March 19, 1885] 



XA TURE 



457 



which the syndics of the Cambridge University Press have 

 intrusted to me to complete and edit. In reading the great 

 number of memoirs relating to the subject I have been much 

 struck by the want of a clear and accurate terminology in both 

 theoretical and practical elasticity. I have been forced to the 

 conclusion that the great discrepancy, which is often to be found 

 between theoretical and practical results, is in some measure due 

 to the want of this terminology (e.g. the extreme looseness of 

 the term " limit of elasticity "). I find it needful for the pur- 

 poses of the above work to adopt such a terminology, but before 

 doing so it would be extremely valuable to have the opinion of 

 some of our leading elasticians on the terms I venture to 

 propose. I should be very glad of any suggestions, through 

 the columns of Nature, towards a definite and uniform 

 terminology. 



I am particularly dissatisfied with the term " limit of super- 

 imposition." It is exceedingly clumsy. Other possible terms 

 are — "limit of superposable stress," " linear limit," and "limit 

 of constant slope," the last two phrases having reference to the 

 fact that the stress-strain curve at this limit ceases to be a straight 

 line. With regard to this limit of superimposition I may remark 

 that it may arise from one of two causes — (i) The strain com- 

 ponents become so large that we cannot neglect the squares of 

 small quantities, or the stress components can no longer be 

 taken proportional to those of strain. This might happen before 

 permanent set. (2) Permanent set may arise which does not 

 follow the generalised Hooke's Law. This seems the more 

 probable case, and has been adopted below. Prof. Kennedy 

 tells me that he thinks when a body has been reduced to a state 

 of ease that the superior elastic limit and the limit of super- 

 imposition coincide. 



It has been proposed, I believe, to term that limit of stress 

 at which bars begin locally to "thin down" the limit of vis- 

 cosity. The " limit of uniform strain " is not altogether satis- 

 factory or quite suggestive of this peculiar viscosity. " State of 

 maximum stress " might perhaps serve the purpose, were one 

 quite sure that this state always coincides with the viscous limit. 

 In the following remarks I have been much assisted by Prof. 

 J. Thomson's epoch-making paper in the Cambridge and Dublin 

 Mathematical yourtial for 1 848, and even more by Prof. Alex. 

 I!. W. Kennedy's paper on Riveted Joints in the Proceedings of 

 the Institution of Mechanical Engineers, April, 1881 (especially 

 pp. 208-213). 



We have first to distinguish between two classes of materials. 

 In the one we may suppose the particles to be in a state of in- 

 ternal stress before any external force is applied ; in this case 

 any, the least, external stress will probably produce permanent set. 

 If this stress be removed and then reapplied, after one or two 

 trials it will cease to produce permanent set, or at least the per- 

 manent set will be extremely small as compared with the elastic 

 strain. We thus need a term to mark that state of the body 

 when external stress does not produce permanent set owing to 

 the existence of internal stress. This might perhaps be termed 

 the state of ease. Many discordant results with regard to the 

 constants of elasticity are not improbably due to the fact that 

 the ratio of stress to strain has been measured before the ma- 

 terial has been reduced to a state of ease. In the second class 

 of materials we may suppose this state of ease to exist before the 

 application of any stress. Supposing a body to be in its state 

 of ease, there will then exist two limits, one on one side, and 

 one on the other of the unstrained shape, which may be termed 

 the inferior and the superior limits of perfect elasticity. Any 

 external stress which does not produce a strain exceeding these 

 limits will not give rise to permanent set. These inferior and 

 superior limits of perfect elasticity mark, as a rule, the range 

 covered by the usual mathematical theory. Within these limits 

 it is generally safe to assume that the components of internal 

 stress are proportional to the components of strain. In some 

 cases, i.e. cast iron, where, however, it is difficult to produce 

 the state of ease, this does not seem to be accurate — the stress 

 and strain components appear never to be proportional. 



In most materials the range of perfect elasticity is not large. 

