The big idea: Young's modulus E = tensile stress σ ÷ tensile strain ε.
Stress σ = F ÷ A — the force divided by the ORIGINAL cross-sectional area, in pascals. Strain ε = ΔL ÷ L — the extension divided by the original length, with no units at all.
The test piece beside the graph, with each quantity measured on the thing it comes from.
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| On the graph | What it is | What it tells a designer |
|---|---|---|
| Gradient of the straight part | Young's modulus, E | Stiffness — how much the part will deflect under a given load |
| End of the straight part | Yield strength | The stress after which the part is permanently changed and must be retired |
| The peak | Ultimate tensile strength | The highest stress it will carry — the breaking figure on a specification |
| The end of the curve | Fracture | Where it actually breaks, and how much it stretched before it did |
| The area underneath | Toughness | Energy absorbed before fracture, which is what matters for an impact |
Typical values, for checking an answer: Steel about 200 GPa, aluminium about 70 GPa, most timbers 10-15 GPa, most polymers 1-3 GPa, rubber well under 0.1 GPa.
An answer in kilopascals or terapascals has a unit error in it, and this list is the fastest way to catch one.
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Four steps, in this order
Area, in square metres
A 10 mm × 5 mm bar is 50 mm², which is 50 × 10⁻⁶ m². Converting first is what stops the answer being out by a million.
Stress = F ÷ A
3,000 N ÷ 50 × 10⁻⁶ m² = 60 × 10⁶ Pa, which is 60 MPa.
Strain = ΔL ÷ L
0.6 mm ÷ 200 mm = 0.003. Both lengths in the same unit, and the answer has no units.
E = σ ÷ ε
60 × 10⁶ ÷ 0.003 = 20 × 10⁹ Pa = 20 GPa — a stiff timber or a filled polymer, which is the sanity check.
The two errors that cost the marks: Leaving the area in square millimetres, which makes the stress a million times too small.
And taking the gradient past the yield point, where the line is no longer straight and the value is meaningless.
How this is tested — calculating Young's modulus and interpreting stress-strain graphs. It comes up two ways:
Paper 1 — multiple choice
- Read the yield or ultimate strength off a graph.
- Calculate a stress, a strain or a modulus.
Paper 2 — analysing a product
- Calculate Young's modulus from test data.
- Interpret a stress-strain graph for a named product.
The trap: Working in millimetres. Convert the area to square metres before dividing, and state the unit at every stage.
A tie bar of cross-section 8 mm × 4 mm and length 500 mm carries 4,800 N and extends by 0.75 mm. Apply the formulae to find the stress, the strain and Young's modulus, and identify a likely material.
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