  We consider the Lame system of linear elasticity with periodically
distributed inclusions whose elastic parameters have high contrast compared to
the background media. We develop a unified method based on layer potential
techniques to quantify three convergence results when some parameters of the
elastic inclusions are sent to extreme values. More precisely, we study the
incompressible inclusions limit where the bulk modulus of the inclusions tends
to infinity, the soft inclusions limit where both the bulk modulus and the
shear modulus tend to zero, and the hard inclusions limit where the shear
modulus tends to infinity. Our method yields convergence rates that are
independent of the periodicity of the inclusions array, and are sharper than
some earlier results of this type. A key ingredient of the proof is the
establishment of uniform spectra gaps for the elastic Neumann-Poincare operator
associated to the collection of periodic inclusions that are independent of the
periodicity.
