356 Mr. L. N. G-. Filon. On the Mastic Equilibrium of 



equal to the diameter — the exact ratio, 7r/3, being chosen so as to 

 simplify the arithmetic as far as possible. 



As in the preceding example, tables of the stresses are given for a 

 large number of points in the cylinder. From these the principal 

 stresses and the principal stretch were calculated ; and again from 

 these, by interpolation, curves were drawn showing the loci of points 

 in the cylinder where the greatest stress, the greatest stretch, or the 

 greatest stress-difference had the same value. 



The curves show that, whatever theory of yielding is adopted, 

 namely, the greatest-stress theory of Navier and Lame, or the greatest 

 strain-theory of St. Venant, or the greatest stress difference (or. greatest 

 shear) theory which has more recently been put forward, failure of elas- 

 ticity will begin to take place round the perimeter of the plane ends. 



Thus, in the case of the stress, consider the regions where the stress is 

 greater than a certain value S. When S is nearly equal to the greatest 

 stress these regions are thin annuli round the ends. As S diminishes 

 the regions become made up, partly of such annuli (of increasing 

 thickness), partly of a closed region round the centre of the cylinder. 

 When S reaches a certain critical value, S , these two regions join on 

 to one another. The regions where the stress is less than So consist of 

 caps at the two ends and of cylindrical shells, forming the " skin" of 

 the cylinder. 



The regions of least stress consist only of caps or buttons of material 

 at the two ends. 



The variations of the principal stretch and of the principal stress- 

 difference can be described in the same general teims. 



For materials like stone and cement, which have no very definite 

 yield point, the elastic distribution will give at least an indication of 

 the state of stress almost up to the point of rupture, and if it be 

 assumed that the latter takes place over the regions of greatest stress, 

 or greatest strain, or greatest shear, according to the particular theory 

 we adopt, the results above show that the fracture will start from the 

 perimeter of the ends, and that caps or buttons, which may have an 

 approximately conical shape, will probably be cut off at the ends. 



The fact that yielding first occurs at the perimeter, when the stress 

 exceeds 1/T686 of the limiting stress for uniform pressure, leads to 

 the conclusion that the strength of a cylinder under this system of 

 stress is considerably less than the strength of a cylinder uniformly 

 compressed. This result apparently contradicts the fact that the 

 strength of stone and cement, when tested between lead plates, which 

 allow of expansion, is very much less than when tested between mill- 

 board which does not allow of expansion, a fact which has led Pro- 

 fessor Perry to state that the true strength of such materials is about 

 half their published strength. (' Applied Mechanics,' p. 345.) 



The contradiction, however, seems to be explained by a remark of 



