58 44 



^. - ik /tO-O-78^1) [67] 



where h Is the thickness of the diaphragm. For the same steel and for water 

 this becomes 



T, = 0.061 a /V + . 1 00 -^ milliseconds [68] 



provided a and h are expressed In the same unit. If there Is liquid on both 

 sides of the diaphragm and of the baffle, having a density p^ on one side and 

 P2 on the other, p^ Is to be replaced In Equation [67] by p, + p^. 



If the diaphragm Is mounted In one side of a gas-filled box only 

 slightly larger In diameter, the coefficient O.78 In Equation [67] Is changed 

 to something like OA, and 0.100 In Equation [68] to roughly O.O5. 



The effect of the elastic range Is discussed In the Appendix. 



DEFLECTION FORMULAS FOR A DIAPHRAGM 



From a survey of the preceding analytical results It appears that 

 only limited progress has been made as yet toward an exact treatment of the 

 hydrodynamlcal side of the problem that Is presented by the Impact of a shock 

 wave upon a diaphragm. The situation Is somewhat better as regards the be- 

 havior of the diaphragm Itself, although even here complexities and uncer- 

 tainties are encountered because of work hardening, Increase of stress at 

 high strain rate and thinning of the diaphragm. It is not the purpose of 

 this report to attempt an accurate theory of the plastic deformation of a 

 diaphragm. Simplified assumptions as to its behavior will be adopted in 

 order to obtain a few approximate formulas possessing a limited usefulness. 



Let the yield stress a be constant. For steel this is more nearly 

 true at high strain rates than at low rates. Let both the elastic range and 

 the thinning be neglected. Actually, the thinning may extend to l/3 or even 

 2/5, but its effect is at least in the opposite direction to that of work 

 hardening. With these assumptions the fundamental equation for plastic de- 

 flection can be written in the simple form, 



E = ahAA [69] 



where E is the net energy delivered to the diaphragm, h is its thickness and 

 AA is its Increase in area due to plastic flow. 



For a circular diaphragm deflected into a spherical form, AA = nz^ 

 in terms of the central deflection z; this formula is almost correct also for 

 the parabololdal form. For a circular cone,* AA = nz'^/2. Profiles for these 



Shapes between spherical and conical are often produced by underwater explosion; they are nearly 

 hyperboloidal, as illustrated in Figure 2zV. Certain observations indicate that in the course of the 

 damaging process nearly conical shapes may occur momentarily. 



