Coefficient of Thermal Expansion
Reference for linear thermal expansion ΔL = α × L₀ × ΔT.
How solids grow or shrink with temperature — critical for engineering joints and tolerances.
Linear Expansion
A solid object grows in length when heated and contracts when cooled. The change in length ΔL is proportional to the original length L₀, the temperature change ΔT, and a material constant α called the linear coefficient of thermal expansion.
Volumetric Expansion
For isotropic solids (those that expand equally in all directions), the volumetric expansion coefficient β is approximately three times the linear coefficient α.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| α (alpha) | Linear thermal expansion coefficient | 1/K or /°C (often ppm/°C) |
| β (beta) | Volumetric thermal expansion coefficient | 1/K or /°C |
| L₀ | Original length | m |
| V₀ | Original volume | m³ |
| ΔT | Change in temperature | K or °C |
| ΔL, ΔV | Change in length, change in volume | m, m³ |
Typical Values
| Material | α (× 10⁻⁶ /°C) |
|---|---|
| Invar (Fe-Ni alloy) | ~1.2 |
| Quartz (fused) | ~0.5 |
| Borosilicate glass | ~3.3 |
| Tungsten | ~4.5 |
| Soda-lime glass | ~9 |
| Steel (structural) | ~12 |
| Concrete | ~12 |
| Copper | ~17 |
| Brass | ~19 |
| Aluminum | ~23 |
| Polyethylene | ~150 |
| PVC | ~80 |
Example — Steel Bridge in Summer
A 100 m steel bridge section experiences a 40°C temperature swing from winter to summer. How much does it expand?
α for steel ≈ 12 × 10⁻⁶ /°C
ΔL = α × L₀ × ΔT = 12 × 10⁻⁶ × 100 × 40
= 0.048 m = 4.8 cm
The bridge expands by 4.8 cm
This is why bridges and overpasses include expansion joints — gaps that absorb thermal length changes without distorting the structure.
Example — Thermal Stress (Constrained Expansion)
If the bridge could not expand freely, what compressive stress would develop?
Strain prevented = α × ΔT = 12 × 10⁻⁶ × 40 = 4.8 × 10⁻⁴
Stress = E × ε = 200 GPa × 4.8 × 10⁻⁴
Compressive stress = 96 MPa
That stress is approximately one third of structural steel's yield strength. Without expansion joints, the bridge would either buckle or experience large permanent deformation.
When to Use It
- Sizing expansion joints in piping, bridges, rail, and large structures
- Predicting bimetallic strip deflection in thermostats and circuit breakers
- Calculating fits between dissimilar materials (shaft into hub, glass-to-metal seals)
- Compensating for measurement drift in precision instruments
- Predicting glass-ceramic or refractory failure under thermal cycling
Bimetallic Strips and CTE Mismatch
A bimetallic strip is two metals with different α values bonded together. When heated, the metal with larger α expands more, causing the strip to curve. The same principle drives thermal stress failures in glass-to-metal seals and ceramic-to-metal joints — if α values don't match within a few ppm/°C, cycling will eventually crack the joint.