Thermal strain ($e^{\mathrm{th}}={\alpha }_v(T-T_0)I/3$)

Thermal strain is the strain that heating would produce without mechanical stress. In isotropic linear thermoelasticity it is a pure dilation: a volumetric expansion coefficient ${\alpha }_v$ corresponds to a linear expansion coefficient ${\alpha }_v/3$. The constitutive law acts on the mechanical strain $e-e^{\mathrm{th}}$. Spatially varying thermal strain need not be globally compatible with a stress-free displacement.