﻿Isotropic Spheres antler Uniform Surface- Pressure, Ll 



Three-layer Compound Shell. 



§ 4. Wo now proceed to consider the effects arising from the 

 application of the pressures p and p' over the inner and outer 

 surfaces of the three-layer compound shell (e . a . c . a.^ . b . a . a). 

 In this a layer (c .o^.b) of material k l9 n x is intercalated 

 between the layers {e . a . c) and (b . a . a) both composed of 

 material Jc, n. 



The strains and stresses that would exist if the shell were 

 all of the material k, n are given by (4)-(8). Our principal 

 object is to find the additions made to the strains and stresses 

 in (e . a . c) and (b . a . a) in consequence of the existence of 

 the layer (c.^.b), which we shall generally denote the 

 " altered layer." To do this we have only to find the incre- 

 ments 8p b and Bp c to the pressures p b , p c at the surfaces r=b 

 and r = c of the altered layer. The changes in the strain and 

 stress in (b . a . a) arise solely from the action of the pressure 

 8p b over r = b, while the changes in the strain and stress in 

 (e . a . c) arise solely from the action of the pressure Sp c over 

 r = c. When 8p b and Sp c are known, the corresponding 

 strains and stresses may be written down at once from 

 (4)-(8). 



The values of 8p b and Bp c are easily found as follows: — The 

 shell (b . a . a) is in equilibrium under the pressures p b + 8p b and 

 p' over its inner and outer surfaces, while the shell (c . a x . b) is 

 in equilibrium under the pressures p c + 8p c and p b -f 8p b over 

 its inner and outer surfaces. We thus know by (5) the 

 values of the strain sf in the two layers in terms of these 

 pressures. But s'~u/r, and u is necessarily continuous at 

 the common surface r=b of the two media ; thus we have 

 s' continuous, whence 



sP?{ &»+**(£ + hi) ~*><m + s) } = 



a u vn 



^ { iP^P^tk * jk)-^ +Sp Hm + ££)} ■ (14) 



Similarly from the contin uit y of s ' at r=c we find 



These two equations determine j* + $P» and ^ c + fy c in terms 



6 3 



