Stress Theory
When is [math]\displaystyle{ \sigma= \frac{F}{A} }[/math] Valid?
The equation [math]\displaystyle{ \sigma= \frac{F}{A} }[/math] is valid only if the stress is uniformly distributed over the cross-section of the bar. This condition is realized if the axial force F acts through the centroid of the cross-sectional area. Uniform stress exists throughout the length of the bar except near the ends. The stress distribution at the end of a bar depends upon how the load F is transmitted to the bar. If the load is distributed uniformly over the end, the stress pattern at the end is the same as everywhere else. However, it is more likely that the load is transmitted through a pin or a bolt, producing high localized stresses called stress concentrations. As a practical rule, the formula [math]\displaystyle{ \sigma= \frac{F}{A} }[/math] may be used with good accuracy at any point within a prismatic bar that is at least as far away from the stress concentration as the largest lateral dimension of the bar. Of course, even when the stress is not uniformly distributed, the equation [math]\displaystyle{ \sigma= \frac{F}{A} }[/math] may still be useful because it gives the average normal stress on the cross section. [1]
True vs. Engineering Stress
Beyond the yield point, molecular flow causes a substantial reduction in the specimen cross-sectional area A, so the true stress [math]\displaystyle{ \sigma_t= \frac{F}{A} }[/math] actually borne by the material is larger than the engineering stress computed from the original cross-sectional area ([math]\displaystyle{ \sigma_e= \frac{F}{A_0} }[/math]). The load must equal the true stress times the actual area ([math]\displaystyle{ F =\sigma_t A }[/math]), and as long as strain hardening can increase σt enough to compensate for the reduced area A, the load and therefore the engineering stress will continue to rise as the strain increases. [2] Eventually, however, the decrease in area due to flow becomes larger than the increase in true stress due to strain hardening, and the load begins to fall. This is a geometrical effect, and if the true stress rather than the engineering stress were plotted no maximum would be observed in the curve. The strain hardening rate is the slope of the stress-strain curve, also called the tangent modulus. At the UTS the differential of the load F is zero, giving an analytical relation between the true stress and the area at necking:
[math]\displaystyle{ F =\sigma_t A \rightarrow dF = 0 =\sigma_t dA + A d\sigma_t \rightarrow -\frac{dA}{A} = \frac{d\sigma_t}{\sigma_t} }[/math]
This expression states that the load and therefore the engineering stress will reach a maximum as a function of strain when the fractional decrease in area becomes equal to the fractional increase in true stress.