329 



III. APPROXIMATE THEORY OP CIRCULARLY CUI4PED PLATES 

 CONSTRAINED TO A R4RAB0LIC PROFILE 



1, Introduction 



Since experiments on the plastic deflection of clamped plates 

 (Modugno gauges, etc.) by underwater explosion waves show the final pix)- 

 file to be very nearly parabolic in most cases, it seems likely that a 

 simple theory using a parabolic constraint should give satisfactory re- 

 sults. This supposition has been well-confirmed experimen tally as will 

 be shown in Part V. 



Since in this treatment- it is possible to confine oiir attention, 

 to the center of the plate ('"•*), a more rigorous treatment of the water 

 loading including retardation effects is tractable. In Fart IV, where the 

 plate deformation is treated without constraints on the profile, a compara- 

 bly rigorous treatment of the water loading would be excessively difficult - 

 and consequently only the incompressive approximation is considered there. 



Since a constrained mechanical system behaves as though it were 

 stiffer and more massive, one -.vill expect the treatment of this Fart to 

 give smaller deflections and longer deflection times than the other^se 

 equivalent unconstrained treatment, 



2, Equations of Motion 



If we substitute the expression for a parabolic profile 



xCr,-t)= x.Ct)\:i- (r/RjM , (1) 



where z^ C^] is the central deflection and R^ the radius of the plate, into 

 £q, (27) of Part I (witti *=i , and consequently trx^'^ and p« pfO , 



20 



