  Bounds are obtained on the volume fraction in a two-dimensional body
containing two elastically isotropic materials with known bulk and shear
moduli. These bounds use information about the average stress and strain
fields, energy, determinant of the stress, and determinant of the displacement
gradient, which can be determined from measurements of the traction and
displacement at the boundary. The bounds are sharp if in each phase certain
displacement field components are constant. The inequalities we obtain also
directly give bounds on the possible (average stress, average strain) pairs in
a two-phase, two-dimensional, periodic or statistically homogeneous composite
