188 



xi=j ' ^2 =2S»1, 



2 



where S , it will be remembered is 4 ^ . Then, provided S is large enough 



Z = vt =J^kL - H) 



a- s vi-U^X a/ (a) 



(b) (34) 





V 



It can be shown that this approximation is uniform over the range of U 



from K to 7c ^"^ ^^^ ^^"g^ °^ § ^^°'^ 1 to j but the nearer X is to 1, 

 the larger d must be . Now we may distinguish several possibilities. If 

 X = 0, which is the case of a material which tes strictly no elastic strain 

 range, we find that the shape of the deformed profile, described by (34) (a) , 

 is conical near the center, with a center deflection identical with that 



given by the elementary theory. At the edge, the slope is zero (as required by (30) (b). 



the 



The thickness distribution is identical with that given by elementary approximation, 



2 

 while from (34) (c), the radial velocity increases very rapidly from zero to Z_ 



As K. increases the solution approaches the elementary approximation, the slope 

 at the edge increasing, until when H.= 9b , the shape is exactly conical as in (29). 

 As K increases further, the slope at the edge becomes greater, the center deflection 

 remaining the same, however, as does the thickness distribution. The radial velocity 

 U, at the bending wave, decreases, again very rapidly, from )^v to ^ . These 

 results all seem physically quite reasonable and might have been expected on such 

 grounds. ^ calculation indicates that there is actually very little other effect 

 of different K values on the profile, as long as S is very large. 



Apparently, then, we may conclude that the elementary approximation 

 is even more reliable than could have been hoped for, as long as there is no 

 work hardening $ indeed, under the conditions noted, the entire discussion con- 

 cerning equations (29) is applicable. In the next section we consider some effects 



of work hardening. 



- 25 - 



