A glass bangle dropped on a hard floor breaks into pieces at once. A copper wire from an old electric cable can be bent, twisted and drawn out long before it breaks. An electrician bends the same wire back and forth to snap it off. Three behaviours, and all of them are elastic at first.
Beyond Hooke's law
The chapter on Stress, Strain and Hooke's Law followed a wire up to its elastic limit. Now keep pulling. In a testing machine, a specimen of the metal is stretched steadily, and the force and the extension are recorded until it breaks. The force divided by the original area gives the stress, and the extension divided by the original length the strain.
Follow the steel curve of Fig 6.1 from the origin.
- From O to A the line is straight. Stress is proportional to strain, and A is the proportional limit.
- From A to B the curve bends a little, but the wire still returns to its length when unloaded. B is the elastic limit.
- At C, the yield point, the metal begins to flow. The strain grows a lot with little or no extra stress.
- The metal then hardens, and the stress rises again to its greatest value at D, the ultimate tensile stress.
- Beyond D one part of the specimen thins quickly, called necking, and it breaks at E, the fracture point.
Beyond the elastic limit the wire no longer comes back. Unloaded from a point past B, it shortens along a line parallel to OA and stops with some strain left. This is the permanent set of the chapter on Stress, Strain and Hooke's Law. The region from B to E is the plastic region. Metals are shaped in it: drawn into wire, rolled into sheet, pressed into car bodies.
A steel wire is pulled until it breaks. Put the points of its stress-strain curve in the order they are reached, first to last.
Why the curve falls before the break
After D the curve comes down, as if the metal were getting weaker. It is not. The stress on the graph is the force divided by the original area. In the neck the real area shrinks fast, so the testing machine needs less force to keep stretching. The force falls, and so does the plotted stress, while the stress inside the thin neck keeps rising until it breaks.
Between the ultimate stress D and the fracture point E, the stress on a steel wire's stress-strain curve falls. Why?
Ductile, brittle and rubbery
Choose the copper chip of Fig 6.1. Copper has no sharp yield point, and its plastic region is very long. It can be drawn into fine wire and hammered into sheet before it breaks. Such a material is ductile. Gold, silver, aluminium and mild steel are ductile too.
Now choose glass. The line rises straight and simply stops: the glass breaks at or near its elastic limit, with almost no plastic region. Such a material is brittle. Cast iron, glass and most stones are brittle. A brittle solid gives no warning, which is why a glass bangle breaks at once while a copper wire bends first.
Rubber belongs to a third group, the elastomers. Choose the rubber chip. It stretches to several times its length and still returns, yet its curve is not a straight line. And it returns along a lower curve than it went out on, enclosing a loop. This is elastic hysteresis.
The area under the loading curve is the work done in stretching. The area under the unloading curve is the work the rubber gives back. The loop between them is the energy that stays behind as heat, in every cycle. That is why a car tyre, flexing each time it rolls round, grows warm on a long drive. It is also why rubber mounts are used to damp the shaking of machines.
Which statements about real materials are true?
Elastic fatigue
Bend a paper clip back and forth a few dozen times and it snaps, although no single bend was hard. A metal loaded and unloaded many times, even at stresses well below its breaking stress, grows tiny cracks that spread with each cycle. This loss of strength under repeated loading is elastic fatigue. It is why old bridges are declared unsafe after decades of traffic, and why aircraft parts are replaced after a set number of flights.
A car's rubber tyres grow warm on a long drive, even on a cool day. Which property of rubber explains it?
In the middle of the nineteenth century, railway axles in Germany kept breaking. Each one had been designed to carry far more than its load. August Wöhler, a railway engineer, built machines that loaded axles again and again, millions of times. He found that the number of cycles an axle survived fell as the stress rose. Below a certain stress, steel axles seemed to last for ever. His results, published around 1870, founded the study of fatigue.
Designing a crane rope
A crane at a port must lift a container of \(10\) tonnes, a load of \(10^5\,\text{N}\). The steel of its rope has an elastic limit of \(3 \times 10^8\,\text{Pa}\). The rope must never pass that limit, or it will take a permanent set and weaken. So the area must be at least \(10^5/(3 \times 10^8) = 3.3 \times 10^{-4}\,\text{m}^2\), a radius of about \(1\,\text{cm}\).
No engineer stops there. Loads are sometimes jerked, as the lesson on elastic energy showed, and ropes wear and suffer fatigue. So the working stress is kept a safety factor below the limit, often about \(10\). With a factor of \(10\), the area is ten times as large and the radius about \(3.3\,\text{cm}\). The rope is made of many thin wires twisted together, so that it bends easily round the drum.
A crane rope must lift \(5\) tonnes without its stress passing \(2.5 \times 10^8\,\text{Pa}\). Take \(g = 10\,\text{m/s}^2\). Find the least area of cross-section of the rope, in cm².
Beams bend, and the I-section
A beam resting on two supports and loaded in the middle bends. Its upper layers are squeezed and its lower layers are stretched. A layer through the middle is neither. The largest strains are at the top and bottom faces, and the middle carries almost none.
Take a beam of width \(b\), depth \(d\) and length \(l\) with a load \(W\) at the middle. Its sag is \(\delta = Wl^3/(4bd^3Y)\), a result quoted here. The depth enters as its cube. Doubling the depth cuts the sag eight times, while doubling the width only halves it. So beams are laid on edge, deep and narrow.
The I-section, the shape of railway rails and steel girders, carries the idea further. Its top and bottom flanges hold most of the metal, where the strain is largest. A thin web joins them in the middle, where little metal is needed. The girder is almost as stiff as a solid block of the same depth, at a fraction of the weight.
The rock at the base of a mountain carries the weight of the rock above it. Take the density of rock as \(3000\,\text{kg/m}^3\), \(g = 10\,\text{m/s}^2\), and the elastic limit of rock as about \(3 \times 10^8\,\text{Pa}\). Estimate the greatest height of a mountain, in km, before its base begins to flow.
Mount Everest, at about \(8.8\,\text{km}\), puts a stress near \(8848 \times 3000 \times 10 \approx 2.7 \times 10^8\,\text{Pa}\) on the rock at its base. That is close to the stress at which rock begins to flow. A mountain much higher would slowly sink and spread under its own weight. The limit is set by the strength of rock and by \(g\). That is why mountains on Mars, with a smaller \(g\), can be far higher.
Return to the bangle and the wire. The glass bangle is brittle: its curve stops near the elastic limit, and a sharp knock is enough. The copper wire is ductile: past its yield point it bends and stays bent. Only the back-and-forth of fatigue, or a long neck, finally breaks it. One curve, read to its end, tells which is which.
- Thinking the metal weakens after the ultimate stress.The plotted stress uses the original area; the true stress in the neck still rises.
- Treating the proportional limit and the elastic limit as the same point.Hooke's law ends at the proportional limit; the wire still returns up to the elastic limit, a little beyond.
- Believing a load below the breaking stress can be repeated for ever.Repeated loading causes elastic fatigue; working stresses carry a safety factor.
- Making a beam stiffer by making it wider.The sag falls as the cube of the depth and only as the first power of the width.