REAL STRESS DIAGRAMS. 159 



simple change of scale. But this is not so. If the speci- 

 men is at all ductile, longitudinal strains beyond the elastic 

 limit are accompanied by corresponding alterations in the 

 lateral dimensions, and as the length increases, the area of 

 the cross-section diminishes, and the contrary takes place 

 when the length is diminished, as in a compression speci- 

 men. Within the elastic limit the stress on the bar is 

 sensibly proportional to the load, but when this point is 

 passed, the stress varies at a rate different from that of the 

 load. In a tension test the area of the cross section 

 diminishes as the load increases, and as the stress at any 

 point is the load divided by the area, the stress per unit 

 area increases more rapidly than the load. Hence in this 

 case the load is not itself a measure of the stress. 



A true stress-strain curve could be obtained by measuring 

 the lateral dimensions of the bar at each increase of load, 

 calculating the areas at these points, and dividing the load 

 by the area in each case to obtain the stress, and plotting 

 the diagram from these results. Such a plan, however, 

 would be extremely laborious. The same end can be 

 gained by first plotting the load-strain diagram, and by 

 means of a simple geometrical construction, obtaining the 

 true stress-strain curve from this. 



The determination of this curve for a tension test of 

 mild steel is shown on Fig. 75. Here the load-strain 

 diagram A B C D is plotted as before. Then the line 

 A, at the extreme left of the diagram, from which the 

 loads have been measured, is produced to G, A G being 

 made equal to the length of the specimen previous to the 

 test, O A being the extended length. Both these must be 

 to the same scale. Take any point on the load-strain curve 

 C, and draw through it a horizontal line C H. Also draw a 

 vertical through this same point to meet the base line in a 

 point K. Join G and K, and produce this to cut the line 

 H C produced in E. Then E will be a point on the real 

 stress diagram. 



The proof of this depends on the assumptions that the 

 bar remains parallel during the plastic deformation, and 

 also that the volume of the bar remains constant. These 

 are both approximately true. 



Let A = the area of the original cross-section of the 



bar. 



,, a = the cross-section at the load H C. 

 ,, L = G A = the original length. 

 ,, I = H A = the extension corresponding to the 

 loadHC. 



