= Hill-Mandel energy identity
{c}
{title2=$\langle\sigma:\varepsilon\rangle=\langle\sigma\rangle:\langle\varepsilon\rangle$}
For symmetric <stress> in <static elastic equilibrium> without <body force>, write $E=\langle\varepsilon\rangle$ and $S=\langle\sigma\rangle$. 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.
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