﻿Isotropic Sp/wes under Uniform Surface-Pressure. 163 



The quantity S is termed the stress difference, and its 

 greatest value, viz. 



S = f(/~p)^, (10) 



occurring at the inner surface, is called the maximum stress 

 difference. 



Here it is assumed that 



U>2n (11) 



If this relation failed to hold, a bar of the material would 

 increase in radius when exposed to longitudinal traction, and 

 I am not aware that such a striking phenomenon has yet been 

 observed in an isotropic material. 



The signs, whether positive or negative, of the strains, the 

 dilatation, and the stresses for pressure over one only of the 

 two surfaces are given in Table I. 



Table I. 



when when 



/3M* r /3&U 





ee. 



+ + 



p alone ) 

 acting J 



p' alone ) 

 acting J 



We shall employ the notation 8s &c. to denote increments 

 in s &c. answering to increments of any size 8p or 8p f in p 

 and p' . The expressions for s &c. being linear in p and p' } 

 the above table shows at once the signs of 8s/8p &c. when 

 one only of the surface-pressures is altered. 



The two most important quantities are probably A and 



(— rr). The value of A is constant, the volume expanding 

 uniformly under pressure at the inner surface, and contracting 

 uniformly under pressure at the outer surface. The radial 

 stress is always a pressure. 

 If p' = 0, then _ 



p— ( — rr) _ r z —e % a % 



while if p = 0, 





p 







r B 



a 3 - 



-«•' 



/■ 



-(- 

 p' 



•rr) 



— 



a 3 ~ 



r 3 



e z 





r* 



a 3 



-e* 



Thus, in both cases, as we recede from the surface where 



M2 



