Definition

Young's modulus

Y = \frac{F / A}{Δ L / L _{0}}

Statement

The ratio of tensile (or compressive) stress to strain, describing a material's stiffness along its length.

Symbols and units

$Y$	Young's modulus	Pa
$F / A$	stress	Pa
$Δ L / L_{0}$	strain	—

When it applies

Tensile or compressive loading, within the elastic limit.

Don't use when

Valid only within the elastic limit — beyond it deformation is permanent and stress is no longer proportional to strain; also does not describe shear deformation, which uses the shear modulus instead.

Derivation

Within the elastic limit, stress is observed to be proportional to strain (Hooke's law extended to materials); the constant of proportionality is Young's modulus:

Y = \frac{stress}{strain} = \frac{F / A}{Δ L / L _{0}}

Worked examples

Elongation of a steel wire

A steel wire (Y ≈ 2.0×10¹¹ Pa), A = 1.0×10⁻⁶ m², L₀ = 2.0 m, under F = 200 N:

Δ L = \frac{F L _{0}}{Y A} = \frac{( 200 ) ( 2.0 )}{( 2.0 \times 1 0 ^{11} ) ( 1.0 \times 1 0 ^{- 6} )} = 2.0 mm

Watch out for

Valid only within the elastic limit — beyond it the material deforms permanently, and stress is no longer proportional to strain. Shear modulus and bulk modulus follow the same ratio for shearing and volume deformation respectively.

Connects to

Stress and strain