40 THE PLASTIC STATE. 



We have not at present sufficient knowledge of these 

 plastic properties to enable us to deduce any very definite 

 laws respecting them, but a few of the relations existing 

 between stress and strain under plastic and semi-plastic 

 conditions may be quoted here. 



With the same notation as before (Fig. 4), let a bar be 

 stretched to a length (L + /) from an original length L ; and 

 let its original area of cross section be A, and the final 

 cross sectional area a. Let us consider two stages : one at 

 the beginning of the plastic state, and the other at or near 

 the point of fracture. Assuming the volume to remain 

 constant during the process and the cross section to remain 

 the same along the whole length of the bar, we have 

 (Initial volume) = L x A, and 

 (Final volume) = (L + l) a. 

 But the volume being practically constant, 

 L x A = (L + I) a 



L a 



or, t r / = IT 



L + I A 



or, 



This may be written = ^ (XXVIII.) 



or, Extension Reduction of area 



Stretched length ~ Original area 

 The percentage of reduction of area, is usually calcu- 

 lated for a bar after test, and is 



Reduction of area 



rr^- T 

 Original area 



and the percentage of elongation is usually reckoned as 

 Extension 



Original length 

 which is not the same as the ratio given above, where the 

 extension is given on the stretched and not the original 

 length. 



So that from this we gather that so long as the bar 

 remains parallel, and the volume is constant, the per- 

 centage of reduction of area, calculated on the original 

 area, is always less than the percentage of elongation 

 reckoned on the original length. 



