175 



Part 2. Equations of Lotion in the Flat Central Region 



The component velocity of any particle in the flat central region, 

 perpendicular to the original diaphragm plane, is supposed to have the 

 constant value v until the bending wave sweeps inward over it. This normal 

 motion is quite independent of the radial motion, which in general night 

 conceivably be much more coKplicated. The radial motion is of importance, 

 for upon it depends, in part, the distribution of thinning in the diaphragm. 

 Let us now consider this radial motion. 



As mentioned previously, in order to avoid the mathe.-iiatical 



complication of non uniform thinning in the flat central region and the 



necessity of attempting to solve non-lineur partial differential equations 



of motion, we introduce a system of constraints which serve to maintain a 



unifor.'ii thickness H throughout this region. Consequently, at time t + dt, 



a disc \/hich was of radius R - dL, and uniform thiclcness H at time t, has 



stretched into one of radius R - dL ■♦• Udt and uniform thiclaiess H + Hdt. 



Since its volume is conserved we are led to the equation 



i ♦ 2 H - (15) 



H R 



A similar consideration of an interior disc of momentary radius 



r <^ at time t, and initial radius r , shows that 



r. ^^E: 



i-o Vh 



r » g r 



To ^ To VH (16) 



so that r_ is a function of time only. liquations (16) iiri.ll be termed 



To 

 the "constraint equations" for the flat central region, 



Kovj the forces exerted on each material particle in the central 



region are due to the constraints and to the plastic stresses. We are 



supposing that the only normal plastic stresses in the cmitral fegion are 



radial and circumferential stress components O'j. and CTq resoectively. On 



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