Thermal Expansion and Thermal Stress
Thermal strain, constrained thermal stress, CTE values and temperature conversion formulae.
Equations
Thermal strain (free expansion): ε_th = α · ΔT = α · (T − T_ref) Thermal stress (fully constrained): σ_th = E · α · ΔT Thermal stress (partially constrained, stiffness ratio n): σ_th = n · E · α · ΔT where n = k_bolt / (k_bolt + k_member) for bolted joints Temperature conversions: T(K) = T(°C) + 273.15 T(°F) = (9/5) · T(°C) + 32 T(°C) = (5/9) · (T(°F) − 32) ΔT(K) = ΔT(°C) (same increment) ΔT(K) = (5/9) · ΔT(°F)
Typical CTE Values (×10⁻⁶/°C)
| Material | α (×10⁻⁶/°C) | Direction | Notes |
|---|---|---|---|
| Aluminium 2024-T3 | 22.9 | Isotropic | |
| Aluminium 7075-T6 | 23.6 | Isotropic | |
| Titanium Ti-6Al-4V | 8.6 | Isotropic | |
| Steel 4340 | 11.7 | Isotropic | |
| Stainless 17-4 PH | 10.8 | Isotropic | |
| Inconel 718 | 13.0 | Isotropic | |
| CFRP (UD, fibre dir.) | −0.1 to 0.5 | 1 (fibre) | Near-zero or negative |
| CFRP (UD, transverse) | 26-30 | 2 (transverse) | Matrix-dominated |
| CFRP [0/±45/90]s | 2-4 | Laminate | Quasi-isotropic |
| GFRP (UD, fibre dir.) | 6-8 | 1 (fibre) |
Variables
| Symbol | Definition | Units |
|---|---|---|
| α | Coefficient of thermal expansion (CTE) | ×10⁻⁶/°C (or /°F) |
| ΔT | Temperature change from reference | °C (or K) |
| T_ref | Reference (stress-free) temperature | °C |
| ε_th | Thermal strain | — |
| σ_th | Thermal stress | MPa |
| E | Young's modulus | GPa |
Notes and Limitations
- A uniform temperature change in an unconstrained structure produces zero stress
- Thermal stress arises from constraint: external (supports) or internal (mixed materials, non-uniform T)
- For composites, CTE is directional — the mismatch between α₁ and α₂ creates internal residual stress on cooldown from cure
- CTE may vary with temperature — use temperature-dependent values for large ΔT
Related Knowledge
See the Thermal Analysis Knowledge category for thermal-stress theory. See How to Perform a Thermal Stress Analysis for the practical workflow.