Hill-Mandel energy identity ($⟨\sigma :\epsilon ⟩=⟨\sigma ⟩:⟨\epsilon ⟩$)

For symmetric stress in static elastic equilibrium without body force, write $E=⟨\epsilon ⟩$ and $S=⟨\sigma ⟩$. The difference between microscopic and macroscopic work is $|\Omega {|}^{-1}{\int }_{\partial \Omega }(\sigma -S)n\cdot (u-Ex)$. It vanishes for affine boundary displacement, uniform boundary traction, or matching periodic displacement fluctuations and opposite tractions on paired faces.