Considère Criterion and Uniform Elongation

Armand Considère was a French engineer who, in an 1885 lecture on the use of iron and steel in construction, set out the condition for a tensile bar to start necking. The argument is one line of calculus and it is worth following, because almost everything people believe about tensile ductility follows from it.

A tensile test is a competition. As the bar stretches, the metal hardens — that raises the load it can carry. At the same time the cross-section shrinks — that lowers it. While hardening wins, the load rises and the deformation stays spread evenly along the gauge length, because any spot that thins locally hardens locally and stops thinning. The instant the shrinking wins, the situation inverts: a spot that thins now carries more stress than its neighbours, thins faster still, and runs away. That runaway is the neck. Setting the load to its maximum, dL = 0 with L = σA and constant volume so dA/A = −dε, gives dσ/dε = σ. Necking begins where the slope of the true stress–strain curve equals the stress itself.

Now put a Hollomon power law into that condition. If σ = Kεⁿ then dσ/dε = nKεⁿ⁻¹ = nσ/ε. Setting that equal to σ gives ε = n. The strain at the onset of necking is exactly the strain-hardening exponent. It is one of the cleanest results in the whole of mechanical metallurgy, and it explains why a sheet metal's n value is treated as a formability number rather than a curve-fitting parameter. It also explains why the ultimate tensile strength of a metal is not really a material property at all — it is the point where a geometric instability happens to catch up with a hardening rate.

The engineering elongation that a tensile machine records at that instant follows from the strain conversion: e = exp(ε) − 1, so e_u = exp(n) − 1. A steel with n = 0.22 necks at 24.6 % engineering elongation. That is the figure a forming operation actually gets to use.

Uniform elongation is not total elongation, and confusing the two is the standard mistake on a mill certificate. The "% elongation in 50 mm" reported on a test certificate is measured after fracture, and it includes everything the neck did after maximum load — which on a ductile steel is often as much again as the uniform part. Total elongation is therefore a mixture of a material property and a specimen geometry: a short gauge length gives a bigger number, because the neck's contribution is spread over less length, which is exactly why gauge length must be quoted with any elongation figure. Uniform elongation has no such dependence, and it is the number that predicts whether a part can be formed, because once a neck appears the part is scrap regardless of what the material does afterwards.

Two extensions worth knowing. If the material is rate-sensitive — flow stress rising with strain rate, quantified by an exponent m — then a thinning neck strains faster, hardens by that alone, and the neck is stabilised. A high m is why superplastic alloys reach elongations of several hundred percent with no meaningful strain hardening at all, and why hot forming is more forgiving than cold. And in sheet under biaxial tension the criterion changes: the local necking that matters in a press is governed by the forming limit diagram rather than by this uniaxial condition, though n is still the dominant variable in where that diagram sits.