Thermal Expansion Mismatch of Dissimilar Materials
Theory: the difference in desired expansion becomes stress
Overview
On an automotive board, after temperature-cycling tests, only the larger chip components' solder joints cracked. The small ones were fine — why?
Thermal expansion mismatch. Ceramic components expand about 6.5 ppm per kelvin, glass-epoxy boards about 17 ppm. Every temperature swing moves the board more than the component, and the solder absorbs the difference. The displacement grows with distance from the component centre, so larger parts distort their end joints more. Fatigue life depends on roughly the square of strain, so size differences become large life differences.
Mismatch strain and film stress
$\Delta\alpha$ is the expansion-coefficient difference and $\Delta T$ the temperature change from the stress-free (joining) state. A thin film on a thick substrate is forced to follow the substrate, developing biaxial in-plane stress; $E_f/(1-\nu_f)$ is the effective biaxial stiffness.
Bimetal curvature (Timoshenko)
Bonding two layers of similar thickness produces bending instead of just stress. $h$ is total thickness, $t_1, t_2$ layer thicknesses and $E_1, E_2$ Young's moduli. Bimetal thermostats exploit this bending.
Solder-joint strain
$L_{DNP}$ is the distance from the component centre (neutral point) to the joint and $h$ the solder height. Some displacement is relieved by component and board bending, so this is an upper-bound estimate. Fatigue life $N_f$ falls roughly with the second power of strain (a Coffin–Manson-type relation).
Railway rails and pendulum clocks
One of the first deliberate uses of differential expansion was the 18th-century clockmaker Harrison's gridiron pendulum: iron and brass rods arranged so their thermal elongations cancelled, keeping pendulum length constant. It was invented for accurate marine timekeeping, and Harrison also devised an early form of the bimetal strip. The opposite — failing to relieve expansion — is familiar from rails buckling in summer. How to relieve or exploit expansion differences remains a common problem from electronics to bridges.
Worked examples
Example 1: material pairs and mismatch strain
| Pair | Δα | Mismatch strain at 100 K |
|---|---|---|
| Aluminium / steel | 11 ppm/K | 0.11% |
| Copper / alumina | 10 ppm/K | 0.10% |
| Silicon / FR-4 (in-plane) | 14.4 ppm/K | 0.14% |
| Steel / concrete | 2 ppm/K | 0.02% |
0.1% looks small, but in steel it corresponds to about 200 MPa. Reinforced concrete works partly because steel and concrete expand almost identically.
Example 2: copper film on alumina
A copper film (E = 120 GPa, ν = 0.34) on alumina cooled by 100 K sees about 182 MPa in-plane. Annealed copper yields at a few tens of MPa, so the film deforms plastically; repeated cycling accumulates plastic strain and causes delamination or cracking — why copper-clad ceramic substrates in power modules are a reliability concern.
Example 3: aluminium–steel bimetal
Bonding 0.5 mm each of aluminium (70 GPa) and steel (200 GPa) and heating by 100 K gives a curvature of 1.53 1/m (radius 0.65 m); a 100 mm cantilever tip moves about 7.7 mm. With equal moduli it would be 1.65 1/m, so the stiffness difference matters only about 7%.
Example 4: chip-component solder joints
| Distance from centre | Height 0.05 mm | Height 0.10 mm |
|---|---|---|
| 1.6 mm | 5.5% | 2.8% |
| 3.2 mm | 11.1% | 5.5% |
| 6.0 mm | 20.8% | 10.4% |
Doubling the component size doubles the strain.
Yes. And if life scales with strain to about the −2.3 power, doubling strain cuts life to about a fifth. For a ceramic part on FR-4 at −40 to 125 °C, the formula ignores component and board bending and is an upper bound, but it captures the trend. Remedies include avoiding or splitting large parts, taller joints, better-matched substrates, and underfill to share the load. Detailed strain and life assessment by FEA comes afterwards.
How to approach the analysis
Procedure and cautions
- Set the stress-free temperature — near solidification for solder, cure temperature for adhesives. Using room temperature is a common error.
- Include temperature-dependent expansion; resins' coefficients jump several-fold above the glass transition.
- Solder creeps with temperature and time; use a nonlinear material model.
- Bond edges are stress singularities whose values rise with refinement; compare using a fixed criterion rather than raw peak values.
“Assembled at room temperature, yet warped”
A board that seemed fine at assembly is warped from the start — usually because of soldering or bonding. Solder solidifies above 200 °C, and the expansion difference acts as it cools to room temperature, so at room temperature it has already seen nearly a 200 K change. Adhesives likewise set their shape at the cure temperature. Taking room temperature as stress-free misses this initial warp and stress entirely; the right starting point is the process temperature history.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Stress-free at room temperature | Initial warp and stress missed | Use the joining temperature |
| Constant expansion coefficient | Wrong above resin Tg | Use temperature dependence |
| Elastic solder | Wrong strain and life | Creep model |
| Taking raw edge peaks | Mesh-dependent values | Compare at fixed location or volume average |
| Treating large parts like small ones | Life overestimated | Evaluate by distance from centre |
I'd like to compute thermal expansion and stress myself.
Try the thermal expansion calculator for expansion, the bimetal thermostat tool for bimetals and the thermal stress simulator for thermal stress. Related topics are covered in bimetallic strips, solder-joint reliability, thermal cycling and thermal residual stress.