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