THE PLASTIC STATE. 



In compression, let 



A = original area of the cross section ; 

 AI= final area of the cross section ; 

 L = original length of the specimen ; 

 I = the shortening or strain. 



Then 



Origin 



A X L = 



Original volume = final volume. 



,-- 



'L-i A 



1 - Al ~ A 



(XXIX.) 



L-i A 



or 



The percentage of ^ ( The percentage of 

 shortening reckoned on >- = -< increase of area, calculated 

 final length. ) \_ on the original area. 



1 6. Torsion. When a shaft is subjected to a torsional or 

 twisting moment, so long as the shaft remains perfectly 

 elastic, we have seen (VIII.) that the twisting moment 



where r is the external radius of the shaft. This is for 

 solid shafts. Where they are hollow, we have 



T W-r* 



: T ; ~TT~ 



As in the case of a prismatic bar under a tensile load, 

 when the twisting moment exceeds a certain amount, the 

 limit of elasticity is passed and, at first, the outer layers 

 are strained beyond their elastic state, and this passing 

 from the elastic to the semi-plastic state gradually proceeds 

 from the outer surface inwards, until the whole cross 

 section of the bar has passed beyond the elastic stage. 

 When the elastic limit has been passed, the above equation 

 is no longer applicable, and the intensity of the stress, 

 which, under the elastic conditions, was proportional to 

 the distance from the axis of the shaft, becomes more 

 nearly distributed over the whole area. Experiments show 

 that after the elastic limit has been passed, the shaft 

 begins to assume a condition of partial plasticity, which, 

 as the twisting proceeds, approaches the state of perfect 

 plasticity, but never actually reaches it. 



The above are the relations between stress and strain 

 under the elastic conditions. 



