180 



to be solved subject to q(0) - 0, q(0) = 0. VJe find that 



t 



4(t) = ao^.si.f][^ 



y 



q(t) - a Cr ii-V) 



E 



1 - cos i.\l E t 



^i/^iiriTy 



Vvhen q is a maxiimn, the stresses in the diaphragm reach the yield point, 

 and from that time on, plastic flow takes over. This happens at a time 



the velocity distribution may be written 



} t -T a V Pv2 E 



(23) 



The time for complete plastic deformation of a diaohragm may be estimated 

 roughly, considering the bending wave to travel along a radius vrith a 



speed ^m-\\ <^ , 



It roaches the center at a time 



the ratio of tg to to is thus 



^ q E / 



for some steels,* Similarly, elastic considerations give the radial velocity 

 at the periphery, 





Elastic 



■'f^if^ 



.675 V 



* Typical values of parameters for obtaining these quick estimates are: 

 cr' = 6 X 10^ Ib/in^ p = .75 x lO""' Ib.sec^in^ h = .05 in 

 E = 3 X lO''' lb/in v = 10^ in/sec ■ = 5 in 



i^= .3 



-17- 



