Anomalous expansion of water and thermal stress

Water does not follow the usual rule between 0 °C and 4 °C: it contracts on heating in this range. From 4 °C upwards it expands on heating like other liquids.

So water has its largest density at 4 °C. Its volume is least there and the density falls on either side.

Consequence for lakes and ponds in winter: water cooled at the surface sinks until the whole body reaches 4 °C. Colder water then stays on top, and the lake freezes from the surface downwards.

The ice layer and the cold water above the depths act as insulators, so the water at the bottom stays near 4 °C and water life survives the winter. If water behaved normally, lakes would freeze from the bottom up.

Thermal stress: a rod whose ends are held fixed cannot expand when heated, so its supports compress it. The compressive strain equals the expansion that was prevented, αΔT.

Stress = Y × strain = YαΔT. For a steel rail 5 m long and 40 cm² in section, warmed by 10 °C with α = 1.2 × 10⁻⁵ K⁻¹ and Y = 2 × 10¹¹ N m⁻²: strain 1.2 × 10⁻⁴, stress 2.4 × 10⁷ N m⁻², force 2.4 × 10⁷ × 40 × 10⁻⁴ ≈ 10⁵ N. This is why rails and bridges have expansion gaps.

A blacksmith fixes an iron ring on the wooden rim of a horse-cart wheel. At 27 °C the ring's diameter is 5.231 m and the rim's is 5.243 m; with α = 1.20 × 10⁻⁵ K⁻¹ the ring must be heated to about 218 °C. It fits hot and grips the rim tightly as it cools.

Working for the ring: ΔL = 5.243 − 5.231 = 0.012 m = α L ΔT, so ΔT = 0.012/(1.20 × 10⁻⁵ × 5.231) ≈ 191.2 K, and T = 27 + 191.2 ≈ 218 °C.