Thermal Expansion Calculator

Solids expand on heating because the interatomic potential well is asymmetric. The repulsive wall on the short-separation side is steeper than the attractive tail, so as vibrational amplitude grows the time-averaged separation shifts outward. A perfectly harmonic solid would vibrate more vigorously but would not expand at all, which is why thermal expansion is fundamentally an anharmonic effect. The coefficient of linear expansion is defined as alpha = (1/L)*(dL/dT) at constant pressure, and the working formula delta-L = alpha*L0*delta-T is the first-order expansion of that definition. Area and volume coefficients follow as 2*alpha and 3*alpha only for isotropic materials; anisotropic crystals such as graphite and calcite have direction-dependent coefficients, and some are negative along one axis. Typical room-temperature values in units of 1e-6 per K are 0.55 for fused quartz, 1.2 for Invar, 3.3 for borosilicate glass, 8.5 for soda-lime glass, 11.8 for iron, about 12 for both structural steel and concrete, 14.2 for gold, 16.5 for copper, 19 for brass, 23.1 for aluminium and roughly 200 for polyethylene. The near-match between steel and concrete is what makes reinforced concrete viable. Two limitations matter. Alpha is itself temperature dependent, falling toward zero as absolute zero is approached in step with the heat capacity, so a single value is only valid over a modest range. And when expansion is restrained the material develops thermal stress of order E*alpha*delta-T: steel with a Young modulus of 200 GPa held rigid through a 50 K rise sees about 120 MPa, comparable to the yield strength of mild steel.

Thermal expansion was exploited long before it was quantified. Galileo's thermoscope of about 1592 read temperature from the expansion of trapped air, and Daniel Fahrenheit's mercury-in-glass thermometer of 1714 depended on mercury expanding considerably more than the glass containing it. John Harrison built the gridiron pendulum around 1726, pairing brass and steel rods so their opposing expansions cancelled and the effective pendulum length stayed constant as workshop temperature drifted, one of the earliest deliberate engineering compensations for the effect. Careful quantitative measurements of the expansion of mercury and of solids were made by Pierre Louis Dulong and Alexis Therese Petit around 1817. The decisive advance for precision work came in 1896, when Charles-Edouard Guillaume at the International Bureau of Weights and Measures near Paris discovered that a nickel-iron alloy of about 36 percent nickel had an almost vanishing expansion coefficient. He named it Invar, and its use in length standards, surveying tapes and clock pendulums earned him the 1920 Nobel Prize in Physics. On the theoretical side, Eduard Grueneisen showed early in the twentieth century that the expansion coefficient is proportional to the heat capacity, tying the effect directly to lattice vibrations. Modern civil practice reflects the same physics in expansion joints and in continuously welded rail, which is pre-stressed at a chosen neutral temperature to avoid buckling.