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Universal bounds on the electrical and elastic response of two-phase (andmultiphase) ellipsoidal or parallelopipedic bodies have been obtained byNemat-Nasser and Hori. Here we show how their bounds can be improved andextended to bodies of arbitrary shape. Although our analysis is for two-phasebodies with isotropic phases it can easily be extended to multiphase bodieswith anisotropic constituents. Our two-phase bounds can be used in an inversefashion to bound the volume fractions occupied by the phases, and forelectrical conductivity reduce to those of Capdeboscq and Vogelius when thevolume fraction is asymptotically small. Other volume fraction bounds derivedhere utilize information obtained from thermal, magnetic, dielectric or elasticresponses. One bound on the volume fraction can be obtained by simply immersingthe body in a water filled cylinder with a piston at one end and measuring thechange in water pressure when the piston is displaced by a known small amount.This bound may be particularly effective for estimating the volume of cavitiesin a body. We also obtain new bounds utilizing just one pair of (voltage, flux)electrical measurements at the boundary of the body.

journal of the mechanics and physics of solids

Universal bounds on the electrical and elastic response of two-phase  bodies and their application to bounding the volume fraction from boundary  measurements