Stress and Strain

Force and extension describe what happened to one particular object. Stress and strain describe what happened to the material, independently of the size and shape of the sample.

Tensile stress — force per unit cross-sectional area:

σ = F / A

Measured in pascals, the same unit as pressure.

Tensile strain — extension per unit original length:

ε = x / L

A ratio of two lengths, so strain has no unit. It is often quoted as a percentage or as a small decimal such as 6.0 × 10⁻⁴.

Example: why bother? A thick steel cable and a thin steel wire have completely different force–extension graphs, but identical stress–strain graphs, because they are the same material. Stress and strain strip out the geometry and leave the property of the substance itself.

Breaking stress (ultimate tensile strength) — the maximum stress a material can withstand before it fractures. It is a property of the material, which is precisely why it is quoted as a stress rather than a force: a force at which a sample breaks would depend on how thick that sample was.

Why thin wires break more easily — not because the material is weaker, but because the same force acting over a smaller area produces a larger stress. Halving the cross-sectional area doubles the stress for the same load.

Example: a wire of cross-sectional area 2.0 × 10⁻⁶ m² carrying 80 N is under a stress of 80 ÷ (2.0 × 10⁻⁶) = 4.0 × 10⁷ Pa. If the diameter were halved, the area would fall by a factor of four and the stress would rise to 1.6 × 10⁸ Pa — the same load, four times the stress.

Working with diameters — measurements are usually of diameter, so remember A = πd²/4, or πr² with r as half the diameter. Since the area depends on the square of the diameter, the percentage uncertainty in the diameter doubles when it becomes the uncertainty in the area, which makes the diameter the measurement most worth taking carefully.