Lower bounds on the stress and strain fields inside random two-phase elastic media

Summary The heterogeneous media under consideration are isotropic composites consisting of two well-ordered elastic isotropic phases and subjected to uniform macroscopic loading. By extending a method due to Lipton [2], lower bounds on the stress and strain fields inside each phase are explicitly established in terms of the phase volume fractions and properties. These bounds on the second moments turn out to be optimal, since they are achieved by the relevant stress and strain fields inside the finite-rank laminates which, constructed by Francfort and Murat [6], attain the Hashin-Shtrikman lower and upper bounds on the elastic bulk and shear moduli.