Thermal Expansion

As discussed previously almost all solids expand when heated. The equation governing the change in a single dimension (linear expansion) is given by

DL / L = a DT

where a is called the coefficient of thermal expansion. The units for this coefficient are inverse °C or °C^-1. Typical values for metals are 10^-6 to 10^-5. For liquids they are 10 times bigger.

Here is an interesting video demonstration that shows linear thermal expansion at work:

This, by the way is the second way we have encountered that one can put strain on a material. Here we are dealing with thermal strain, whereas we has previously encountered mechanical strain.

The thermal coefficients b and g for a 2-dimensional or 3-dimensional expansion can be easily derived from the equation for linear expansion:

Area expansion:

A = L² = L₀²(1+aDT)² = A₀(1+2aDT+a²DT²) A₀(1+2aDT)
We neglect terms with higher powers of a, because a << 1.
=> DA := A-A₀ = A₀2aDT => DA/A = 2aDT
You could also set up the area expansion using a generic coefficient of area expansion,
DA/A = b DT
When you combine the last derived equation and this definition equation, then you get:

b = 2 a

Volume expansion:

V = L³ = L₀³(1+aDT)³ = V₀(1+3aDT+...)
=> DV := V-V₀ = V₀3aDT => DV/V = 3aDT

You could also set up the volume expansion using a generic coefficient of volume expansion,
DV/V = g DT
When you combine the last derived equation and this definition equation, then you get:

g = 3 a