A microstructure diagram for known bounds in conductivity

Two important analytical means—theoretical bounds and homogenization techniques—have gained increasing attention and led to substantial progress in material research. Nevertheless, there is a lack of relating material microstructures to an entire theoretical bound and exploring the possibility of generating multiple microstructures for each property value. This paper aims to provide a microstructure diagram in relation to “bound B” constructed by translation and Weiner bounds. The inverse homogenization technique is used to seek for the optimal phase distribution within a base cell model to make the effective conductivity approach the “bound B” in two- or three-phase material cases. The design shows that the “bound B” is exactly attainable for two-phase composites even with single-length-scale microstructures. Although the multiphase translations bounds are well known to be asymptotically attainable on some parts, they still appear too roomy to be attained by single-length-scale composites. Our results showed a certain improvement in the attainability of single-length-scale structural composites when compared with new bounds established by [V. Nesi: Proc. R. Soc. Edinburgh Sect. A125, 1219 (1995)], [V. Cherkaev: Variational Methods for Structural Optimization (Springer Verlag, New York, 2000)], and (N. Albin et al.: Proc. R. Soc. London Ser. A463, 2031 (2007)]. Applicability of the translation bounds to the composites with high-contrast conductivities of phase compositions is also studied in this paper. Finally, we explore the multiple solutions to the optimal microstructures and categorize them into three classes in line with their topological resemblance, namely, spatially identical, unidirectionally identical, and bidirectionally different solutions.