Young's modulus

The ratio of stress to strain within the proportional region OA is a property of the material, called a modulus of elasticity.

Young's modulus Y = tensile (or compressive) stress σ ÷ longitudinal strain ε = (F/A)/(ΔL/L) = FL/(AΔL). Experiment shows the same strain for equal tensile and compressive stress.

Strain has no unit, so Y has the unit of stress, N m⁻² or Pa. Metals have large Y; wood, bone, concrete and glass have rather small Y.

To stretch a wire of cross-section 0.1 cm² by 0.1% takes 2000 N for steel, and 690 N, 900 N and 1100 N for aluminium, brass and copper. Steel is therefore the most elastic of the four, which is why heavy machines and structures use it.

Steel rod, radius 10 mm, length 1.0 m, pulled with 100 kN, Y = 2.0 × 10¹¹ N m⁻²: stress = 3.18 × 10⁸ N m⁻², elongation = 1.59 mm, strain = 1.59 × 10⁻³ ≈ 0.16%.

Copper wire 2.2 m and steel wire 1.6 m, both 3.0 mm in diameter, joined end to end: same tension and area, so ΔLc/ΔLs = (Ys/Yc)(Lc/Ls) = 2.5 with Yc = 1.1 × 10¹¹ and Ys = 2.0 × 10¹¹ N m⁻². A total stretch of 0.70 mm splits as 0.50 mm + 0.20 mm, and the load is about 1.8 × 10² N.

Human pyramid: 280 kg in all, the bottom performer 60 kg, so his legs carry 220 kg, or 2156 N, 1078 N per thighbone. With L = 0.5 m, radius 2.0 cm (A = 1.26 × 10⁻³ m²) and Y(bone) = 9.4 × 10⁹ N m⁻², each bone shortens by 4.55 × 10⁻⁵ m, a fractional change of 0.0091%.