799 particles may now be regarded as being all completely relaxed. Let, next, one end of the bar be fixed, and the other be made to revolve by torsion, till the particles at the circumference of the bar are strained to the utmost extent of which they can admit, without undergoing a permanent alteration in their mutual con nexion. 1 In this condition, equal elements of the cross section 01 the bar afford resistances proportional to the distances of the elements from the centre of the bar ; since the particles are dis placed from their positions of relaxation through spaces which are proportional to the distances of the particles from the centre. The couple which the bar now resists, and which is equal to the sum of the couples due to the resistances of all the elements of the section, is that which is commonly assumed as the measure of the torsional strength of the bar. For future reference, this couple may be denoted by L, and the angle through which it has twisted the loose end of the bar by 0. 12. " The twisting of the bar may, however, be carried still farther, and during the progress of this process the outer particles will yield in virtue of their ductility, those towards the interior successively reaching their elastic limits, until, when the twisting has been sufficiently continued, all the particles in the section, except those quite close to the centre, have been strained beyond their elastic limits. Hence, if ve suppose 2 that no change in the hardness of the substance composing the material has resulted from the sliding of its particles past one another, and that therefore all small elements of the section of the bar afford the same resist ance, no matter what their distances from the centre may be, it is easy to prove that the total torsional resistance of the bar is | of what it was in the former case ; or, according to the notation already adopted, it is 3 now | L. 13. "If, after this, all external stress be removed from the bar, it will assume a position of equilibrium, in which the outer particles will be strained in the direction opposite to that in which it was twisted, and the inner ones in the same direction as that of the twisting, the two sets of opposite couples thus produced among the particles of the bar balancing one another. It is easy to show that the line of separation between the particles strained in one direction and those in the other is a circle whose radius is if of the radius of the bar. The particles in this line are evidently subject to no strain 4 when no external couple is applied. The bar 1 " I here assume the existence of a definite elastic limit or a limit within which, if two particles of a substance be displaced, they will return to their original relative positions when the disturbing force is removed. The opposite conclusion, to which Mr Hodgkinson seems to have been led by some interesting experimental results, will be considered at a more advanced part of this paper." 2 [Note added October 1877.] This supposition may be true for some solids ; it is certainly not true for solids generally. A piece of copper or of iron taken in a soft and unstrained condition certainly becomes " harder" when strained beyond its first limits of elasticity, that is to say, its limits of elasticity become wider; and a similar result will probably be found in ductile metals generally. Thus the resistance of the outer elements will be greater than those of the inner dements in the case described in the text, until the torsion has been pushed so far as to bring about the greatest hardness in all the elements at any consider able distance from the axis. It may be that before this condition has been attained the hardening of the outer elements will have been overdone, and they miy have legun to lose strength, and to have become friable and fissured. The principle set forth in the text is not, however, vitiated by the incorrectness of a supposition introduced merely for the sake of numerical illustration. 3 "To prove this, let r be the radius of the bar, i) the utmost force of a unit of area of the section to resist a strain tending to make the particles slide past one another, or to resist a shearing strain, as it is commonly called. Also, let the section of the bar be supposed to be divided into an infinite number of con centric annular elements, the radius of any one of these being denoted by x and its area by l-irxdx, "Now, when only the particles at the circumference are strained to the utmost, and when, therefore, the forces on equal areas of the various elements are pro portional to the distances of the elements from the centre, we have r/ for the force of a unit of area at the distance of x from the centre. Hence the total tan gential force of the element is and the couple due to the same element is =x . 2-n-xdx . ?/ =2-7ri) . and therefore the total couple, which has been denoted above by L, is -*, I fhatis Next, when the bar has been twisted so much that all the particles in its section afford their utmost rcsistance, we have the total tangential force of the element 2-irxdx , r , and the couple due to the same clement =x . 2-TTxdx . ?; = 2T Hence the total couple due to the entire section is But this quantity is of the value of L in formula (a). That is the couple which the bar resists in this case is $ L, or of that which it resisted in the former case.
