381 



in favor of the strain theory advocated in this report over the alterna- 

 tive one based upon normal motion. A theory based upon normal motion 

 would give unreasonable values of Jj^ abotit one-tenth as large as those 

 obtained here. 



The treatment presented in Sees. 2 and 3 may seem overly elab- 

 orate in relation to the crude use maae of it in Sec. A, However, it is 

 hoped that the results presented in Sec. 3 may be used more fully and 

 satisfactorily at some future time when a more successful attack is made 

 upon the hydro dynamical problem. 

 2 . Solution f or impulsively loaded infinite plastic manbrane 



The equation giving the first approximation to the vertical dis- 

 placement of an infinite thin diaphragm under no external forces may be 

 written 



z =«:z^i) + (oc-), 



where z is the vertical displacement, o< the expansion parameter, r the 

 radial coordinate, t the time, and c/ = -]/a^ /jj the velocity of trans- 

 verse plastic waves in the plate. The initial velocity distribution is 

 assumed to be of the form 



if Cr,0) - «ce(|j . (2) 



where R is some representative linear dimension of the system. The 

 initial displacement is, of course, taken to be zero. 



For convenience, the foregoing equations are put into nondimen- 

 sional form. By introducing the variables x = r/R, y = z /R, and V = 



64 



