800 ELASTICITY cause oi an important fact observed by Mr Eaton Hodgkinson in Ms valuable researches in regard to the strength of cast iron (Report of the British Association for 1837, p. 362). 1 He found, that, contrary to what had been previously supposed, a strain, however small in comparison to that which would occasion rupture, was sufficient to produce a set, or jfermanent change of form, in the beams on which he experimented. Now this is just what should be expected in accordance with the principles which I have brought forward ; for if, for some of the causes already pointed out, various parts of a beam previously to the application of an external force have been strained to the utmost, when, by the application of such force, however small, they are still farther displaced from their positions of relaxation, they must necessarily undergo a permanent alteration in their connexion with one another, an alteration permitted by the ductility of the material; or, in other words, the beam as a whole must take a set. 18. " In accordance with this explanation of the fact observed by llr Hodgkinson, I do not think we are to conclude with him that the maxim of loading bodies within the elastic limit has no founda tion in nature. It appears to me that the defect of elasticity, which he has shown to occur even with very slight strains, exists only when the strain is applied for the first time ; or, in other words, that if a beam has already been subjected to a considerable strain, it may again be subjected to any smaller strain in the same direction with out its taking a set. It will readily be seen, however, from Mr Hodgkinson s experiments, that the term elastic limit, as com monly employed, is entirely vague, and must tend to lead to erroneous results. 19. "The considerations adduced seem to me to show clearly that there really exist two clastic limits for any material, between which the displacements or deflexions, or what may in general be termed the changes of form, must be confined, if we wish to avoid giving the material a set, or, in the case of variable strains, if we wish to avoid giving it a continuous succession of sets which would gradually bring about its destruction ; that these two elastic limits are usually situated one on the one side and the other on the opposite side of the position which the material assumes when subject to no external strain, though they may be both on the same side of this position of relaxation ; 2 and that they may there fore with propriety be called the superior and the inferior limit of the change of form of the material for the particular arrangement which has been given to its particles ; that these two limits are not fixed for any given material, but that, if the change of form be continued beyond either limit, two new limits will, by means of an alteration in the arrangement of the particles of the material, be given to it in place of those which it previously possessed ; and lastly, that the processes employed in the manufacture of materials are usually such as to place the two limits in close contiguity with one another, thus causing the material to take in the first instance a set from any strain, however slight, while the interval which may afterwards exist between the two limits, alrd also, as was before stated, the actual position assumed by each of them are determined by the peculiar strains which are subsequently applied to the material. 20. " The introduction of new, though necessary, elements into the consideration of the strength of materials may, on the one hand, seem annoying from rendering the investigations more com plicated. On the other hand, their introduction will really have the effect of obviating difficulties, by removing erroneous modes of viewing the subject, and preventing contradictory or incongruous results from being obtained by theory and experiment. In all investigations, in fact, in which we desire to attain or to approach nearly to truth, we must take facts as they actually are, not as we might be tempted to wish them to be for enabling us to dispense with examining processes which are somewhat concealed and intricate but are not the less influential from their hidden character." 21. Passing now to homogeneous matter (sec. 38), homogeneously strained (chap. ii. of Math. Theory below), 1 " For further information regarding Mr Hodgldnson s views and experiment* Jlec liis communications in tlie Transactions of the Sections of the Jiritish Association for the years 1843 (p. 23) and 1844 (p. 25), and a work by him, en titled Experimental Researches on the Strength and (/liter Properties of Cast Iron, 8vo., 184fi." 2 " Thus if the section of a beam be of some such form as that shovji in either of the accompanying figures, the one rib or the two ribs, as the case may be, being very weak in comparison to the thick part of the beam, it may readily occur that the tn-o elastic limits of deflexion may be situated both on the same side of the position assumed by the beam when free from external force. For if the beam has been sup ported at its extremities and loaded at its middle till the rib A B has yielded by its ductility so as to muke all its particles exert their utmost tension, and if the load be now gradually removed, the particles at B may come to te compressed to the utmost before the load lias been entirely re- uwred." Fig. 1. we aro met by physical questions of great interest regarding limits of elasticity. Supposing the solid to be homogeneously distorted in any particular way to nearly the limit of its elasticity for this kind of distortion, will the limits be widened or narrowed by the superposition of negative or positive pressure equal in all directions produc ing a dilatation or a condensation ] It seems probable that a dilatation would narrow the limits of elasticity, and a condensation widen them. This, however, is a mere guess : experiment alone can answer the question. Take again a somewhat less simple case. A wire is stretched by a weight to nearly its limits of longitudinal elasticity ; a couple twisting it is applied to its lower end Will this either cause the weight to run down and give the wire a permanent set, or break it ? Probably, yes ; but experi ment only can decide. The corresponding question with reference to a column loaded with a weight may have the same answer, but not necessarily so. Experiment again is wanting. A wire hanging stretched by a light weight, merely to steady it, is twisted to nearly its limit of torsional elasticity by a couple of given magnitude applied to its lower end the stretching weight is increased Will this cause it to yield to the couple and take a permanent set ] Probably, yes. [Certainly yes, for steel piano-forte wire experimented on by Mr M Farlane to answer this question since it was first put in type fur the present article.] If so, then the limits of torsional elasticity of a wire bearing a heavy weight are widened by diminishing or taking off the weight; and no doubt it will follow continuously that a column twisted by opposing couples at its two ends will have its limits of torsional elasticity widened by the application of forces to its two ends, pressing them towards one another. Experiments to answer these questions would certainly reward the experimenter with definite and interest ing results. 22. NARROWNESS OF LIMITS OF ELASTICITY Solids. The limit of elasticity of metals, stones, crystals, woods, are so narrow that the distance between any two neighbour ing points of the substance never alters by more than a small proportion of its own amount without the substance either breaking or experiencing a permanent set, and there fore the angle between two lines meeting in any point of the substance and passing always through the same matter is never altered by more than a small fraction of the radian, 3 before the body either breaks or takes a permanent set. By far the widest limits of elasticity hitherto discovered by experiment, for any substance except cork, india-rubber, jellies, are those of steel pianoforte wire. Take, for example, the piano-forte wire at present in use for deep-sea soundings. It is No. 22 of the Birmingham wire gauge, its density is 7 727, it weighs 34 gramme per centimetre, or G 298 kilogrammes per nautical mile of 1852 3 metres, and therefore its sectional area and diameter are 0044 square centimetre and -0244 centimetre. It bears a weight of 106 kilogrammes, which is equal in weight to about 31 kilometres of its length, and when this weight is alternately hung on and removed the length of the wire varies by ^ of its amount. While this elongation takes place there is a lateral shrinking, as we shall see (section 47), of from to -jSj. of the same amount. 23. Consider now in the unstrained wire two lines through the substance of the wire at right angles to one another in any plane through or parallel to the axis of the wire in directions equally inclined to this line. When the wire is pulled lengthwise the two vertical angles bisected by the length of the wire become acute, and the other two obtuse by a small difference, as illustrated in the diagram (fig. 2), 3 The radian is the angle whose arc is equal to radius ; it is equal
to 57.29