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Elasticity

55. Practical Application of Saint-Venant's Theory

The theory above described is exact provided the forces applied to the loaded end, which have W for resultant, are distributed over the terminal section in a particular way,not likely to be realized in practice; and the application to practical problems depends on a principle due to Saint-Venant, to the effect that, except for comparatively small portions of the beam near to the loaded and fixed ends, the resultant only is effective, and its mode of distribution does not seriously affect the internal strain and stress. In fact, the actual stress is that due to forces with the required resultant distributed in the manner contemplated in the theory, superposed upon that due to a certain distribution of forces on each terminal section which, if applied to a rigid body, would keep it in equilibrium; according to Saint-Venant's principle, the stresses and strains due to such distributions of force are unimportant except near the ends. For this principle to be exactly applicable it is necessary that the length of the beam should be very great compared with any linear dimension of its cross-section; for the practical application it is sufficient that the length should be about ten times the greatest diameter.

56. In recent years the problem of the bending of a beam by loads distributed along its length has been much advanced. It is now practically solved for the case of a load distributed uniformly, or according to any rational algebraic law, and it is also solved for the case where the thickness is small compared with the length and depth, as in a plate girder, and the load is distributed in any way. These solutions are rather complicated and difficult to interpret. The case which has been worked out most fully is that of a transverse load distributed uniformly along the length of the beam. In this case two noteworthy results have been obtained. The first of these is that the centralline in general suffers extension. This result had been found experimentally many years before. In the case of the plate girder loaded uniformly along the top, this extension is just half as great as the extension of the central-line of the same girder when free at the ends, supported along the base, and carrying the same load along the top. The second noteworthy result is that the curvature of the strained centralline is not proportional to the bending moment. Over and above the curvature which would be found from the ordinary relation curvature of central-line =bending moment =flexural rigidity, there is an additional curvature which is the same at all the cross-sections. In ordinary cases, provided the length is large compared with any linear dimension of the cross-section, this additional curvature is small compared with that calculated from the ordinary formula, but it may become important in cases like that of suspension bridges, where a load carried along the middle of the roadway is supported by tensions in rods attached at the sides.

57. When the ordinary relation between the curvature and the bending moment is applied to the calculation of the deflection of continuous beams it must not be forgotten that a correction of the; kind just mentioned may possibly / be requisite. In the usual method of treating the problem such corrections are not considered, and the ordinary relation is made the basis of the theory. In order to apply this relation to the calculation of the deflection, it is necessary to know the bending moment at every point; and, since the pressures of the supports are not among the data of the problem, we require a method of determining the bending moments at the supports either by calculation or in some other way. The calculation of the bending moment can be replaced by a method of graphical construction, due to Mohr, and depending on the two following theorems: (i.) The curve of the central-line of each span of a beam, when the bending moment M is given,' is identical with the catenary or funicular curve passing through the ends of the span under a (fictitious) load per unit length of the span equal to MÆI, the horizontal tension in the funicular being unity.

(ii.) The directions of the tangents to this funicular curve at the ends of the span are the same for all statically equivalent systems of (fictitious) load.

When M is known, the magnitude of the resultant shearing stress at any section is dM/dx, where x is measured along the beam.

' The sign of M is shown by the arrow-heads in fig. 19, for which, with y downwards, =0.

FIG. 17.

FIG. 18.

58. Let l be the length of a span of a loaded beam (fig. 19), M1 and M2 the bending moments at the ends, M the bending moment at a section distant x from the end (M 1), M' the bending moment at M FIG. 19.

the same section when the same span with the same load is simply supported; then M is given by the formula M =M'+Mi l l - i-M2l, and thus a fictitious load statically equivalent to MÆI can be easily found when M' has been found. If we draw a curve (fig. 20) to pass through the ends of the span, so that its ordinate represents the value of M'ÆI, the corresponding fictitious loads are statically equivalent to a single load, of amount represented by the area of the curve, placed at the point of the span vertically above the centre of gravity of this area. If PN is the ordinate of this curve, and if at the ends of the span we erect ordinates in the proper sense to represent M i ÆI and M 2 ÆI, the bending moment at any point is represented by the length PQ. 1 For a uniformly distributed load the curve of M' is a parabola M' = zwx (l -x), where w is the load per unit of length; and the statically equivalent fictitious load is i,swl3ÆI placed at the middle point G of the span; also the loads statically equivalent to the fictitious loads M i (l- x) /lEI and M2x/lEI are zMilÆI and zM2lÆI placed at the points g, g' of trisection of the span. The funicular polygon for the fictitious loads can thus be drawn, and the direction of the central-line at the supports is determined when the bending moments at the supports are known.

59. When there is more than one span the funiculars in question may be drawn for each of the spans, and, if the bending moments at the ends of the extreme spans are known, the intermediate ones can be determined. This determination depends on two considerations: (I) the fictitious loads corresponding to the bending moment at any support are proportional to the lengths of the spans which abut on that support; (2) the sides of two funiculars that end at any support coincide in direction. Fig. 21 illustrates the method for the case of a uniform beam on three supports A, B, C, the ends A and C being freely supported. There will be an unknown bending moment Mo at B, and the system' of fictitious loads is 112wAB3ÆI at G the middle point of AB, 1 1 ,swBC 3 ÆI at G' the middle point of BC, -zMoABÆI at g and - zMoBCÆI at g', where g and g' are the points of trisection nearer to B of the spans AB, BC. The centre of FIG. 21.

gravity of the two latter is a fixed point independent of Mo, and the line VK of the figure is the vertical through this point. We draw AD and CE to represent the loads at G and G' in magnitude; then D and E are fixed points. We construct any triangle UVW whose sides UV, UW pass through D, B, and whose vertices lie on the verticals gU, VK, g'W; the point F where VW meets DB is a fixed 1 The figure is drawn for a case where the bending moment has the same sign throughout.

2 Mo is taken to have, as it obviously has, the opposite sense to that shown in fig. 19.

point, and the lines EF, DK are the two sides (2, 4) of the required funiculars which do not pass through A, B or C. The remaining sides (1, 3, 5) can then be drawn, and the side 3 necessarily passes through B; for the triangle UVW and the triangle whose sides are C 2, 3, 4 are in perspective.

The bending moment Mo is represented in the figure by the vertical line BH where H is on the continuation of the side 4, the scale being given by BH _ zMoBC CE-izwBC3, this appears from the diagrams of forces, fig. 22, in which the oblique lines are marked to correspond to the sides of which they are parallel.

