True Stress Strain Curve

The true stress - strain curve does not exhibit a maximum, even though the load reaches a maximum value and the nominal stress curve then begins to fall. The reason for this can be seen by considering the incremental conditions using the equation for cold working.
Assuming that the cross sectional area of the specimen at any instant is A, the true stress will be given by:

This shows that the load will reach a maximum value when the slope of the stress strain-curve, which is the rate of work-hardening, equals the magnitude of the applied stress. Physically this means that the increase in strength due to work-hardening just balances the decrease in area. Further increase in strain causes instability, leading to the formation of a neck and finally to fracture. However the work hardening continues, there is no change in the true stress-strain curve at this point.

Experimental work shows that this relationship is approximately true.

David J Grieve, 24th October 2008.