= Two-dimensional isotropic polycrystal conductivity
{title2=$H(c)=\frac{2ab+c(a+b)}{a+b+2c}$}
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 $H'(c)=(a-b)^2/(a+b+2c)^2$, 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.
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