AkCalculators

Thermal expansion — concepts, formulas and engineering notes

Thermal expansion quantifies how materials change dimension with temperature. Engineers and scientists use simple linear relations for everyday calculations; more advanced problems require temperature-dependent coefficients or stress analysis for constrained components. This article covers definitions, formulas for different geometries, common coefficients, worked examples and design tips.

Definitions and formulas

Linear expansion: ΔL = α L₀ ΔT where α is the linear coefficient (units 1/°C or 1/K). The final length L = L₀ (1 + α ΔT) for small ΔT. For anisotropic materials α depends on direction.

Area expansion: For an isotropic surface, the area change is approximately ΔA = 2 α A₀ ΔT (thus area coefficient ≈ 2α).

Volumetric expansion: ΔV = β V₀ ΔT where β is the volumetric coefficient. For isotropic solids β ≈ 3α; for liquids β is often measured directly and can be much larger than solids.

Temperature dependence and integration

If α varies with temperature, the correct expression is ΔL = L₀ ∫_{T1}^{T2} α(T) dT. For moderate ranges where α is approximately constant the simple linear form is adequate. For precision work, use tabulated α(T) or fitted functions and integrate numerically.

Typical coefficients

Representative values at room temperature: aluminum ≈ 23×10⁻⁶ /°C, steel ≈ 11–13×10⁻⁶ /°C, copper ≈ 16.5×10⁻⁶ /°C, glass (borosilicate) ≈ 3.3×10⁻⁶ /°C. Liquids and polymers can have much larger values and often show stronger temperature dependence.

Worked examples

Example 1 — Linear. A 5 m steel rail (α = 11.5×10⁻⁶ /°C) heats from 0°C to 40°C (ΔT = 40°C). ΔL = α L₀ ΔT = 11.5e-6 × 5 × 40 = 0.0023 m = 2.3 mm. Expansion joints must accommodate this movement in track design.

Example 2 — Volumetric. A 100 L container of liquid with β = 7×10⁻⁴ /°C is heated by 30°C. ΔV = β V₀ ΔT = 7e-4 × 100 × 30 = 2.1 L (significant — account for expansion in storage tanks).

Design considerations

• Allow for thermal clearances in mechanical assemblies (bolted joints, rails, piping).
• Use materials with matched α for bonded assemblies or differential expansion will cause stress and possible failure.
• Constrained expansion produces thermal stress: σ = E α ΔT for a fully constrained long bar (simple estimate), where E is Young's modulus; use proper mechanics for partial constraints.

Measurement and accuracy

Measure α using high-precision dilatometers or reference published data for the material and temperature range. For large ΔT or near phase transitions, material properties (α, E) can change and must be treated accordingly.

Limitations

Simple linear formulas assume small strains and no phase change. For composites, anisotropic crystals, or high-temperature ranges, rely on directional coefficients, integration, or finite-element thermal–mechanical analysis.