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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.20401 |
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Table of Contents:
- We consider the problem of isotropic effective conductivity $σ_e(σ_1,\ldots,σ_n)$ in two-dimensional three- and four-phase symmetric composites with a partial isotropic conductivity $σ_j$ of the $j$-th phase. The upper $Ω(σ_1,\ldots,σ_n)$ and lower $ω(σ_1,\ldots,σ_n)$, $n=3,4$, bounds for effective conductivity, found by the algebraic approach, are universal (independent of the composite micro-structure) and possess all algebraic properties of $σ_e(σ_1,\ldots,σ_n)$ 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 $σ_e(σ,\ldots,σ)=σ$ and satisfy Dykhne's ansatz. Their comparison with previously known numerical calculations, asymptotic analysis, and exact results for isotropic effective conductivity $σ_e(σ_1,\ldots,σ_n)$ of two-dimensional three- and four-phase composites showed complete agreement. The bounds $Ω(σ_1,\ldots,σ_n)$ and $ω(σ_1,\ldots,σ_n)$ in both cases $n=3,4$ are stronger than the currently known variational bounds.