1849.] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



41 



ON THE STRENGTH OF MATERIALS. 



On the Strength of ilateriaU. as Influenced by the Existence or 

 non-Ejcistence of certain Mutual Strains among the Particles com- 

 posing them. By James Thomson, Jun., M.A., College, Glasgow. 

 — [From the Cambridge and Dublin Mathematical Journal, Novem- 

 ber, 184.8.] 



My principal object in the following paper is to show that the 

 absolute strength of any material composed of a substance possess- 

 ing ductility (and few substances, if any, are entirely devoid of 

 this property), may vary to a great extent, according to the state 

 of tension or relaxation in which the particles have been made to 

 exist when the material as a wliole is subject to external strain. 



Let, for instance, a cylindrical bar of malleable iron, or a piece 

 of iron wire, be made red hot, and then be allowed to cool. Its 

 particles may now be regarded as being all completely relaxed. 

 Let next the 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- 

 nection.* In this condition, equal elements of the cross-section 

 of 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 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 e. 



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 assum- 

 ing successively the condition of greatest tension ; until, when the 

 twisting has been sufficiently continued, all the particles in the 

 section, except those quite close to the centre, will have been 

 brought to afford their utmost resistance. Hence, if we suppose 

 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 resistance, no matter what their distances 

 from the centre may be — it is easy to prove that the total resist- 

 ance of the bar is now | of what it was in the former case ; or, ac- 

 cording to the notation already adopted, it is now |L. 



To prove this, let r be the radius of the bar, ri the utmost force of a uni' 

 of area of the section to resist a strain tending to make the particles shde 

 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 concentric annular elements; the radius of any one of these 

 being denoted by x, and its area by 2xxdx. 



Now, when only the particles at the circumference are strained to the 

 utmost ; and when, therefore, the forces on equal areas of the various ele- 

 ments are proportional to the distances of the elements from the centre, we 



X 



have ij— for the force of a unit of area at the distance x from the centre, 

 r 



Hence the total tangential force of the element is 



X 



= 2 T xdx .Tj — ; 

 r 



and the couple due to the same element is 



X 1 



= j:.27r.r(iar.r)— =27r n — . x^dx: 

 r r 



and therefore the total couple, which has been denoted above by L, is 



1 />r 



•=2irr)- / x^dx, — that is 

 Vo 



L = ^ IT 7) r^ (a). 



Next, when the bar has been twisted so much that all the particles in its 

 section afford their utmost resistance, we have the total tangential force of 

 the element =2Trxdx.n; 



and the couple due to the same element =x.2iTxdx.Ti = 2irT].x^ dx. 

 Hence the total couple due to the entire section is 



= 2 7rij/ X'dx = f 7r7;r3. 

 -^ 



* I here assume the existence of a defluite " elastic limit," or a limit witliin which if 

 two particles of a snlistarce be displaced, they will return to their original relative posi- 

 tions when the disturbing force is removed. The opposite conclusion, to which Mr. 

 Horigkinson seeiris to have been led by some Interesting experimental results, will be 

 considered at a more advanced part of this paper. 



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 resists in the 

 former case. 



If, after this, all external strain be removed from the bar, it will 

 assume a position of equilibrium, in whicli tlie outer particles will 

 be strained in the direction opposite to that in which it was twisted 

 and the inner ones in the same 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 the one direction, and 

 those in the other, is a circle whose radius is | of the radius of the 

 bar. The particles in this line are evidently subject to no strain* 

 when no external couple is applied. Tlie bar with its new mole- 

 cular arrangement may now be subjected, as often as we please,f to 

 the couple i L, without undergoing any farther alteration ; and 

 therefore its ultimate strength to resist torsion, in the direction of 

 the couple L, has been considerably increased. Its strength to 

 resist torsion in the opposite direction has, however, by the same 

 process, been much diminished ; for, as soon as its free extremity 

 has been made to revolve backwards through an angle of | 9 from 

 the position of equilibrium, the particles at the circumference will 

 have suffered the utmost displacement of which they can admit 

 without undergoing permanent alteration. Now it is easy to prove 

 that the couple required to produce a certain angle of torsion is the 

 same in the new state of the bar as in the old.J Hence the ulti- 

 mate 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. The wire in its netp 

 state has twice as much strength to resist torsion in the one direction as 

 it has to resist it in the other. 



Principles quite similar to the foregoing, operate in regard to 

 beams subjected to cross strains. As, however my chief object at 

 present is to point out the existence oi^ 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 ma- 

 terials, 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 purposes it is 

 found advisable to adopt. I shall therefore merely make a few 

 remarks on this subject. 



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 strengtli to resist cross-strain, in the 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 de- 

 pend on the comparative ultimate resistances afforded by the sub- 

 stance 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-strains, wliich are occasioned by variations in its molecular 

 arrangement, are much greater even than those which have already 

 been pointed out as occurring in the strength of bars subjected to 

 torsion. 



What has been already stated is quite sufficient to account for 

 many very discordant and perplexing results which have been ar- 

 rived 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, are used in the manufacture 



* Or at least they are subject to no strain of torsion either in the one direction or iu 

 the other; though they may perhaps be subject to a strain of compression or extension 

 in the direction of the length of the bar. This, however, does BOt fail to be considered 

 in the present investigation. 



+ This statement, if not strictly, is at least extremely nearly true : since from the ex- 

 periments made by Mr. Fairbairn and Mr. Hodgkinson on cast-iron (see various Reporis 

 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. 



t To prove this, let the bar be supposed to be divided into an infinite number of ele- 

 mentary concentric tubes (like the so-called annual rings of growth in trees) ; to twist 

 each of these tubes through a certain angle, the same couple will be 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 ele- 

 mentary tubejs be or be cot relaxed, wheu the bar as a whole 1« free from external strain. 



