The Young Modulus

The Young modulus — the ratio of tensile stress to tensile strain, for a material within its limit of proportionality:

E = stress ÷ strain = σ / ε

Since strain has no unit, the Young modulus has the same unit as stress: the pascal. Values for metals are large — steel is about 2.1 × 10¹¹ Pa — which is why they are usually quoted in GPa.

What it means — the Young modulus measures stiffness: how much stress is needed to produce a given strain. A large Young modulus means a material that resists being stretched. It is a property of the material, so every steel sample has the same Young modulus regardless of its dimensions.

Stiffness is not strength — a stiff material resists deformation; a strong material resists breaking. Glass has a high Young modulus and is stiff, yet it is brittle and breaks easily. The two properties are independent and describe different things.

In terms of measurable quantities —

E = FL / (Ax)

which is what you actually calculate from an experiment: force, original length, cross-sectional area and extension.

Measuring E for a wire — a long, thin wire is clamped at one end and loaded at the other, with a marker on the wire read against a fixed rule. The diameter is measured with a micrometer at several points along the wire and averaged, since wires are rarely perfectly uniform.

Why the wire is long and thin — a long wire gives a larger extension for the same strain, making the extension easier to measure accurately. A thin wire gives a larger stress for the same load, so smaller masses produce measurable strains. Both choices reduce the percentage uncertainty in the final answer.

Getting E from a graph — plot stress against strain and take the gradient of the straight section. Using a gradient rather than a single pair of readings averages out random error and shows immediately where the straight-line region ends.

Example: a stress of 4.0 × 10⁷ Pa producing a strain of 2.0 × 10⁻⁴ gives E = (4.0 × 10⁷) ÷ (2.0 × 10⁻⁴) = 2.0 × 10¹¹ Pa.