  A two phase elastic composite with weakly compressible elastic inclusions is
considered. The homogenised two-scale limit problem is found, via a version of
the method of two-scale convergence, and analysed. The microscopic part of the
two-scale limit is found to solve a Stokes type problem and shown to have no
microscopic oscillations when the composite is subjected to body forces that
are microscopically irrotational. The composites spectrum is analysed and shown
to converge, in an appropriate sense, to the spectrum of the two-scale limit
problem. A characterisation of the two-scale limit spectrum is given in terms
of the limit macroscopic and microscopic behaviours.
