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

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. (2007 a J. Mech. Phys. Solids 55 , 1513–1553).