190 



^ »1 may be made. It is clear that for the typical values we have been 



utilizing this is the case (e.g, , — ^80). Thus (36) (a) immediately yields 



h 



r-i^n .1 =^ . (37) 



After some manipulation, the solutions, to this order of approximation, can 



o R 2 CT 



be rewritten with p, or — , as the independent variable. Thus, with c = "7" > 



2 

 as before, and d = — , we have 

 P 



U = ^[l-*p (a) 



h e "^'L'' r J (b) (38) 



Z = vt = --^ [1 - (()] dR (c) 



2^2 ^ 



where (T s 2 — ; — log — . It is interesting to note that as d-^-O, that is, 

 qU a 



the work-hardening stress becomes smaJJL, the above solution reverts to the 



elementary approximation (29), since 9 vanishes like d in the limit. One 



major difference stands out between the above and the previous solutions - at 



the center of the deformed diaphragm, (— ) vanishes. That is, the apex of 



dR 



the conical shape is rounded off, in complete qualitative agreement vjith the 

 observations, and with the surmise made in an earlier section. It is also 

 of interest to observe that the center deflection of the deformed diaphragm 

 is somewhat less than that of a diaphragm whose material does not work-harden, 

 but which has the same yield limit and initial conditions. These observations 

 are illustrated by the diagrams of Figure 5, in which are compared the deformed 

 diaphragm profiles for two materials, one of which does not work-harden. The 

 profiles of the diaphragms whose material work-hardens were calculated by 

 integrating numerically (38) (c) (which, incidentally, may be put in the form 

 of an incomplete I -function). It is probable, that by taking account of 



27- 



