CIRCULAR CYLINDERS ITNDER CERTAIN PRACTICAL SYSTEMS OF LOAD. 21 I 
of elastic theory give in such a case an indication of the state of stress when the 
specimen Ijreaks. 
There are three distinct theories, both of rupture and of failure of elasticity. 
According- to Lame and Navtee, failure occurs when the greatest stress at any point 
exceeds a certain limiting value. This is also often taken as the criterion of absolute 
rupture. According to Saint-Venant, the maximum strain, and not the maximum 
stress, is that which determines failure. Finally, a theory has lately been put forward 
by various elasticians to the effect that failure occurs when the greatest sliear at ajiy 
point, that is, the greatest principal stress-difference, exceeds a definite amount. 
Professor Perry has proposed another criterion, suggested l)y the angle at which 
compressed cylinders shear (see ‘ Applied Mechanics,’ pp. 344-345), namely, that 
rupture takes place when s — ixp exceeds a certain value, where s is the shear across 
any element of area at a point, p is the normal pressure on tins element of area, and 
/i is a constant depending on the material. This theory, however, need not concern 
us so much, as it appears more specially applicable to tbe final breakdown of ductile 
materials long after they have become plastic. On tlie other hand, it has been 
shown hy Mr. J. J. Guest (‘ Phil. Mag.,’ July, I9U0) that the beginning of plasticity 
was very probably determined l)y the maximum stress-difierence. 
Let us now proceed to apply tliese three criteria, namely, those of the maximum 
stress, maximum strain, aud maximum stress-difference to the cylinder in the present 
example, and see what results they give us, on the hypothesis that for materials like 
stone and cement, plastic yielding and rupture are simultaneous. 
Consider fir.st the greatest stress theory. This would make failure of elasticity 
first begin to occur round the perimeter of the plane ends, and that as soon as 
] '68035 Q > a certain limiting value Sq. If the j)ressure l)e uniformly apj)lled, and 
the ends expand, we get failure of elasticity when 
Q > So. 
Hence the apparent strength of a cylinder tested in this way would be about '593 
of the streugtli of a cylinder tested under a distribution such as is usually assumed. 
Further, if we consider the regions where the stress is greater than a given value S, 
we find that they consist of separate spaces, which join on to each other as S 
diminishes, the critical value for which this occurs being given by 8 = I'Ol Q. The 
regions of greatest stress consist therefore of a central core, which spreads out into a 
sort of hollow cone near the ends. If then we suppose fracture to occur over regi(.)ns 
of greatest stress, we see why it is that the material breaks oft* in conical pieces at 
the ends. 
Consider now the greatest strain theory. Let Tj, To, Tb be the tiiree principal 
stresses, and .Sj, So, So the corresponding stretches. 
= glT -. 
A 
3X “I" 2^ 
2 E 2 
(T,+T, + T3) 
Then 
