Isotropic conductivity of two-dimensional three- and four-phase symmetric composites: duality and universal bounds

We consider the problem of isotropic effective conductivity ${\sigma }_e({\sigma }_1,\dots ,{\sigma }_n)$ in two-dimensional three- and four-phase symmetric composites with a partial isotropic conductivity ${\sigma }_j$ of the $j$-th phase. The upper $\mathrm{\Omega }({\sigma }_1,\dots ,{\sigma }_n)$ and lower $\omega ({\sigma }_1,\dots ,{\sigma }_n)$, $n=3,4$, bounds for effective conductivity, found by the algebraic approach, are universal (independent of the composite micro-structure) and possess all algebraic properties of ${\sigma }_e({\sigma }_1,\dots ,{\sigma }_n)$ that follow from physics: first-order homogeneity, full permutation invariance, Keller’s self-duality, positivity, and monotony. The bounds are compatible with the trivial solution ${\sigma }_e(\sigma ,\dots ,\sigma )=\sigma$ and satisfy Dykhne’s ansatz. Their comparison with previously known numerical calculations, asymptotic analysis, and exact results for isotropic effective conductivity ${\sigma }_e({\sigma }_1,\dots ,{\sigma }_n)$ of two-dimensional three- and four-phase composites showed complete agreement. The bounds $\mathrm{\Omega }({\sigma }_1,\dots ,{\sigma }_n)$ and $\omega ({\sigma }_1,\dots ,{\sigma }_n)$ in both cases $n=3,4$ are stronger than the currently known variational bounds.