For isotropic material, and for small expansions, the linear thermal expansion coefficient is one third the volumetric coefficient. To derive the relationship, let’s take a cube of steel that has sides of length L. The original volume will be V = L3,and the new volume, after a temperature increase, will be:
V+ΔV=(L+ΔL)3=L3+3L2ΔL+3L(ΔL)2+(ΔL)3≈L3+3L2ΔL=V+3VΔLLV+ΔV=(L+ΔL)3=L3+3L2ΔL+3L(ΔL)2+(ΔL)3≈L3+3L2ΔL=V+3VΔLL.
The approximation holds for a sufficiently small ΔLΔL compared to L. Since:
ΔV / V=3ΔL / L
(and from the definitions of the thermal coefficients), we arrive at:
αV=3αLαV=3αL.