Hooke's law and the stress-strain curve

Hooke's law

Hang loads on a wire, one after another, and measure each stretch. For small loads, twice the load gives twice the stretch. Robert Hooke found this in 1676:

The constant is a modulus of elasticity. For stretching it is Young's modulus, $Y$, which we meet in the next lesson.

The stress-strain curve

Now keep pulling until the wire breaks. A metal such as steel gives a curve like this.

• O to A: a straight line. Hooke's law holds; A is the proportional limit.
• A to B: no longer quite straight, but the wire still comes all the way back. B is the elastic limit.
• B to C: past the elastic limit, part of the stretch stays when you let go. At C, the yield point, the wire starts to flow.
• C to D: the wire stretches a lot for little extra stress. D is the ultimate tensile strength, the greatest stress it can take.
• D to E: one part gets thinner (it 'necks'), and the wire breaks at E.

Try loading and unloading a steel wire.

You should find that below the elastic limit the wire comes back to zero, but past it a permanent stretch stays, because letting go follows a line with the same slope as the start.

Watch out. The elastic limit and the proportional limit are close, but not the same point.

Ductile and brittle

A ductile material, such as copper or mild steel, has a long stretch between D and E, so it can be drawn into wires. A brittle material, such as glass or cast iron, breaks soon after its elastic limit. Rubber is different again: it stretches to several times its length and still comes back, but it does not obey Hooke's law.

A worked example

A steel wire $1.2\ \text{mm}$ across has an elastic limit of $250\ \text{MPa}$. What is the largest mass it can hold and still come back?

1. Area. $A = \pi(0.60\ \text{mm})^2 = 1.131\ \text{mm}^2$.
2. Largest force. $250\ \text{N/mm}^2 \times 1.131\ \text{mm}^2 = 283\ \text{N}$.
3. Mass. $\dfrac{283\ \text{N}}{9.8\ \text{m/s}^2} = 28.9\ \text{kg}$.

Think first. Engineers never load a wire that close to its limit; they use a safety factor, as we see in the last course.

Practise with new numbers

Try it in Python

The program fits a straight line to a wire's first few readings and finds where it stops obeying Hooke's law.

Remember these

• Hooke's law: stress $\propto$ strain, within the proportional limit.
• On the curve: proportional limit, elastic limit, yield point, ultimate strength, breaking point.
• Past the elastic limit, some stretch stays.
• Ductile: long plastic region. Brittle: breaks soon after the elastic limit.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A load of $3\ \text{kg}$ stretches a spring-like wire by $0.9\ \text{mm}$. What does $5\ \text{kg}$ do, within the limit?

2. A wire is loaded past its yield point and then unloaded. Does it return to its first length?

3. Why can copper be drawn into wire but glass cannot?

4. Does rubber obey Hooke's law over its whole stretch?