Hashin-Shtrikman conductivity variational principle

For a scalar reference $b$, choose a trial polarization $q$ and solve $b\Delta v+\nabla \cdot q=0$ with zero Dirichlet data. The trial functional isA reference above all phase conductivities gives $J(u)\le {\mathcal{F}}_b(q)$; one below all of them gives $J(u)\ge {\mathcal{F}}_b(q)$. If the reference equals a phase conductivity, admissible polarization vanishes in that phase and the inverse is taken only on the contrast subspace.

For uniformly distributed rotations of a two-dimensional conductivity tensor with positive principal values $a\le b$, the Hashin-Shtrikman conductivity variational principle with isotropic comparison $cI$ yields $H(c)I$. Averaging a rotated diagonal tensor replaces it by half its trace times the identity. Since $H^{\prime }(c)=(a-b{)}^2/(a+b+2c{)}^2$, the admissible comparison endpoints give $a(a+3b)/(3a+b)\le C_{\mathrm{eff}}\le b(3a+b)/(a+3b)$ within the statistically isotropic polycrystal model.

These are the optimal volume-fraction bounds for three-dimensional statistically isotropic two-phase conductivity with $0<a_1<a_2$. Constant phase polarization trial fields reduce the variational principle to a scalar quadratic optimization. The upper trial functional is minimized; the lower one is maximized.