191 



strain-rate effects, which have not yet been incorporated into plasticity- 

 theory, an even more marked rounding off would occur at the center than is 

 evident from Figure 5. 



One should note that, as a result of neglecting the constraint 



forces in the approximate solution for a work-hardening material the initial 



2 

 value of U is completely deter.iiined, and equal to — , as in the elementary 



theory. A more general solution including the constraint terms would of 



course allow this initial value to be given arbitrarily. Although such a 



solution has not been investigated, it is likely that U would quickly approach 



the value given by (38) (a), after performing its initial task of increasing 



or decreasing the slope of the diaphragja profile near the edge. 



Conclusions 



In partial conclusion, before sumiiing up the accomplishments of 

 this theory, it might be well to point out some of its shortcoinings. First, 

 the constraint forces imposed on the center flat portion are seen actually 

 to yield an infinite constraint tension or stress just at the center of the 

 diaphragm. However, other idealized theories have frequently led to sLiiilar 

 paradoxes; e,g., the pressure (which is actually a constraint stress) imposed 

 by the assumption of incompressibility is infinite at a sharp corner in the 

 theory of potential flow of a liquid. Second, the imposition of rigidity on 

 the material behind the bending wave is an obvious artificiality; finally, 

 the assumption that the bending wave is an actual discontinuity in slope is 



-28- 



