179 



Part L. The Specification of the Initial State 



As the initial elastic stress wave front sweeps inward from the 

 edge it may accelerate the material particles, leaving any real diaphragm 

 in a state of plastic flow with the material flowing radially outwards. 

 However, for the ideal material with which we have been dealing, with a 

 zero elastic strain range, the elastic stress wave would travel at an 

 infinite speed from the periphery to the centar. Consequently, although 

 the discontinuity in stress across the front of such a wave is quite high, 

 jumping from zero to the yield stress <J'= Cr(0), the actual inpulse 

 delivered to any material particle swept over by the front would be zero. 

 This seems to be possibly a rather extreme case, however, and since the 

 theory does permit the assumption of an initial radial velocity distribu- 

 tion of a certain kind, a rough estimate of the possible magnitude of this 

 effect will be made as follows. 



In accordance with the constraints placed on the diaphragm 

 material, \k shall assume that the motion, generated in the elastic deforma- 

 tion, can be approximated by one of a nornal type with a linear radial 

 velocity distribution ifrtiich is zero at the center and greatest at the 

 periphery. Then the radial coordinate r of a particle, initially at r , is 



r = r (1 ♦ q(t}) 

 where q is the normal coordinate of tlie constrained raction, and "a" is the 

 radius of the diaphragm. The elastic stresses for such a motion are 



Or = CT^ = _E g_ 



^ 1 - V^ a 



where E is Young's modulus and V is Poisson's ratio. The motion begins at a 



tlTie t = 0, when the stress at the edge rises suddenly to the yield stress. C. 



Hence the equation of notion is, after calculating the equivalent mass from 



the kinetic energy, 



pa q ♦ _E_ q - CJ 



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