Andrej Cherkaev, Grzegorz Dzierzanowski
The paper establishes tight lower bound for effective elastic stress energy of two-dimensional three-phase anisotropic composites. It is assumed that the materials are mixed with fixed volume fractions and that one of the phases is degenerated to void, i.e. the effective material is porous. The bound expands the Hashin-Shtrikman and translation bounds to multiphase structures. It is derived using a combination of the translation method and addtional inequalities on the stress fields in materials; similar technique was used by Cherkaev (Mech. Mater. 41, 411-433, 2009) for isotropic multiphase composites. It is shown that in the high-porosity regime the obtained energy estimate is exact and it is realized on microstructures in the form of high-rank laminates whose optimal geometrical parameters are explicitly found. To this end, the field matching method is elaborated for the plane elasticity problem, following Albin et. al. (J. Mech. Phys. Solids 55, 1513-1553, 2007) and Cherkaev and Zhang (Int. J. Solids Struct. 48, 2800-2813, 2011) where similar technique was developed for conductivity case. Corresponding formulae solving the G-closurs problem of a three-phase composite are also derived. Conjectures regarding low-porosity regimes are presented. However, full discussion of this issue is postponed to a separeate publication.
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http://arxiv.org/abs/1302.2729
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