-5- 111 



Th8 only relevant a priori theory on this subject is that the ratios of the stress compononts 

 in a plastic body are related to the ratios of strain components by the same relationship that 

 applies to viscous fluids, namely /li = V, From (20) and (22) it will be seen that, in terms of 

 a and /3, this would ^ive 



{2a- l)/(2 -a) 



(23) 



This relationship is also shown in Figure 1. It has been used by von Mises and others in deriving 

 theoretical solutions of problems in plasticity. 



Strain and displacement of particles . 



The form of the expression (12) for the radial displacement of particles in a circular 

 diaphragm suggests the possibility that a solution may be found by assuming for the two components 

 of displacement f, 77 parallel to the axes x and y the forms 



f = Ax (1 

 V 



-2, 2 2,,2> 



< /a - y /b ) 



By (1 - x^/a^ - y^/0^) 



(2*) 



This assumption satisfied the coHOition that § - T) = at the edge of the plate. The components 

 of strain In the surface of the distorted plate are then 



11 * -Vl 



3 X * UBx 



3 77 I Vdz 



3 y 2 l^y 



3x2 ^2 



3y' 



2^ y 



(25) 



Since it has been assumed that O" /:r = a is independent of x and y, the unique experimental 



relationship between a and /3 ensures that /? = ^v'^v ^^^^^ "^ indiipendent of x and y. Substituting 



for £ and € from (25) in the equation € = fie the condition that B may be independent of x 



X y X y 2 ? 



and y is found by equating the constant terms and the coefficients of x and y . Hence 



B(-3/a^ + l/a^) + 2h„2//S , 



B(-l/b^ + S/b^) 



2h.'/b 



2,k'* 



(26) 



and therefore 



/3 = bVa 



K 2/ 2 

 2,2 



(27) 



Having determined fi from (27) and hence a from the curve Figure 1, cr and a can be connected with 

 P, the testing machine measured yield stress, by the formulae 



P a, CT = P 



y 



if Mohr's theory is used 



""x ' Po/v l-a + a^, CTy = PyV I -a + a^ according to 



von Mises' theory. 



(28) 

 (29) 



This completes the solution of the problem, all thf: necessary conditions being satisfied. 

 Substituting from (26) or (29) in (18) the following .jxprcssions are derived connecting the prossu'.-, 

 the rraximum displacement h and the yield stress:- 



P = 



