The strain equation

The strain equation is tensile strain = ΔL ÷ L: the extension of a material divided by its original length. Because both are lengths, strain has no units; it is often given as a decimal or a percentage. A 2.0 m wire that stretches by 1.5 mm has a strain of 0.0015 ÷ 2.0 = 7.5 × 10⁻⁴, or 0.075%. Strain goes with tensile stress, force ÷ area, and the ratio of the two is the Young modulus.

The strain equation and the stress equation

AQA's A-level data booklet gives tensile strain = ΔL ÷ L, tensile stress = F ÷ A and Young modulus = tensile stress ÷ tensile strain. Its specification covers tensile strain and stress, elastic strain energy, breaking stress and stress-strain curves.

Why use strain instead of extension?

A long wire stretches more than a short wire of the same material under the same force. Dividing by the original length removes that effect, so strain describes the material rather than the particular sample. Stress does the same for thickness: a thicker wire needs more force for the same stress.

The Young modulus

The Young modulus measures how stiff a material is. A high value means a large stress is needed for a small strain. It only applies while the material obeys Hooke's law, in the straight-line part of a stress-strain graph. AQA's specification asks students to use stress-strain graphs to find the Young modulus, and its required practicals include determining the Young modulus by a simple method.

Here x is the strain in units of 10⁻³ and y is the stress in MPa. The gradient, stress ÷ strain, is the Young modulus: here 250 MPa per 10⁻³, which is 2.5 × 10¹¹ Pa. A real material's graph stays straight only up to its limit of proportionality.

Strain equation worked examples

Example 1: finding strain

A wire 2.0 m long stretches by 1.5 mm. Find the strain.

Convert: 1.5 mm = 1.5 × 10⁻³ m.

As a percentage, that is 0.075%. There are no units.

Example 2: finding stress

The wire has a diameter of 1.0 mm and carries a load of 150 N. Find the stress.

The cross-section is a circle of radius 0.50 mm = 0.50 × 10⁻³ m:

Example 3: the Young modulus

Using Examples 1 and 2:

Example 4: finding an extension

A wire of length 1.5 m and cross-sectional area 2.0 × 10⁻⁷ m² is made of a material with E = 1.2 × 10¹¹ Pa. It carries a load of 30 N. Find its extension.

Stress = 30 ÷ 2.0 × 10⁻⁷ = 1.5 × 10⁸ Pa. Strain = stress ÷ E = 1.5 × 10⁸ ÷ 1.2 × 10¹¹ = 1.25 × 10⁻³.

The wire stretches by about 1.9 mm.

Example 5: elastic strain energy

How much energy is stored in the wire from Examples 1 and 2?

AQA's booklet gives energy stored = ½FΔL, the area under the force-extension graph while Hooke's law applies:

Here is a quick check.

The Hooke's Law Calculator finds force, extension and the spring constant, the force-extension side of the same behaviour. Our guide to the spring constant equation covers F = kΔL.

Try it

Hooke's Law Calculator

Find the spring constant, force and extension

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Reading a stress-strain graph

• Limit of proportionality: the end of the straight-line section. Up to here, stress is proportional to strain.
• Elastic limit: beyond this, the material does not return to its original length when the load is removed.
• Plastic region: the material stretches permanently, and the curve flattens.
• Breaking stress (ultimate tensile stress): the highest stress the material can withstand before it breaks.

A brittle material, such as glass, follows a straight line almost to the point of fracture, with little or no plastic region. A ductile material, such as copper, has a long plastic region before it breaks. AQA's specification includes describing plastic behaviour, fracture and brittle behaviour from force-extension graphs.

Common mistakes

• Giving strain units. Strain is a ratio of two lengths, so it has none.
• Extension in mm, length in m. Convert both to the same unit before dividing.
• Using the diameter as the radius when working out the cross-sectional area.
• Using the new length instead of the original length on the bottom of the strain equation.
• Using E beyond the limit of proportionality. The Young modulus only applies in the straight-line region.

Practise materials in a short timed paper

A-level Physics: Year 1 gives you a short timed paper every day, with 8 questions worth 25 to 35 marks across the course, so materials and the Young modulus turn up regularly but not in every paper.

Read next

Physics · MechanicsStress strain graph: the key points, gradient and areaStress-strain graphs for A-level Physics: the limit of proportionality, elastic limit, plastic region, breaking stress, Young modulus and area.