The curve was rising the whole time

The ductility that depends on the ruler takes a tensile test apart along one axis: the elongation a certificate reports is a material strain plus a fixed extension divided by whatever gauge length the laboratory used. This page takes the same test apart along the other axis, where the trouble is worse and much older.

Both denominators on a tensile curve are wrong. Stress is plotted as force over the original area and strain as extension over the original length, and a specimen being pulled apart has neither of them any more. The consequences look like properties of the steel and are properties of the arithmetic.

The falling branch is the most-reproduced feature of the most-reproduced graph in materials engineering, and it is a division artefact.

The two corrections, and why only one of them is difficult #

The strain correction is a definition. Adding successive increments of extension, each divided by the length the specimen has at the time, gives $\varepsilon = \ln(L/L_0)$ — the natural strain, which has the property that a stretch followed by another stretch adds rather than compounds. It is 0.22 where the engineering strain is 0.246.

The stress correction is a measurement, and only up to a point. Below the ultimate load a metal deforms at constant volume and uniformly, so $A_0L_0 = AL$ and the actual area follows from the extension without anybody having to measure it: $\sigma = S(1+e)$. That is the whole of the true curve up to necking, and it needs nothing the machine does not already record.

Past necking the volume is still constant and the deformation is no longer uniform, so the length ratio says nothing about the area at the neck. From that instant on the true curve requires a measurement of the specimen rather than of the test, which is where the difficulty starts and where most published curves stop.

Which free body produced the number #

The free body is a slice through the specimen, cut normal to its axis, and what crosses the cut is the load.

Before necking there is nothing else to say: the slice is in uniaxial tension, the stress is uniform over it, and the stress is the load over the area. That is the state every quoted strength refers to.

After necking the slice is not in uniaxial tension. The neck has curved the specimen’s profile, so the material outside the neck is restraining the material inside it from contracting, and the restraint is a radial and circumferential tension that no external force applied. A free body of the central core therefore has a hydrostatic tension in it that a free body of the same material in a plain bar does not.

That matters because yielding does not care about hydrostatic stress and fracture does. The flow stress — what the material would need in simple tension — is lower than the axial stress the machine is dividing out, by a factor that depends only on the neck’s geometry.

Bridgman’s result is that the correction can be computed from the profile alone — no material constant enters it — which is why it survived. What it needs is the shape of the neck, which is a photograph, a shadowgraph or a pair of callipers, and which no load cell, extensometer or data logger contains. A flow curve past the ultimate load is a photographic measurement, and that is the honest reason there are so few of them.

Considère, and the exponent that is also an elongation #

Between the two corrections sits the moment they are both about, and it has an exact criterion.

A specimen necks when the load stops rising. Write the load as $F = \sigma A$, differentiate, and use constant volume:

$$\frac{d\sigma}{d\varepsilon} = \sigma$$

which is Considère’s construction: the material necks where the true stress curve’s own slope has fallen to the value of the stress. The rung below this one names it in a sentence and does not draw it, because up there it was an explanation of why localisation happens. Here it is an equation with an answer in it.

The crossing is at 0.22 because the hardening law used here is $\sigma = K\varepsilon^n$ with $n = 0.22$, and substituting it into Considère gives $n\varepsilon^{n-1}K = K\varepsilon^n$, so $\varepsilon = n$ — exactly, with no other constant surviving.

That is the most useful line on the page. The strain at which a metal necks is its hardening exponent, so the uniform elongation is not an independent property to be measured but a number that can be read off a log-log plot of the flow curve. And the strength follows from the same constant: the true stress at necking is $Kn^n$ and the engineering ultimate is that discounted by the area change, $Kn^ne^{-n}$.

One number, two headlines #

Which makes the trade every steel specification argues about into a piece of arithmetic.

Two things about that figure need saying, and the second is the qualification the first requires.

Within one hardening law, strength and ductility are the same parameter. A material that hardens slowly reaches a high load early and necks early; one that hardens quickly is still hardening at large strains and necks late. Nothing has to be traded off by anybody: the trade is in the shape of the curve.

