167 



tilts each annular element as it sweeps over it, removing its kinetic 

 energy. This is then transmitted to the central region where it is 

 ultimately converted to plastic work of stretching and thinning. In this 

 region, the diaphragm is still nearly flat, and therefore, in the absence 

 of any nor.-nal force components it retains its uniform normal velocity; 

 its initial radial velocity distribution, however, may be altered, both by 

 dissipation of energy in plastic work, and by the probably non-uniform 

 distribution of tension resulting from non-uniform radial flow, thinning, 

 and work- hardening, 



Because no notion is observed in the diaphragm material behind the 

 bending wave, it is surmised that this material has been unloaded and 

 returned to the elastic state. Indeed the sonev/hat conical shape assumed 

 by the diaphragm suggests that the stresses in a particle momentarily behind 

 the bending wave, although possibly near "the yield point" of the material, 

 quickly subside. 



As the bending wave sweeps inward, it speeds up. This is possibly 

 due to a rise in the stress ahead of the wave because of work-hardening 

 effects in the plastic material in the flat central region. This speeding 

 up accounts at least partially for the rounding off of the diaphragm 

 profile at the apexj in addition it is probable that the bending wave has 

 a finite radius of curvature, that is, it iva^y have a finite "length" in the 

 radial direction, vrfiich would also help account for this rounding off, 



Basic Suppositions 



The exact non-linear partial differential equations of motion 

 describing the dynamic plastic deformation of the metal in a diaphragm such 

 as we have been considering are extremely complex. 3ven if it were possible 

 to solve these with existing mathematical techniques the solution would 



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