Bonded Doubler: Thermal Expansion Mismatch
A repair scheme calls for a 0.60 m × 0.40 m × 1.6 mm 2024 aluminium alloy doubler to be bonded to a carbon-fibre skin. The film adhesive cures at 120 °C; the assembly then cools to the 20 °C shop temperature before the aircraft flies. Data: 2024 has α = 23 × 10⁻⁶ K⁻¹, E = 73 GPa, yield 325 MPa. The quasi-isotropic CFRP skin has α = 3 × 10⁻⁶ K⁻¹ and is stiff enough to be treated as rigid for a first estimate. The stress note requires that the doubler carry no more than 30 % of its yield stress as residual thermal stress. The aircraft will later cruise with a skin temperature of −55 °C, and the doubler must still have margin for flight loads.
1. Treating the skin as rigid, estimate the differential contraction between doubler and skin over the 0.60 m length during cooldown, and the resulting residual stress in the doubler.
Read the worked answer
Temperature change: ΔT = 20 − 120 = −100 K.
Free contractions over 0.60 m:
- Aluminium: 23 × 10⁻⁶ × 100 × 0.60 = 1.38 mm
- CFRP: 3 × 10⁻⁶ × 100 × 0.60 = 0.18 mm
Differential: 1.38 − 0.18 = 1.20 mm over 600 mm, i.e. a constrained strain of 1.20/600 = 2.0 × 10⁻³.
Stress: σ = Eε = 73 × 10⁹ × 2.0 × 10⁻³ = 146 MPa (tension in the doubler: the skin holds it stretched relative to its free length).
This is an upper bound. Real adhesive shear lag spreads the mismatch over the bond length, and the skin is not infinitely stiff, so the true value is lower - but the rigid-skin figure is the number the stress note asks for as a first pass.
2. Does the design meet the '30 % of yield' criterion? If not, propose a change and support it with numbers.
Read the worked answer
Limit: 0.30 × 325 = 97.5 MPa.
The estimate of 146 MPa exceeds this by about 50 %, so on the stated criterion the scheme is not acceptable as written.
Option - lower cure temperature. With an 80 °C cure, ΔT = 60 K:
- strain = 20 × 10⁻⁶ × 60 = 1.2 × 10⁻³
- σ = 73 × 10⁹ × 1.2 × 10⁻³ = 87.6 MPa - below 97.5 MPa, criterion met.
Other routes: a lower-modulus adhesive to reduce shear lag, a tapered/stepped patch to spread the load, or a proper bonded-joint analysis that shows the rigid-skin bound is far too pessimistic. Note that reducing patch thickness does not help here, because the constrained thermal stress is independent of thickness. The decision is an engineering one - the number, not the paperwork, drives it.
3. At cruise the skin is at −55 °C. Using the 20 °C shop state as the datum, what additional thermal stress appears, and is the doubler still acceptable against yield?
Read the worked answer
Additional cooling: ΔT = −55 − 20 = −75 K, with the same α mismatch of 20 × 10⁻⁶ K⁻¹.
Additional constrained strain = 20 × 10⁻⁶ × 75 = 1.5 × 10⁻³ → σ = 73 × 10⁹ × 1.5 × 10⁻³ = ≈110 MPa extra tension (same sign: the aluminium always wants to shrink more than the skin).
Cumulative (rigid-skin model throughout): 146 + 110 ≈ 256 MPa, about 79 % of the 325 MPa yield.
Only ~69 MPa (≈ 21 % of yield) would then remain for flight loads; with a typical 1.5 limit factor this is not enough margin. Because the rigid-skin assumption is doubly conservative at two temperatures, the correct conclusion is not 'reject' but 'the simple model cannot clear this - a shear-lag analysis or the 80 °C cure is required before release'.