Necking and the ultimate stress

It is important to distinguish between the nominal stress, which is the load at any time during deformation divided by the initial cross-sectional area, and the true stress, which is the load divided by the actual cross-section at any time. The cross-section of the sample decreases with increasing extension, so the true stress may be increasing when the apparent or conventional stress or load remains constant or even decreasing. This has been discussed very well by Nadai [2] and Orowan [3]. 

This gives a stress-strain curve of the form shown. The load reaches its maximum value at the instant the extension of the sample ceases to be uniform. At this elongation the specimen begins to neck and consequently the load falls, as shown 

It is instructive to plot the true tensile stress at any elongation rather than the nominal stress This is given by a = P/A, where A is the actual cross-section at any time. We now assume, as is usual for plastic deformation, that the deformation takes place at constant volume. Then Al = A lo, and if we put // /o = A, where X is the extension ratio. 

The argument above is based on the assumption that the stress-strain curve completely defines the material behaviour. In reality, with polymers, stress depends on the strain rate and because necking is associated with a local increase in strain rate the issue is more complex. A strong dependence of stress on strain rate can inhibit necking even when the nominal stress reaches a maximum the existence of the maximum is a necessary condition for necking, but not a sufficient one. Necking in rate-dependent materials has been discussed recently by Sweeney et al. [4]. 

In mathematical terms, the existence of a stress maximum as the condition necessary for necking is 

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