189 



Considerations of Work Hardening 



In line vvitb the result g of the last section, it seems apropiys, 

 when introducing the complication of vjorl: hardening, to neglect the radial 

 .T.otion, insofar as its inertial effects are concerned. This v/as done in the 

 development of the elei.ientar/ approximation in the previous section, as it is 

 in th.e following. 



Now the stress function of a naterial which ijork hardens .-ray be written 

 cr (log ^)= (X * U)log^ (35) 



where as before (f is the .^rield stress and CO is the 'jork hardening stress per 

 unit natural strain in a tensile test, and has a value of approyj.:nately 

 LOO X ICr lb/in for some medium steels. 3uch a stress-strain relation is 

 not at all inconsistent ;vith many empirical data. 



After introducing (35) into the f^eneral equations (B) , with the 

 constraint forces deleted, the differential equations of "otion can be written 

 in terms of tne^non-diinensional notation with the addition of 

 f = ^ 



If, furthermore, we let 



5=y£-i.£iogi 



the equations take the simple form 



^^ ♦ 2^5 -1 = (a) 



A =-l_ _£ldf (b) (36) 



At . ± (c) 



where for the time being i , not ^ , may be considered to be the independent 

 variable. Although these equations can be solved explicitly with but one 

 indicated quadrature remaining (from (36) (c)), the solutions are so complicated 

 that it is more instructive to consider a special case in which the approximation 



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