Thermal expansion

Thermal expansion is the change in a part's size when its temperature changes. Most materials grow as they heat and shrink as they cool. A part that can't move freely builds up stress instead.

The formula

$$\Delta L = \alpha\, L\, \Delta T$$

$\alpha$ is the coefficient of thermal expansion, $L$ the original length and $\Delta T$ the temperature change. Volume grows by about three times that fraction.

If the part is held at both ends and can't grow, the expansion turns into stress.

$$\sigma = E\,\alpha\,\Delta T$$

$E$ is Young's modulus. The stress doesn't depend on the part's length.

Coefficients for common materials

Material	α, 10⁻⁶ per K
Aluminium	23
Copper	17
Stainless steel 304	17
Carbon steel	12
Titanium Ti-6Al-4V	8.6
Borosilicate glass	3.3
Invar	1.2
Carbon fibre, along the fibre	slightly below 0

A worked example

A 6061-T6 aluminium spacer, 200 mm long and 20 mm square, heats from 20 °C to 70 °C.

• Free to move, it grows $23\times10^{-6} \times 200 \times 50 =$ 0.23 mm.
• Clamped between two rigid plates, it can't grow. The stress is $69{,}000 \times 23\times10^{-6} \times 50 \approx$ 79 MPa in compression.

That is about 29% of its yield strength, from a 50 K change and no load at all. The spacer is short and stocky enough not to buckle. A long, slender rail held the same way would bow sideways well before it reached that stress.

Where it matters

• Joints between different materials. Aluminium bolted to steel grows twice as much, so the joint slips, loosens or cracks.
• Electronics. Solder joints fatigue as a board and its chips expand by different amounts every power cycle.
• Clearances. Shafts, bearings and housings need gaps that work at both cold and hot.
• Composites. Parts cure hot and cool down, which leaves stress in the plies.

Common mistakes

• Forgetting the constraint. A part at one even temperature that expands freely has no thermal stress. The same part bolted down can yield. A part with a temperature gradient is stressed even when it is free.
• Using one coefficient over a wide range. It rises with temperature for most metals.
• Choosing the wrong stress-free temperature. For a cured composite or a weld it is the process temperature, not room temperature.

Thermal expansion using Navier AI

Prism runs structural FEA on Ferro, its built-in solver, with CalculiX, code_aster and Elmer available from inside the app.

Our composite layup study treats each plate as stress-free at its 453 K cure. On cooling, a ply contracts across its fibres by 28 × 10⁻⁶ per kelvin and barely moves along them. The 160 K cool-down alone leaves the cross-ply and quasi-isotropic plies at 81% of their transverse strength, before any load arrives. Download Prism to run your own.

References

• W. D. Callister and D. G. Rethwisch, Materials Science and Engineering, Wiley.
• R. G. Budynas and J. K. Nisbett, Shigley's Mechanical Engineering Design, McGraw-Hill.