In the application of the method to more complicated cases there are two systems of fixed points corresponding to F, by means of which the sides of the funiculars are drawn.

60. Finite Bending of Thin Rod. - The equation curvature = bending moment = flexural rigidity may also be applied to the problem of the flexure in a principal plane of a very thin rod or wire, for which the curvature need not be small. When the forces that produce the flexure are applied at the ends only, the curve into which the central-line is bent is one of a definite family of curves, to which the name elastica has been given, and there is a division of the family into two species according as the external forces are applied directly to the ends or are applied to rigid arms attached to the ends; the curves of the former species are characterized by the presence of inflections at all the points at which they cut the line of action of the applied forces.

We select this case for consideration. The problem of determining the form of the curve (cf. fig. 23) is mathematically identical with the problem of determining the motion of a simple circular pendulum oscillating through a finite angle, as is seen by comparing the differential equation of the curve EI d +W sin qh= 0 with the equation of motion of the pendulum d24, l d t- +g s i n c6= 0.

The length L of the curve between two inflections corresponds to the time of oscillation of the pendulum from rest to rest, and we thus have L ((WÆI) =2K, where K is the real quarter period of elliptic functions of modulus sin la, and a is the angle at which the curve cuts the line of action of the applied forces. Unless the length of the rod exceeds (EI/W) it will not bend under the force, but when the length is great enough there may be more than two points of inflection and more than one bay of the curve; for n bays (n+I inflections) the length must exceed nor ,/ (EI/W). Some of the forms of the curve are shown in fig. 24.

For the form d, in which two bays make a figure of eight, we have L I/ (WÆI) =4.6, a =130° approximately. It is noteworthy that whenever the length and force admit of a sinuous form, such as a or b, with more than two inflections, there is also possible a crossed form, like e, with two inflections only; the latter form is stable and the former unstable.

61. The particular case of the above for which a is very small is a curve of sines of small amplitude, and the result in this case has been applied to the problem of the buckling of struts under thrust. When the strut, of length L', is ??M2 A FIG. 22.

the funiculars to FIG. 23.

FIG. 24.

maintained upright at its lower end, and loaded at its upper end, it is simply contracted, unless L' 2 W> 41r 2 EI, for the lower end corresponds to a point at which the tangent is vertical on an elastica for which the line of inflections is also vertical, and thus the length must be half of one bay (fig. 25, a). For greater lengths or loads the strut tends to bend or buckle under the load. For a very slight excess of L'2W above 41r 2 EI, the theory on which the above discussion is founded, is not quite adequate, as it assumes the central-line of the strut to be free from extension or contraction, and it is probable that bending without extension does not take place d IC when the length or the force FIG. 25. exceeds the critical value but slightly. It should be noted also that the formula has no application to short struts, as the theory from which it is derived is founded on the assumption that the length is great compared with the diameter (cf. § 56): The condition of buckling, corresponding to the above, for a long strut, of length L', when both ends are free to turn is L' 2 W>7r 2 EI; for the central-line forms a complete bay (fig. 25, b); if both ends are maintained in the same vertical line, the condition is L' 2 W > 47r 2 EI, the central-line forming a complete bay and two half bays (fig. 25, c). 62. In our consideration of flexure it has so far been supposed that the bending takes place in a principal plane. We may remove this restriction by resolving the forces that tend to produce bending into systems of forces acting in the two principal planes. To each plane there corresponds a particular flexural rigidity, and the systems of forces in the two planes give rise to independent systems of stress, strain and displacement, which must be superposed in order to obtain the actual state. Applying this process to the problem of §§ 48-54, and supposing that one principal axis of a cross-section at its centroid makes an angle 0 with the vertical, then for any shape of section the FIG. 26.

neutral surface or locus of unextended fibres cuts the section in a line DD', which is conjugate to the vertical diameter CP with respect to any ellipse of inertia of the section. The centralline is bent into a plane curve which is not in a vertical plane, but is in a plane through the line CY which is perpendicular to DD' (fig. 26).

63. Bending and Twisting of Thin Rods

When a very thin rod or wire is bent and twisted by applied forces, the forces on any part of it limited by a normal section are balanced by the tractions across the section, and these tractions are statically equivalent to certain forces and couples at the centroid of the section; we shall call them the stress-resultants and the stresscouples. The stress-couples consist of two flexural couples in the two principal planes, and the torsional couple about the tangent to the central-line. The torsional couple is the product of the torsional rigidity and the twist produced; the torsional rigidity is exactly the same as for a straight rod of the same material and section twisted without bending, as in SaintVenant's torsion problem (§ 42). The twist T is connected with the deformation of the wire in this way: if we suppose a very small ring which fits the cross-section of the wire to be provided with a pointer in the direction of one principal axis of the section at its centroid, and to move along the wire with velocity v, the pointer will rotate about the central-line with angular velocity Tv. The amount of the flexural couple for either principal plane at any section is the product of the flexural rigidity for that plane, and the resolved part in that plane of the curvature of the central line at the centroid of the section; the resolved part of the curvature along the normal to any plane is obtained by treating the curvature as a vector directed along the normal to the osculating plane and projecting this vector. The flexural couples reduce to a single couple in the osculating plane proportional to the curvature when the two flexural rigidities are equal, and in this case only.

The stress-resultants across any section are tangential forces in the two principal planes, and a tension or thrust along the central-line; when the stress-couples and the applied forces are known these stress-resultants are determinate. The existence, in particular, of the resultant tension or thrust parallel to the central-line does not imply sensible extension or contraction of the central filament, and the tension per unit area of the crosssection to which it would be equivalent is small compared with the tensions and pressures in longitudinal filaments not passing through the centroid of the section; the moments of the latter tensions and pressures constitute the flexural couples.

64. We consider, in particular, the case of a naturally straight spring or rod of circular section, radius c, and of homogeneous isotropic material. The torsional rigidity is 4 Eirc 4 /(r +a); and the flexural rigidity, which is the same for all planes through the central-line, is 4E7rc 4; we shall denote these by C and A respectively. The rod may be held bent by suitable forces into a curve of double curvature with an amount of twist T, and then the torsional couple is Cr,and the flexural couple in the osculating plane is A/p,where p is the radius of circular curvature. Among the curves in which the rod can be held by forces and couples applied at its ends only, one is a circular helix; and then the applied forces and couples are equivalent to a wrench about the axis of the helix.

