﻿Rotation of Slightly Elastic Bodies. 37 



And the radial extension is 

 p — r = <j \_a x r -\-b l ji — ? ,3 /8] 



o 



The maximum radial traction occurs where 



(« 2 -'- 2 )(>' 2 -i 2 )/>' 2 



is greatest, namely, at r 2 = ab. The greatest radial traction 

 therefore occurs on the cylinder whose radius is the geo- 

 metric mean of the radii of the bounding surfaces of the 

 cylinder. Its value here is 



pco%2\ + -V)(a - by/MX + 2fi). 



The transverse tension is given by 



To = q [ [\ + 2/z) Vl jr + 2\ dih/dr] 



= 7 /4[(2X + 3/x)(a 2 + 6 2 ) + (2\ + 3^)a 2 6 2 /r 2 -(\ + 2^> 2 ] 



= (2X+3^)|[a 2 + 6 2 -a 2 & 2 /^-(\ + 2^)r 2 /(2\+3/ A )]. 



The maximum value of this tension occurs on the cylinder 

 with radius r 2 given by 



r 2 = abi/{(2\ + 3fi)/(\ + 2fi)}, 



and there its value is 



7l2X ^ 3 ^[a 2 H-6 2 -2a/W(^ + L>)/(2X + 3^)]. 

 Finally, the tension parallel to the length of the cylinder is 



_ fa 



•2 \ + 3/x 

 X-\- a, 



<r 



+ 1 



2 )-2, 2 ]. 



The longitudinal tension T 3 is therefore greatest on the 

 inner boundary. Here its value is 



q\ r(2\ + 3yu,Vr+ (fi-2\)h 



i\ n2\ + 3fju)a*+(fjL~2\ )(Sl 

 4 L X + /x J' 



In the well-known case of a solid cylinder, the longitudinal 

 tension has its maximum on the axis, and its value there is 



(2X-h3yLt)^ 2 X/4:(X+u). 