 An external stress, which is by no means nearly equal to that 

 which is required to produce rupture will give rise to a per- 

 manent set. Thus permanent set in some materials will com- 

 mence at a stress only $ to | of the stress that those materials 

 are capable of standing. Thus beyond the limit of elasticity we 

 have first a range of stresses, which produce strains partly elastic 

 and partly permanent. The strain in this range might still remain 

 proportional to the stress ; the permanent is yet small as com- 



pared with the elastic part of the strain. This range is bounded, 

 however, by a stress for which the strain ceases to be propor- 

 tional to the stress. In other words, the "generalised' 

 Hooke's law is no longer applicable. Up to this point, if we 

 are merely desirous of finding the strain produced by any system 

 of statical stress, the mathematical equations of elasticity will 

 apply, supposing, as seems probable, that the elastic constants 

 do not alter, owing to the permanent set. Those equations 

 would not of course be valid if we wished to find the strain in 

 the body, if the stress were altered, nor would they suffice to 

 treat vibratory motions capable of producing permanent set. 

 This limit, which is that at which the ut tcnsio sic vis principle 

 ceases, requires a name. It might perhaps be termed the limit 

 of superimposition. That is to say, if a certain addition to this 

 limiting stress produced a certain increase of strain, and a second 

 addition another increase, these increments of stress, if superim- 

 posed would not produce the sum of the strain increments. It 

 might at first sight appear more direct to term it the modular 

 limit, or the limit of Hooke's law, but it would seem that, after 

 this limit is passed, Hooke's law, probably with the same 

 modulus, applies to so much of the strain as iselastic strain ; in 

 fact at the limit of superimposition it is the permanent set part 

 of the strain which ceases to obey Hooke's law. In some mate- 

 rials the limits of perfect elasticity and of superimposition may 

 coincide. At the latter limit the permanent set is still in some 

 cases only one-twenty-fifth of the total strain. Neither of the 

 limits above considered is commercially treated as the limit of 

 elasticity. This is the point at which the material " breaks 

 down," that is to say, the stress being continually increased, a 

 strain is obtained which would be preserved by replacing the 

 stress by one very much less. The material is unable to balance 

 the stress upon it. If the stress be maintained the strain will 

 suddenly increase by a considerable amount (without the stress 

 being increased). This remarkable limit, it has been suggested by 

 Mr. Tweddell, should be termed the limit of fatigue. The limit 

 of fatigue being past, a small proportion of the strain, namely, 

 so much as corresponds to the modulus, is elastic, the greater 

 part is permanent set. 



In the case of bars of iron subjected to longitudinal pull, if 

 the stress be increased beyond the limit of fatigue, another 

 limiting strain is reached, namely, one at which local contraction 

 begins, or the bar commences to draw out at some point, i.e. 

 the strain ceases to be uniform. The material now begins to act 

 as if it were " viscous," and it would be convenient to describe 

 this state as that of viscosity, had not this name been appropriated 

 to that permanent set which may be produced by the application 

 for a long period of a stress well within the limits of perfect 

 elasticity. Closely associated, if not the same, with this limit op 

 uniform strain is the state of maximum load. From this point 

 onwards, as the strain increases the load decreases, till the 

 breaking load is reached with a magnitude below that of the 

 maximum load. To distinguish one from the other requires 

 a special manipulation. As a rule, what is meant by the 

 absolute cr breaking strength is probably the maximum load, for 

 if this load was allowed to remain, the bar would break under 

 it. It might perhaps be convenient, however, to speak of one 

 as the maximum and the other as the terminal load. With the 

 terminal load the "elastic life "of the material is concluded. 

 It must be remembered that owing to the bar locally thinning 

 down, the stress per unit area at the terminal load is greater 

 than the stress per unit area at the maximum load. 



Such are the limits for which it is needful that a terminology 

 should be established. I shall be extremely glad if any of the 

 readers of Nature, who happen to be elasticians, will suggest 

 a more concise phraseology. Karl Pearson 



University College, February 14 



Civilisation and Eyesight 



I have been interested in Lord Rayleigh's note on "Vision," 

 and would offer my mite on the subject. 



I have no doubt that brilliancy of image and power of 

 distinguishing largely depend on definition. The brilliancy 

 does so for the same reason as that which induces an artist to 

 1 eighten colour-effects by sharp contrasts. In the same way, if 

 we seek to decide if two colours are alike, we place them in 

 immediate contact with a sharp edge. Details are best seen 

 with a telescope when the images are sharp and untroubled. 

 When slight tremors are in the air, and the image is rapidly 

 displaced in all directions, so that what we see is the resultant 



