Thermal Expansion

Why thermal expansion matters

Nearly every structure and machine on Earth sees a working temperature range far wider than most people assume. A steel bridge deck can sit at −20 °C on a clear winter night and reach +50 °C in direct summer sun; a jet-engine turbine disc swings from ambient to over 600 °C in seconds; a silicon die on a copper heat-sink cycles by tens of degrees every time a laptop wakes. Because the coefficient of linear expansion α of the materials involved almost never matches, those temperature swings turn into dimensional changes, relative motion, and — if the motion is blocked — enormous internal forces.

• Civil structures. Bridges, rails, pipelines, and buildings must be free to grow and shrink, or must be deliberately restrained and designed for the resulting stress.
• Precision machines. A coordinate-measuring machine or a lithography stepper drifts out of tolerance if its frame breathes with the room; metrology labs hold 20 °C to ±0.1 °C for exactly this reason.
• Electronics packaging. Solder joints fatigue and crack because silicon (α ≈ 2.6 ×10⁻⁶ /K) is bonded to substrates and boards with much higher α — the coefficient-of-thermal-expansion (CTE) mismatch is a leading cause of chip-package failure.
• Power and process plant. Steam pipes, boilers, and heat exchangers move centimetres as they warm; missing that motion snaps supports and nozzles.
• Everyday sensing. The same effect that is a nuisance is also a feature: bimetallic strips turn temperature directly into mechanical motion for thermostats and breakers.

How it works, step by step

Thermal expansion is fundamentally about the shape of the interatomic potential well. If atoms sat in a perfectly symmetric (harmonic) well, heating would increase their vibration amplitude but not their average position, and there would be no expansion. Real bonds are anharmonic: the repulsive wall on the short-range side is steeper than the attractive tail on the long-range side. As temperature rises, atoms spend more time on the shallow far side, so the average spacing grows. Expansion is the macroscopic sum of that atomic-scale asymmetry.

1. Start with free length. Take an unrestrained bar of original length L at reference temperature T₀.
2. Raise temperature by ΔT. Each unit length grows by the fractional strain ε = α·ΔT, so the whole bar grows by ΔL = α·L·ΔT.
3. Extend to area and volume. A square face of side L becomes (L + αLΔT)²; expanding and dropping the tiny (αΔT)² term gives ΔA ≈ 2α·A·ΔT. The same logic in three dimensions gives ΔV ≈ 3α·V·ΔT, so the volumetric coefficient β ≈ 3α for isotropic solids.
4. Now block the motion. Clamp both ends so the bar physically cannot lengthen. The thermal strain still "wants" to occur, but the constraint forces a compensating mechanical strain of −α·ΔT.
5. Read off the stress. By Hooke's law that mechanical strain implies a stress σ = E·(−α·ΔT), a compression of magnitude E·α·ΔT. Length has cancelled out entirely — a short stub and a long beam of the same material develop the same restrained stress for the same ΔT.
6. Combine differentials. Bond two materials with different α (a bimetallic strip) and each layer is partly stretched and partly compressed to reconcile their lengths; the internal moment bends the strip toward the low-α side.

The governing equations, with every symbol defined:

ΔL = α · L · ΔT

σ = E · α · ΔT

• ΔL — change in length (m)
• L — original length at reference temperature (m)
• α — coefficient of linear thermal expansion (1/K, i.e. per kelvin)
• ΔT — temperature change, T − T₀ (K, numerically identical to °C for a difference)
• σ — thermal (axial) stress when expansion is fully prevented (Pa); compressive when heated, tensile when cooled
• E — Young's modulus of the material (Pa)

Coefficients of thermal expansion — reference values

Values below are approximate linear coefficients near room temperature; α itself rises slowly with temperature, so handbooks always quote a range. To convert a fractional strain into a temperature difference, remember 1 ×10⁻⁶ /K means one micrometre of growth per metre per kelvin.

Material	α (×10⁻⁶ /K)	E (GPa)	Restrained σ per 100 °C (MPa)
Invar (Fe–36Ni)	1.2	141	≈ 17 (comp.)
Fused silica	0.5	73	≈ 3.7
Borosilicate glass	3.3	64	≈ 21
Silicon	2.6	170	≈ 44
Titanium (Ti-6Al-4V)	8.6	114	≈ 98
Glass (soda-lime)	9	70	≈ 63
Concrete	10–12	30	≈ 33
Structural steel	12	200	≈ 240
Stainless steel (304)	17	193	≈ 328
Copper	17	117	≈ 199
Brass	19	110	≈ 209
Aluminium	23	69	≈ 159

Two lessons jump out of the last column. First, restrained stress scales with the product E·α, not with α alone — aluminium expands nearly twice as much as steel yet builds less restrained stress because its modulus is far lower. Second, steel's E·α is high enough that a modest 100 °C rise generates ~240 MPa, comfortably past the yield strength of mild steel (~250 MPa), which is exactly why constrained steel structures buckle or yield rather than "just get a bit tight."

Worked example — a steel rail on a hot day

Consider a 25-metre length of continuous welded rail, structural steel with α = 12 ×10⁻⁶ /K and E = 200 GPa, installed at a neutral temperature of 25 °C and heated to a 55 °C rail-head temperature on a summer afternoon (ΔT = 30 K).

If it were free to move, its growth would be:

ΔL = α·L·ΔT = (12 ×10⁻⁶)(25 m)(30 K) = 9.0 ×10⁻³ m = 9.0 mm

Because it is clamped (continuous welded rail has no joints), that 9 mm cannot appear as motion, so it appears as compressive stress:

σ = E·α·ΔT = (200 ×10⁹)(12 ×10⁻⁶)(30) = 72 ×10⁶ Pa = 72 MPa compression

The corresponding compressive force in a rail of cross-section A ≈ 7,700 mm² is P = σ·A = (72 MPa)(7,700 mm²) ≈ 555 kN — over 55 tonnes pushing along the rail. This is why rail engineers install continuous welded rail at a controlled "rail neutral temperature," anchor sleepers heavily, and monitor for buckling ("sun kinks") on the hottest days: the stress is fixed by ΔT from neutral, but whether it produces a harmless squeeze or a lateral buckle depends on lateral track restraint.

Common misconceptions and failure modes

• "Longer parts build more thermal stress." False — restrained σ = E·α·ΔT contains no length term. A long bar moves more if free, but a short and long bar of the same material reach the same stress if both are fully clamped.
• "Only heating is a problem." Cooling a restrained member puts it in tension of the same magnitude, which is more dangerous for brittle materials like concrete and cast iron that are weak in tension.
• "β equals α." The volumetric coefficient is ≈ 3α, and the area coefficient ≈ 2α. Mixing them up gives errors of 2–3× in fluid-expansion and gap-fit calculations.
• "Water expands normally." Water is anomalous: between 0 and 4 °C it contracts as it warms (density peaks at 4 °C), which is why ice floats and lakes freeze top-down. Most CTE intuition from solids does not carry over to it.
• "CTE mismatch only matters at high temperature." In electronics, a 40 °C swing across a silicon–copper interface fatigues solder over thousands of thermal cycles — Coffin–Manson low-cycle fatigue, not a single overstress.
• "Glass breaks from heat, not from gradients." Thermal shock cracking comes from a temperature gradient: the hot inner surface expands against a cold outer skin, creating local tensile stress. Low-α borosilicate resists it far better than soda-lime glass.
• "An expansion joint can be undersized a little." If the gap closes before the temperature peaks, the joint locks solid and the whole restrained stress slams back in — designers size the gap for the full expected ΔT plus installation tolerance.