Thermal Expansion Reference Cheat Sheet

Key Facts

• Linear thermal expansion is calculated with $\Delta L = \alpha L_0 \Delta T$, where $\Delta L$ is the change in length.
• The final length after heating or cooling is $L = L_0 + \Delta L = L_0(1 + \alpha \Delta T)$.
• The temperature change is $\Delta T = T_f - T_i$, and it has the same size in $^\circ\mathrm{C}$ and $\mathrm{K}$.
• For isotropic solids, the area expansion coefficient is approximately $\beta \approx 2\alpha$.
• Area expansion is modeled by $\Delta A = \beta A_0 \Delta T$, so $A = A_0(1 + \beta \Delta T)$.
• For isotropic solids, the volume expansion coefficient is approximately $\gamma \approx 3\alpha$.
• Volume expansion is modeled by $\Delta V = \gamma V_0 \Delta T$, so $V = V_0(1 + \gamma \Delta T)$.
• If expansion is completely prevented, thermal stress may be estimated with $\sigma = Y\alpha\Delta T$, where $Y$ is Young's modulus.

Vocabulary

Thermal expansion

Thermal expansion is the change in a material's size caused by a change in temperature.

Coefficient of linear expansion

The coefficient of linear expansion, $\alpha$, tells how much length changes per unit length for each $1^\circ\mathrm{C}$ or $1\ \mathrm{K}$ temperature change.

Area expansion

Area expansion is the change in surface area of a material as its temperature changes.

Volume expansion

Volume expansion is the change in the space occupied by a solid, liquid, or gas as temperature changes.

Thermal stress

Thermal stress is the internal force per unit area that occurs when a material is prevented from expanding or contracting.

Expansion joint

An expansion joint is a designed gap or flexible connection that allows materials to expand and contract safely.