Stress–Strain Curves, Elasticity & Failure

The start of the curve makes three claims that first-year answers often run together. The proportional limit is where the graph stops being straight: below it, σ = Eε exactly. The elastic limit is the largest stress after which the specimen still returns to its original length, a statement about unloading rather than about shape. Rubber shows how far apart the two can sit: its curve bends almost from the start, yet it recovers from strains of several hundred per cent. Yield is the start of plastic flow, planes of atoms slipping permanently past one another. In most metals all three fall within a fraction of a per cent of strain, too close to separate cleanly in a test, which is why engineers quote a defined yield strength instead. Annealed low-carbon steel adds a twist: at yield its stress dips slightly and the bar stretches at nearly constant stress for a while before hardening begins.

The only way to find out how much of a strain is permanent is to take the load off. Below the elastic limit, the specimen retraces its loading line back to zero. Beyond it, the stress falls along a new straight line with the same slope E, parallel to the original elastic line, because the atomic bonds unload just as they loaded. The recovered strain is σ/E and what remains is the permanent set, εₚ = ε − σ/E. Load a steel wire with E = 200 GPa to a strain of 3.00 %, where the stress is 380 MPa, and let go: it springs back 380 MPa ÷ 200 GPa = 0.19 % and keeps 2.81 %, about 94 % of what it had. Reload it and it climbs that same unloading line elastically, not yielding until it again reaches 380 MPa. That is cold working: the wire now behaves elastically up to 380 MPa, well above its original yield strength, and has spent part of its ductility to get there.

Many alloys, aluminium and stainless steel among them, have no sharp yield point: the curve bends away from its straight line gradually, so any stress picked by eye is arbitrary. The offset method replaces judgement with a construction. Draw a line of slope E starting from ε = 0.002 on the strain axis; the stress where it crosses the curve is the 0.2 % offset yield strength σ_y. The number has a precise meaning: unload from that point, down a line of slope E, and the specimen keeps exactly 0.2 % of permanent strain. For an aluminium alloy with E = 70 GPa whose curve passes through 280 MPa at a total strain of 0.60 %, the offset line reaches 70 000 MPa × (0.0060 − 0.002) = 280 MPa at that strain, so σ_y = 280 MPa, made of 0.40 % elastic strain and 0.20 % plastic. Two tenths of a per cent is small enough to leave a part serviceable and large enough to measure reliably with an extensometer.

Past yield, slip piles up defects that obstruct further slip, so the stress needed to keep stretching the metal rises. The true stress σₜ = F/A tracks that honestly because it uses the current area; at constant volume A = A₀L₀/L, which gives σₜ = σ(1 + ε) until necking starts. A copper specimen at an engineering strain of 0.30 and 220 MPa carries a true stress of 286 MPa at a true strain of ln 1.30 = 0.262. The load F = σₜA is the product of a rising factor and a falling one. It peaks when dF = A dσₜ + σₜ dA = 0, and at constant volume dA/A = −dεₜ, so the peak comes where dσₜ/dεₜ = σₜ. Beyond that point a slightly thinner section carries a higher stress than its neighbours but can no longer harden fast enough to compensate, so it keeps thinning while the rest of the bar stops stretching. That local collapse is the neck, and the peak engineering stress is the ultimate tensile strength σᵤ.

Ductility is measured after fracture, on the broken halves fitted back together: the percentage elongation (L_f − L₀)/L₀, or the percentage reduction of area at the neck. A material that fractures with less than about 5 % elongation is usually called brittle; glass, cast iron, ceramics and concrete in tension fail on or near their elastic line, often suddenly and with no neck to warn anyone. The energy absorbed per unit volume is the area under the curve, and it separates the two classes far more sharply than strength does. Steel that averages about 350 MPa over 20 % strain absorbs roughly 350 × 10⁶ Pa × 0.20 = 70 MJ m⁻³. Window glass with a fracture stress near 50 MPa and E = 70 GPa stores only σ²/(2E) = 1.8 × 10⁴ J m⁻³ before it breaks, about four thousand times less, although its stiffness matches aluminium's. Even the steel's elastic share, σ_y²/(2E) = 0.16 MJ m⁻³ for σ_y = 250 MPa, is under 1 % of its total: a ductile metal's toughness is almost all plastic work.

A ductile part has failed its purpose once it yields, even if it has not broken, so its allowable stress is σ_y/n. A structural steel with σ_y = 250 MPa and n = 1.67 may carry 150 MPa, keeping the working point on the straight line where every strain springs back. Brittle materials give no plastic warning and scatter widely in strength, so designs divide σᵤ by a larger n. The factor also has to cover what the tensile test never saw. A small circular hole in a wide plate triples the local stress, so 150 MPa far from the hole becomes 450 MPa at its edge, past yield. Loads repeated millions of times grow fatigue cracks at stresses well below σ_y. Some steels turn brittle in the cold, which is how welded Liberty ships cracked in cold seas in the 1940s. A slender strut in compression buckles long before its material yields. The curve reports the material's limits; the design has to supply the rest.