CTE Mismatch Thermal Stress

Thermal stress at a bonded bi-material interface from a temperature swing — the first-order constrained-strain estimate for die-attach, solder, and coating stacks.

Inputε_free = (α₂ − α₁)·ΔT , σ₁ = E'₁·ε_free·E'₂/(E'₁+E'₂) , E' = E/(1−ν)

CTE material 1 (ppm/°C)CTE material 2 (ppm/°C)Temperature change ΔT (°C)or enter °F — converts to °CYoung's modulus 1 (GPa)or enter Mpsi — converts to GPaYoung's modulus 2 (GPa)or enter Mpsi — converts to GPaPoisson ratio 1 — optionalPoisson ratio 2 — optional

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The engineering

When two bonded materials with different coefficients of thermal expansion see a temperature swing, they want to grow by different amounts but the interface won't let them. The frustrated strain (α₂−α₁)·ΔT gets converted to stress by an effective modulus, with the stiffer partner carrying the larger share. This card gives the constrained-strain first cut for die-attach, solder joints, and coating stacks — the number you check before deciding a joint needs the full Suhir shear-lag treatment.

The biaxial modulus E/(1−ν) is the right one for a thin layer bonded over an area, not the plain uniaxial E — using E alone under-predicts stress by 20–40% for typical Poisson ratios. Remember the mismatch is what matters, not the absolute CTEs: silicon on copper (Δα ≈ 14 ppm/°C) is brutal, silicon on molybdenum (Δα ≈ 2) is gentle. This closed form ignores edge/peeling concentrations, which is exactly where real solder joints crack — treat it as a lower bound on the trouble.

Where this math comes from

The bi-material bending problem goes back to Stoney's 1909 thin-film stress formula and Timoshenko's 1925 bimetal-thermostat analysis, which treated two bonded strips as a beam that curls when heated. Those gave curvature and average stress but glossed over what happens at the free edge, where microelectronics packages actually fail.

Ephraim Suhir, working at Bell Labs in the 1980s, reformulated the problem with an interfacial compliance / shear-lag model that captured the peaking of shear and peeling stress near the edges of an assembly. His 1986 ASME paper became the reference for solder-joint and die-attach reliability, and the simple constrained-strain estimate this card computes is the sanity check that precedes his full treatment.

1909George StoneyRelates film stress to substrate curvature — the first bonded-layer stress formula.
1925Stephen TimoshenkoBimetal thermostat analysis: two bonded strips bending under ΔT.
1986Ephraim SuhirInterfacial shear-lag model for bonded assemblies — edge stress peaking, the microelectronics reference.
2000IPC / JEDECCTE-mismatch fatigue models (Coffin-Manson family) standardized for solder-joint reliability.

See the full timeline of the math behind every calculator →

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