Stress, Strain and Hooke's Law: Quick Revision

A brief note on Stress, Strain and Hooke's Law. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Elastic and plastic: an elastic body returns to its shape when the force goes; a plastic one does not.
• Stress $= \dfrac{F}{A}$, the restoring force per unit area, in pascal. Strain $= \dfrac{\Delta L}{L}$, with no unit.
• Hooke's law: within the proportional limit, stress is in proportion to strain.
• The stress-strain curve: proportional limit, elastic limit, yield point, ultimate strength, breaking point. Past the elastic limit some stretch stays.
• Young's modulus: $Y = \dfrac{FL}{A\,\Delta L}$, a property of the material. Ductile materials flow a long way before breaking; brittle ones do not.

The facts at a glance

A worked example

A brass wire $2.8\ \text{m}$ long and $0.90\ \text{mm}$ across holds a $45\ \text{N}$ load. ($Y = 9.0 \times 10^{10}\ \text{Pa}$.)

1. Area. $A = \pi(0.45 \times 10^{-3}\ \text{m})^2 = 6.36 \times 10^{-7}\ \text{m}^2$.
2. Stress. $\dfrac{45\ \text{N}}{6.36 \times 10^{-7}\ \text{m}^2} = 7.07 \times 10^{7}\ \text{Pa}$.
3. Strain. $\dfrac{7.07 \times 10^{7}\ \text{Pa}}{9.0 \times 10^{10}\ \text{Pa}} = 7.86 \times 10^{-4}$.
4. Stretch. $7.86 \times 10^{-4} \times 2.8\ \text{m} = 2.20 \times 10^{-3}\ \text{m}$, that is $2.20\ \text{mm}$.

Practice with new numbers

A program to try

The program works out the brass wire of the worked example.

Mistakes to avoid

• The diameter used as the radius. Halve it first; the area goes as $r^2$.
• Millimetres left in a formula in metres. Change the stretch and the diameter to metres.
• Force for stress. A wire breaks at a stress, so a thicker wire holds more.
• Hooke's law past the proportional limit. It holds only on the straight part.
• $Y$ depending on the size of the wire. It belongs to the material.

Check yourself

1. A load gives a wire a stress of $5.0 \times 10^{7}\ \text{Pa}$ and a strain of $2.5 \times 10^{-4}$. What is $Y$?

2. Two wires of the same metal have the same length; one has three times the diameter. The same load hangs on each. Compare their stretches.

3. Which point on the stress-strain curve marks the greatest stress a wire can bear?

4. Is glass ductile or brittle?

5. A wire stretches $0.8\ \text{mm}$ under a load. The same metal, twice as long and with the same diameter, carries the same load. How far does it stretch?