And $K$ is the other parameter, which is what a metallurgist actually moves. Raising $K$ at constant $n$ raises the strength and leaves the uniform elongation untouched, because the necking strain has no $K$ in it. So the real design space is two-dimensional and the figure is a slice through it — which is why “strong steels are brittle” is a statement about the correlation between $K$ and $n$ in real alloys rather than a consequence of anything above.

What this changes about the numbers used elsewhere #

Nothing that is used elastically, and rather a lot that is used plastically.

A strength quoted for design is an engineering one and should stay that way. The check it feeds — a member’s tension capacity, a bolt’s — divides a force by an original area, so the number and the calculation are wrong in the same direction and the answer is right. Substituting a true stress into a nominal-area check would be an unconservative error of exactly $e^{\varepsilon}$.

A plastic hinge’s rotation is a different matter. A section yielding from the outside in is computed with an elastic-plastic material law, and the strain it reaches at the extreme fibre is a few per cent — well before any of this. The hardening exponent enters as the slope of the plateau’s tail, and the reason rotation capacities are quoted with such wide bands is that the tail is precisely where the engineering curve stops describing the material.

And a forming calculation cannot use the engineering curve at all. Deep drawing, cold-rolling and the springback of a formed section all involve strains past the ultimate load, so they run on the true curve — which is why $n$ and $K$ are the two numbers a press shop asks for and a structural certificate does not carry.

The slope is a stiffness, and it is used as one #

There is a second reason to want the true curve, and it has nothing to do with strength.

The quantity Considère’s construction compares against the stress is $d\sigma/d\varepsilon$, the tangent modulus — and that same derivative is what governs a column buckling in the inelastic range. A column that has yielded before it was loaded buckles at a load set by the tangent modulus rather than by Young’s modulus, because the fibres that are about to bend further are on the slope of the curve rather than on its elastic part.

Which slope, though? The engineering curve’s derivative and the true curve’s differ by a factor that grows through the plastic range, and past the ultimate load they differ in sign. A tangent modulus taken off the engineering curve at 20 per cent strain is negative; the material’s is 1,020 MPa and positive. Any buckling calculation using the first would report a column that has already failed.

At small plastic strains — which is where every inelastic column check lives, a fraction of a per cent — the difference between the two slopes is under two per cent and nobody has ever needed to care. The point is not that the correction matters there. It is that the same derivative is being asked for by two different subjects, one of which uses it at strains of 0.002 and the other at strains of 0.2, and only one of them can take the engineering curve at face value.

The same is true of the plateau that structural plasticity runs on. A section yielding from the outside in and what is left after the first fibre yields both assume a horizontal plastic branch, and the material has no horizontal branch at all — the flat part of an engineering curve is a Lüders plateau, which is a travelling front rather than a stress-strain relation, and after it the material hardens steeply. Structural plasticity works because it stops at strains where the hardening has not yet supplied anything worth counting, and it says so by ignoring it.

What a certificate is actually asserting #

Reading the four numbers on a mill certificate against the two curves is a short exercise and it changes what they mean.

The yield stress is a point on both curves and the corrections are negligible there — 0.2 per cent strain is a factor of 1.002. It is the one honest number on the certificate.

The tensile strength is the peak of the engineering curve and therefore a compound of a material property and a divisor. Its usefulness is that it is reproducible and that everything using it divides by the original area too; its meaninglessness is that no material anywhere is under 431 MPa at the moment that number is recorded.

The elongation is the gauge-length problem of the rung below: a material strain plus a neck extension divided by a ruler.

And the yield-to-tensile ratio — the number a seismic specification puts a cap on — is $f_y/(Kn^ne^{-n})$, which is a way of asking for a minimum hardening exponent without measuring one. A ratio of 0.85 on this steel is a demand that $n$ exceed about 0.1, which is a demand that the material still be hardening when the first section of a member has yielded. That is exactly what a plastic hinge needs and exactly what a steel stronger in a millisecond has less of, and it is specified in the one form that requires no test the mill was not already doing.

