The tensile test and the stress–strain curve

A tensile test takes a machined specimen with a known cross-section and a marked gauge length, grips it at both ends, and pulls it apart at a slow, steady rate. Two things are recorded throughout: the load applied, and the extension of the gauge length.

Those two raw measurements are then converted into quantities that do not depend on the size of the specimen:

• Nominal stress σ = P/A₀, where A₀ is the original cross-sectional area
• Nominal strain ε = δ/L₀, where L₀ is the original gauge length

Dividing by the original dimensions is what makes the result a property of the material rather than a property of that particular test piece. A specimen twice as thick carries twice the load at every stage, and the stress–strain curve comes out identical.

For a ductile structural steel the curve has a very recognisable shape, and each part of it is telling you about a different physical process.

The linear region. Stress is proportional to strain. Atoms are being pulled slightly further apart and spring straight back when released. The slope of this line is Young's modulus. Note that E is measured here and nowhere else — it is a property of the elastic region only.

The limit of proportionality and the elastic limit. These are close together and often treated as the same point. Below the elastic limit the specimen returns to its original length on unloading. Above it, some permanent set remains.

Yielding. In mild steel the curve flattens dramatically: the specimen extends substantially at roughly constant load. Whole planes of atoms are sliding over one another. This flat plateau is the reason steel is such a forgiving structural material — it deforms visibly and at constant load long before it breaks.

Strain hardening. After the plateau the curve rises again. The sliding planes have become tangled and obstruct one another, so more stress is needed to keep deforming the material.

The ultimate point and necking. The curve reaches a maximum — the ultimate tensile strength — and then falls. Nothing has weakened. What has happened is that deformation has localised into a neck, and the real area there is shrinking faster than the material is hardening. Because we insist on dividing by the original area, the plotted stress goes down.

Fracture. The neck thins until the specimen parts, usually with a cup-and-cone fracture surface.

Not every material behaves this way, and the differences matter more than the similarities.

Brittle materials — cast iron, concrete in tension, glass — have no plateau and almost no plastic region. They follow a roughly straight line and then break, with very little warning and very little energy absorbed. A structure that fails in a brittle mode gives you nothing: no sag, no cracking, no time.

Materials with no defined yield point — high-strength steel, aluminium alloys, stainless steel — curve smoothly from elastic to plastic with no flat portion at all. There is no obvious stress to call "the yield". So a convention is used instead: the proof stress.

To find the 0.2% proof stress, draw a line parallel to the elastic slope, offset along the strain axis by 0.002. Where it cuts the curve is the proof stress. By construction, loading the material to that stress and then unloading leaves exactly 0.2% permanent strain. It is a definition, not a physical event — but it is a repeatable one, which is what design needs.

Ductility is measured after the test, from the broken halves fitted back together, and is reported two ways:

• Percentage elongation = (L_f − L₀)/L₀ × 100
• Percentage reduction in area = (A₀ − A_f)/A₀ × 100

Elongation depends on the gauge length used, because most of the extension is concentrated in the neck — so a quoted elongation is meaningless unless the gauge length is quoted with it. Reduction in area does not have that problem, but is harder to measure.

Ductility is not a strength. It buys you something different and arguably more valuable: warning, and the ability of a structure to redistribute load away from an overstressed region instead of tearing there.

It is worth naming the classification the tensile test implies, because it drives structural choices more than any single strength figure.

Ductile materials — structural steel, aluminium alloys, copper — have a large plastic region. They deform visibly before failing, they absorb a great deal of energy, and they redistribute load away from an overstressed region. Every method in this course that relies on redistribution assumes ductility.

Brittle materials — cast iron, glass, concrete in tension, most ceramics — have almost no plastic region. They fail at small strain with little warning and absorb little energy. Their tensile strength is also highly variable, because it is governed by the largest flaw present rather than by the bulk material.

Materials that behave differently in tension and compression. Concrete is the structural example: strong in compression, weak and unreliable in tension. That single asymmetry is the reason reinforced concrete exists, and the reason cracked-section analysis discards the concrete below the neutral axis.

Time-dependent materials — timber, polymers, and concrete over years — deform appreciably under sustained load. Their short-term test results do not describe their long-term behaviour.

The strain-hardening region seen after the yield plateau is what gives ductile materials their reserve. Dislocations become tangled and obstruct one another, so more stress is required to continue deforming. The ratio of ultimate to yield stress is a rough measure of how much reserve there is: around 1.4 for a typical mild steel, and closer to 1.1 for a high-strength steel — which is one reason very high-strength steels are treated more cautiously.