Materials: Stress, Strain and Young's Modulus

Stress, strain and the Young modulus

When a material is stretched, we describe its behaviour using stress, strain and the Young modulus.

Stress

Force per unit cross-sectional area:

stress = force ÷ area      (σ = F ÷ A)   — units: pascals (Pa) = N/m²

Strain

The fractional change in length (no units):

strain = extension ÷ original length   (ε = ΔL ÷ L)

Young modulus

A measure of stiffness — the ratio of stress to strain in the linear region:

Young modulus E = stress ÷ strain = (F L) ÷ (A ΔL)   — units: Pa

A larger Young modulus means a stiffer material.

Force–extension graphs

• Hooke's law region: straight line — extension ∝ force (up to the limit of proportionality).
• Elastic limit: beyond this the material won't return to its original length.
• Elastic deformation: returns to original shape when the force is removed.
• Plastic deformation: permanent change of shape.
• The area under a force–extension graph = work done / elastic strain energy stored (= ½Fx in the linear region).

Material properties

• Brittle (e.g. glass): little plastic deformation, snaps at the elastic limit.
• Ductile (e.g. copper): can be drawn into wires (large plastic region).
• Strong: high breaking stress; stiff: high Young modulus; tough: absorbs a lot of energy before breaking.

Stress–strain graphs

Show intrinsic material behaviour (independent of the sample's dimensions): the gradient of the linear part = the Young modulus; the area under = energy per unit volume.

Worked example

A wire of length 2.0 m and cross-sectional area 1.0 × 10⁻⁶ m² extends 1.0 mm under a 50 N force. Find the Young modulus.

• Stress = F/A = 50 ÷ 1.0×10⁻⁶ = 5.0 × 10⁷ Pa.
• Strain = ΔL/L = 0.001 ÷ 2.0 = 5.0 × 10⁻⁴.
• E = stress/strain = 5.0×10⁷ ÷ 5.0×10⁻⁴ = 1.0 × 10¹¹ Pa. ✓

Common mistakes

• Confusing stress (force/area) with strain (extension/length).
• Forgetting strain has no units and stress is in pascals.
• Reading the Young modulus from a force–extension graph — it's the gradient of a stress–strain graph.

Exam tips

• Learn the three equations and their units.
• Distinguish elastic vs plastic deformation and brittle/ductile behaviour.
• Remember energy stored = area under the force–extension graph.

Key facts to remember

• Stress = F/A (Pa), strain = ΔL/L (no units), Young modulus E = stress/strain (stiffness).
• Force–extension: Hooke's law (linear) up to the limit of proportionality; elastic (returns) vs plastic (permanent); area = energy stored.
• Brittle vs ductile; the Young modulus is the gradient of the linear part of a stress–strain graph.