Why do TBCs develop thermal stress?

Ceramic zirconia (expansion about 10.5×10⁻⁶/K) and the superalloy beneath (about 15×10⁻⁶/K) expand differently. Cooling from a stress-free high temperature, the metal contracts more, squeezing the thin ceramic film in-plane. A 1000 K cool-down gave 281 MPa compression in porous zirconia.

Why are TBCs porous or columnar?

To lower in-plane modulus and accommodate mismatch strain. Under the same conditions dense zirconia (200 GPa) would carry 1,125 MPa — four times more — with more stored strain energy and easier spallation. Pores and columnar gaps let the coating shrug off strain.

A major cause is growth of the thermally grown oxide (TGO, alumina) on the bond coat. Alumina expands even less and is stiff, so on cooling it carries about 3.5 GPa compression. Stored energy grows with thickness, reaching 124 J/m² at 5 µm; when the interface can no longer bear it, spallation follows.

Thermal Barrier Coating: Thermal Stress

Category: Coupled Analysis – Thermal-Structural | Revised 2026-10-01

CAE visualization for thermal barrier coating theory - technical simulation diagram

Thermal Barrier Coating (TBC) Analysis

Theory: thermal mismatch in thin films

Overview

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A TBC's job is blocking heat. What's the structural concern?

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Spallation. A TBC stacks several materials with different expansion coefficients. Each shutdown cool-down, the metal shrinks more than the ceramic and squeezes the thin film; in service there's a large temperature drop within the film too. Meanwhile an oxide slowly grows on the bond coat, carrying high compression that tries to peel the interface. Let's compute layer-by-layer thermal stresses and stored strain energy.

Mismatch stress and strain energy

$$ \sigma_f = \frac{E_f\,(\alpha_s - \alpha_f)\,\Delta T}{1 - \nu_f},\qquad U = \frac{\sigma_f^2\,h\,(1-\nu_f)}{E_f} $$

For a thin film on a thick substrate, equally restrained in both in-plane directions. $U$ is the elastic energy stored per unit film area, proportional to thickness $h$; if it exceeds the interface's resistance to delamination (interface toughness), the film can spall.

Stress from in-service temperature gradient

$$ \sigma_{grad} = \pm\frac{E_f\,\alpha_f\,\Delta T_{grad}}{2(1-\nu_f)} $$

Coffee Break Trivia

Inventing the columnar structure

Early TBCs spalled quickly under repeated thermal cycling. In the 1970s and 80s, methods were developed to deliberately leave fine cracks and pores in plasma-sprayed coatings, and to deposit coatings by electron beam as rows of columns. Gaps between columns open and close to accommodate mismatch with the metal. Balancing insulation with spallation resistance became the centre of TBC materials development.

Worked example: a TBC cooled by 1000 K

Superalloy expansion 15×10⁻⁶/K; zirconia (YSZ) 10.5×10⁻⁶/K, Poisson 0.2; oxide (alumina) 8×10⁻⁶/K, 380 GPa, Poisson 0.25.

Stress by layer

Layer	Young's modulus	Thickness	Compressive stress	Stored strain energy
YSZ (porous, sprayed)	50 GPa	300 µm	281 MPa	380 J/m²
YSZ (dense)	200 GPa	300 µm	1,125 MPa	1,519 J/m²
Oxide (alumina)	380 GPa	5 µm	3,547 MPa	124 J/m²

Assuming an interface toughness of 100 J/m², the film thickness reaching it is 79 µm for porous YSZ, 20 µm for dense YSZ and 4 µm for the oxide. Real TBCs have even lower in-plane modulus from pores and cracks, and creep relaxes stress at temperature, so they outlast this simple estimate.

In-service gradient

With a 217 K drop across a 0.3 mm TBC (from the heat-transfer analysis), a bending-restrained film sees ±71 MPa — compression at the surface, tension at the bond-coat side.

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A mere 5 µm oxide stores a third as much energy as the 300 µm ceramic layer, sixty times thicker.

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Alumina is stiff and mismatched with the superalloy, so on cooling it carries a very large 3.5 GPa compression. Energy goes with stress squared, so even a thin layer stores a lot — and the oxide thickens slowly with service, storing ever more. That energy concentrates at the weakest place: the ceramic–bond-coat interface. That's why TBC life is often set by oxide growth, and life prediction combines oxide growth rate with the number of thermal cycles.

Analysis tips

Give each layer's expansion and modulus as functions of temperature.
Account for sprayed coatings' anisotropy (in-plane vs through-thickness).
Track stress relaxation by creep at temperature and stress build-up on cooling over cycles.
Interface roughness peaks and valleys concentrate stress; assess with local models.
Include oxide growth as a function of time.

Coffee Break Trivia

“It lasted in the furnace test”

One TBC showed long life in constant-temperature furnace tests but spalled early in real service with frequent start-stops. At constant temperature the oxide merely grows slowly; each shutdown cools the system, repeatedly loading the oxide and ceramic in compression and gradually extending small interface cracks. Testing was changed to match service thermal cycles and assess life by both cycle count and oxide thickness.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Bulk ceramic properties	Stress overestimated	Measured coating data
Omitting the oxide	Spallation missed	Model TGO as a layer
Isothermal life only	Early field failure	Assess thermal cycles
Flat interface	Local stress missed	Model roughness
Ignoring creep	Wrong cool-down stress	Include creep

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