Let a be the angle and r the radius of the helix, so that p is r sec 2 a; and let R and K be the force and couple of the wrench (fig. 27). Then the couple formed by R and an equal and opposite force at any section and the couple K are equivalent to the torsional and flexural couples at the section, and this gives the equations for R and K R=A sia a cos' a CT cos a K The thrust across any section is R sin a parallel to the tangent to the helix, and the shearing stress-resultant is R cos a at right angles to the osculating plane.

When the twist is such that, if the rod were simply unbent, it AR FIG. 27.

r 2 r ' K = A cos 3 a r -PCT sin a. would also be untwisted, T is (sin a cos a)/r, and then, restoring the values of A and C, we have Eirc a R = 1 + a sin a cos'a, 4r2 E7rc 1+0cos2a K= 4r 1+a cos a.

65. The theory of spiral springs affords an application of these results. The stress-couples called into play when a naturally helical spring (a, r) is held in the form of a helix (a', r'), are equal to the differences between those called into play when a straight rod of the same material and section is held in the first form, and those called into play when it is held in the second form.

Thus the torsional couple is C (sin a' cos a' sin a cos a) r' r and the flexural couple is A (cos:, a' cos' a) ' r The wrench (R, K) along the axis by which the spring can be held in the form (a', r) is given by the equations s C in a' cos' aCos a' a'Cos a' sin acos R=A' (cos2a r' r) - r' r' - cos' a sin a' cos a' sin a cos a K=Acos a r > +Csina (si r) When the spring is slightly extended by an axial force F, = - R, and there is no couple, so that K vanishes, and a', r' differ very little from a, r, it follows from these equations that the axial elongation, ox, is connected with the axial length x and the force F by the equation _ E?rc sin a 8x F 47 1-1-a cos' a x ' and that the loaded end is rotated about the axis of the helix through a small angle 4aFxr cos a Eirc 4 ' the sense of the rotation being such that the spring becomes more tightly coiled.

66. A horizontal pointer attached to a vertical spiral spring would be made to rotate by loading the spring, and the angle through which it turns might be used to measure the load, at any rate, when the load is not too great; but a much more sensitive contrivance is the twisted strip devised by W. E. Ayrton and J. Perry. A very thin, narrow rectangular strip of metal is given a permanent twist about its longitudinal middle line, and a pointer is attached to it at right angles to this line. When the strip is subjected to longitudinal tension the pointer rotates through a considerable angle. G. H. Bryan (Phil. Mag., December 1890) has succeeded in constructing a theory of the action of the strip, according to which it is regarded as a strip of plating in the form of a right helicoid, which, after extension of the middle line, becomes a portion of a slightly different helicoid; on account of the thinness of the strip, the change of curvature of the surface is considerable, even when the extension is small, and the pointer turns with the generators of the helicoid.

If b stands for the breadth and t for the thickness of the strip, and T for the permanent twist, the approximate formula for the angle 0 through which the strip is untwisted on the application of a load W was found to be The quantity br which occurs in the formula is the total twist in a length of the strip equal to its breadth, and this will generally be very small; if it is small of the same order as t/b, or a higher order, the formula becomes 1WbT(1+ a)Æt 3, with sufficient approximation, and this result appears to be in agreement with observations of the behaviour of such strips.

67. Thin Plate under Pressure

The theory of the deformation of plates, whether plane or curved, is very intricate, partly because of the complexity of the kinematical relations involved. We shall here indicate the nature of the effects produced in a thin plane plate, of isotropic material, which is slightly bent by pressure. This theory should have an application to the stress produced in a ship's plates. In the problem of the cylinder under internal pressure (§ 77 below) the most important stress is the circumferential tension, counteracting the tendency of the circular filaments to expand under the pressure; but in the problem of a plane plate some of the filaments parallel to the plane of the plate are extended and others are contracted,. so that the tensions and pressures along them give rise to resultant couples but not always to resultant forces. Whatever forces are applied to bend the plate, these couples are always expressible, at least approximately in terms of the principal curvatures produced in the surface which, before strain, was the middle plane of the plate. The simplest case is that of a rectangular plate, bent by a distribution of couples applied to its edges, so that the middle surface becomes a cylinder of large radius R; the requisite couple per unit of length of the straight. edges is of amount C/R, where C is a certain constant; and the requisite couple per unit of length of the circular edges is of amount Cu/R, the latter being required to resist the tendency to anticlastic curvature (cf. §47). If normal sections of the plate are supposed drawn through the generators and circular sections of the cylinder, the action of the neighbouring portions on any portion so bounded involves flexural couples of the above amounts. When the plate is bent in any manner, the curvature produced at each section of the middle surface may be regarded as arising from the superposition of two cylindrical curvatures; and the flexural couples across normal sections through the lines of curvature, estimated per unit of length of those lines, are C(r/R i +a/R 2) and C(i/R 2 +a/R 1), where R 1 and R2 are the principal radii of curvature. The value of C for a plate of small thickness 2h is 3Eh /(i - a). Exactly as in the problem of the beam (§§ 48, 56), the action between neighbouring portions of the plate generally involves shearing, stresses across normal sections as well as flexural couples; and the resultants of these stresses are determined by the conditions. that, with the flexural couples, they balance the forces applied to bend the plate.

68. To express this theory analytically, let the middle plane of the plate in the unstrained position be taken as the plane of (x, y), and let normal sections at right angles to the axes of x and y be drawn through any point. After strain let w be the displacement of this point in the direction perpendicular to the plane, marked p in fig. 28. If the axes of x and y were parallel to the lines o£ curvature at the point, the flexural couple acting across the section normal to x (or y) would have the axis of y (or x) for its axis; but when the lines of curvature are inclined to the axes of co-ordinates, the flexural couple across a section normal to either axis has a component about that axis as well as a component about the per portion of the plate to the right upon the portion to the left, across the element, gives rise to a couple about the middle line pendicular axis. Consider an element ABCD of the section at right angles to the axis of x, contained between two lines near together and perpendicular to the middle plane. The action of the (y) of amount, estimated per unit of length of that line, equal /0 w 1 to C t e., - T fl =G1, say, and to a couple, similarly estimated, about the normal (x) of amount - C (1 - a)o xa y say. The 2E1 3 / 30 t' B = Wbr(1 ?-a) corresponding couples on an element of a section at right angles to the axis of y, estimated per unit of length of the axis of x, are of amounts - C (ay2+ T ax), =G 2 say, and -H. The resultant S1 of the shearing stresses on the element ABCD, estimated as before, is given by the equation S1= ax 1- ?yI (cf. § 57), and the corresponding resultant S2 for an element perpendicular to the axis of y is given by the equation S2 = - a x aye2 If the plate is bent by a pressure p per unit of area, the equation of equilibrium as, as2 i s ax + c3y - p or, in terms of w, This equation, together with the special conditions at the rim, suffices for the determination of w, and then all the quantities here introduced are determined. Further, the most important of the stress-components are those which act across elements of normal sections: the tension in direction x, at a distance z from the middle plane measured in the direction of p, is of amount 3C2 a 2 w (32w 2 h 3 ax2 +a ayz , and there is a corresponding tension in direc tion y; the shearing stress consisting of traction parallel to y on planes x=const., and traction parallel to x on planes y=const., is of amount 3C(1a)z a2w these tensions and shearing stresses are 2h axay' equivalent to two principal tensions, in the directions of the lines of curvature of the surface into which the middle plane is bent, and they give rise to the flexural couples.

