Thermal stress ($\tau =C:(e-e^{\mathrm{th}})$)

Thermal stress is stress generated when thermal expansion or contraction is constrained or incompatible. A uniformly heated isotropic body whose entire boundary is fixed has $e=0$ and $\tau =-K{\alpha }_v(T-T_0)I$. For a nonuniform temperature field, interior displacement generally remains nonzero even with a fixed outer boundary; equilibrium and compatibility must be solved together. Cancelling the divergence of a thermal stress term with a body force is not the same as making that stress vanish.