P S , P T respec tively, then p represents the curvature (or bending) resolved in the plane S PT, cr that in R P T, while T represents the twist, Now, if the elastic forces constitute a conservative system, the Expres- amount of work done on an element of the body corresponding to a sions J oi length 5s of the central line is to be calculated entirely from its force an&amp;lt; change of form. It must therefore be expressible in the form couple i terms of where w is a function of p, a; r couples producing the bending are which must be such that the ail( j twist. dw, dw -r- and -r- , dp da while that producing the twist is dw dr&quot; These again, are functions of p, a, r, and they must, on account of the principle of superposition, be linear and homogeneous. For, within the limits to which we have restricted ourselves, the doubling alike of bending and twist must involve the doubling of each of the couples. Thus w must be a homogeneous function of p, &amp;lt;r, T of the second degree. Hence we may assume w = (Ap 2 -f Bo- 2 + Or 2 + 2Dpo- + 2E&amp;lt;rr + 2Frp) , where A, B, C, D, E, Fare quantities depending on the form of the section of the wire and the nature of its material at each point. This gives dp d&amp;lt;r du; -- CT. Hence, when the couples are assigned, the amounts of bending and twist are at once calculated from them. But the expression above is much more general than we require for the limited case we are considering. For, if the only couples applied to a portion of the prism or cylinder considered be in planes perpendicular to its length, twist only will be produced. Thus, for = 0, = 0, we ought to have also p = 0, &amp;lt;r = 0. Hence E and F both vanish and we have simply w = J(Ap 2 + 2Dp&amp;lt;r + Bo- 2 + CT&quot;). This may be reduced, by properly selecting the planes originally drawn through the elastic central line, to the form w = i(Ap 2 + Bo-- + CT - ). Now we see that div dw -3- = Ap, -^- dp dff 274. In a prismatic or cylindrical wire of homogeneous isotropic material, the elastic central line is thus a torsion axis simply. Equal and opposite couples, applied to the ends of such a wire, in planes perpendicular to its length, produce twist in direct proportion to the moments of the couples. There are two planes perpendicular to one another, and passing through this line, such that, if equal and opposite couples in either of these planes be applied at any parts of the wire, the portion between is bent into a circular arc in that plane. These are the principal planes of flexure. The quantities A and B which, when multiplied by the amount of bending in either of these planes, give the moment of the corresponding couple are called the prin cipal &quot; flexure rigidities &quot; of the wire. When they are equal Flexure (as in the case of a wire of circular, square, equilateral rigidity. triangular, &c., section) any plane through the axis is a principal plane of flexure. C is the torsional rigidity of Torsinm the wire. lu general, when the wire is fixed at one end &quot;gi^ty. and a couple applied at the other, the wire assumes the form of a circular helix. The exceptions (or rather parti cular cases) are : (a) when the plane of the couple con tains the elastic central line, and there is mere flexure, without twist ; (b) when the plane of the cuuple is perpen dicular to the wire, and there is twist simply. 275. As an example of the preceding theory, take first