Where the model stops #

Hollomon’s law is a fit, not a mechanism. Real metals depart from a single power law at both ends: at small strains the yield point and the Lüders plateau are nothing like $K\varepsilon^n$, and at large ones the exponent drifts. It is a good fit over the plastic range that matters, and the exactness of $\varepsilon_u = n$ belongs to the law rather than to the steel.

Constant volume is an assumption with an exception. It is excellent for metals, whose plastic deformation is shear on slip planes and changes no volume. It is wrong for concrete, for soils and for polymers with crazing, in each of which the plastic strain has a volumetric part — so the whole chain from $\sigma = S(1+e)$ onwards is a statement about metals rather than about materials.

The neck’s profile is idealised. Bridgman’s correction assumes the neck is a circular arc and the strain across the minimum section is uniform. Both are approximations, and the second fails at large strains where the core of the specimen has strained rather more than its surface.

And the specimen is round. A flat coupon necks differently — it forms a diffuse neck first and a local one afterwards — so the correction is not the same function and the sharpness has two radii rather than one.

What the pictures cannot show #

The curves stop where the material fractures, and every figure here draws that as an endpoint. It is not one: fracture is a separate event with a separate criterion, and the true strain at fracture is the number that measures a metal’s real ductility — $\ln(A_0/A_f)$ from the broken halves, which for a mild steel is around 1.0 against the 0.25 the certificate reports.

That factor of four is worth carrying. The most-quoted measure of ductility understates the true one by about four times, for the same reason everything else on this page: it is measured on the original section, over a gauge that includes material which stopped straining at the ultimate load.

The figures also cannot show rate. Every curve here is quasi-static, and the same steel pulled a thousand times faster has a higher yield stress, a higher $K$ and usually a lower $n$ — which is to say a different point on the trade in the fourth figure, arrived at without changing the alloy.

The assumption the figure rests on #

That the material is the same everywhere in the specimen.

It is nearly true of a machined round bar from the middle of a plate, and less true of everything a structure is made of. A rolled section is anisotropic; a welded region has a strip whose curve is a different curve; a reinforcing bar has a rib pattern and a work-hardened surface. In every one of those the specimen necks wherever the material is weakest rather than wherever the stress is highest, and Considère’s competition is decided by a gradient in $K$ rather than by the rate of hardening.

That is the same structure of argument as the localisation that ends the rung below: once deformation concentrates, what governs is the worst region rather than the average one, and every quantity measured over a length becomes a statement about the specimen as well as about the material.

The history, which is three men and eighty years #

Considère published his construction in 1885, in a paper about the stability of columns in bridges — the necking criterion arrives as an aside in an argument about buckling, which is not the accident it looks like: both are questions about whether a deformation that has started will arrest itself.

Ludwik and Hollomon supplied the power law between 1909 and 1945, and its value was that it made the criterion algebraic instead of graphical. Bridgman’s correction came in 1944 out of high-pressure physics rather than metallurgy, which is why it is expressed in terms of hydrostatic stress: he had spent thirty years measuring what pressure does to materials and recognised that a neck manufactures its own.

Eighty years separate the criterion from the correction, and in between sits the whole of plasticity theory. What the sequence shows is that the descending branch was understood as an artefact almost from the beginning, and is still drawn without comment in every introductory text — because the engineering curve is what the machine produces and the true curve is what the material did, and only one of them is free.

The ladder from here #

Later rungs on this anchor: the true fracture strain and reduction of area as the ductility measure the section actually has, and why it appears on no certificate. Lüders bands and the upper yield point, where the curve is not a function of strain at all and the specimen deforms in a travelling front. Anisotropy of rolled plate, where $n$ and $K$ differ with direction and the through-thickness one is the direction nobody tested. The forming limit diagram, which is Considère’s argument run in two dimensions and is where the whole of sheet metalworking lives. Rate and temperature as the two axes this page held fixed. And the same stability argument applied to a structure rather than a specimen, where a load that stops rising is a limit point rather than a neck.