Optimal lower bounds on the hydrostatic stress amplification inside random two-phase elastic composites

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Composites made from two linear isotropic elastic materials are subjected to a uniform hydrostatic stress. It is assumed that only the volume fraction of each elastic material is known. Lower bounds on all r th moments of the hydrostatic stress field inside each phase are obtained for r ⩾ 2 . A lower bound on the maximum value of the hydrostatic stress field is also obtained. These bounds are given by explicit formulas depending on the volume fractions of the constituent materials and their elastic moduli. All of these bounds are shown to be the best possible as they are attained by the hydrostatic stress field inside the Hashin–Shtrikman coated sphere assemblage. The bounds provide a new opportunity for the assessment of load transfer between macroscopic and microscopic scales for statistically defined microstructures.