363 



4. The Initial Phase of the Plastic Deformation of a Thin Plate 



In Sections 2 and 3 of this Part, the equation of motion of a 

 thin plate, loaded by an exponential e^qjlosion wave, was solved by expansion 

 of the deflection in a series of Bessel functions. Since the series does 

 not converge very rapidly in the early phase of the motion, aiother type of 

 solution is presented here, in which the propagation of the transverse mem- 

 brane wave from the edge to the center of the diaphragml is explicitly treated. 

 At the same time, the solution given here is less general than our previous 

 one, since it is only possible to carry it through in the approximation of 

 \aiiform water loading at all points on the diaphragm. 



As surmised by Bohnenblust and von Karman, by analogy with the 

 case of the string, the center portion of the plate remadns flat out to the 

 point reached by the incoming transverse plastic wave at the given time. 

 For impulsive loading, there is a kink in the profile at this point. For 

 loading with a finite duration, the second derivative, but not the first, is 

 discontinuous. If the nomuiiform distribution of water loading were properly 

 taken into account, the center portion would not, of course, be exactly flat. 



The equation of motion of the plate loaded with a shock wave of 

 free field pressure p^ (t) may, in the incon^jressive approximation, be written 



where x = t/Rq 



and is an integral operator defined by 



(22) 



50 



