Bounds on the volume fraction of 2-phase, 2-dimensional elastic bodies   and on (stress, strain) pairs in composites
  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 
Graeme Walter Milton and Loc Hoang Nguyen
math.AP cond-mat.mtrl-sci physics.class-ph