176 



4n elementBry segment, of width dr and thickness H, which is a distance r 

 from the center of the diaphragm where it subtends an angle dS, these 

 stresses exert a net force.- 



TT^^'^r^ 



-<r.] 



(r <^) -CA Hdedr . 



But from the constraint expressed by (l6), it is clear that the radial 



and tangential strains, and hence the radial and tangential strain velocities 



are equal and functions of the time only. It follows from plasticity theory 



that the radial and tangential plastic stress components are equal and are 



functions of the tine only, (vide Part 3). Hence the plastic stresses 



exert no net accelerating force on the particles in the flat central region 



interior to the bending wave. The only possible accelerating forces in this 



region are the constraint forces; that is, we cay define the out*fard accelerating 



force on the element of mass pHr dr dQ by a differential quantity d TdG « 



'•' " * dr dO» except for elements just ahead of the bending wave. Combining 



(16) vfith the equation of motion of this element, we find that 



pH^r^ d2 / 1 \ - ^ r (17) 



At the bending wave itself we must have 



RH Sj^ = liH cr. ♦ r^ (18) 



where ) □ is the value of I when r •= R, and is to be so chosen that the 

 constraint forces do no work. This means that the rate of v/orking by the 

 whole set of constraint forces is zero, viz., 



-r-^ dPdO - ajtfj^ U = J 

 if this integration is carried out, using (li>), (16), and (17), we see that 



Ht = P H^ r! d^ / 1 \ (19) 



^ dt2 \^ 



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