69. In the special example of a circular plate, of radius a, supported at the rim, and held bent by a uniform pressure p, the value of w at a point distant r from the axis is 64 C (a2 -r2) (1+-- r a2 -r2, and the most important of the stress components is the radial tension, of which the amount at any point is 'd2(3+a)pz(a2-r)/h3; the maximum radial tension is about s (a/h) 2 p, and, when the thickness is small compared with the diameter, this is a large multiple of p. 70. General Theorems. - Passing now from these questions of flexure and torsion, we consider some results that can be deduced from the general equations of equilibrium of an elastic solid body.

The form of the general expression for the potential energy (§ 27) stored up in the strained body leads, by a general property of quadratic functions, to a reciprocal theorem relating to the effects produced in the body by two different systems of forces, viz.: The whole work done by the forces of the first system, acting over the displacements produced by the forces of the second system, is equal to the whole work done by the forces of the second system, acting over the displacements produced by the forces of the first system. By a suitable choice of the second system of forces, the average values of the component stresses and strains produced by given forces, considered as constituting the first system, can be obtained, even when the dis tribution of the stress and strain cannot be determined.

Taking for example the problem presented by an isotropic body of any form' pressed between two parallel planes distant 1 apart (fig. 29), and denoting the resultant pressure by p, we find that the diminution of volume -Sv is given by the equation -Iv = 1p/3 k, where k is the modulus of compres _ sion, equal to 3E/(I -20). Again, FIG. 29. take the problem of the changes produced in a heavy body by different ways of supporting it; when the body is suspended from one or more points in a horizontal plane its volume is increased by Sv =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 1 The line joining the points of contact must be normal to the planes.

vertical pressures at one or more points in a horizontal plane the volume is diminished by Iv =Wh'/3k, where h' is the height of the centre of gravity above the plane; if the body is a cylinder, of length 1 and section A, standing with its base on a smooth horizontal plane, its length is shortened by an amount Sl = Wl/2EA; if the same cylinder lies on the plane with its generators horizontal, its length is increased by an amount Sl = aWh'ÆA.

71. 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, we may show that the stress at a distance r from the point consists of (I) a radial pressure of amount 2-a F cos o 1 - a 4 7r r2 (2) tension in all directions at right angles to the radius of amount 1 -2a F cos() 2 (1a) 47r r 2 ' (3) shearing stress consisting of traction acting along the radius dr on the surface of the cone B = const. and traction acting along the meridian do on the surface of the sphere r = const. of amount 1 -2a F sin° 2(1-a) 47r r 2 ' where () is the angle between the radius vector r and the line of action of F. The line marked T in fig. 30 shows the direction of the tangential traction on the spherical surface.

Thus the lines of stress are in and perpendicular to the meridian plane, and the direc tion of one of those in the meridian plane is inclined to the radius vector r at an angle tan i (2_4lltanO) .

The corresponding displacement at any point is compounded of a radial displacement of amount 1+a F cos ° 2 (1- a) 41rE r and a displacement parallel to the line of action of F of amount (34a)(I+a) F 2(1 - a) 41rE 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.

72. The stress system considered in § 7 1 is equivalent, on the plane through the origin at right angles to the line of action of F, to a resultant pressure of magnitude 1F at the origin and a radial traction of amount (1a F 2, and, by the application 2(1-a) 4?rr of this system of tractions to a solid bounded by a plane, the displacement just described would be produced. There is also another stress system for a solid so bounded which is equivalent, on the same plane, to a resultant pressure at the origin, and a radial traction proportional to I /r 2, but these are in the ratio 27r: r 2, instead of being in the ratio 47r(I-6):(I-20)r 2 The second stress system (see fig. 31) consists of: (I) radial pressure F'r-2, (2) tension in the meridian plane across the radius vector of amount F'r 2 cos0/(I + cos 0), (3) tension across the meridian plane of amount F'r 2 /(t + cos B), (4) shearing stress as in § 71 of amount F'r 2 sin o/(I + cos 0), and the stress across the plane boundary consists of a resultant pressure of magnitude 27rF' and a radial traction of amount F'r 2. If 2 - p ax 4+ ay 4 + ax2ay2 C. a 4 w a4w FIG. 30.

FIG. 31.

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 - (I - 20-) W/21rF', 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 cos' 0/27rr'. The corresponding displacement consists of (I) a horizontal displacement radially outwards from the vertical through the origin of amount W(1+0-) sin 8 (cos o - 1-20.

271Er 1 -cos of (2) a vertical displacement downwards of amount W(1+x) {2(1-6) ±cos20}.

27rEr 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 o - =, impinging directly with relative velocity v, the patches that come into contact are circles of radius (2J6)s(V)ir, 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) (2025r 2) 1'5 r (5 12 v'"74".

