CIRCULAR CYLINDERS UNDER CERTAIN PRACTICAL SYSTEAIS OF T.OAD. 221 
Construction’), wliere blocks show less strength wlien three sheets of lead are 
Introduced between the compressing planes and the test piece than when one sheet 
only is introduced, the lateral flow being greater in the first case owing to the larger' 
amount of lead. 
It would seem, therefore, as if the true strength of a cylinder were really greater 
than its strength as tested either between millboards or between lead sheets, and 
not, as Professor Perry states in his ‘ Applied Mechanics,’ equal to the strength 
shown in the lead test—this test, as we see, leading to results that are not definite, 
hut vary with the expansion of the lead. The millboard test, liowever, wdrich is 
advocated by Uxwix, shordd give a constant value, although it is not tire value whicli 
would hold for a cylinder under uniform pressure. 
(tZ.) Diagrams 12-14 suggest an explanation of the fret that, when short cylinders 
are strongly compressed between very hard surfaces, pieces are sometimes cut out at 
the ends of an approximately conical shape. The same occurs when spherical pieces 
of metal, such as hall-hearings, are compressed between parallel plates, ’flris is usually 
explained by saying that the material Irreaks along the planes of principal shear. On 
the other hand, it may be argued simply that rupture should take place over the 
regions of greatest stress. These are near the perimeter at the ends, and gradually 
close ill upon the centre, forming hollow caps. 
Further, in the case of the lead tests, where P > Q, tliis state of things is 
reversed, and the material should give way from tlie inside, so that we should expect 
it to split axially, and possibly along meridian planes as well. That this is wliat 
really occurs can he verified by referring to the figures in the chapter on testing of 
stone in Unwin’s ‘ Testing of Materials of Construction.’ 
(e.) The results both of this and of tlie first jirohlem show us liow unrelialfie any 
experiments on short cylinders must he, which have in view tlie determination, by 
tensile strain, of either Young’s modulus or Poisson’s ratio. Thus any I’esults 
obtained in such a case without the dimensions and tlie mode of application of the 
stress being exactly specified, would not justify us in general in drawing any 
conclusions as to whether a given material possesses or not uniconstant Isotrojiy. 
§ 28. The Third Problem. Case of Torsion. Expressions for the 
Displacement and Stresses. 
T now proceed to consider a case where u and ir are zero, that is, where we have 
to deal with the solution in v, which we have seen is independent of the others. 
AVe have in the notation of § :I 
(y: rU) r = <». 
Hence, excluding K-functions, since the solution must he finite and continuous at 
tlie origin, we have 
V = S (A;, sin + U; cos Jez) {h?'). 
