CTE-Graded Design (Functionally Graded Materials)
Theory: stress from expansion mismatch
Overview
I brazed a ceramic plate to metal and it cracked on cooling. How should I design it?
Bond at high temperature and cool, and the metal tries to shrink more than the ceramic. For alumina and nickel the expansion coefficients differ by about 5×10⁻⁶ /K, so 500 K of cooling leaves a 0.26% strain mismatch, taken up abruptly at the interface — hence large stresses. One remedy is a functionally graded material (FGM): an interlayer whose composition changes gradually so the coefficient transitions smoothly. But unless you calculate what it reduces and what it doesn't, the result can disappoint.
Laminate equations
For a free plate on cooling, strain is linear through the thickness (stretch $\varepsilon_0$ and curvature $\kappa$), fixed by zero net force and moment. Because it shrinks equally in both directions, $E/(1-\nu)$ replaces Young's modulus.
Within the graded layer the metal volume fraction $V_m$ varies through the thickness and properties are averaged by a rule of mixtures (a simple approximation). $p = 1$ is linear, $p < 1$ metal-rich and $p > 1$ ceramic-rich.
A concept born of spaceplane heat
The idea of functionally graded materials emerged in Japan in the mid-1980s, from research aimed at spaceplane airframes. Making parts from combined ceramic and metal — over 1,700 °C outside, cooled inside, a difference of more than 1,000 K — ran into delamination at the bonded surface. The response was to vary composition continuously so the interface itself disappears. The concept has since spread to thermal barrier coatings, cutting tools and biomaterials.
Worked example: alumina bonded to nickel
Alumina (E 380 GPa, ν 0.22, α 8.1×10⁻⁶ /K) and nickel (E 200 GPa, ν 0.31, α 13.3×10⁻⁶ /K) plates were joined and cooled 500 K. A direct bond (1 mm each) was compared with 0.75 mm each plus a 0.5 mm graded layer, using the laminate equations above (elastic, no edge effects).
| Configuration | Largest interface stress jump | Ceramic at interface | Ceramic surface | Warp (50 mm) |
|---|---|---|---|---|
| Direct bond | 991 MPa | −590 MPa | +342 MPa | 0.60 mm |
| 1-step interlayer | 584 MPa | −396 MPa | +266 MPa | 0.57 mm |
| 3 steps | 323 MPa | −389 MPa | +293 MPa | 0.58 mm |
| 9 steps | 136 MPa | −385 MPa | +303 MPa | 0.59 mm |
| Continuous (p = 1) | ≈ 0 | −382 MPa | +308 MPa | 0.59 mm |
| Continuous (p = 2) | ≈ 0 | −328 MPa | +348 MPa | 0.58 mm |
The jump drops nicely. But the tension at the ceramic surface doesn't fall, and the warp is the same?
That's the key point. Warp and surface stress are set by how unevenly the high- and low-expansion material is distributed across the whole plate. The graded layer only smooths the transition and hardly changes that top–bottom imbalance, so bending is almost unchanged. What grading addresses is the abrupt jump at the interface — the cause of delamination. And going ceramic-rich (p = 2) reduced interface compression but raised surface tension from 342 to 348 MPa. Alumina's flexural strength is around 300–400 MPa, so if surface cracking is the problem you need other measures: change the thickness ratio, or add ceramic on the other side to make the stack symmetric.
Mind the assumptions
This calculation is elastic, and the peak nickel stress in the direct bond was 401 MPa. Annealed nickel yields well below that, so in reality the metal yields and stresses relax. The laminate equations also give mid-plate stresses; at the free edge, interface stresses concentrate and grow (delamination usually starts at the edge). Grading pays off most in softening that edge concentration, so final decisions rest on 2D/3D finite element analysis that includes the edges.
Design workflow
- Collect the temperature change from bonding to service and each material's expansion coefficient, modulus and strength.
- Estimate interface jump, surface stress and warp for a direct bond with the laminate equations.
- If the problem is the interface jump (delamination), consider grading; if it's surface cracking or warp, consider thickness ratio or symmetry.
- Compare step count, composition profile and graded-layer thickness with finite element analysis including the edges, adding metal yielding and creep.
- Verify the graded material's own properties (deviation from the rule of mixtures, porosity) on test coupons.
“More steps, but no fewer cracks”
Cracking appeared in a ceramic–metal joint, so the graded layer was increased from three to nine steps. Delamination fell, but cracks at the ceramic surface appeared just as before. As the table shows, more steps barely change surface tension. The cause was not delamination but bending stress at the surface from warping. Thickening the ceramic to shift the neutral plane, or adding a ceramic layer on the back for symmetry, cuts surface tension sharply. Identify which stress is responsible from the crack location and orientation first.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Expecting grading to fix everything | Warp and surface cracks remain | Identify the responsible stress |
| Judging from mid-plate stress only | Edge concentration missed | FE analysis including edges |
| Metal kept elastic | Stress overestimated | Include yielding and creep |
| Trusting the rule of mixtures | Graded properties off | Verify on coupons |
| Using bonding temperature directly as ΔT | Stress overestimated | Start from where stress begins to lock in |
I'd like to try related calculations.
The thermal expansion calculator, thermal stress calculator and bimetal simulator let you check expansion, thermal stress and warp. For background, see bimetallic strips, thermal expansion mismatch, thermal barrier coatings and solder joint reliability.