Young Modulus
A steel guitar string and a rubber band can be stretched by the same strain, yet the force needed to do it differs by a factor of hundreds. That difference in stiffness — how strongly a material resists being deformed — is captured by a single number: the Young modulus.
Linking stress and strain
For many materials, over a certain range of loading, stress and strain turn out to be directly proportional to one another: double the stress, and the strain doubles too. This region of proportional behaviour is called the elastic region (covered in full in the next lesson), and within it, the ratio of stress to strain is constant for a given material. That constant is the Young modulus, given the symbol $E$:
Since stress is measured in pascals and strain is dimensionless, the Young modulus has the same units as stress: pascals (Pa), or more commonly in engineering contexts, gigapascals (GPa).
Crucially, $E$ is a genuine material property. Unlike raw stiffness (force per unit extension for a specific sample, which depends on the sample's length and cross-section), the Young modulus is the same for any two pieces of the same material regardless of their size or shape, because stress and strain have already been normalised for geometry in the previous lesson. This is exactly why $E$ is quoted in material data tables and used by engineers to compare materials directly, rather than having to account for the dimensions of whatever sample happened to be tested.
The Young modulus describes how much a material resists elastic deformation — how stiff it is — not how much load it can withstand before breaking or yielding. A material can have a very high Young modulus (very stiff, e.g. glass) while still being brittle and failing at relatively low stress. Stiffness and strength are related but distinct ideas, and this topic returns to strength in later lessons.
Typical values across engineering materials
The Young modulus varies enormously between different classes of material — often by several orders of magnitude — which is part of why it's such a useful quantity for material selection.
| Material | Approximate Young modulus, E |
|---|---|
| Rubber | ~0.01–0.1 GPa |
| Wood (along grain) | ~10 GPa |
| Glass | ~70 GPa |
| Aluminium | ~70 GPa |
| Concrete | ~30 GPa |
| Steel | ~200 GPa |
Notice the scale of the spread: steel is roughly 2000 times stiffer than rubber for the same applied stress. This is why a steel beam bends imperceptibly under loads that would visibly stretch a rubber band — it isn't (only) about strength, it's fundamentally about how each material's internal structure resists elastic deformation.
Worked example
This calculation — finding $E$ from measured stress and strain data, or using a known $E$ to predict extension under a given load — is one of the most common calculations in this topic, and it underlies how engineers size structural components before they're ever built.
The Young modulus can be found experimentally by measuring stress and strain together (the subject of the "Tensile Testing" lesson later in this topic) and finding the gradient of the resulting straight-line region — this is exactly what the next lesson on elastic deformation formalises. But $E$ only describes behaviour within that straight-line elastic region. Push a material beyond it, and the neat proportional relationship between stress and strain breaks down — which is the subject of the two lessons that follow.
Key takeaways
- The Young modulus $E = \sigma/\varepsilon$ is the constant of proportionality between stress and strain in a material's elastic (linear) region.
- $E$ has units of pascals (Pa), since strain is dimensionless — commonly quoted in GPa for engineering materials.
- $E$ is a true material property: independent of a specimen's size or shape, unlike raw stiffness (force per unit extension).
- Typical values span enormous ranges — from around 0.01–0.1 GPa for rubber up to around 200 GPa for steel — reflecting how dramatically stiffness varies between material classes.
- Given $E$, the applied stress (or force and area), and original length, the extension of a component can be predicted directly.
- A high Young modulus means high stiffness, not necessarily high strength — the two are related but distinct properties.
- $E$ only describes behaviour in the elastic region; beyond it, stress and strain are no longer simply proportional, which the next two lessons cover in detail.