Theory of Elasticity. 



281 



zero at all distances. The strains'* are a circular (tangential) 

 strain varying from contraction at the inner surface to a some- 

 what smaller extension at the outer surface, while the strains 

 in the other two directions are simply the corresponding lateral 

 strains at the inner and outer surfaces and intermediate values 

 between. The primary stresses satisfy, as we should expect, 

 the proper equations of equilibrium. The state of primary 

 stress and strain will be very clear if we conceive it as similar 

 to that which exists when an outer tube of a gun is shrunk 

 onto an inner tube slightly too large to enter it when cold. 

 But of course in the case just discussed there is no discontinuity 

 such as would exist in the practical case mentioned above. As 

 to the practical possibility of producing such a state of strain 

 as has just been considered, nothing need here be said, for 

 we are merely interested in considering the consequences 

 of such a state of stress and strain could it be attained. It is 

 of great interest to note however that a somewhat similar state 

 of primary stress and strain with its consequent advantages in 

 increased elastic resistance would be obtained by systematically 

 and uniformly overstraining a tube by application of interior 



The primary strains are 



(e) = 



(P) 



(Q) 



(Q)+(R) py V! 



[(r — r ) (r — r^) — a (r 2 — r ri)] (radial 



<'>= i 



E E r 2 (r 2 - r{>) 



strain — extension inside changing to contraction outside) 



E ~ E r\r^-r^) 

 tial strain — contraction inside changing to extension outside) 



[.(r 2 — ron)— a(r— r ) (r— n)] (tangen- 



(9) = 



(R) 



(P) + (Q) Pi 



ToTx 



t^-'] 



E E r\r£-ri>) 



sion inside changing to contraction outside) 

 At the inner boundary r == ri they take the values 



(axial strain — exten- 



(Pi) = o 



(QiY- 



(JRi) - 



7*0+ fi 



(«l) = 



CA) = 



Pi 



r 



r 



(9i) = +° 



E ro+ri 

 Pi r 



E r + rj 

 At the outer boundary r = r they take the values 



(radial strain — actually extension) 

 (tangential strain - actually contraction) 

 (axial strain— actually extension) 



(Po) =0 



(«o) = 



Pi 



(Ro) = (g ) = 



We note that 4^ + 

 or 



equilibrium. 



E r +ri 

 Px_ r x 

 E r +r 



Pi l T\ 



(radial strain — actually contraction) 

 (taugential strain— actually extension) 

 (axial strain— actually contraction) 



E r + r 

 (P) ~ ( ^ = and that [p 1 ^ Z q satisfying the equations of 



