A microstructure diagram for known bounds in conductivity
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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. A 125, 1219 (1995)], [V. Cherkaev: Variational Methods for Structural Optimization (Springer Verlag, New York, 2000)], and (N. Albin et al.: Proc. R. Soc. London Ser. A 463, 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.
I. INTRODUCTION
The behavior of a composite material depends not only on the physical properties of its constituents but also upon the microstructure layouts of the phases, like lamellae, sphere, and needle. These two sets of parameters provide us with a great space to customize different material behaviors. Unfortunately, it is by no means easy to determine which material compositions and what microstructures could optimally lead to a specific physical property. In this context, two important means, namely, the theoretical bounds and the inverse homogenization methods, prove useful to characterize and devise sophisticated composite materials. For a given composite, it is often cumbersome to gain insight into the microstructure and precisely determine its effective (bulk) properties. As an alternative, the mathematical bounds that linearly or nonlinearly coma)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/JMR.2008.0101 798 J. Mater. Res., Vol. 23, No. 3, Mar 2008 http://journals.cambridge.org Downloaded: 14 Mar 2015
bine the individual properties of compositions in terms of their volume fractions have been developed to provide a proper estimation to the admissible range of variations. In effect, the bounds have served as an i
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