"Or at least they are subject to no strain of torsion, either in the one direction
other; though they may be subject to a strain of compression or ex- with its now molecular arrangement may now be subjected, a,? often as we please, 5 to the couple L without undergoing any farther alteration. Its strength to resist torsion, in the direction of the couple L has therefore been considerably increased. Its strength to resist torsion in the opposite direction has, however by the same process, been much diminished; for as soor as its free extremity has been made to revolve backwards through an angle 6 of -? f from the position of equilibrium, the particles of the circumference will have suffered the utmost distortion cf which they can admit without undergoing permanent altera tion. Now, it is easy to prove that the couple required to pro duce a certain angle of torsion is the same in the new state of the bar as in the old. 7 Hence the ultimate strength of the bar when twisted backwards is represented by a couple amounting to only | L. But, as we have seen, it is L when the wire is twisted forwards. That is, then, The wire in its new state has twice as much strength to resist torsion in one direction as it has to resist it the other. 14 " Principles quite similar to the foregoing, are applicable in regard to beams subjected to cross strain. As, however, my chief object at present is to point out the existence of such principles, to indicate the mode in which they are to be applied, and to show their great practical importance in the determination of the strength of materials, I need not enter fully into their application in the case of cross strain. The investigation in this case closely resembles that in the case of torsion, but is more complicated on account of the different ultimate resistances afforded by any material to tension and to compression, and on account of the numerous varieties in the form of section of beams which for different pur poses it is found advisable to adopt. I shall therefore merely make a few remarks on this subject. 15. " If a bent bar of wrought iron or other ductile material be straightened, its particles will thus be put into such a state that its strength to resist cross strain, in tho direction towards which it has been straightened, will be very much greater than its strength to resist it in the opposite direction, each of these two resistances being entirely different from that which the same bar would afford were its particles all relaxed when the entire bar is free from ex ternal strain. The actual ratios of these various resistances depend on the comparative ultimate resistances afforded by the substance to compression and extension, and also, in a very material degree, on the form of the section of the bar. I may, however, state that in general the variations in the strength of a bar to resist cross strain, which are occasioned by variations in its molecular arrange ment, are much greater even than those which have already been pointed out as occurring in the strength of bars subjected to torsion. 16. "What has already been stated is quite sufficient to account for many very discordant and perplexing results which have been arrived at by different experimenters on the strength of materials. It scarcely ever occurs that a material is presented to us, either for experiment or for application to a practical use, in which the particles are free from great mutual strains. Processes have already been pointed out by which we may at pleasure produce certain peculiar strains of this kind. These, or other processes producing somewhat similar strains, arc used in the manufacture of almost all materials. Thus, for instance, when malleable iron has received its final conformation by the process termed cold swaging, that is, by hammering it till it is cold, the outer particles exist in a state of extreme compression, and the internal ones in a state of extreme tension. The same seems to be the case in cast iron when it is taken from the mould in which it has been cast. The outer portions have cooled first, and have therefore contracted, while the inner ones still continued expanded by heat. The inner ones then contract as they subsequently cool, and thus they, as it were, pull the outer ones together. That is, in the end the outer ones are in a state of compression and the inner ones in the opposite condition. 17. "The foregoing principles may serve to explain the true tension in the direction of the length of the bar." [That they are so is proved by experiments made for the present article by Mr Thomas Gray in October 1377]. "This, however, does not fall to be considered in the investigation of the text." s "This statement, if not strictly, is at least extremely nearly true, since from the experiments made by Mr Fairbairn and Mr Ilodgkinson on cast-iron (see various fit-ports of the British Association), we may conclude that the metals are influenced only in an extremely slight degree by time. Were the bars composed of some substance, such as sealing wax, or hard pitch, possessing a sensible amount of viscidity, the statement in the text would not hold good." 6 [iVote added October 1877.] This assumes that the limits of elasticity in a substance which has already been strained beyond its limits of elasticity are equal on the two sides of the shape which it has when in equilibrium without disturbing force a supposition which may be true or may not be true. Ex periment is urgently needed to test it; for its truth or falseness is a matter of much importance in the theory of elasticity. 7 "To prove this, let tl.e bar be supposed to be divided into an infinite number of elementary concentric tubes (like the so-called annual rings of growth in trees). To twist cich of these tubes through a certain angle, the same couple wiirbe required, whether the tube is already-subject to the action of a couple of any moderate amount in either direction or not. Hence, to twist them all, or, what is the same thing, to twist the whole bar, through a certain angle, the same couple will be required whether the various elementary tubes be or be not
relaxed, when the bar ua a whole is free from external strain."