178 



an isotropic stress state. The factor of proportionality, L , is 



Y 



the ratio of the octahedral shear stress 



to the octahedral shear strain 



and, when the Liaterial is yielding plastically, the ratio is a function 

 of y, having a form determined empirically from a tensile test for example. 

 According to our previous supposition, in a tensile test, the tensile stress 

 Oip, is related to the natural longitudinal strain, c^^ by 



In a tensile test f " V 2 (f , while y = V2 c , Hence, we may 



3 "^ 2 ^ 



rewrite the above relation as 



Ten - j/ilcr(V2^ r) 

 3 



which is now applicable to a more general stress state than that occuring 

 in a tensile test. 



Let us nov/ apply these laws to the central flat region of the 

 diaphragm. In this case 



£^ - £p = log ]_r_, ^2 = ^e " ^°8_j:_, f = £* - log h . 



Pr^ Tq 3 » h, 



where CT' is the principal stress in the thickness direction, and £ , Cat 



and £,, are the natural (logarithmic) strains in the directions indicated by 



H 



the subscripts. As a consequence of (16), t, = Cp » - — Co» so we find 

 that 



cr = cr^ (21) 



r 9 



CT = (^(log _h_) (22) 



^ H 



ixquality (21) has already been used in the previous section in 

 obtaining (1?). 



-1^ 



and 



