1892.] Mr Chree, On Long Rotating Circular Cylinders. 289 



The longitudinal strain, -r- , is everywhere negative, i.e. a 



compression. It has the same constant value as wjz, and so 

 is given in Table I. The transverse strain, u/r, is everywhere 

 positive, i.e. an extension, and is never algebraically less than 



ft If 



the radial sti'ain -r- . Its greatest value, s. which is found at 



dr ° 



the axis of a solid cylinder or the inner surface of a hollow 

 cylinder, is thus the greatest strain. For a hollow cylinder it 

 is the quantity 8a' /a' of equation (11) and Table III. In a solid 

 cylinder it is given by 



'-<**-3*&*j ^ 



whence answering to 77 = 0, 77 — "25 and 77 = 5 we obtain 



s + (co 2 pa 2 /E) = -37o, -2916 and '125 respectively. 



The radial strain is most conveniently dealt with by means 

 of the formula 



8E(l-fj) du 



<o 2 p T dr JKX) { h 



where for a solid cylinder 



r~ z f (r) = a 2 (3 - 5 V ) - Sr (1 - 2 V ) (1 + 77) (14a), 



and for a hollow cylinder 



f(r) = - a 2 a" (1 + 77) (3 - 277) + (a 2 + a' 2 ) r 2 (3 - 577) 



- 3r 4 (1 - 277) (1 + 77) (146). 



For the sign of -=- we need only consider that of /(r). 



§ 10. In a solid cylinder f(r) is positive inside and negative 

 outside the surface 



rW (3 -577) -{3 (1-277X1 + 77)} (15); 



but the radius of this surface exceeds a when 77 > 3. 



Thus in a solid cylinder when 77 > - 3 the radial strain is every- 

 where an extension ; when, however, 77 < '3 there is a cylindrical 

 surface, viz. (15), outside of which it is a compression. When 

 77 = or *3 the radial strain vanishes over the surface of the 

 cylinder, and elsewhere is an extension ; but for intermediate 

 values of 77 the region wherein this strain is a compression has 

 a small but finite thickness. For a given value of a this thick- 

 ness has a maximum value of "03775a approximately when 77 = "2. 