For two steel spheres of the size of the earth impinging with a velocity of I 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 p, strained by the mutual gravitation of its parts, the stress at a distance r from the centre consists of (I) uniform hydrostatic pressure of amount r,gpa(3 -0)/(I - a), (2) radial tension of amount gp(r 2 /a) (3 - v)/(I - v), (3) uniform tension at right angles to the radius vector of amount i o gp (r 2 / a) (I +30-)/(I where g is the value of gravity at the surface. The corresponding strains consist of (I) uniform contraction of all lines of the body of amount sak -lgpa (3 - cr)/(I - v), (2) radial extension of amount k1 gp(r 2 /a) (I +a)/(I - if), (3) extension in any direction at right angles to the radius vector of amount 11k-igp(r21a) (i +cr)/(I - a), where k is the modulus of compression. The volume is diminished by the fraction gpa/5k of itself. The parts of the radii vectores within the sphere r =a{(3-0-)/(3+30)}'° 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 a, and the radial extension at the surface nearly 6, and these fractions can by no means be regarded as " small." 76. In a spherical shell of homogeneous isotropic material, of internal radius r l and external radius r0, subjected to pressure Po on the outer surface, and p i on the inner surface, the stress at any point distant r from the centre consists of (I) uniform tension in all directions of amount pir13 - par°3, (2) radial pressure of amount p3 - r ' r0331 3, r r l r (3) tension in all directions at right angles to the radius vector of amount 1 pi - Yo r03r13 2 r0' - r1 3 r3 The corresponding strains consist of (I) uniform extension of all lines of the body of amount 1 p,r13 - p,r03 3k 1 r ° 3 - r13 (2) radial contraction of amount 1 pl - p° rc3r13 2µ r0 3 - r1 3 r3 (3) extension in all directions at right angles to the radius vector of amount 1 Pi - p 0 r03r13 4µ r ° 3 - r 1 3 r' where ,c.c is the modulus of rigidity of the material, = ZE/(i +o-). The volume included between the two surfaces of the body is increased by the fraction k r, 1(rc' ')3 of itself, and the volume within the inner surface is increased by the fraction 3 (P1 - p0) r 0 3 p i r 1 3 p - ,r03 4,u r°3 - r1 3 + ( r0'' - 1r,') 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 pro(1 r03 3k+ r 13); the greatest tension is the transverse tension at the inner surface, and its amount is p(2ro'+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 (I) uniform tension in all directions at right angles to the axis of amount por2 r02 - r12 (2) radial pressure of amount 2 - - r l r r0 p2 r02212 ' (3) hoop tension numerically equal to this radial pressure.

The corresponding strains consist of (I) uniform extension of all lines of the material at right angles to the axis of amount 1 - a p012 - p0r02 E r02 - r ig (2) radial contraction of amount 1-1-a i - po r02r12 E r 2 - r 2 r 2 ' (3) extension along the circular filaments numerically equal to this radial contraction, (4) uniform contraction of the longitudinal filaments of amount 2 v pir12 - por02 E r02 - r12 For a shell subject only to internal pressure p the greatest extension is the circumferential extension at the inner surface, and its amount is e r12 E roe - rl2 the greatest tension is the hoop tension at the inner surface, and its amount is p(r02+ r12)/(r02 - r12).

78. When the ends of the tube, instead of being free, are closed by disks, so that the tube becomes a closed cylindrical vessel, the longitudinal extension is determined by the condition that the resultant longitudinal tension in the walls balances the resultant normal pressure on either end. This condition gives the value of the extension of the longitudinal filaments as (pi r12 - po r o 2) /3 k (r02 - r12), where k is the modulus of compression of the material. The result may be applied to the experimental determination of k, by measuring the increase of length of a tube subjected to internal pressure (A. Mallock, Proc. R. Soc. London, lxxiv., 1904, and C. Chree, ibid.). 79. The results obtained in § 77 have been applied to gun construction; we may consider that one cylinder is heated so as to slip over another upon which it shrinks by cooling, so that the two form a single body in a condition of initial stress. We take P as the measure of the pressure between the two, and p for the pressure within the inner cylinder by which the system is afterwards strained, and denote by r' the radius of the common surface. To obtain the stress at any point we superpose the system consisting of radial pressure pr12 and hoop tension r 2 r ° r l r r 2 +r2 upon a system which, for the outer cylinder, consists p r2 2 - rle P Y, Y ?, r' 2 r 2 - 2 /2 ±y.2 of radial pressure Pre and hoop tension Pr r2 r02_ r, 2, and ro for the inner cylinder consists of radial pressure P r 2 rig a n d hoop tension Pr 2 r2 ,± r1 2. The hoop tension at the inner surface r r rl is less than it would be for a tube of equal thickness without initial stress in the ratio P 2r' 2 r02 - r12 1 - p 2+ 2 r'2 - r12.1.

This shows how the strength of the tube is increased by the initial stress. When the initial stress is produced by tightly wound wire, a similar gain of strength accrues.

80. In the problem of determining the distribution of stress and strain in a circular cylinder, rotating about its axis, simple solutions have been obtained which are sufficiently exact for the two special cases of a thin disk and a long shaft.

Suppose that a circular disk of radius a and thickness 2l, and of density p, rotates about its axis with angular velocity w, and consider the following systems of superposed stresses at any point distant r from the axis and z from the middle plane: (I) uniform tension in all directions at right angles to the axis of amount 8 w2pa2(3+a), (2) radial pressure of amount gc02pr2(3+a), (3) pressure along the circular filaments of amount 8w 2 pr 2 (1 +30.), (4) uniform tension in all directions at right angles to the axis of amount sw2p(12-3z2)o-(I +0_)/(1 - o-).

The corresponding strains may be expressed as (I) uniform extension of all filaments at right angles to the axis of amount 1-6,1 w 2 pa2(3-F v), (2)' radial contraction of amount i_ c23 2 s pr?, (3) contraction along the circular filaments of amount 2 E gw2pr2' (4) extension of all filaments at right angles to the axis of amount E 2 p (l2_3 z 2)?(1+ cr), (5) contraction of the filaments normal to the plane of the disk of amount kw 2 pa 2 (3 +U) - Zw 2 pr 2 (1 + v) ?- sw2p (l2= 2), (1+ ?). The greatest extension is the circumferential extension near the centre, and its amount is (3 +a) (1 +0-)c,„„,12.

8E 6E The longitudinal contraction is required to make the plane faces of the disk free from pressure, and the terms in 1 and z enable us to avoid tangential traction on any cylindrical surface. The system of stresses and strains thus expressed satisfies all the conditions, except that there is a small radial tension on the bounding surface of amount per unit area sw2p(12 - 3z2)a(I +o-)/(I - v). The resultant of these tensions on any part of the edge of the disk vanishes, and the stress in question is very small in comparison with the other stresses involved when the disk is thin; we may conclude that, for a thin disk, the expressions given represent the actual condition at all points which are not very close to the edge (cf. § 55). The effect to the longitudinal contraction is that the plane faces become slightly concave (fig. 32).

81. The corresponding solution for a disk with a circular axle-hole (radius b) will be obtained from that given in the last section by superposing the following system of additional stresses: (I) radial tension of amount g c 2pb2(1 - a2/r2) (3+v), (2) tension along the circular filaments of amount 8 w 2 pb 2 (1 +a 2 /r 2) (3 + 6) The corresponding additional strains are (I) radial contraction of amount ? (1 +0-)622 - (1 - Q) w2pb2, (2) extension along the circular filaments of amount (2 38+E ? (1 -I-v)a +(1 - v) w2pb2. (3) contraction of the filaments parallel to the axis of amount 0-()w2pb2 4+ Again, the greatest extension is the circumferential extension at the inner surface, and, when the hole is very small, its amount is nearly double what it would be for a complete disk.

