Thermal Stress

When a solid body is heated or cooled, it tends to expand or contract. In the absence of any constraint, this dimensional change is purely kinematic — no forces are generated and no stress develops. Thermal stress appears as soon as this free thermal deformation is resisted, either by external supports or by internal incompatibility within the body itself.

Fully constrained bar

The simplest case is a bar of length L, elastic modulus E, and linear thermal expansion coefficient \alpha, with both ends fixed to rigid walls. When the temperature rises by \Delta T, the bar tries to elongate by \delta_\text{thermal} = \alpha L \Delta T. The walls prevent this elongation entirely. The bar behaves as though it were compressed back by the same amount, developing a compressive thermal stress:

The negative sign indicates compression for a temperature increase. If the bar is cooled instead (\Delta T < 0), the stress is tensile. The magnitude can be substantial: for structural steel (E = 200 \, \text{GPa}, \alpha = 12 \times 10^{-6} \, \text{K}^{-1}), a temperature change of 100 \, \text{K} produces \sigma = 240 \, \text{MPa} — approaching the yield strength of mild steel.

Thermal gradients

In most engineering components, temperature is not uniform: one surface is hotter than another, or heat is generated internally. A temperature gradient produces non-uniform thermal expansion within the body, which cannot be accommodated without internal compatibility stresses even if the component is externally unconstrained. In a flat plate with a temperature difference \Delta T between its two faces, the hotter face tries to expand more than the cooler face. The plate responds by bending (if free) or by developing a stress distribution (if constrained). In a thick-walled cylinder subjected to a radial temperature gradient — such as a pressure vessel or a gun barrel — the thermal stresses are superimposed on the mechanical pressure stresses, and the combined loading governs the design.

Dissimilar materials

When two materials with different thermal expansion coefficients are bonded together — a bimetallic strip, a ceramic coating on a metal substrate, a solder joint between copper and silicon — a temperature change causes them to try to expand by different amounts. Since they are bonded, compatibility must be maintained, and interface stresses develop. The mismatch strain is:

This mismatch drives shear and peel stresses at the interface, which are the primary cause of delamination and cracking in composite structures, electronic packages, and thermal barrier coatings.

Thermal fatigue

When temperature cycles are repeated — as in engine components, turbine blades, nuclear reactor parts, and soldered electronic assemblies — the cyclic thermal stresses drive fatigue damage. Thermal fatigue differs from isothermal mechanical fatigue in that the strain amplitude is thermally imposed and the material properties (yield strength, ductility, creep resistance) change with temperature throughout the cycle. Low-cycle fatigue is the dominant failure mode in most thermal fatigue applications, because the constrained thermal strains often exceed the elastic range and produce plastic deformation in each cycle.

Design considerations

Thermal stress management is a primary concern in the design of high-temperature equipment. Key strategies include: allowing free thermal expansion through expansion joints, bellows, and sliding supports in piping systems; grading material compositions to reduce thermal expansion mismatch in coatings and joints; using materials with low thermal expansion coefficients in precision structures; and designing cooling circuits to limit temperature gradients in turbine blades and electronic devices. In structures where thermal stresses cannot be avoided, the design must account for them explicitly in both static strength checks and fatigue life predictions.

Common mistakes

A common mistake is calculating free thermal expansion and treating it as stress without checking the actual constraint. A freely expanding body develops thermal strain but no thermal stress. Another is using constant material properties over a large temperature range even though elastic modulus, yield strength, ductility, and fatigue resistance may change significantly. A strong thermal-stress review states temperature field, constraint condition, expansion coefficient, material properties versus temperature, load combination, cycle count, gradients, interfaces, and whether plasticity or fatigue must be included.