Glass Tempering Thermal Stress Analysis
Theory: freezing order leaves stress
Overview
I hear tempered glass is several times stronger than ordinary glass. How does heat treatment alone do that?
Glass breaks because tensile stress concentrates at tiny surface flaws. Leave compression in the surface beforehand, and external loads can't open the flaws until they overcome that compression. The method: heat the glass above 600 °C and quench both faces with blown air. The surface solidifies first; the still-soft centre cools and contracts later, pulling the solidified surface into compression and leaving the centre in tension. The magnitude depends on quench strength and thickness.
Instant-freeze model
The simplest model assumes the temperature profile $T_f(z)$ at the instant the centre cools to the glass transition temperature $T_g$ is locked in. As the whole plate then cools uniformly to room temperature, regions that were hotter at freezing — the centre — contract more and end in tension. For a parabolic profile, surface stress is −2 times centre stress.
The Biot number, built from half-thickness and conductivity $k$, compares surface cooling with internal conduction. With glass conductivity only about 1 W/mK, even air quenching gives Biot numbers near 1, producing large temperature differences through the plate.
Prince Rupert's drops
Drop molten glass into water and it freezes into a tadpole shape. These little objects, called 'Prince Rupert's drops' and a sensation in 17th-century England, survive hammer blows to the thick head, yet snapping the thin tail shatters the whole drop instantly. Strong surface compression balances strong internal tension — the same mechanism as tempered glass. Industrial air tempering was commercialised in France in the early 20th century and spread to car windows and more. Tempered glass breaking into small granules is also the sudden release of that stored tensile energy.
Worked example: thickness and quench strength
Soda-lime glass (E 70 GPa, α 9×10⁻⁶ /K, ν 0.22, k 1.0 W/mK) was cooled from 650 °C by 30 °C air on both faces; residual stress was computed by freezing the profile when the centre reached 560 °C (radiation ignored). Classes follow ASTM C1048: 24–52 MPa surface compression heat-strengthened, 69 MPa or more fully tempered.
| Thickness | h | Biot | ΔT at freeze | Surface | Centre | Class |
|---|---|---|---|---|---|---|
| 4 mm | 100 | 0.20 | 48 K | −26 MPa | +13 MPa | Heat-strengthened |
| 4 mm | 400 | 0.80 | 154 K | −82 MPa | +43 MPa | Fully tempered |
| 6 mm | 100 | 0.30 | 70 K | −37 MPa | +19 MPa | Heat-strengthened |
| 6 mm | 200 | 0.60 | 124 K | −66 MPa | +34 MPa | Heat-strengthened |
| 6 mm | 400 | 1.20 | 203 K | −107 MPa | +56 MPa | Fully tempered |
| 10 mm | 100 | 0.50 | 107 K | −57 MPa | +30 MPa | Heat-strengthened |
| 10 mm | 200 | 1.00 | 180 K | −95 MPa | +50 MPa | Fully tempered |
With the same cooling, thinner plates end up with less stress.
Right. Stress follows the through-thickness temperature difference, which is set by the Biot number — quench strength times half-thickness. In a thin plate, heat reaches the centre quickly, so large differences are hard to build. A 10 mm plate reaches the tempered class at h = 200, but 4 mm needs over 300. That's why thin tempered glass is difficult; at 2–3 mm, chemical strengthening — ion exchange putting larger ions into the surface — replaces air tempering. Note also that every surface-to-centre ratio is about −2, showing the profile is nearly parabolic.
Model limitations
The instant-freeze model ignores gradual stress relaxation through the glass-transition range, so it overestimates residual stress. Real design uses models handling viscoelasticity and structural relaxation, such as Narayanaswamy's, calibrated to measured surface stress. Radiative cooling also contributes in real furnaces and is not negligible at high temperature.
Analysis workflow
- Obtain the surface heat transfer distribution from nozzle layout and airflow (CFD or correlations).
- Compute temperature history by transient through-thickness heat transfer, including radiation.
- Compute residual stress with a viscoelastic, structural-relaxation model.
- Edges and holes alter in-plane stress; verify in 2D/3D.
- Verify with surface stress meters or fragmentation tests (particle counts).
“Tempered glass with a mottled look”
Viewing building tempered glass through polarised sunglasses sometimes reveals a mottled pattern. Called strain patterns or quench marks, they trace slight residual stress variation where heat transfer differed with the nozzle layout. Usually not a strength issue, they can prompt appearance complaints. Computing the heat transfer distribution from the nozzle layout lets you study oscillation amplitude and airflow adjustments that fade the pattern.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Using instant-freeze values directly | Stress overestimated | Correct with viscoelastic model and measurement |
| Same quench for thin as thick | Insufficient tempering | Set cooling by Biot number |
| Ignoring radiation | Early cooling underestimated | Include radiation |
| Uniform heat transfer | Variation and patterns missed | Derive distribution from nozzle layout |
| Ignoring edges | Edge-initiated failure | Check in 2D/3D |
I'd like to try related calculations.
The thermal stress calculator, Biot number calculator and thermal expansion calculator let you check thermal stress, Biot number and expansion. For background, see thermal shock analysis, thermal residual stress and CTE-graded design.
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