337 



Eq« (16), we find that the first case is realized vriien <*ij'6 is small rela- 

 tive to unity, and the second case when u)*© is large relative to unity. 

 In other words, impulse or peak pressure is the decisive damage factorj ac- 

 cording to vrinether the plastic period of the plate is large or small relative 

 to the duration parameter of the incident wave, 



IV. TH^ PLASTIC DEFORl^lATION OF A THIN, CIRCULAR 

 PLATE WITH AN UNCONSTRAINED PROFILE 



1, Introduction 



In Part III an analysis of the plastic deformation of a thin cir- 

 cular plate by an underwater a)q>losion wave was presented for the case of the 

 profile constrained to a parabolic shape. Here, a more exact solution is ob- 

 tained without artificial constraints by expanding the deflection in a series 

 of Bessel functions. By this means^ proper account is taken of the frequency 

 shifts and interaction of the deflection modes arising from the water load- 

 ing. Because of the excessive difficulty in treating the retardation effects 

 in the water loading for this case only the incompressive approximation is 

 considered. 



Detailed calculations are made for the case in which the first two 

 terms of the Bessel function series are retained, corresponding to the funda- 

 mental mode and one overtone. The final deflections so obtained do not differ 

 greatly from those calculated with the parabolic profile. The times of de- 

 flection are about 20^ lower than in the parabolic treatment (vdth incompres- 

 sive water loading). The maximum central deflection is about 2"^% higher for 

 impulsive loading and not significantly different for loading by a wave of 

 duration long compared with the period of the fundamental mode. 



28 



