Elasticity, stress and strain

Bodies that change shape

Pull a rubber band and it stretches; let go and it comes back. Pull a steel wire just as hard and you see nothing, but it has stretched too, by a tiny amount, and it also comes back.

A body that returns to its shape and size when the force is removed is elastic. One that keeps its new shape, like clay, is plastic. Steel is elastic up to a limit, which we meet in the next lesson.

Inside a solid, the atoms are held in place by forces that act like tiny springs. Pull the solid and the atoms move apart a little; the springs pull back. That pull back is the restoring force.

Stress

A load that a thick rope holds easily can snap a thin thread. What matters is the force on each square metre of the cross-section. That is the stress:

Its unit is $\text{N/m}^2$, the pascal (Pa). The numbers are large, so we often use $\text{MPa} = 10^{6}\ \text{Pa}$. A handy fact: $1\ \text{N/mm}^2 = 1\ \text{MPa}$.

There are three kinds of stress.

• Longitudinal: the force is at right angles to the face. A pull gives tensile stress, a push compressive stress.
• Shear: the force acts along the face, as when you push the cover of a thick book sideways.
• Hydraulic: a pressure acts on every face, as on a stone at the bottom of a lake.

Strain

Strain is the fractional change, so it has no unit.

• Longitudinal strain: $\dfrac{\Delta L}{L}$.
• Shear strain: the angle $\theta$ through which the sides tilt (in radians).
• Volume strain: $\dfrac{\Delta V}{V}$.

For metals, strains are tiny, usually well under $0.01$.

See how much the diameter matters.

You should find that half the diameter gives four times the stress: the area goes as $d^2$.

Watch out. A wire breaks when its stress is too large, not its force. That is why a thicker rope holds more.

A worked example

A fishing line $0.50\ \text{mm}$ across holds a $3.0\ \text{kg}$ fish. The $1.6\ \text{m}$ line stretches by $2.4\ \text{cm}$. Find the stress and the strain.

1. Area. $A = \pi(0.25 \times 10^{-3}\ \text{m})^2 = 1.96 \times 10^{-7}\ \text{m}^2$.
2. Stress. $\dfrac{F}{A} = \dfrac{3.0\ \text{kg} \times 9.8\ \text{m/s}^2}{1.96 \times 10^{-7}\ \text{m}^2} = 1.50 \times 10^{8}\ \text{Pa} = 150\ \text{MPa}$.
3. Strain. $\dfrac{\Delta L}{L} = \dfrac{0.024\ \text{m}}{1.6\ \text{m}} = 0.015$.

Think first. Nylon stretches far more than steel, so a strain of $0.015$ is reasonable for a fishing line, though huge for a steel wire.

Practise with new numbers

Try it in Python

The program works out the fishing line of the example.

Remember these

• Stress $= \dfrac{F}{A}$, in pascal; $1\ \text{N/mm}^2 = 1\ \text{MPa}$.
• Strain is a fraction with no unit: $\dfrac{\Delta L}{L}$, the angle $\theta$, or $\dfrac{\Delta V}{V}$.
• Three kinds: longitudinal (tensile or compressive), shear, hydraulic.

Check yourself

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

1. A pillar with a cross-section of $0.50\ \text{m}^2$ carries $2.0 \times 10^{6}\ \text{N}$. What is its stress, and of which kind?

2. Two wires of the same metal carry the same load. One is three times as thick. How do their stresses compare?

3. Why does strain have no unit?

4. A sheet of rubber on a table is pushed sideways along its top face. What kind of stress acts in it?