Hashin-Shtrikman bounds and their attainability for multi-phase composites

Abstract

In this paper, we consider the problem of characterizing the set of the effective tensors of multi-phase composites, including those of conductive materials and elastic materials. We first present a novel derivation of the Hashin-Shtrikman (HS) bounds for multi-phase composites and the associated attainment condition. The attainment condition asserts that the HS bound is attainable if, and only if, there exists a second gradient field that is constant in all but the matrix phases. By restricting and constructing such second gradient fields, we obtain a series of sufficient conditions such that the HS bounds are attainable or unattainable. These attainability and unattainability results appear new for a generic situation. For special situations, our attainability and unattainability results recover the results of Milton (1981 Phys. Rev. Lett. 46, 542-545), Sigmund (2000 J. Mech. Phys. Solids 48, 397-428), Gibiansky & Sigmund (2000 J. Mech. Phys. Solids 48, 461-498) and Albin et al. (2007a J. Mech. Phys. Solids 55, 1513-1553). © 2010 The Royal Society.