3.4MechanicsStretch

The Young Modulus

A thick steel cable and a thin steel wire stretch by very different amounts under the same force — but that difference is entirely down to their dimensions, not the steel itself. Strip away the effects of size altogether, using stress and strain instead of raw force and extension, and you’re left with a number that describes the material alone: the Young modulus.

40 min Video by Launchpad Learning Module 3.4 Materials

Stress, Strain & Young’s Modulus — A-level PhysicsWatch the full walkthrough before the notes below.

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What you'll be able to do

Define stress and strain
Define the Young modulus and use $E = \frac{σ}{ε}$
Interpret a stress–strain graph, including the elastic limit and breaking stress
Calculate the Young modulus from experimental data

Stress and strain

$Stress$ is the force applied per unit cross-sectional area of a material, measured in pascals (Pa). $Strain$ is the extension produced, expressed as a fraction of the material’s original length — strain has no units, since it is a ratio of two lengths.

σ = \frac{F}{A} ε = \frac{Δ x}{x _{0}}

Stress (σ) and strain (ε).

1Use

σ = \frac{F}{A}

σ = \frac{15}{2.5 \times 1 0 ^{- 7}}

Answer

σ = 6.0 \times 1 0^{7}

The Young modulus

The $Young modulus$ , $E$ , is the ratio of stress to strain for a material, within its elastic (Hooke’s law) region — a property of the material itself, independent of the sample’s particular dimensions. A larger Young modulus means a stiffer material, requiring more stress to produce the same strain.

E = \frac{σ}{ε} = \frac{F x _{0}}{A Δ x}

The Young modulus, measured in pascals.

1Stress:

σ = \frac{F}{A} = \frac{24}{3.0 \times 1 0 ^{- 7}} = 8.0 \times 1 0^{7}

Pa.

2Strain:

ε = \frac{Δ x}{x _{0}} = \frac{0.0012}{1.5} = 8.0 \times 1 0^{- 4}

3Young modulus:

E = \frac{σ}{ε} = \frac{8.0 \times 1 0 ^{7}}{8.0 \times 1 0 ^{- 4}}

Answer

E = 1.0 \times 1 0^{11}

Pa (100 GPa)

Tip — Because the Young modulus describes the material only, a thin wire and a thick cable of the exact same material have the SAME Young modulus — even though the thick cable is obviously much harder to stretch in absolute terms.

Stress–strain graphs

A stress–strain graph for a ductile material typically starts as a straight line through the origin (Hooke’s law region), with gradient equal to the Young modulus. Beyond the $elastic limit$ , the graph curves as the material begins to deform plastically; it eventually reaches the $breaking stress$ (ultimate tensile stress), the maximum stress the material can withstand before fracturing.

Tip — On a stress–strain graph, the Young modulus is read from the GRADIENT of the initial straight-line section only — never from a point on the curved region beyond the elastic limit.

Equation recap

σ = \frac{F}{A}

Stress.

ε = \frac{Δ x}{x _{0}}

Strain.

E = \frac{σ}{ε}

The Young modulus.

Common mistakes to avoid

Believing the Young modulus depends on a sample’s length or cross-sectional area.

The Young modulus is a property of the MATERIAL alone — stress and strain are specifically designed to cancel out the effects of a sample’s particular dimensions.

Reading the Young modulus from a point on the curved section of a stress-strain graph.

The Young modulus is the gradient of the initial straight-line (Hooke’s law) region only, before the elastic limit is reached.

Giving strain units (like "m" or "m/m").

Strain is a ratio of two lengths, so it has no units — it may be expressed as a plain number or a percentage, but never with a length unit.

Key takeaways

Stress σ = F/A; strain ε = Δx/x₀ (strain has no units).
Young modulus E = σ/ε, a property of the material alone, independent of a sample’s dimensions.
On a stress-strain graph, the Young modulus is the gradient of the initial straight-line section.
Beyond the elastic limit, the graph curves; breaking stress is the maximum stress before fracture.

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The Young Modulus

40 min Video by Launchpad Learning Module 3.4 Materials