Stress, strain and the Young modulusEdexcel International A Level Physics: Revision notes

Tensile stress is the force per unit cross-sectional area:

σ = F / A, in pascals (Pa = N m⁻²)

Tensile strain is the extension per unit original length:

ε = ΔL / L, with no unit (a ratio, sometimes given as a percentage)

For a compression the same definitions apply, using the compressive force and the decrease in length.

For a wire of diameter d, the area is A = πd²/4. Convert mm to m before calculating: 0.80 mm = 0.80 × 10⁻³ m.

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

E = σ / ε = (F/A) / (ΔL/L), in Pa

Unlike stiffness k, E depends only on the material, not on the size or shape of the sample. A large E means a stiff material (steel is about 2 × 10¹¹ Pa).

Worked example: a wire of length 2.50 m and diameter 0.80 mm carries 120 N and extends by 3.0 mm. A = π(0.40 × 10⁻³)² = 5.0 × 10⁻⁷ m²; σ = 120 / 5.0 × 10⁻⁷ = 2.4 × 10⁸ Pa; ε = 3.0 × 10⁻³ / 2.50 = 1.2 × 10⁻³; E = 2.4 × 10⁸ / 1.2 × 10⁻³ = 2.0 × 10¹¹ Pa.

A stress-strain graph has stress on the vertical axis and strain on the horizontal axis.

• In the straight section the gradient is the Young modulus.
• The graph curves after the limit of proportionality. A ductile metal then shows elastic limit, yield and a long region of plastic deformation.
• The breaking stress is the stress at which the material fractures.
• A brittle material, such as a ceramic or glass, gives a straight line up to fracture with no plastic region, so it breaks suddenly.

Because stress and strain remove the effect of size, the stress-strain graph is the same for any sample of the same material, whereas the force-extension graph depends on the dimensions of the sample.

Use a long, thin wire (large extension for a given load, small area so large stress), clamped at one end and passing over a pulley with masses on the free end.

1. Measure the original length L from the clamp to a marker with a metre rule.
2. Measure the diameter with a micrometer at several points and in perpendicular directions; take the mean and find A = πd²/4.
3. Add masses in equal steps, measuring the extension from the marker against a scale each time.
4. Plot force against extension: a straight line within the limit of proportionality, gradient = EA/L.
5. E = gradient × L / A.

Safety: wear goggles and use a box of sand under the masses in case the wire snaps. Do not exceed the elastic limit.