Bimetallic Strips: How Two Metals Turn Heat into Motion

Atoms that vibrate harder take up more room

Heat a solid and its atoms don't just vibrate faster around their lattice positions — because the interatomic bond potential is not a perfect symmetric well (it is steeper on the compression side than the stretching side, an anharmonic potential), the time-averaged atomic spacing actually increases as vibration amplitude grows. That asymmetry, summed over billions of bonds, is what thermal expansion is: a real, average lengthening of the material, not just faster jiggling in place. It is quantified by the coefficient of thermal expansion α, the fractional length change per degree of temperature change, ΔL/L₀ = α·ΔT.

Different materials have very different α because their bond stiffness and lattice structure differ: steel expands at roughly 12×10⁻⁶ per °C, aluminium at about 23×10⁻⁶ per °C — nearly double — and Invar, a nickel-iron alloy engineered specifically to minimise this effect, sits below 2×10⁻⁶ per °C. A bimetallic strip bonds two of these together, and it is precisely that mismatch in α that makes it useful.

Why two bonded metals bend

If you heated the two metal strips separately and let them expand freely, the higher-α strip would simply become longer than the lower-α one. Bonded together along their whole length, neither strip is free to do that — the interface forces them to stay the same length at every point, so the mismatch in how much each wants to expand is instead resolved by the whole assembly curving: the low-expansion strip ends up on the inside (concave side) of the curve, the high-expansion strip on the outside (convex side), because bending shortens the inner arc and lengthens the outer one relative to the strip's straight length, matching what each metal was already trying to do.

The Timoshenko formula

Stephen Timoshenko derived the exact curvature of a bonded bimetallic strip in 1925, accounting for both strips' thicknesses and elastic moduli. In the common case of two strips of equal thickness t and comparable stiffness, the radius of curvature simplifies to a compact, widely quoted form:

1/ρ ≈ 6(α2 - α1)·ΔT / (h·[3(1 + m)² + (1 + m·n)(m² + 1/(m·n))])
  simplified case (equal thickness t, similar stiffness):
1/ρ ≈ 3(α2 - α1)·ΔT / (2t)

  ρ  = radius of curvature of the bent strip
  α1, α2 = expansion coefficients of the two metals (α2 > α1)
  ΔT = temperature change
  h  = total strip thickness,  t = thickness of each layer
  m  = t1/t2 (thickness ratio), n = E1/E2 (stiffness ratio)

  tip deflection δ ≈ L² / (2ρ)     for a strip of free length L

If a bimetallic strip is instead rigidly clamped at both ends so it cannot bend or expand at all, the mismatch shows up as pure internal stress rather than curvature: for a single constrained material, σ = E·α·ΔT, Young's modulus times the coefficient of expansion times the temperature change — the stress a bridge expansion joint, a railway rail, or a poorly designed sealed pipe run must be engineered to survive.

Where the bend is put to work

Thermostats are the classic application: a bimetallic strip or coil is wound so that heating bends it toward or away from an electrical contact, opening or closing a circuit at a temperature set by the pre-tension — no electronics required, just geometry. The same principle drives circuit-breaker overload trips (excess current heats a strip resistively until it bends far enough to snap the contact open), oven and iron thermostats, and some fire-sprinkler and thermal-fuse mechanisms. Because the response is a passive mechanical consequence of the temperature difference, these devices work reliably for decades without power or maintenance.

Frequently asked questions

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