Pour boiling water into a cold drinking glass and there is a real chance it will crack with an audible snap. The glass was not struck and nothing was dropped into it. The damage came entirely from temperature — specifically from the difference in temperature between the inner surface, which heated almost instantly, and the cooler bulk of the wall behind it. That mismatch is thermal shock, and it limits ceramics, glass, and brittle castings far more often than mechanical loads do.
This article explains how a temperature change turns into mechanical stress, works through a numerical example for a quenched glass plate, and lists the assumptions that make thermal-shock estimates easy to misuse.
Why this calculation matters
Brittle materials have an awkward property: they are strong in compression but weak in tension, and they have almost no ability to yield and relieve a stress concentration. A metal part facing a sudden temperature change can deform plastically and shed some of the stress. A ceramic or glass part cannot. Whatever stress thermal shock generates is carried elastically right up to the point of fracture.
That is why thermal shock governs the service limits of kiln furniture, turbine ceramics, glass cookware, brake rotors, electronic substrates, and refractory linings. Components like these often see their worst loading not during steady operation but during startup, shutdown, or an accidental quench. If you size a brittle part only for its mechanical duty and ignore transient temperature gradients, you can be surprised by a crack that appears the first time the part is heated or cooled quickly. Estimating the thermal stress up front tells you whether a given temperature swing is survivable.
The core formula
When a solid surface changes temperature suddenly, it tries to expand or contract. The bulk behind it, still at its original temperature, has not moved yet and holds the surface back. That constraint is what produces stress.
For the most severe idealized case — a surface whose temperature changes instantly by delta_T while the bulk is fully restrained — the thermal stress is:
sigma = E * alpha * delta_T
Here E is the elastic modulus of the material, alpha is its coefficient of thermal expansion, and delta_T is the magnitude of the temperature change. The product alpha * delta_T is the thermal strain the surface "wants" to undergo; multiplying by the modulus E converts that blocked strain into stress.
The sign matters. Sudden cooling makes the surface try to contract while the bulk holds it stretched, so the surface goes into tension — and tension is what cracks brittle materials. Sudden heating puts the surface into compression, which brittle materials tolerate far better. This is why a cold-water quench is the dangerous direction for hot glass and ceramics.
The equation above is an upper bound. Real surfaces do not change temperature instantly; heat takes time to flow in or out. The fraction of the ideal stress that is actually reached depends on how fast the surface exchanges heat relative to how fast the material conducts it inward — a competition captured by the Biot number. A low Biot number means a gentle, gradual exchange and a stress well below E*alpha*delta_T; a high Biot number means a near-instant surface change and a stress close to the full value.
A worked example
Consider a flat plate of ordinary glass that is suddenly quenched, so its surface temperature drops by delta_T = 80 K. Take the elastic modulus as E = 70 GPa and the coefficient of thermal expansion as alpha = 9e-6 per kelvin. Assume a severe quench, so the surface stress approaches the fully restrained upper bound.
Step 1 — find the thermal strain. The strain the surface is prevented from undergoing is:
alpha * delta_T = 9e-6 * 80 = 7.2e-4
That is a strain of 720 microstrain — small in absolute terms, but brittle materials fail at small strains.
Step 2 — convert strain to stress.
sigma = E * alpha * delta_T
sigma = 70e9 * 7.2e-4
sigma = 5.04e7 Pa
sigma = 50.4 MPa
Step 3 — compare against strength. The tensile strength of ordinary glass is around 50 MPa. The computed surface stress of 50.4 MPa sits right at that limit. A quench of this severity is therefore enough to crack the plate — there is no margin left.
The lesson is sobering. An 80 K temperature swing feels modest; it is no larger than the difference between iced water and a warm room. Yet for glass that swing is sufficient to reach the fracture stress. This is exactly why thermal-shock failures so often catch people off guard.
Common mistakes
Assuming the worst-case formula always applies. sigma = E*alpha*delta_T is the fully restrained upper bound. A slow temperature change, a thin part, or a poor surface heat-transfer coefficient all reduce the actual stress below this value. Use the upper bound for a conservative first check, then refine with the Biot number if the part fails that check.
Ignoring the direction of the temperature change. Heating and cooling are not symmetric for brittle materials. Cooling puts the surface in tension and is the dangerous case; heating puts it in compression. Quoting a single delta_T without saying whether the surface is being heated or cooled hides the most important fact.
Forgetting that thermal shock is a transient. The stress is largest while the temperature gradient through the wall is steepest, which is early in the transient. Once the part reaches a uniform temperature the thermal stress relaxes toward zero. Checking only the final steady state misses the peak entirely.
Treating handbook strength as a hard number. The strength of brittle materials is statistical and sensitive to surface flaws, scratches, and edge condition. A nominal 50 MPa is a central estimate, not a guarantee, so a stress that merely "equals" the strength should be read as a likely failure, not a safe one.
Try the interactive NovaSolver calculator
Running the upper-bound stress by hand is quick, but judging real severity means accounting for how fast heat actually moves. The Thermal Shock & Fatigue Calculator — R-Value & Coffin-Manson Life on NovaSolver lets you set the temperature difference delta_T and the Biot number and returns the thermal shock resistance R, the surface thermal stress, and a Coffin-Manson fatigue life, with material presets ranging from ceramics to carbon steel so you can compare candidates side by side.
Related calculators
- Thermal Expansion Calculator — quantify the dimensional change behind thermal stress before the part is even restrained.
- Stress-Strain Curve Explorer — see how a material responds to stress, and why brittle materials offer no plastic relief.
- Transient Heat Conduction Calculator — track how a temperature gradient develops and decays through a wall over time.
You can browse the rest in the thermal tools hub.
Closing note
Thermal shock is a reminder that you do not need a mechanical load to break a part — a temperature change alone can do it. The core idea is compact: a blocked thermal strain alpha*delta_T, scaled by the elastic modulus E, becomes a stress. For brittle materials, the dangerous version of that stress is tensile, it appears during the transient, and it peaks when the surface temperature changes fastest. Estimate the upper-bound stress first, watch the direction of the temperature change, and treat any result near the material's strength as a warning rather than a pass.
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