  We analyse a problem of two-dimensional linearised elasticity for a
two-component periodic composite, where one of the components consists of
disjoint soft inclusions embedded in a rigid framework. We consider the case
when the contrast between the elastic properties of the framework and the
inclusions, as well as the ratio between the period of the composite and the
framework thickness increase as the period of the composite becomes smaller. We
show that in this regime the elastic displacement converges to the solution of
a special two-scale homogenised problem, where the microscopic displacement of
the framework is coupled both to the slowly-varying "macroscopic" part of the
solution and to the displacement of the inclusions. We prove the convergence of
the spectra of the corresponding elasticity operators to the spectrum of the
homogenised operator with a band-gap structure.
