Most solids change dimensions when their temperature changes. Over a sufficiently small temperature range, the fractional change in length is approximately proportional to the temperature change.

For an initial length $L_0$, define the coefficient of linear thermal expansion $\alpha$ by

$$\boxed{\Delta L=\alpha L_0\Delta T}.$$

Equivalently,

$$\frac{\Delta L}{L_0}=\alpha\Delta T.$$

The coefficient $\alpha$ has units of inverse temperature, such as $\mathrm{K^{-1}}$. Because a temperature difference has the same numerical size in kelvins and degrees Celsius, either scale can be used for $\Delta T$.

If $\alpha>0$, heating produces expansion and cooling produces contraction. Some materials can have negative expansion coefficients over particular temperature ranges, so the sign must come from the material behavior rather than from the word “heating” alone.

Worked example: a rail

A steel rail has

$$L_0=20.0,\mathrm m,$$

$$\alpha=12\times10^{-6},\mathrm{K^{-1}},$$

and experiences a temperature increase of

$$\Delta T=40,\mathrm K.$$

Its free expansion is

$$\Delta L =(12\times10^{-6})(20.0)(40) =9.6\times10^{-3},\mathrm m.$$

Thus

$$\boxed{\Delta L=9.6,\mathrm{mm}}.$$

Expansion joints allow such dimensional changes without forcing large internal stresses into the structure.

Area expansion

For an isotropic solid, every linear dimension expands by approximately the same fractional amount $\alpha\Delta T$.

Consider a rectangle with sides $L$ and $W$. After heating,

$$L'\approx L(1+\alpha\Delta T),$$

$$W'\approx W(1+\alpha\Delta T).$$

Its new area is

$$A'=LW(1+\alpha\Delta T)^2.$$

For small expansions, the term $(\alpha\Delta T)^2$ is negligible, so

$$A'\approx A(1+2\alpha\Delta T).$$

Therefore

$$\boxed{\frac{\Delta A}{A}\approx2\alpha\Delta T}.$$

Volume expansion

For an isotropic solid with three expanding dimensions,

$$V'\approx V(1+\alpha\Delta T)^3.$$

Neglecting second- and third-order small terms,

$$V'\approx V(1+3\alpha\Delta T).$$

Thus

$$\boxed{\frac{\Delta V}{V}\approx\beta\Delta T},$$

where the volumetric expansion coefficient satisfies

$$\boxed{\beta\approx3\alpha}$$

for an isotropic solid in the small-expansion regime.

Holes expand too

A hole in a uniformly heated plate expands as though it were filled with the same material and that material expanded with the rest of the plate.

For example, if a circular hole initially has diameter $D_0$, then approximately

$$\Delta D=\alpha D_0\Delta T.$$

Heating the plate therefore makes the hole larger, not smaller.

This follows from uniform geometric scaling: every distance between material points increases by the same fractional amount.

Limits of the linear model

The relations above assume that the expansion coefficient is approximately constant over the temperature interval and that dimensional changes are small. Over a large temperature range, $\alpha$ can vary with temperature and the expansion should be accumulated more carefully.

Thermal expansion describes the deformation a body would undergo if it were free to change size. If supports or neighboring materials prevent that free expansion, mechanical stresses can develop; that is a separate elasticity problem built on the same expansion strain.