True Stress from Engineering Stress

A tensile machine knows two things: the load in the grips and the extension of the gauge length. It knows the cross-section of the bar you told it about before the test, and it has no way of learning that the bar has got thinner since. So it divides load by the ORIGINAL area and calls the result stress, and divides extension by the ORIGINAL length and calls the result strain. Those are engineering values, and the curve drawn from them is a fiction useful for comparing materials and useless for describing what the metal is doing.

True stress divides the load by the area the bar actually has at that instant. Since plastic flow conserves volume, A₀L₀ = AL, so A = A₀/(1 + e) and σ_t = σ_e(1 + e). True strain is defined as the integral of dL/L rather than the ratio ΔL/L₀, which gives ε_t = ln(1 + e). Both conversions rest on constant volume, so they are meaningless below yield — elastic deformation changes volume, and the elastic part of the curve should not be converted at all.

Two properties make true strain worth the trouble. It is additive: strain the bar 10 % and then 10 % again, and the true strains add to give exactly the true strain of the combined operation, while the engineering strains do not. And it is symmetric in tension and compression: squeezing a specimen to half its height is a true strain of −0.693, and stretching it to double is +0.693, whereas the engineering values are −0.5 and +1.0. Any calculation that combines deformation steps — a rolling schedule, a multi-pass draw — has to be in true strain or the arithmetic simply does not work.

The most useful thing this conversion explains is why the engineering curve turns over at the ultimate tensile strength while nothing about the metal softens. The true curve rises monotonically all the way to fracture. The engineering curve peaks and falls because the load is being divided by an area that no longer exists, and past maximum load the section is shrinking faster than the metal is hardening. The UTS is therefore not a strength in any material sense; it is the coordinate of a geometric instability. That is Considère's criterion restated, and it is why the UTS moves if you change the specimen shape while the true flow curve does not.

The conversion has a hard limit and it is the neck. Both formulas assume the deformation is UNIFORM along the gauge length, and the instant a neck forms that stops being true — the neck is straining rapidly while the rest of the bar has stopped. Past maximum load the true stress has to be computed from the measured minimum diameter of the neck itself, and even then it needs the Bridgman correction, because the neck's geometry sets up a triaxial tensile stress state that raises the axial stress above what uniaxial flow would need. Uncorrected post-necking "true stress" values are systematically high, and any flow curve fitted through them is wrong at the end where it matters most.

One practical note. Extension measured at the crossheads includes the compliance of the whole load train — grips, load cell, frame — and at small strains that can be a substantial fraction of what is recorded. Use an extensometer clipped to the gauge length, or a digital image correlation field, if the strain values are going anywhere near a flow-curve fit.