A vegetable seller hangs a bag of onions on a spring balance, and the pointer moves along a scale. Look at the scale: the marks for 1 kg, 2 kg and 3 kg are equally far apart. The maker did not have to test every load. One load was tested, and the rest of the marks were drawn with a ruler.
Loading a wire, one mass at a time
Take a steel wire \(2\,\text{m}\) long with a cross-section of \(1\,\text{mm}^2\). Hang it from a firm support, and hang a slotted-weight hanger from its lower end. A pointer at the end of the wire moves along a millimetre scale. Add masses of \(5\,\text{kg}\), that is \(50\,\text{N}\), one at a time, and read the extension after each.
Watch the first five masses in Fig 2.1. Each \(50\,\text{N}\) adds exactly \(0.5\,\text{mm}\), so the points lie on a straight line through the origin. At \(100\,\text{N}\) the extension is \(1\,\text{mm}\). The load is proportional to the extension, \(F = k\,\Delta L\), with \(k = 100\,\text{N/mm} = 10^5\,\text{N/m}\).
Now turn load and extension into stress and strain. At \(100\,\text{N}\) the stress is \(100/10^{-6} = 10^8\,\text{Pa}\), and the strain is \(10^{-3}/2 = 5 \times 10^{-4}\). Their ratio is \(2 \times 10^{11}\,\text{Pa}\). At \(200\,\text{N}\) both double, and the ratio stays the same. This is Hooke's law.
Within the proportional limit, the stress in a body is proportional to the strain. The ratio of stress to strain is a constant for the material, called its modulus of elasticity.
The modulus \(E\) belongs to the material, not to one wire. A thicker or longer wire of the same steel has a different \(k\), but the same \(E\). The next three lessons name the three moduli. Young's modulus is for stretching, the bulk modulus for squeezing from all sides, and the shear modulus for sliding.
The spring of a balance stretches by \(2\,\text{cm}\) under a load of \(10\,\text{N}\). It obeys Hooke's law. How far does it stretch under \(25\,\text{N}\), in cm?
Where the straight line ends
Hooke's law has a limit. In Fig 2.1, past \(250\,\text{N}\) the points leave the dashed straight line. At \(300\,\text{N}\) the extension is \(3.2\,\text{mm}\) instead of \(3\,\text{mm}\), and at \(350\,\text{N}\) it is \(5.5\,\text{mm}\) instead of \(3.5\,\text{mm}\). The largest stress for which stress stays proportional to strain is the proportional limit. For this wire it is \(250/10^{-6} = 2.5 \times 10^8\,\text{Pa}\).
A second limit matters more in practice. The elastic limit is the largest stress from which the body still returns to its original length when unloaded. For steel it lies just above the proportional limit, so the two are often treated as one. Loaded only below the elastic limit, a wire always comes back.
The proportional limit is the largest stress up to which stress is proportional to strain. The elastic limit is the largest stress from which the body returns to its original size and shape when the load is removed.
The rest of the curve
A graph of stress against strain carried all the way to breaking has a few named points. Fig 2.2 shows them for a ductile metal and for a brittle solid. The next chapter, Elastic Energy, Thermal Stress and Real Materials, studies the whole curve; here we only name its points.
From O to A the curve is the straight line of Hooke's law. Its slope, stress over strain, is the modulus, so a steeper straight part means a stiffer material. Between A and B the body is still elastic, but stress is no longer proportional to strain. At B, the yield point, the material begins to flow: the strain grows a lot for little extra stress. The highest point C is the ultimate tensile strength, the largest stress the wire can bear. Beyond C one part of the wire thins into a neck, less stress is needed, and the wire breaks at D, the fracture point.
A metal like copper or mild steel has a long stretch between B and D. It can be drawn into wires and hammered into sheets, and it is called ductile. Glass or cast iron breaks at D very soon after its elastic limit, with almost no plastic stretch. It is called brittle. In the brittle chip of Fig 2.2 the line stays nearly straight to the break.
