Hashin–Shtrikman bounds on the shear modulus of a nanocomposite with spherical inclusions and interface effects

The recently developed variational framework for polarization methods in nanocomposites is applied to the determination of a lower-bound on the shear modulus of a nanocomposite with monosized, spherical inclusions. This bound explicitly accounts for linear elastic effects in the matrix–inclusion interface. Even if the polarization fields involved in its derivation are much more intricate, this bound is closely related to the classical Hashin–Shtrikman bound, with which it coincides when surface stresses are disregarded. More strikingly, when surface stresses are not disregarded, it also coincides with previously established Mori–Tanaka estimates. This result provides firm ground for the practical use of these estimates, for example for design purposes.