 Solution

Choose the affine trial function $v(x)=\lambda \cdot x$. It has the required trace and constant gradient, so the Dirichlet principle givesFor the macroscopic isotropic effective medium, the corresponding affine-boundary energy is $|\Omega |a^{\ast }|\lambda {|}^2$. Dividing by $|\Omega ||\lambda {|}^2$ for a nonzero imposed gradient yields the Voigt boundMore generally the same calculation is the tensor inequality in the Loewner order $a^{\ast }⪯⟨a⟩$. For a finite unresolved specimen the calculation bounds its apparent affine-boundary response; identification with the bulk effective conductivity uses the macroscopic scale separation.