May 16, 2026 · 4 min read · By Berkay Kuşkıra
Constrained thermal expansion makes real stress with no external load on the part. Why a free part sees nothing, what alpha-delta-T-E buys you as a sanity check, CTE mismatch in joints, and why you can't always just add it to the mechanical case.
I get handed load cases as a tidy list, and somewhere down it, if I’m lucky, is a thermal case. Half the time it isn’t there at all and I have to go ask whether the part sees a delta-T worth caring about, and the loads engineer looks at me like I’ve asked something odd, because there’s no force in it. That’s exactly the trap. Thermal stress doesn’t show up on a free body diagram of external forces, so it’s easy to leave off the list, and then it bites you at a fitting where two materials with different expansion coefficients are bolted together and nobody ran the case.
So this is the explainer I end up giving juniors maybe twice a year.
Heat up a bar sitting free on a bench. It gets longer by alpha-times-delta-T per unit length, every fibre expands the same, and there’s no stress anywhere. None. The thing is straining its head off and the stress is zero, because nothing is stopping it. This is the bit that catches people: strain without stress. We spend our whole training tying stress to strain through E and then thermal comes along and breaks the link, because free thermal strain costs nothing.
Now clamp both ends. Same delta-T, but the bar would like to be longer and the walls won’t let it, so the constraint pushes back with exactly the force needed to squash it back to length. That force over area is your stress, and the held-rigid value is
sigma = alpha * delta-T * E
a genuine stress, the material can yield from it, buckle from it, crack from it, with not one newton applied to the part. It’s the upper bound (real boundaries give a little, so the real stress is some fraction), but it’s the number I reach for first to find out whether I even care. Aluminium: alpha around 23 microstrain per degree C, E about 70 GPa. A 100 degree swing fully constrained is 23e-6 times 100 times 70000, call it 160 MPa. That’s not a rounding error, that’s a real chunk of your allowable spent before any flight load shows up. Steel has a lower alpha but a higher E, so it lands in a similar postcode, which surprises people who assume steel is somehow immune.
The uniformly heated, uniformly constrained bar is the textbook version and you rarely get it that clean. The real damage is differential: two things that want to expand by different amounts, tied so they can’t.
The classic is a metal fitting on a composite. CFRP has a near-zero (sometimes slightly negative) CTE along the fibres, aluminium is up around 23. Bolt an aluminium lug to a carbon spar, run it down to a cold soak at altitude, and the aluminium contracts a long way while the composite barely moves, so the fasteners and the interface eat the difference: bearing at the holes, prying, shear in the fasteners, all of it thermal and all of it there before the wing carries a single g. Titanium gets used at these interfaces partly because its alpha (around 8 to 9) sits much closer to the composite, so the mismatch comes down (galvanic compatibility with carbon is the other reason). Same story anywhere two materials share a boundary and see a delta-T: a bonded joint cooling from cure locks in residual stress, a steel bush in an aluminium housing, anywhere near hot structure where a gradient alone does the damage as one part of a panel grows into material that’s holding it back.
Here’s where I have to be careful, and where I’ve seen it done too casually. If everything’s linear elastic, thermal stress is just another field and you superpose it: run the mechanical case, run the thermal case, add the tensors, check the combined state. Fine, and a lot of the time you live in that world.
But superposition is a linear trick and thermal cases love to go nonlinear. The properties move (E, alpha, yield all drift with temperature), so the “thermal case” you’d superpose isn’t even computed at the same stiffness as the mechanical one, and you really want the combined case solved hot, not two linear runs glued together. Yielding breaks it too: if the thermal stress alone pushes the notch root plastic, more load redistributes instead of stacking, and thermal-then-mechanical lands somewhere different from the reverse, so you solve them together with the loads ramped sensibly. And a compressive thermal stress feeds straight into a buckling problem, which is nonlinear by nature, so that combined field goes into the stability solution as one thing.
None of this means thermal is hard maths. The alpha-delta-T-E ballpark takes ten seconds and tells you whether to bother. It means thermal is easy to forget and occasionally easy to get wrong by being lazy about the combination. So when I get that tidy list, the first thing I do is ask what temperatures the part sees and whether anything constrains it or sits next to a different material at the boundary. Usually it’s nothing. Sometimes it’s the case that was quietly governing the whole time.