82. In the problem of the rotating shaft we have the following stress-system: (1) radial tension of amount acw2p(a2 - r2)(3-2c)/(1 - cr), (2) circumferential tension of amount 8w2p{a2(3-2c)l(1 - cr) - r2(I+2c)/(I - (7)}, (3) longitudinal tension of amount w 2 p(a 2 - 2r 2) c/(I - C). The resultant longitudinal tension at any normal section vanishes, and the radial tension vanishes at the bounding surface; and thus the expressions here given may be taken to represent the actual condition at all points which are not very close to the ends of the shaft. The contraction of the longitudinal filaments is uniform and equal to 2w 2 pa 2 cÆ. The greatest extension in the rotating shaft is the circumferential extension close to the axis, and its amount is aw2pa2 (3- 5v)Æ (1 - a). The value of any theory of the strength of long rotating shafts founded on these formulae is diminished by the circumstance that at sufficiently high speeds the shaft may tend to take up a curved form, the straight form being unstable. The shaft is then said to whirl. This occurs when the period of rotation of the shaft is very nearly coincident with one of its periods of lateral vibration. The lowest speed at which whirling can take place in a shaft of length 1, freely supported at its ends, is given by the formula w 2 p = 4 Ea t (7r/1) 4. As in § 61, this formula should not be applied unless the length of the shaft is a considerable multiple of its diameter. It implies that whirling is to be expected whenever w approaches this critical value.

83. When the forces acting upon a spherical or cylindrical body are not radial, the problem becomes more complicated. In the case of the sphere deformed by any forces it has been completely solved, and the solution has been applied by Lord Kelvin and FIG. 32.

Sir G. H. Darwin to many interesting questions of cosmical physics. The nature of the stress produced in the interior of the earth by the weight of continents and mountains, the spheroidal figure of a rotating solid planet, the rigidity of the earth, are among the questions which have in this way been attacked. Darwin concluded from his investigation that, to support the weight of the existing continents and mountain ranges, the materials of which the earth is composed must, at great depths (1600 kilometres), have at least the strength .of granite. Kelvin concluded from his investigation that the actual heights of the tides in the existing oceans can be accounted for only on the supposition that the interior of the earth is solid, and of rigidity nearly as great as, if not greater than, that of steel.

84. Some interesting problems relating to the strains produced in a cylinder of finite length by forces distributed symmetrically round the axis have been solved. The most important is that of a cylinder crushed between parallel planes in contact with its plane ends. The solution was applied to explain the discrepancies that have been observed in different tests of crushing strength according as the ends of the test specimen are or are not prevented from spreading. It was applied also to explain the fact that in such tests small conical pieces are sometimes cut out at the ends subjected to pressure.

85. Vibrations and Waves

When a solid body is struck, or otherwise suddenly disturbed, it is thrown into a state of vibration. There always exist dissipative forces which tend to destroy the vibratory motion, one cause of the subsidence of the motion being the communication of energy to surrounding bodies. When these dissipative forces are disregarded, it is found that an elastic solid body is capable of vibrating in such a way that the motion of any particle is simple harmonic motion, all the particles completing their oscillations in the same period and being at any instant in the same phase, and the displacement of any selected one in any particular direction bearing a definite ratio to the displacement of an assigned one in an assigned direction. When a body is moving in this way it is said to be vibrating in a normal mode. For example, when a tightly stretched string of negligible flexural rigidity, such as a violin string may be taken to be, is fixed at the ends, and vibrates transversely in a normal mode, the displacements of all the particles have the same direction, and their magnitudes are proportional at any instant to the ordinates of a curve of sines. Every body possesses an infinite number of normal modes of vibration, and the frequencies (or numbers of vibrations per second) that belong to the different modes form a sequence of increasing numbers. For the string, above referred to, the fundamental tone and the various overtones form an harmonic scale, that is to say, the frequencies of the normal modes of vibration are proportional to the integers I, 2, 3,. .. In all these modes except the first the string vibrates as if it were divided into a number of equal pieces, each having fixed ends; this number is in each case the integer defining the frequency. In general the normal modes of vibration of a body are distinguished one from another by the number and situation of the surfaces (or other loci) at which some characteristic displacement or traction vanishes. The problem of determining the normal modes and frequencies of free vibration of a body of definite size, shape and constitution, is a mathematical problem of a similar character to the problem of determining the state of stress in the body when subjected to given forces. The bodies which have been most studied are strings and thin bars, membranes, thin plates and shells, including bells, spheres and cylinders. Most of the results are of special importance in their bearing upon the theory of sound.

86. The most complete success has attended the efforts of mathematicians to solve the problem of free vibrations for an isotropic sphere. It appears that the modes of vibration fall into two classes: one characterized by the absence of a radial component of displacement, and the other by the absence of a radial component of rotation (§ 14). In each class there is a doubly infinite number of modes. The displacement in any mode is determined in terms of a single spherical harmonic function, so that there are modes of each class corresponding to spherical harmonics of every integral degree; and for each degree there is an infinite number of modes, differing from one another in the number and position of the concentric spherical surfaces at which some characteristic displacement vanishes. The most interesting modes are those in which the sphere becomes slightly spheroidal, being alternately prolate and oblate during the course of a vibration; for these vibrations tend to be set up in a spherical planet by tide-generating forces. In a sphere of the size of the earth, supposed to be incompressible and as rigid as steel, the period of these vibrations is 66 minutes.

87. The theory of free vibrations has an important bearing upon the question of the strength of structures subjected to sudden blows or shocks. The stress and strain developed in a body by sudden applications of force may exceed considerably those which would be produced by a gradual application of the same forces. Hence there arises the general question of dynamical resistance, or of the resistance of a body to forces applied so quickly that the inertia of the body comes sensibly into play. In regard to this question we have two chief theoretical results. The first is that the strain produced by a force suddenly applied may be as much as twice the statical strain, that is to say, as the strain which would be produced by the same force when the body is held in equilibrium under its action; the second is that the sudden reversal of the force may produce a strain three times as great as the statical strain. These results point to the importance of specially strengthening the parts of any machine (e.g. screw propeller shafts) which are subject to sudden applications or reversals of load. The theoretical limits of twice, or three times, the statical strain are not in general attained. For example, if a thin bar hanging vertically from its upper end is suddenly loaded at its lower end with a weight equal to its own weight, the greatest dynamical strain bears to the greatest statical strain the ratio 163:1; when the attached weight is four times the weight of the bar the ratio becomes 1.84:1. The method by which the result just mentioned is reached has recently been applied to the question of the breaking of winding ropes used in mines. It appeared that, in order to bring the results into harmony with the observed facts, the strain in the supports must be taken into account as well as the strain in the rope (J. Perry, Phil. Mag., 1906 (vi.), vol. ii.).

