^Tii /'. W. B/'idgman — Crystalline Cylinders. 



We first sul vi' supposing that. on the infinitely distant ends nor- 

 mal to the axis there is such a stress as to make the displace- 

 ment along - the s axis independent of s, and later add such a 

 normal stress on the ends as to satisfy the actual conditions there. 

 This modification of the first solution is always exceedingly 

 easy to make. 



Cylindrical coordinates are, of course, the natural choice for 

 this problem. The equations are copied here for convenience of 

 reference ; they may be found in any work on elasticity. The 

 displacements radially, circiimferentially, and axially are denoted 

 respectively by u„ vg, v z . The strains are then given by : 



9tl r 1 9l(g U,. 



9u 

 6zz ~~Bi> 



l 9u z Bt(g, 9",. Bu z 

 r do az Bz dr 



__ Bug Vg 1 BU, 



Br r r 39 



and the stress equations' of equilibrium are ; 



Brr 1 9r~0 3r7 rr — 66 _ 

 ~Br~ + VJd9z ~ 



3rd 1 366 36l f6 _ 



Jr + V~36 + Jz + 2 7 ~° 



Br + r 36 + 9z + r 



Considerable simplification might be made at once in these equa- 

 tions, which are general, because the solutions in which we are 

 interested are independent of z. 



These equations will now be applied to the simpler groups of 

 crystals. 



Cubic Crystals.— The first task is to obtain the stress-strain 

 relations in cylindrical coordinates. The equations in rectangu- 

 lar coordinates are given in Love ; the transformations, which 

 may be made by familiar methods, give the following results: 



rr — (a — c cos 46)e rr + {b + c cos 40) egg + c lt e zz + e sin 46e r g 



06 = (h + c cos 40)e rr + (a — c cos 40)e 99 -f c„e zz + c sin 40e r9 



r~6 = c sin 46(e rr — e gg ) -f- — - + c cos 46je r9 

 zz — c, „(<?,.,. + e 9 g) + c„e zz 



zr — c AA e z 



