177 



Equation (17) nay now be integrated to yield 



as the quantity defining the distribution of constraint forces in the flat 

 central region. 



It is interesting to note (and a chec!: on the work) that if one now 

 equates the total rate of decrease of ld.netic energy for the whole diaphragm 



to the rate at which energy is being absorbed in plastic working of the material, 



. 2 



VIZ. -<J(^H«R , the result xng relation, after a bit of manipulation, is identical 



with (7), as it should be. 



Part 3. The Application of Plasticity Theory 



To complete the discussion in the previous section it is 



necessary nov< to introduce certain stress and strain relationships in the 



flat central region. Because the rates of change of the principal strains 



are pronortional to the principal strains themselves, as a result of the 



constraint relation (16), it is not important to formulate the plastic flow 



relationships in a completely general fashion.* It suffices to vn'ite these 



laws in terms of the principal stresses (T , (Jl, O' and principal natural 



♦ '- 3 



strains £^, C , £ in the form 



^ ^ e, + €^ i- Ej = o 



^1-^3 . ^2-cr3 _ t(x) ^ 

 €i - c^ €2 - £3 r * 



The first states the law of conservation of volume, while the second, as a 

 consequence of the first, states that the principle strains (more generally 

 the strain rates) are proportional tc the deviations of the stresses from 



* For such formulations, s.;e, for example: G.H. Handeljnan, C.C.Lin, 'ii. Prager, 

 "On the mechanical behaviour of metals in the strain-hardening range", 

 Quart. Anpl. ilath l^ 397-407 (192*7). In the present report, the plastic 



flow laws reduce to a forr. identical i/ith that of the defor.nation theory of 



plasticity, 



-lU- 



