Beyond Hooke: Stress–Strain Curves
Stretch too far and the wire never comes back.
Lesson at a glanceShow the key points
- Metal wire: stress ∝ strain up to the proportional limit (strain ~ 0.01); still elastic up to the elastic limit (yield point).
- Beyond it the behaviour is plastic: on unloading a permanent set remains. The wire breaks at the fracture point (breaking stress).
- Ductile materials deform a lot between elastic limit and fracture; brittle ones break soon after the elastic limit.
- Vulcanised rubber stays elastic up to several times its length, but is never linear, and unloading follows a lower curve: elastic hysteresis.
- The energy lost in a hysteresis loop becomes heat; rubber pads use it to absorb vibrations.
Hooke’s law is only the opening chapter. Keep pulling a wire and it stops springing back, stretches like toffee, and finally snaps. Rubber does something different again.
A metal wire, pulled further and further
Stretch a metal wire steadily and plot stress against strain. Several landmarks appear:
Past the elastic limit the strain grows rapidly and the behaviour is plastic: remove the load and the wire shortens along a line roughly parallel to the original straight part, but keeps a permanent extension, the permanent set. Keep loading and the wire eventually breaks at the fracture point (d); the stress there is the breaking stress.
- Materials that deform a lot between the elastic limit and fracture are ductile (copper, mild steel: they can be drawn into wires).
- Materials that break soon after the elastic limit are brittle (cast iron, glass).
A steel wire of diameter 1.0 mm has a breaking stress of about 1.0 × 10⁹ N/m². Roughly what load breaks it? Why do lab instructions say to use at most half of this?
Rubber
Vulcanised rubber behaves quite differently. It stays elastic even when stretched to several times its original length (eight times, say) and still returns completely. In that sense it is “more elastic” than steel. But the stress needed for a given strain is far smaller than in steel, so in the sense of strong restoring forces steel is much more elastic.
Two features stand out:
- There is almost no region where stress is proportional to strain.
- On unloading, the curve doesn’t retrace the loading curve: it lies below it, though the sample still ends at its natural length.
- 1Metal wire: proportional limit → elastic limit (yield point) → plastic flow with permanent set → fracture at the breaking stress.
- 2Ductile materials stretch a lot before breaking; brittle ones break soon after the elastic limit.
- 3Rubber: elastic to huge strains, never linear, with a hysteresis loop whose area becomes heat.
Check yourself
Keep exploring
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