140 



of a rectangular plastic plate under uniform pressure, p, acting on one side, is studied. Though 

 the distribution of the plastic strain in the plane of the plate is not uniform the plate behaves, 

 so far as displacement nortral to the surface is concerned, like a soap film or membrane with uniform 

 tension. This uniform tension is related to the yield stress measured in a tensile test by a 

 factor which differs according to whether v. Mises' or Mohr's theory of plasticity is used and 

 varies slightly with the ratio of length to breadth of the plate, but with a plate for which length/ 

 breadth = 3/2 v. Mises" theory gives a result which is 1.8 per cent too snail and Mohr's t.S per 

 cent tco big as compared with a rough theory which simply assumes that the plate behaves like a 

 mef«brane stretched to uniform tension Pd in all directions, where P is the yield point of the steel 

 and d the thickness of the plate. 



Using this theory for a plate Ahose dimensions are 2b x U/3b (b is 3 feet in the particular 

 plate to which we are now applying the analysis) I find the following formulae:- 



Maximum displacement h = 0.179 -pg 

 Mean displacement y = 0.0870 £g 



(tl) 



so that y/h = 0.1185 



The energy absorbed by the plate during its displacement is 

 E = [ 5.76 P(y)^ d/b^] (area of plate) 



so that work done on unit volume of the material of the plate is 



W = 5.76 y^ p/b^ 

 Trme taken for di sjilacement to reach maximum^ 



M 



(U4) 



The vibrations of a membrane stretched with tension Pd on a rectangular frame of dimensions 

 2b, and 2b- have been studied by Rayleigh who showed that the period of the fundamental Is 



where ps is the density of the membrant;. 4 plastic sheet is like a membrane during the time when 

 it reaches its maximum displacement it only recovers by the amount of elastic recovery which is in 

 the case here considered small compared with the plastic displacement. Thus a possible movement 

 of a plastic sheet is identical with the first ^period of the analogous elastic membrane. Thus 

 if the plastic sheet is given initially the norrrial velocity distribution corresponding witn the 

 fundamental period it will come to rest in time 



^R" 



(*5) 



In the case of a plate 6 feet x n feet and density 7.8 this time of displacement r Is 

 T = 2.07 X 10"^ seconds if P is 30 tons/square inch 



T = 2.54 X 10 -^ seconds if P is 20 tons/square inch 



(46) 



This time T Is marked on Figure 3 so that the relationship between it and the durations of 

 the pressures may be understood. It will be seen that T is very long compared with the duration of 

 the pressure wave but that the duration of the long continued pressure associated with the exfanslon 

 and vertical motion of the bubble is longer than r. 



This is convenient for it necessarily implies that it is justifiable to use the method of 

 the report "The pressure and impulse of submarine explosion waves on plates' in discussing the 



displacement 



