Q.Define the coefficient of linear expansion ($\alpha$), the coefficient of superficial (areal) expansion ($\beta$), and the coefficient of cubical (volume) expansion ($\gamma$) of a solid. State, without derivation, the relation connecting $\alpha$, $\beta$ and $\gamma$ for an isotropic solid.

The coefficient of linear expansion $\alpha$ is the fractional increase in length per unit rise in temperature: $L = L_0(1+\alpha\,\Delta T)$, so $\alpha = \dfrac{\Delta L}{L_0\,\Delta T}$.

The coefficient of superficial (areal) expansion $\beta$ is the fractional increase in area per unit rise in temperature: $A = A_0(1+\beta\,\Delta T)$, so $\beta = \dfrac{\Delta A}{A_0\,\Delta T}$.

The coefficient of cubical (volume) expansion $\gamma$ is the fractional increase in volume per unit rise in temperature: $V = V_0(1+\gamma\,\Delta T)$, so $\gamma = \dfrac{\Delta V}{V_0\,\Delta T}$.

For an isotropic solid, since area scales as (length)$^2$ and volume as (length)$^3$, these are related by $\beta = 2\alpha$ and $\gamma = 3\alpha$.

Define each coefficient from its governing expansion equation, then note that area/volume scale as the square/cube of a linear dimension to obtain the $\beta=2\alpha$, $\gamma=3\alpha$ relations.

• Reversing the relations (writing $\alpha = 2\beta$ instead of $\beta = 2\alpha$).
• Assuming these simple ratio relations hold for a non-isotropic (anisotropic) material, where they do not in general apply.