1. Physical Origin of Thermal Expansion in Solids

All solid materials possess the property of expanding in volume as temperature increases. From a microscopic perspective, this is explained by lattice vibration. Atoms inside a solid vibrate while maintaining an equilibrium energy state due to interatomic bonding forces. When energy is applied and the temperature rises, the vibration amplitude of the atoms increases. Due to the asymmetry of the non-linear interatomic potential energy curve, the average distance between the centers of vibration increases. This is the physical principle observed macroscopically as thermal expansion.

• Linear Expansion: Expansion that occurs in objects where the longitudinal dimension is dominant, such as rods or bars.
• Volume Expansion: Expansion of the entire volume of a three-dimensional shape. For isotropic solids, the coefficient of volume expansion is approximately three times the coefficient of linear expansion (β ≈ 3α).

2. Thermal Expansion Design Equations and Displacement Derivation

The linear expansion relationship of solids can be predicted with high precision using a linear approximation, provided the temperature range is not excessively large.

① Basic Formula for Linear Expansion:

ΔL = α × L₀ × ΔT  [mm]

Here, α is the coefficient of linear expansion (K⁻¹ or °C⁻¹), L₀ is the initial length (m) at the initial temperature, and ΔT is the change in temperature (T₂ - T₁, °C).

② Final Length Calculation:

L = L₀ + ΔL = L₀ × (1 + α × ΔT)  [m]

③ Thermal Stress: If both ends are constrained by rigid walls and heated so that expansion is impossible, a compressive strain equivalent to the expansion displacement occurs, generating a very strong internal force. The formula for thermal stress under bilateral constraint is as follows:

σ_t = E × α × ΔT  [MPa]

Here, E is the Young's Modulus of the metal. When designing mechanical components, installing a thermal gap or an expansion joint is essential to prevent damage caused by this thermal stress.

3. Thermal Expansion Characteristics by Key Metallic Materials and Design Guide

Various industrial metals have different coefficients of linear expansion due to differences in crystal structures and bonding forces. When designing, the potential for bimetallic warping (bimetallic effect) when joint structures of dissimilar metals undergo temperature changes must be carefully considered.

• Aluminum: At approximately 23.0 × 10⁻⁶ /°C, its thermal expansion is extremely high, making gap design crucial in lightweight engineering.
• Brass: At approximately 19.0 × 10⁻⁶ /°C, it possesses a high expansion rate characteristic of copper alloys.
• Copper: At approximately 17.0 × 10⁻⁶ /°C, longitudinal contraction and expansion must be accounted for during electrical wire design and piping installation.
• Carbon Steel: At approximately 11.5 × 10⁻⁶ /°C, it is relatively low and stable, and its similarity to the expansion coefficient of concrete forms the basis for reinforced concrete structures.