Breaking stress and the safety factor
The stress at which a wire breaks is its breaking stress, a property of the material. The largest load a wire can hold is therefore the breaking stress times the area. Engineers never load a wire near that. They divide the breaking load by a safety factor, often 4 to 10, and use only that much.
A steel wire has a cross-section of \(1\,\text{mm}^2\), and the breaking stress of the steel is \(5 \times 10^8\,\text{Pa}\). Take \(g = 10\,\text{m/s}^2\). (a) What load breaks the wire? (b) With a safety factor of 5, what mass may hang from it?
(a) \(F = \sigma A = 5 \times 10^8 \times 10^{-6} = 500\,\text{N}\).
(b) The safe load is \(500/5 = 100\,\text{N}\), which is the weight of \(10\,\text{kg}\). A thicker wire of the same steel holds more in proportion to its area, not its diameter.
On a stress-strain curve, up to which point is stress proportional to strain?
Which statements about ductile and brittle materials are true?
Loaded past the limit, and unloaded
Play Fig 2.1 to the end. From \(350\,\text{N}\) the masses come off one by one. The wire shortens by \(0.5\,\text{mm}\) for every \(50\,\text{N}\) removed, just as it lengthened on the straight part. So the unloading line is straight and parallel to the loading line OP. It reaches zero load at \(2.0\,\text{mm}\), not at zero.
The wire is now \(2\,\text{mm}\) longer than before, with no load at all. The strain that remains after the load is removed is the permanent set. Some bonds did not just stretch; planes of atoms slipped past one another into new places. Springs return, but slipped atoms do not slip back.
A wire is loaded beyond its elastic limit and then unloaded. On the stress-strain graph, what does the unloading path look like?
A spring balance is once overloaded far beyond its range. Afterwards, with nothing on its hook, the pointer rests below the zero mark. Explain why, and what that does to its readings.
The overload stressed the spring beyond its elastic limit, so the coils took a permanent set: with no load the spring stays longer than before. The pointer therefore rests below zero. Within its elastic range the spring still stretches in proportion to the load, but from the new starting length, so every reading is too high by the permanent set unless the balance is reset.
Now return to the vegetable seller's balance. Its spring works well below its elastic limit, where Hooke's law holds. The extension is proportional to the load, so equal loads move the pointer by equal steps, and the marks are evenly spaced. Overload it past the elastic limit, and the spring keeps a permanent set. The pointer no longer rests at zero, so every reading after that is wrong.
In physics, "more elastic" means "returns more exactly and resists deformation more strongly", not "stretches more easily". A rubber band stretches far and returns, but steel needs a far larger stress for the same strain and returns just as fully. So steel is more elastic than rubber; rubber is only more stretchable.
Robert Hooke was a curator of experiments at the Royal Society in London. In 1676 he printed an anagram, "ceiiinosssttuv", to claim a discovery without revealing it. Two years later he gave the answer in Latin: "ut tensio, sic vis", as the extension, so the force. He had tested springs, wires and even a dry wooden beam. He used his law to design spring balances and the balance springs of watches.
A gulel, the Y-shaped catapult of Indian childhood, uses a rubber strip stretched to twice its length or more. Rubber returns from these huge strains, so it is elastic. But its stress-strain graph is a curve almost from the start, so Hooke's law hardly applies. Elastic and proportional are two different properties: rubber has the first without the second.
- Treating the proportional limit and the elastic limit as the same point.The proportional limit ends the straight line; the elastic limit is where the body stops returning. For steel they are close, but they are two definitions.
- Expecting a wire unloaded from beyond the elastic limit to return along the loading curve.It returns along a line parallel to the straight part and keeps a permanent set.
- Scaling a safe load with the diameter.The load scales with the area, \(\pi d^2/4\): twice the diameter holds four times the load.