Self-Dual Symmetric Polynomials and Effective Isotropic Conductivity of Two-Dimensional Composites

We applied an algebraic approach, developed within the framework of the theory of a commutative monoid of self-dual symmetric polynomials, to the problem of effective isotropic conductivity σe(σ1,…,σn) in two-dimensional n-phase symmetric composites with partial isotropic conductivities σj. The upper Ω(σ1,…,σn) and lower ω(σ1,…,σn) bounds for σe(σ1,…,σn), found by the algebraic approach for n=3,4, are universal (independent of the composite microstructure) and possess all algebraic properties of σe(σ1,…,σn) 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 σe(σ,…,σ)=σ and satisfy Dykhne’s ansatz. Their comparison with previously known numerical calculations, asymptotic analysis, and exact results for the effective isotropic conductivity σe(σ1,…,σn) of two-dimensional three- and four-phase composites showed complete agreement. The bounds Ω(σ1,…,σn) and ω(σ1,…,σn) in both cases n=3,4 are stronger than the currently known variational bounds.