Thermal Expansion Calculator
Most materials expand when heated and contract when cooled. Pick linear, area, or volumetric expansion, choose a material preset or enter your own coefficient, set the temperature change, and get the size change, the final dimension, the thermal stress if it cannot move, and the expansion gap to leave.
Formula
Worked example
A 2 m aluminium bar (α = 23×10⁻⁶ /°C) heated by 50 °C: ΔL = 23×10⁻⁶ × 2 × 50 = 0.0023 m, or 2.3 mm, so leave at least a 2.3 mm gap. If that bar were fully clamped instead, the thermal stress would be E·α·ΔT = 69 GPa × 23×10⁻⁶ × 50 ≈ 79 MPa of compression, regardless of its length.
How thermal expansion works (linear, area, and volume)
When a solid is heated, its atoms vibrate more vigorously and on average sit slightly farther apart, so the whole object grows. The linear law captures this with ΔL = α·L₀·ΔT, where α is the coefficient of linear expansion, L₀ is the starting length, and ΔT is the temperature change. Because a flat sheet expands in two directions and a solid block in three, the area coefficient is about twice the linear one (β ≈ 2α) and the volume coefficient about three times (γ ≈ 3α). This calculator lets you pick the mode and applies the right multiple automatically, so the same material coefficient drives length, surface, or volume changes. Cooling simply reverses the sign: a negative ΔT gives a negative change and the object contracts.
Material presets, units, and the expansion gap
Choose a material to auto-fill its linear coefficient, or select Custom to type your own value in parts per million per degree. Soft metals such as aluminium and plastics like PVC expand a lot, while glass, concrete, and Invar barely move, which is why engineers match expansion rates when bonding dissimilar materials. Switch between metric and imperial: because α is defined per degree, the per-°F value equals the per-°C value divided by 1.8, and the temperature change must use the same unit. In linear mode the calculator also reports the expansion gap in millimetres, the absolute growth you should leave as clearance or in an expansion joint so the part can move freely. This is the same reasoning behind expansion joints in bridges, gaps between railway rails, and loops in long pipelines.
Thermal stress when a part cannot move
If a heated or cooled part is fully restrained so it cannot change length, the strain it would have had turns into stress instead. Turn on the thermal stress option and enter the material stiffness, Young's modulus E, to get σ = E·α·ΔT. Crucially this stress does not depend on the length of the part, only on the material, its stiffness, and the temperature change, so even a short clamped bar can develop large forces. Heating a restrained part puts it in compression and cooling puts it in tension, which is why cooling cracks and buckling are common failure modes. The calculator also reports the restraining force per square centimetre of cross section so you can gauge the loads on fixings and supports. These are first-order estimates: real assemblies are rarely perfectly rigid, so treat the stress figure as an upper bound and design in clearance wherever you can.
Linear expansion coefficients and stiffness of common materials
| Material | α (×10⁻⁶ /°C) | E (GPa) |
|---|---|---|
| PVC (plastic) | 60 | 3 |
| Ice | 51 | 9 |
| Aluminium | 23 | 69 |
| Brass | 19 | 100 |
| Silver | 18 | 83 |
| Copper | 17 | 117 |
| Stainless steel | 17 | 193 |
| Gold | 14 | 79 |
| Steel (carbon) | 12 | 200 |
| Concrete | 12 | 30 |
| Glass (window) | 9 | 70 |
| Borosilicate (Pyrex) | 3.3 | 64 |
| Invar alloy | 1.2 | 141 |
Approximate values near room temperature. α in ×10⁻⁶ per °C (divide by 1.8 for per °F); E is Young's modulus.
Frequently asked questions
What is the difference between linear, area, and volumetric expansion?
Linear expansion is the change in a single dimension, length, and uses the coefficient α. Area expansion is the change in a surface and grows at about twice the rate (β ≈ 2α) because two dimensions expand at once. Volumetric expansion is the change in a solid's volume and grows at about three times the rate (γ ≈ 3α). This calculator applies the correct multiple automatically when you pick a mode.
How do I calculate thermal stress?
When a part is fully prevented from changing length, the stress is σ = E·α·ΔT, where E is Young's modulus, α the linear coefficient, and ΔT the temperature change. Notably the result does not depend on the length, so even a short restrained bar can build large stress. Heating a restrained part compresses it and cooling stretches it. Turn on the thermal stress option to compute it.
How big an expansion gap should I leave?
In linear mode the calculator reports the expansion gap in millimetres, equal to the absolute change in length over your temperature range. Leave at least that much clearance, or split it across joints, so the part can grow and shrink freely. For example a 2 m aluminium bar over 50 °C grows about 2.3 mm, so a 2.3 mm gap is the minimum.
How do I convert the coefficient between °C and °F?
A temperature change of 1 °C equals 1.8 °F, so a coefficient per °C is 1.8 times larger than the same coefficient per °F. To convert, divide the per-°C value by 1.8. The calculator does this automatically when you switch to imperial units, so steel's 12×10⁻⁶ /°C becomes about 6.7×10⁻⁶ /°F.