Last updated: 2026-06-15

Young's Modulus

Young's modulus is a single number that captures how stiff a material is — how hard you have to pull to stretch it by a given fraction. Steel resists stretching about three times more strongly than aluminium and thousands of times more than rubber, and that difference is exactly what the modulus measures. This calculator works it out from a simple tension test: a force pulling on a sample of known area and length that stretches by a measured amount.

Where the formula comes from

Pulling on a bar produces two things. The stress is the force shared over the cross-section, $\sigma = F/A$, measured in pascals. The strain is how much the bar stretches relative to its length, $\varepsilon = \Delta L / L$, a pure ratio. In the elastic region — before any permanent bending or breaking — stress and strain stay proportional, and the constant that links them is Young's modulus, $E = \sigma/\varepsilon$. Substituting the two definitions gives $E = (F\,L)/(A\,\Delta L)$.

Formula

Because strain has no units, $E$ carries the units of stress — pascals — and is most often quoted in gigapascals (GPa).

Worked example

A steel rod with a 1 cm² cross-section ($A = 1\times10^{-4}\ \text{m}^2$) and length $L = 2\ \text{m}$ is pulled with $F = 10\,000\ \text{N}$ and stretches by $\Delta L = 1\ \text{mm}$:

That 200 GPa is the textbook value for structural steel.

Typical values

The range across materials is enormous: rubber sits near 0.01–0.1 GPa, wood and bone around 10–15 GPa, aluminium at 69 GPa, steel near 200 GPa, and diamond above 1000 GPa. A higher modulus means the material barely yields under load — useful for a bridge girder, unhelpful for a shoe sole.

Limitations

The modulus only describes the elastic region, where the material springs back to its original length once the load is removed. Past the yield point the relationship stops being linear, the sample deforms permanently, and Young's modulus no longer applies. It also measures stiffness, not strength: a stiff material can still be brittle and snap at a low load. The test assumes a uniform sample loaded purely in tension along its axis, with the force spread evenly over the cross-section.