88. The immediate effect of a blow or shock, locally applied to a body, is' the generation of a wave which travels through the body from the locality first affected. The question of the propagation of waves through an elastic solid body is historically of very great importance; for the first really successful efforts to construct a theory of elasticity (those of S. D. Poisson, A. L. Cauchy and G. Green) were prompted, at least in part, by Fresnel's theory of the propagation of light by transverse vibrations. For many years the luminiferous medium was identified with the isotropic solid of the theory of elasticity. Poisson showed that a disturbance communicated to the body gives rise to two waves which are propagated through it with different velocities; and Sir G. G. Stokes afterwards showed that the quicker wave is a wave of irrotational dilatation, and the slower wave is a wave of rotational distortion accompanied by no change of volume. The velocities of the two waves in a solid of density are' {(X+2,u)/p} and A X and ,u being the constants so denoted in § 26. When the surface of the body is free from traction, the waves on reaching the surface are reflected; and thus after a little time the body would, if there were no dissipative forces, be in a very complex state of motion due to multitudes of waves passing to and fro through it. This state can be expressed as a state of vibration, in which the motions belonging to the various normal modes (§ 85) are superposed, each with an appropriate amplitude and phase. The waves of dilatation and distortion do not, however, give rise to different modes of vibration, as was at one time supposed, but any mode of vibration in general involves both dilatation and rotation. There are exceptional results for solids of revolution; such solids possess normal modes of vibration which involve no dilatation. The existence of a boundary to the solid body has another effect, besides reflexion, upon the propagation of waves. Lord Rayleigh has shown that any disturbance originating at the surface gives rise to waves which travel away over the surface as well as to waves which travel through the interior; and any internal disturbance, on reaching the surface, also gives rise to such superficial waves. The velocity of the superficial waves is a little less than that of the waves of distortion: 0.9554 / (,u/ p) when the material is incompressible 0.9 1 94 (,u/P) when the Poisson's ratio belonging to the material is 4.

89. These results have an application to the propagation of earthquake shocks (see also Earthquake). An internal disturbance should, if the earth can be regarded as solid, give rise to three wave-motions: two propagated through the interior of the earth with different velocities, and a third propagated over the surface. The results of seismographic observations have independently led to the recognition of three phases of the recorded vibrations: a set of " preliminary tremors " which are received at different stations at such times as to show that they are transmitted directly through the interior of the earth with a velocity of about io km. per second, a second set of preliminary tremors which are received at different stations at such times as to show that they are transmitted directly through the earth with a velocity of about 5 km. per second, and a " main shock," or set of large vibrations, which becomes sensible at different stations at such times as to show that a wave is transmitted over the surface of the earth with a velocity of about 3 km. per second. These results can be interpreted if we assume that the earth is a solid body the greater part of which is practically homogeneous, with high values for the rigidity and the resistance to compression, while the superficial portions have lower values for these quantities. The rigidity of the central portion would be about (1.4)f o 12 dynes per square cm., which is considerably greater than that of steel, and the resistance to compression would be about (3.8)10 12 dynes per square cm. which is much greater than that of any known material. The high value of the resistance to compression is not surprising when account is taken of the great pressures, due to gravitation, which must exist in the interior of the earth. The high value of the rigidity can be regarded as a confirmation of Lord Kelvin's estimate founded on tidal observations (§ 83).

90. Strain produced by Heat. - The mathematical theory of elasticity as at present developed takes no account of the strain which is produced in a body by unequal heating. It appears to be impossible in the present state of knowledge to form as in § 39 a system of differential equations to determine both the stress and the temperature at any point of a solid body the temperature of which is liable to variation. In the cases of isothermal and adiabatic changes, that is to say, when the body is slowly strained without variation of temperature, and also when the changes are effected so rapidly that there is no gain or loss of heat by any element, the internal energy of the body is sufficiently expressed by the strain-energy-function (§§ 27, 30). Thus states of equilibrium and of rapid vibration can be determined by the theory that has been explained above. In regard to thermal effects we can obtain some indications from general thermodynamic theory. The following passages extracted from the article " Elasticity " contributed to the 9th edition of the Encyclopaedia Britannica by Sir W. Thomson (Lord Kelvin) illustrate the nature of these indications:- " From thermodynamic theory it is concluded that cold is produced whenever a solid is strained by opposing, and heat when it is strained by yielding to, any elastic force of its own, the strength of which would diminish if the temperature were raised; but that, on the contrary, heat is produced when a solid is strained against, and cold when it is strained by yielding to, any elastic force of its own, the strength of which would increase if the temperature were raised. When the strain is a condensation or dilatation, uniform in all directions, a fluid may be included in the statement. Hence the following propositions: " (1) A cubical compression of any elastic fluid or solid in an ordinary condition causes an evolution of heat; but, on the contrary, a cubical compression produces cold in any substance, solid or fluid, in such an abnormal state that it would contract if heated while kept under constant pressure. Water below its temperature(3.9 0 Cent.)of maximum density is a familiar instance.

" (2) If a wire already twisted be suddenly twisted further, always, however, within its limits of elasticity, cold will be produced; and if it be allowed suddenly to untwist, heat will be evolved from itself (besides heat generated externally by any work allowed to be wasted, which it does in untwisting). It is assumed that the torsional rigidity of the wire is diminished by an elevation of temperature, as the writer of this article had found it to be for copper, iron, platinum and other metals.

" (3) A spiral spring suddenly drawn out will become lower in temperature, and will rise in temperature when suddenly allowed to draw in. [This result has been experimentally verified by Joule (` Thermodynamic Properties of Solids,' Phil. Trans., 1858) and the amount of the effect found to agree with that calculated, according to the preceding thermodynamic theory, from the amount of the weakening of the spring which he found by experiment] " (4) A bar or rod or wire of any substance with or without a weight hung on it, or experiencing any degree of end thrust, to begin with, becomes cooled if suddenly elongated by end pull or by diminution of end thrust, and warmed if suddenly shortened by end thrust or by diminution of end pull; except abnormal cases in which with constant end pull or end thrust elevation. of temperature produces shortening; in every such case pull or diminished thrust produces elevation of temperature, thrust or diminished pull lowering of temperature.

