Stress, Strain and the Young Modulus

Hooke's law tells us how a particular spring behaves under load — but it cannot tell us how the material itself behaves, independent of the object's shape. To say something universal about, say, steel or copper, we must normalise the force by the cross-sectional area and the extension by the original length. The result is a pair of dimensionless(-ish) quantities — stress and strain — and their ratio, the Young modulus, which is a fundamental property of the material.

This lesson covers OCR Module 3.4.2 (Young modulus) and includes the Searle's apparatus practical (PAG 6) and the interpretation of stress-strain graphs.

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Tensile Stress

Tensile stress is the force per unit cross-sectional area acting at right angles to the face of the sample:

σ = F / A

• F = force applied (N)
• A = cross-sectional area perpendicular to the force (m²)
• σ = stress (N m⁻² = pascals, Pa — same unit as pressure)

A typical steel wire might safely carry stresses up to about 2 × 10⁸ Pa (200 MPa). The failure stress of structural steel is around 400 MPa, and that of high-tensile wires can exceed 2 GPa.

Worked Example — Stress in a Wire

A wire of diameter 0.50 mm supports a 4.0 kg load. Find the tensile stress.

• Cross-sectional area A = π r² = π × (0.25 × 10⁻³)² = 1.963 × 10⁻⁷ m²
• Force F = mg = 4.0 × 9.81 = 39.24 N
• Stress σ = F / A = 39.24 / 1.963 × 10⁻⁷ = 2.0 × 10⁸ Pa = 200 MPa

This is getting close to the safe working stress of mild steel — the wire would be near its limit.

Common Exam Mistake: Using the diameter instead of the radius in πr². Halve the diameter first.

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Tensile Strain

Tensile strain is the fractional change in length of the sample:

ε = Δℓ / ℓ₀

• Δℓ = extension (m)
• ℓ₀ = original length (m)
• ε = strain (dimensionless, but often written as a percentage)

A strain of 0.001 (or 0.1%) means the sample has extended by one-thousandth of its original length. A 1 m wire with strain 0.001 has extended by 1 mm.

Worked Example — Strain in a Rod

A 2.40 m steel rod extends by 3.0 mm when loaded. Find the strain.

ε = 0.0030 / 2.40 = 1.25 × 10⁻³ = 0.125%

This is a very small extension in percentage terms — typical of metals before yield.

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The Young Modulus

For a material within its elastic limit, stress is proportional to strain:

σ ∝ ε

The constant of proportionality is called the Young modulus (sometimes "Young's modulus"), denoted E (though in some textbooks Y):

E = σ / ε = (F × ℓ₀) / (A × Δℓ)

• SI unit: Pa, but values are typically large so GPa (10⁹ Pa) is standard.

Typical Values

Material	Young modulus (GPa)
Rubber	0.01–0.1
Polyethylene	0.3
Wood (along grain)	10–15
Bone	14
Concrete	20
Glass	70
Aluminium	69
Brass	100
Copper	117
Steel	200
Tungsten	400
Diamond	1050

A larger Young modulus means a stiffer material — one that resists deformation. Steel (E ≈ 200 GPa) is twice as stiff as copper (117 GPa) and 2000 times as stiff as rubber.

Exam Tip: The Young modulus is an intrinsic property of the material. It does not depend on the shape or size of the specimen. This distinguishes E from the spring constant k, which depends on both the material and the geometry.

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Worked Example — Young Modulus of a Wire

A 1.80 m copper wire of diameter 0.40 mm is stretched by 4.0 kg. The extension is 2.25 mm. Calculate the Young modulus.

• Force F = 4.0 × 9.81 = 39.24 N
• Area A = π × (0.20 × 10⁻³)² = 1.257 × 10⁻⁷ m²
• Stress σ = F/A = 39.24 / 1.257 × 10⁻⁷ = 3.12 × 10⁸ Pa
• Strain ε = 2.25 × 10⁻³ / 1.80 = 1.25 × 10⁻³
• Young modulus E = σ/ε = 3.12 × 10⁸ / 1.25 × 10⁻³ = 2.50 × 10¹¹ Pa = 250 GPa

The literature value for copper is 117 GPa; our measured value is too high by a factor of 2. A real experimental result close to 120 GPa would be satisfactory. OCR exam mark schemes typically allow a wide range.

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