" (5) An india-rubber band suddenly drawn out (within its limits of elasticity) becomes warmer; and when allowed to contract, it becomes colder. Any one may easily verify this curious property by placing an india-rubber band in slight contact with the edges of the lips, then suddenly extending it - it becomes very perceptibly warmer: hold it for some time stretched nearly to breaking, and then suddenly allow it to shrink - it becomes quite startlingly colder, the cooling effect being sensible not merely to the lips but to the fingers holding the band. The first published statement of this curious observation is due to J. Gough (Mem. Lit. Phil. Soc. Manchester, 2nd series, vol. i. p. 288), quoted by Joule in his paper on ` Thermodynamic Properties of Solids' (cited above). The thermodynamic conclusion from it is that an india-rubber band,stretched by a constant weight of sufficient amount hung on it, must, when heated, pull up the weight, and, when cooled, allow the weight to descend: this Gough, independently of thermodynamic theory, had found to be actually the case. The experiment any one can make with the greatest ease by hanging a few pounds weight on a common india-rubber band, and taking a red-hot coal in a pair of tongs, or a red-hot poker, and moving it up and down close to the band. The way in which the weight rises when the red-hot body is near, and falls when it is removed, is quite startling. Joule experimented on the amount of shrinking per degree of elevation of temperature, with different weights hung on a band of vulcanized india-rubber, and found that they closely agreed with the amounts calculated by Thomson's theory from the heating effects of pull, and cooling effects of ceasing to pull, which he had observed in the same piece of india-rubber." 91. Initial Stress. - It has been pointed out above (§ 20) that the " unstressed " state, which serves as a zero of reckoning for strains and stresses is never actually attained, although the strain (measured from this state), which exists in a body to be subjected to experiment, may be very slight. This is the case when the " initial stress," or the stress existing before the experiment, is small in comparison with the stress developed during the experiment, and the limit of linear elasticity (§ 32) is not exceeded. The existence of initial stress has been correlated above with the existence of body forces such as the force of gravity, but it is not necessarily dependent upon such forces. A sheet of metal rolled into a cylinder, and soldered to maintain the tubular shape, must be in a state of considerable initial stress quite apart from the action of gravity. Initial stress is utilized in many manufacturing processes, as, for example, in the construction of ordnance, referred to in § 79, in the winding of golf balls by means of india-rubber in a state of high tension (see the report of the case The Haskell Golf Ball Company v. Hutchinson & Main in The Times of March 1, 1906). In the case of a body of ordinary dimensions it is such internal stress as this which is especially meant by the phrase " initial stress." Such a body, when in such a state of internal stress, is sometimes described as " self-strained." It would be better described as " self-stressed." The somewhat anomalous behaviour of cast iron has been supposed to be due to the existence within the metal of initial stress. As the metal cools, the outer layers cool more rapidly than the inner, and thus the state of initial stress is produced. When cast iron is tested for tensile strength, it shows at first no sensible range either of perfect elasticity or of linear elasticity; but after it has been loaded and unloaded several times its behaviour begins to be more nearly like that of wrought iron or steel. The first tests probably diminish the initial stress.

92. From a mathematical point of view the existence of initial stress in a body which is " self-stressed " arises from the fact that the equations of equilibrium of a body free from body forces or surface tractions, viz. the equations of the type ax z ax y azx a ay x + + az =0, possess solutions which differ from zero. If, in fact, 41, 02, 03 denote any arbitrary functions of x, y, z, the equations are satisfied by putting + 2, ... = and it is clear that the functions 4) i, 02, 03 can be adjusted in an infinite number of ways so that the bounding surface of the body may be free from traction.

93. Initial stress due to body forces becomes most important in the case of a gravitating planet. Within the earth the stress that arises from the mutual gravitation of the parts is very great. If we assumed the earth to be an elastic solid body with moduluses of elasticity no greater than those of steel, the strain (measured from the unstressed state) which would correspond to the stress would be much too great to be calculated by the ordinary methods of the theory of elasticity (§ 75) . We require therefore some other method of taking account of the initial stress. In many investigations, for example those of Lord Kelvin and Sir G. H. Darwin referred to in § 83, the difficulty is turned by assuming that the material may be treated as practically incompressible; but such investigations are to some extent incomplete, so long as the corrections due to a finite, even though high, resistance to compression remain unknown. In other investigations, such as those relating to the propagation of earthquake shocks and to gravitational instability, the possibility of compression is an essential element of the problem. By gravitational instability is meant the tendency of gravitating matter to condense into nuclei when slightly disturbed from a state of uniform diffusion; this tendency has been shown by J. H. Jeans (Phil. Trans. A. 201, 1903) to have exerted an important influence upon the course of evolution of the solar system. For the treatment of such questions Lord Rayleigh (Proc. R. Soc. London, A. 77, 1906) has advocated a method which amounts to assuming that the initial stress is hydrostatic pressure, and that the actual state of stress is to be obtained by superposing upon this initial stress a stress related to the state of strain (measured from the initial state) by the same formulae as hold for an elastic solid body free from initial stress. The development of this method is likely to lead to results of great interest.

Authorities. - In regard to the analysis requisite to prove the results set forth above, reference may be made to A. E. H. Love, Treatise on the Mathematical Theory of Elasticity (2nd ed., Cambridge, 1906), where citations of the original authorities will also be found. The following treatises may be mentioned: Navier, Résumé des icons sur l'application de la mecanique (3rd ed., with notes by SaintVenant, Paris, 1864); G. Lame, Legons sur la theorie mathematique de l'elasticite des corps solides (Paris, 1852); A. Clebsch, Theorie der Elasticitat fester Korper (Leipzig, 1862; French translation with notes by Saint-Venant, Paris, 1883); F. Neumann, Vorlesungen fiber die Theorie der Elasticitdt (Leipzig, 1885); Thomson and Tait, Natural Philosophy (Cambridge, 1879, 1883); Todhunter and Pearson, History of the Elasticity and Strength of Materials (Cambridge, 1886-1893). The article " Elasticity " by Sir W. Thomson (Lord Kelvin) in 9th ed. of Encyc. Brit. (reprinted in his Mathematical and Physical Papers, iii., Cambridge, 1890) is especially valuable, not only for the exposition of the theory and its practical applications, but also for the tables of physical constants which are there given. (A. E. H. L.)